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abstract: 'Whereas Shannon entropy is related to the growth rate of multinomial coefficients, we show that the quadratic entropy (Tsallis 2-entropy) is connected to their $q$-deformation; when $q$ is a prime power, these $q$-multinomial coefficients count flags of finite vector spaces with prescribed length and dimensions. In particular, the $q$-binomial coefficients count vector subspaces of given dimension. We obtain this way a combinatorial explanation for the nonadditivity of the quadratic entropy, which arises from a recursive counting of flags. We show that statistical systems whose configurations are described by flags provide a frequentist justification for the maximum entropy principle with Tsallis statistics. We introduce then a discrete-time stochastic process associated to the $q$-binomial probability distribution, that generates at time $n$ a vector subspace of $\mathbb{F}_q^n$ (here ${\mathbb{F}}_q$ is the finite field of order $q$). The concentration of measure on certain “typical subspaces" allows us to extend the asymptotic equipartition property to this setting. The size of the typical set is quantified by the quadratic entropy. We discuss the applications to Shannon theory, particularly to source coding, when messages correspond to vector spaces.'
author:
- 'Juan Pablo Vigneaux [^1]'
bibliography:
- 'IEEEabrv.bib'
- 'bibtex.bib'
title: Information theory with finite vector spaces
---
Galois fields, combinatorial mathematics, stochastic processes, linear algebra, information theory, non-extensive statistical mechanics, Tsallis entropy, $q$-multinomial coefficients, $q$-binomial distribution, grassmannian, asymptotic equipartition property.
Introduction {#sec:intr}
============
Combinatorial and algebraic characterizations of entropy {#sec:two_faces}
--------------------------------------------------------
The first part of this paper describes a combinatorial interpretation of the quadratic entropy. It provides an explicit example where the lack of additivity of this function can be explained nonaxiomatically.
It is well known that Shannon entropy ${H_{1}}$ is related to the exponential growth of multinomial coefficients. More precisely: given a discrete probability law $(\mu_1,...,\mu_s)$, $$\label{asymptotics}
\lim_n \frac{1}{n} \ln {n\choose \mu_1n, ...,\mu_sn} = -\sum_{i=1}^s \mu_i \ln \mu_i =: {H_{1}}(\mu_1,...,\mu_s).$$
These coefficients have a $q$-analog. Given an indeterminate $q$, the $q$-integers $\{[n]_q\}_{n\in {\mathbb{N}}}$ are defined by $[n]_q := (q^n-1)/(q-1)$, the $q$-factorials by $[n]_q! := [n]_q [n-1]_q \cdots [1]_q$, and the $q$-multinomial coefficients are $${ {n \brack k_1,...,k_s}_q } := \frac{[n]_q!}{[k_1]_q! \cdots [k_s]_q!},$$ where $k_1,...,k_s \in {\mathbb{N}}$ are such that $\sum_{i=1}^s k_i = n$. When $q$ is a prime power, these coefficients count the number of flags of vector spaces $V_1\subset V_2 \subset ... \subset V_n = {\mathbb{F}}_q^n$ such that $\dim V_i = \sum_{j=1}^i k_j$ (here $\mathbb{F}_q$ denotes the finite field of order $q$); we refer to the sequence $(k_1,...,k_s)$ as the *type* of the flag. In particular, the $q$-binomial coefficient ${ {n \brack k}_q } \equiv { {n \brack k,n-k}_q }$ counts vector subspaces of dimension $k$ in ${\mathbb{F}}_q^n$.
In Section \[sec:q\_binomials\] we study in detail the asymptotic behavior of these coefficients. In particular, we show that, given a discrete probability law $(\mu_1,...,\mu_s)$, $$\label{a-asymptotics}
\lim_n \frac{2}{n^2} \log_q { {n \brack \mu_1n,...,\mu_sn}_q } = 1- \sum_{i=1}^s \mu_i^2 =:{H_{2}}(\mu_1,...,\mu_s).$$ The function ${H_{2}}$ is known as *quadratic entropy* [@csiszar2008axiomatic].
More generally, one can introduce a parameterized family of functions $\ln_\alpha: (0,\infty) \to {\mathbb{R}}$, for $\alpha>0$, that generalize the usual logarithm through the formula $$\ln_{\alpha}(x) := \int_1^x \frac{1}{x^\alpha} \d x.$$ The $\alpha$-surprise of a random event of probability $p$ is then defined as $\ln_\alpha(1/p)$, following the traditional definitions in information theory. Given a random variable[^2] $X:\Omega \to S_X$ with law $P$ (a probability on $S_X$), its $\alpha$-entropy ${H_{\alpha}}[X](P)$ is defined as the expected $\alpha$-surprise ${\mathbb{E}}_P\ln_\alpha(1/P(X))$. This $\alpha$-entropy or any real multiple of it can be taken as a generalized information measure. The $1$-entropy is the usual Shannon entropy $${H_{1}}[X](P) = -\sum_{x\in S_X} P(X=x) \ln P(X=x),$$ whereas $\alpha\neq 1$ implies $${H_{\alpha}}[X](P)= \frac{1}{\alpha-1} \left( 1-\sum_{x\in S_X} P(X=x)^\alpha \right).$$ This function appears in the literature under several denominations: it was introduced by Havrda and Charvát [@havrda] as structural $\alpha$-entropy, Aczél and Daróczy [@aczel1975measures] call it generalized information function of degree $\alpha$, but by far the most common name is Tsallis $\alpha$-entropy,[^3] because Tsallis popularized its use in statistical mechanics.
Given a second variable $Y:\Omega \to S_Y$ and a law $P$ for the pair $(X,Y)$, the $\alpha$-entropy satisfy the equations $$\label{cocycle_eqns}
\begin{split}
{H_{\alpha}}[(X,Y)](P) { }={ }& {H_{\alpha}}[X](X_* P) \\
& + \sum_{x\in S_X} P(x)^\alpha {H_{\alpha}}[Y](Y_* P|_{X=x})
\end{split}$$ where $P(x) := P(\{X=x\})$, the symbol $P|_{X=x}$ denotes the conditional law, and $X_*Q$ is the push-forward of the law $Q$ on $S_X\times S_Y$ under the canonical projection $\pi_X :S_X\times S_Y \to S_X$. We have shown in [@vigneaux2017generalized] that ${H_{\alpha}}[\cdot]$ is the only family of measurable real-valued functions that satisfy these functional equations for generic collections of random variables and probabilities, up to a multiplicative constant $K$. The case $\alpha = 1$ is already treated in [@bennequin]. Of course, this depends on a long history of axiomatic characterizations of entropy that begins with Shannon himself, see [@shannon1948; @aczel1975measures; @csiszar2008axiomatic].
In particular, if $X$, $Y$ represent the possibles states of two independent systems (e.g. physical systems, random sources), in the sense that $P(x,y) = X_*P(x) Y_*P(y)$, then $$\label{additivity}
{H_{1}}[(X,Y)](P) = {H_{1}}[X](X_*P)+{H_{1}}[Y](Y_*P).$$ This property of Shannon entropy is called additivity. Under the same assumptions, Tsallis entropy verifies (for $K=1$): $$\label{non-additivity}
\begin{split}
{H_{\alpha}}[(X,Y)](P) { }={ }& {H_{\alpha}}[X](X_*P)+{H_{\alpha}}[Y](Y_*P) \\
&- (\alpha-1) {H_{\alpha}}[X](X_*P){H_{\alpha}}[Y](Y_*P).
\end{split}$$ It is said that Tsallis entropy is nonadditive.[^4] This property is problematic from the point of view of heuristic justifications for information functions, that have always assumed as ‘intuitive’ that the amount of information given by two independent events should be computed as the sum of the amounts of information given by each one separately (this explains the use of the logarithm to define the surprise).
The initial motivation behind this paper was to understand better these generalized information functions of degree $\alpha$. Tsallis used them as the foundation of nonextensive statistical mechanics, a generalization of Boltzmann-Gibbs statistical mechanics that was expected to describe well some systems with long-range correlations. It is not completely clear which kind of statistical systems follow these “generalized statistics”.[^5] There is extensive empirical evidence about the pertinence of the predictions made by nonextensive statistical mechanics [@tsallis-book]. However, very few papers address the microscopical foundations of the theory (for instance, [@ruiz2015emergence; @hanel2011generalized; @jensen2016statistical]). We present here a novel approach in this direction, based on the combinatorics of flags, but only for the case $\alpha = 2$. However, we indicate in the last section how these ideas could be extended to other cases.
There is a connection between the combinatorial and algebraic characterizations of entropy, that we describe in Section \[sec:combinatorial\_shannon\] (Shannon entropy) and Section \[sec:combinatorial\_explanation\] (quadratic entropy). The well known multiplicative relations at the level of multinomial coefficients shed new light on additivity/nonadditivity. In the simplest case, let $(p_0,p_1)$, $(q_0,q_1)$ be two probability laws on $\{0,1\}$; then $$\label{multiplicative_relations}
{n \choose p_0q_0 n,p_0q_1 n,p_1q_0n,p_1q_1n} = {n \choose p_0n} {p_0n \choose p_0q_0n }{p_1n \choose p_1q_0n}.$$ Applying $\frac 1n \ln (-)$ to both sides and taking the limit $n\to \infty$, we recover . Equation remains valid for the $q$-multinomial coefficients, but in this case one should apply $\lim_n \frac{2}{n^2} \log_q(-)$ to both sides to obtain the quadratic entropy: $$\begin{aligned}
\label{non-add-binary}
{H_{2}} & (p_0q_0,p_0q_1,p_1q_0,p_1q_1) \\&= {H_{2}}(p_0,p_1) + p_0^2 {H_{2}}(q_0,q_1) + (1-p_0)^2 {H_{2}}(q_0,q_1) \\
&= {H_{2}}(p_0,p_1) + {H_{2}}(q_0,q_1) - {H_{2}}(p_0,p_1){H_{2}}(q_0,q_1).\end{aligned}$$ Thus, asymptotically, the number of flags $V_{00}\subset V_{01} \subset V_{10}\subset V_{11} = {\mathbb{F}}_q^n$ of type $(p_0q_0 n,p_0q_1 n,p_1q_0n,p_1q_1n)$ can be computed in terms of the number of flags $W_0\subset W_1={\mathbb{F}}^n_q$ of type $(p_0n,p_1n)$ and those flags $W'_0 \subset W_1' = {\mathbb{F}}_q^m$ of type $(q_0m,q_1m)$ |where $m$ can take the values $p_0n$ or $p_1n$| through this nonadditive formula.
A $q$-deformation of Shannon’s theory
-------------------------------------
The second part of this paper builds a generalization of Shannon’s theory where messages are vector spaces. Formula already suggests that the quadratic entropy plays an essential role in it.
In fact, the corresponding formula is of great importance in Shannon’s theory. Consider a random source that emits at time $n\in {\mathbb{N}}$ a symbol $Z_n$ in $S_Z=\{z_1,...,z_s\}$, each $Z_n$ being an independent realization of a $S_Z$-valued random variable $Z$ with law $P$. A *message* (at time $n$) corresponds to a random sequence $(Z_1,...,Z_n)$ taking values in $S_Z^n$ with law $P^{\otimes n}$. The *type* of a sequence ${\mathbf {z}}\in S_Z^n$ is the probability distribution on $S_Z$ given by the relative frequency of appearance of each symbol in it; for example, when $S_Z = \{0,1\}$, the type of a sequence with $k$ ones is $(1-\frac k n) \delta_0 + \frac k n\delta_1$. A “typical sequence” is expected to have type $P$, and therefore its probability $P^{\otimes n}({\mathbf {z}})$ is approximately $\prod_{z\in S_Z} P(z)^{nP(z)} = \exp\{-n {H_{1}}[Z](P)\}$. The cardinality of the set of sequences of type $P$ is ${n \choose P(z_1)n,...,P(z_s)n} \approx \exp\{n {H_{1}}[Z](P)\}$. This implies, according to Shannon, that “it is possible for most purposes to treat the long sequences as though there were just $2^{Hn}$ of them, each with a probability $2^{-Hn}$" [@shannon1948 p. 397]. This result is known nowadays as the asymptotic equipartition property (AEP), and can be stated more precisely as follows [@cover2006elements Th. 3.1.2]: given ${\varepsilon}>0$ and $\delta>0$, it is possible to find $n_0 \in {\mathbb{N}}$ and sets $\{A_n\}_{n\geq n_0}$, $A_n \subset S_Z^n$, such that, for every $n\geq n_0$,
1. $P^{\otimes n}(A_n^c) < {\varepsilon}$, and
2. for every ${\mathbf {z}} \in A_n$, $$\label{eq:class_AEP_equiprobability}
\left|\frac 1n \ln(P^{\otimes n}({\mathbf {z}})) - {H_{1}}[Z](P)\right| < \delta.$$
Furthermore, if $s(n,{\varepsilon})$ denotes $$\min {\{\,|B_n| \, : \, B_n\subset S_Z^n \text{ and } {\mathbb{P}\left((Z_1,...,Z_n) \in B_n\right)} \geq 1-{\varepsilon}\,\} },$$ then $$\lim_n \frac{1}{n} \ln |A_n| = \lim_n \frac{1}{n} \ln s(n,{\varepsilon}) = {H_{1}}[Z](P).$$ The set $A_n$ can be defined to contain all the sequences whose type $Q$ is close to $P$, in the sense that $\sum_{z\in S_Z} |Q(z)-P(z)|$ is upper-bounded by a small quantity; this is known as *strong typicality* (see [@csiszar1981information Def. 2.8]).
Similar conclusions can be drawn for a system of $n$ independent physical particles, the state of each one being represented by a random variable $Z_i$; in this case, the vector $(Z_1,...,Z_n)$ is called a *configuration*. The set $A_n$ can be thought as an approximation to the effective phase space (“reasonable probable” configurations) and the entropy as a measure of its size, see [@jaynes1965gibbs Sec. V]. In both cases |messages and configurations| the underlying probabilistic model is a process $(Z_1,...,Z_n)$ linked to the multinomial distribution, and the AEP is merely a result on measure concentration around the expected type. We envisage a new type of statistical model, such that a message at time $n$ (or a configuration of $n$ particles) is represented by a flag of vector spaces $V_1 \subset V_2 \subset ... \subset V_s = {\mathbb{F}}_q^n$. In the simplest case ($s=2$) a message is just a vector space $V$ in ${\mathbb{F}}_q^n$. While the type of a sequence is determined by the number of appearances of each symbol, the type of a flag is determined by its dimensions or |equivalently| by the numbers $(k_1,...,k_s)$ associated to it; by abuse of language, we refer to $(k_1,...,k_s)$ as the type. The cardinality of the set of flags $V_1 \subset...\subset V_s \subset{\mathbb{F}}_q^n$ that have type $(k_1,...,k_s)$ is ${ {n \brack k_1,...,k_s}_q } \sim C(q) q^{n^2 {H_{2}}(k_1/n,...,k_s/n)/2}$, where $C(q)$ is an appropriate constant.
To build a correlative of Shannon’s theory of communication, it is fundamental to have a probabilistic model for the source. In our case, this means a random process $\{F_i\}_{i\in {\mathbb{N}}}$ that produces at time $n$ a flag $F_n$ that would correspond to a generalized message. We can define such process if we restrict our attention to the binomial case ($s=2$). This is the purpose of Section \[sec:dynamical\_model\].
Let $\theta$ be a positive real number, and let $\{X_i\}_{i\geq 1}$ be a collection of independent random variables that satisfy $X_i\sim {\operatorname{Ber}}\left(\frac{\theta q^{i-1}}{1+\theta q^{i-1}}\right)$, for each $i$. We fix a a sequence of linear embeddings ${\mathbb{F}}_q^1 \hookrightarrow {\mathbb{F}}_q^2 \hookrightarrow ...$, and identify ${\mathbb{F}}_q^{n-1}$ with its image in ${\mathbb{F}}_q^n$. We define then a stochastic process $\{V_i\}_{i\geq 0}$ such that each $V_i$ is a vector subspace of ${\mathbb{F}}_q^i$, as follows: $V_0 = 0$ and, at step $n$, the dimension of $V_{n-1}$ increases by $1$ if and only if $X_n = 1$; in this case, $V_n$ is picked at random (uniformly) between all the $n$-dilations of $V_{n-1}$. When $X_n=0$, one sets $V_n = V_{n-1}$. The $n$-dilations of a subspace $w$ of ${\mathbb{F}}_q^{n-1}$ are defined as $$\begin{split}
{\operatorname{Dil}}_n (w) = {\{\,v\subset {\mathbb{F}}_q^n \, : \, & \dim v - \dim w = 1, \\ & w\subset v\text{ and } v \not \subset {\mathbb{F}}_q^{n-1}\,\} }.
\end{split}$$ We prove that, for any subspace $v\subset {\mathbb{F}}_q^n$ of dimension $k$, ${\mathbb{P}\left(V_n = v\right)}=\frac{\theta^k q^{k(k-1)/2}}{(-\theta;q)_n}$. This implies that ${\mathbb{P}\left(\dim V_n =k\right)} ={ {n \brack k}_q } \frac{\theta^k q^{k(k-1)/2}}{(-\theta;q)_n}$, which appears in the literature as $q$-binomial distribution. (We have used here the $q$-Pochhammer symbols $(a;q)_n := \prod_{i=0}^{n-1} (1-aq^i)$, with $(a;q)_0 = 1$.)
[|C|C|C|]{} **Concept** & **Shannon case** & **$q$-case**\
Message at time $n$ ($n$-message) & Word $w\in\{0,1\}^n$ & Vector subspace $v\subset F_q^n$\
Type & Number of ones & Dimension\
Number of $n$-messages of type $k$ & $\displaystyle{n\choose k} $ & $\displaystyle{ {n \brack k}_q }$\
Probability of a $n$-message of type $k$ & $\displaystyle \xi^k (1-\xi)^{n-k}$ & $\displaystyle\frac{\theta^k q^{k(k-1)/2}}{(-\theta;q)_n}$\
For the multinomial process, the probability $P^{\otimes n}$ concentrates on types close to $P$ i.e. appearances close to the expected value $nP(z)$, for each $z\in S_Z$. In the case of $V_n$, the probability also concentrates on a restricted number of dimensions (types). In fact, it is possible to prove an analog of the asymptotic equipartition partition property; this is the main result of this work, Theorem \[them:AEP\]. It can be paraphrased as follows:
*for every $\delta>0$ and almost every ${\varepsilon}>0$ (except a countable set), there exist $n_0\in {\mathbb{N}}$ and sets $A_n = \bigcup_{k=0}^{ \Delta(p_{\varepsilon})} {\operatorname{Gr}}(n-k,n)$, for all $n\geq n_0$, such that $\Delta(p_{\varepsilon})$ is an integer that just depends on ${\varepsilon}$, ${\mathbb{P}\left(V_n\in A_n^c\right)} \leq {\varepsilon}$ and, for any $v\in A_n$ such that $\dim v = k$, $$\left| \frac{\log_q ({\mathbb{P}\left(V_n = v\right)}^{-1})}{n} - \frac{n}{2}{H_{2}}(k/n) \right| \leq \delta.$$ Moreover, the size of $A_n$ is optimal, up to the first order in the exponential: let $s(n,{\varepsilon})$ denote $$\min{\{\,|B_n| \, : \, B_n\subset {\operatorname{Gr}}(n) \text{ and } {\mathbb{P}\left(V_n \in B_n\right)} \geq 1-{\varepsilon}\,\} },$$ then $$\begin{aligned}
\lim_n \frac{1}{n} \log_q|A_n| &{ }={ } \lim_n \frac{1}{n} \log_q s(n,{\varepsilon}) \\
&{ }={ } \lim_n \frac{n}{2} {H_{2}}(\Delta(p_{\varepsilon})/n) \\
&{ }={ } \Delta(p_{\varepsilon}). \end{aligned}$$*
The set $A_n$ correspond to the “typical subspaces”, in analogy with the typical sequences introduced above. We close Section \[sec:generalized\_info\] with an application of this theorem to source coding.
Notation
--------
The statement $A:=B$ means that $A$ is defined to be $B$, as well as $B=:A$. For us ${\mathbb{N}}=\{0,1,2,3,...\}$. The symbols ${\mathbb{R}}$, ${\mathbb{C}}$ and ${\mathbb{F}}_q$ denote respectively the field real numbers, the field of complex numbers, and the Galois field of order $q$. The multinomial coefficients are denoted by ${n \choose k_1,...,k_s}$, and ${n\choose k} := {n \choose k, n-k}$ are the binomial coefficients. The $q$-multinomial coefficients ${ {n \brack k_1,...,k_s}_q }$ are defined in Section \[subsec:q\_binomials\_def\], together with the $q$-binomial coefficients ${ {n \brack k}_q }$. The discrete interval $\llbracket a, b\rrbracket$ corresponds to ${\mathbb{Z}}\cap [a,b]$ as a subset of ${\mathbb{R}}$. We use Iverson’s convention for characteristic functions [@knuth1992two]: for any proposition $p$, $$[p]:=\begin{cases} 1 & \text{if } p\text{ is true}\\ 0 & \text{if } p\text{ is false} \end{cases}.$$ Therefore the indicator functor of a set $B$ corresponds to $x\mapsto [x\in B]$. The symbol $\log_q$ signifies the usual logarithm in base $q$, as opposed to Tsallis’ deformed $\alpha$-logarithm $\ln_\alpha$. Finally, $f_n\sim g_n$ means $f_n/g_n\to 1$ as $n\to \infty$
Combinatorial characterization of Shannon’s information {#sec:combinatorial_shannon}
=======================================================
Let $X$ be a finite random variable that takes values in the set $S_X=\{x_1,...,x_s\}$. We suppose that, among $N$ independent trials of the variable $X$, the result $x_i$ appears $N(x_i)$ times, for each $i$. Evidently, $\sum_i N(x_i)=N$.
The number of sequences in $(S_X)^N$ that agree with the prescribed counting $(N(x_1),...,N(x_s))$ is given by the multinomial coefficient $$\label{multinomial}
{N \choose \{N(x_k)\}_{k=1}^s } := \frac{N!}{N(x_1)! \cdots N(x_s)!}.$$
But we could also reason iteratively. Let us consider a partition of $\{x_1,...,x_s\}$ in $t$ disjoint sets, denoted $Y_1,..., Y_t$. These can be seen as level sets of a new variable $Y$, taking values in a set $S_Y=\{y_1,...,y_t\}$; by definition, $\{Y=y_t\}=Y_t$. There is surjection $\pi:S_X\to S_Y$ that sends $x\in S_X$ to the unique $y\in S_Y$ such that $x\in \{Y=y\}$. The probability $\nu(x_i) = N(x_i)/N$ on $S_X$ can be pushed-forward under this surjection; the resulting law $\pi_* \nu$ satisfies $\pi_*\nu (y) = \sum_{x\in \pi^{-1}(y)} \nu(x)$. Our counting problem can be solved as follows: count first the number of sequences in $(S_X)^N$ such that $N\pi_*\nu(y_i)$ values correspond to the group $y_i$, for $i\in \{1,...,t\}$. This equals $${N \choose \{N\pi_*\nu(y_i)\}_{i=1}^t }.$$ Then, for each group $\pi^{-1}(y_i)\equiv \{Y=y_i\}$, count the number of sequences of length $N\pi_*\nu(y_i)$ (subsequences of the original ones of length $N$) such that every $x_j \in \pi^{-1}(y_i)$ appears $N(x_j)$ times. These are $${N\pi_*\nu(y_i) \choose \{N(x_j)\}_{x_j \in \pi^{-1}(y_i)} }.$$ In total, the number of sequences of length $N$ such that $x_k$ appears $N(x_k)$ times, for every $k\in \{0,...,s\}$, are $$\label{discrete_cocyle}
{N \choose \{N\pi_*\nu(y_i)\}_{i=1}^t } \prod_{i=1}^t {N\pi_*\nu(y_i) \choose \{N(x_j)\}_{x_j \in \pi^{-1}(y_i)} }.$$ The considerations above give the identity $$\begin{gathered}
\label{multiplicative_identity_multinomials}
{N \choose \{N(x_k)\}_{k=1}^s } = \\ {N \choose \{N\pi_*\nu(y_i)\}_{i=1}^t } \prod_{i=1}^t {N\pi_*\nu(y_i) \choose \{N(x_j)\}_{x_j \in \pi^{-1}(y_i)} }.\end{gathered}$$ This can be rephrased as follows: the multinomial expansion of $(x_1 + \cdots + x_s)^N$ and the iterated multinomial expansion of $( \sum_{y_i} ( \sum_{x_j\in \pi^{-1}(y_i)} x_j ))^N$ assign the same coefficient to $x_1^{N(x_1)}x_2^{N(x_2)}\cdots x_s^{N(x_s)}$.
Equation implies that $$\label{additive_relation_preH}
\begin{split}
\frac{1}{N} &\log {N \choose \{N(x_k)\}_{k=1}^s } { }={ } \frac{1}{N}\log {N \choose \{N\pi_*\nu(y_i)\}_{i=1}^t } \\
& + \sum_{i=1}^t \pi_*\nu(y_i)\frac{1}{N\pi_*\nu(y_i)} \log {N\pi_*\nu(y_i) \choose \{N(x_j)\}_{x_j \in \pi^{-1}(y_i)} }.
\end{split}$$ We can see this as a discrete analog of the third axiom of Shannon. The connection is made explicit by means of the following proposition.
\[log-asymptotics-H1\] Let $N$ be a natural number and $\{N(i)\}_{i=0}^s$ such that $\sum_{i=0}^s N(i)= N$. Suppose that $N(i)/N \to \mu_i \in [0,1]$ as $N\to \infty$, for all $i$. Then $$\lim_{N\to \infty} \frac{1}{N} \ln {N \choose N(0), ...., N(s) } = {H_{1}}(\mu_0,\cdots, \mu_s),$$ where ${H_{1}}$ denotes Shannon entropy: $${H_{1}}(\mu_0,\cdots, \mu_s) = -\sum_{i=0}^{s} \mu_i \ln \mu_i.$$
This is a standard result. See for example [@mori Lemma 4.1].
By convention, $0\ln 0 = 0$. If we take the limit of under the hypotheses of the previous proposition, we obtain $$\begin{gathered}
\label{functional_equations}
{H_{1}}(\mu(x_1),...,\mu(x_s)) = {H_{1}}(\pi_*\mu(y_1),...,\pi_*\mu(y_t)) \\ + \sum_{i=0}^t \pi_*\mu(y_i) {H_{1}}(\mu|_{Y=y_i}(x_1),...,\mu|_{Y=y_i}(x_s)).\end{gathered}$$
Consider now the particular case $X=(Z,Y)$, for certain random variable $Z$ taking values on $S_Z$. We use the notations introduced in Section \[sec:two\_faces\]. Since the support of $\mu|_{Y=y_i}$ is $S_Z\times \{y_i\}$, isomorphic to $S_Z$ by the natural projection $\pi_Z: S_Z\times S_Y \to S_Z$, there is a clear identification of ${H_{1}}(\mu|_{Y=y_i}(x_1),...,\mu|_{Y=y_i}(x_s))$ with $H[Z](Z_* \mu|_{Y=y_i})$. Therefore, reads $$\label{functional_equations_fancy}
{H_{1}}[(Z,Y)](\mu) = {H_{1}}[Y](Y_*\mu) + \sum_{i=0}^t Y_*\mu(y_i) {H_{1}}[Z](Z_* \mu|_{Y=y_i}).$$ This proves combinatorially that Shannon entropy satisfy all the functional equations of the form . The ensemble of these equations |for a given family of finite sets and surjections between them| constitute a cocycle condition in information cohomology (see [@bennequin] and [@vigneaux2017generalized]). These are functional equations that have as unique solution Shannon entropy.
In the following sections, we explore a generalization of the previous argument. We replace the multinomial coefficient with their $q$-deformation, that satisfy the same multiplicative relations. These $q$-multinomial coefficients are related asymptotically to the quadratic entropy.
The $q$-multinomial coefficients {#sec:q_binomials}
================================
We introduce here the combinatorial objects and results used throughout this article. In Section \[subsec:q\_binomials\_def\], we define the $q$-multinomial coefficients, that are associated to the enumeration of flags of finite vector spaces. Section \[sec:asymptotic\_behavior\] studies their asymptotic behavior and establishes the connection with the quadratic entropy. Sections \[sec:combinatorial\_explanation\] and \[sec:max\_ent\] are mutually independent and not essential to understand the rest of the paper: the former uses the asymptotic results to obtain a combinatorial explanation for the nonadditivity of Tsallis $2$-entropy, and the later discuss a combinatorial justification of the maximum entropy principle with Tsallis entropy.
Definition {#subsec:q_binomials_def}
----------
Let $q$ be an indeterminate. Given $(n, k_1,...,k_s)\in {\mathbb{N}}^{s+1}$ such that $\sum_{i=1}^s k_i = n$, the $q$-multinomial coefficient ${{ {n \brack k_1,...,k_s}_q }}$ is defined by the formula $${ {n \brack k_1,...,k_s}_q } := \frac{[n]_q!}{[k_1]_q! \cdots [k_s]_q!}.$$ We have used the notation for $q$-factorials introduced in Section \[sec:intr\].
Throughout this paper, we shall assume that $q$ is a fixed prime power. For such $q$, the $q$-binomial coefficient ${ {n \brack k}_q } \equiv { {n \brack k,n-k}_q }$ counts the number of $k$-dimensional subspaces in ${\mathbb{F}}_q^n$. More generally, given a set of integers $k_1,...,k_s$ such that $\sum_{i=1}^s k_i = n$, the $q$-multinomial coefficient ${ {n \brack k_1,...,k_s}_q }$ equals the number of flags $V_1\subset V_2 \subset \cdots \subset V_{s-1} \subset V_s = {\mathbb{F}}_q^n$ of vector spaces such that $\dim V_j = \sum_{i=1}^j k_i$ [@Prasad2010-1; @Prasad2010-2]. We will say that these flags are of *type* $(k_1,...,k_s)$.
It is possible to introduce a function $\Gamma_q$ as the normalized solution of a functional equation that guaranties that $[n]_q ! = \Gamma_q(n+1)$, see [@askey]. When $q>1$ and $x>0$, this function is given by the formula [@moak]: $$\begin{aligned}
\label{qGamma}
\Gamma_q(x) &= (q^{-1};q^{-1})_\infty q^{{x\choose 2}} (q-1)^{1-x} \sum_{n=0}^\infty \frac{q^{-nx}}{(q^{-1};q^{-1})_n} \\ &= \frac{(q^{-1};q^{-1})_\infty q^{{x\choose 2}} (q-1)^{1-x}}{(q^{-x};q^{-1})_\infty} ,\end{aligned}$$ where we have used the Pochhammer symbol $$(a;x)_n := \prod_{k=0}^{n-1} (1-ax^k), \qquad (a;x)_0 = 1.$$ The equivalent expressions for the function $\Gamma_q$ come from the identity $$\label{eq:q-binomial-theorem}
\frac{(ax;q)_\infty}{(x;q)_\infty} =\sum_{n=0}^\infty \frac{(a;q)_n}{(q;q)_n}x^n \qquad(|q|<1),$$ known as $q$-binomial theorem (see [@kac2001quantum p. 30]).
Recall [@knopp p. 92] that an infinite product $\prod_{i=0}^\infty u_i$ is said to be convergent if
1. there exists $i_0$ such that $u_i\neq 0$ for all $i>i_0$;
2. $\lim_{n\to \infty} u_{i_0+1} \cdots u_{i_0 +n}$ exists and is different from zero.
An infinite product in the form $\prod (1+c_i)$ is said to be absolutely convergent when $\prod (1+|c_i|)$ converges. One can show that absolute convergence implies convergence. Moreover, when the terms $\gamma_i \geq 0$, the product $\prod_i (1 + \gamma_i)$ is convergent if and only if the series $\sum_i \gamma_i$ converges. The convergence of $\sum_i 1/q^i$ gives then the following result, that is used without further comment throughout the paper.
For every $a\in {\mathbb{C}}$, the product $(a;q^{-1})_\infty$ converges. Moreover, if $a\not\in {\{\,q^i \, : \, i\geq 0\,\} }$, then $(a;q^{-1})_\infty \neq 0$.
The $\Gamma_q$ function gives an alternative expression for the $q$-multinomial coefficients $$\label{q-coeff-qgamma}
{ {n \brack k_1,...,k_s}_q } = \frac{\Gamma_q(n+1)}{\Gamma_q(k_1+1) \cdots \Gamma_q(k_s+1)},$$ which in turn extends its definition to complex arguments. We close this subsection with a remark on the unimodality of the $q$-binomial coefficients.
\[lemma:unimodality\_binomial\] For every $n\in {\mathbb{N}}$,
- ${ {n \brack 0}_q } < { {n \brack 1}_q } < \ldots < { {n \brack \floor{n/2}}_q },$
- $ { {n \brack \floor{n/2}}_q } = { {n \brack \ceil{n/2}}_q }, $
- $ { {n \brack \ceil{n/2}}_q } > \ldots { {n \brack n-1}_q } > { {n \brack n}_q }.$
Consider the quotient $$q(n,k):= \frac{{ {n \brack k+1}_q }}{{ {n \brack k}_q }} = \frac{[n-k]_q}{[k+1]_q}.$$ Then, $q(n,k)\geq 1$ iff $q^{n-k}\geq q^{k+1}$ iff $k \leq \frac{n-1}{2}$, with equality just in the case $k=\frac{n}{2}-\frac 12=\floor{n/2}$ (when $n$ is odd).
Asymptotic behavior {#sec:asymptotic_behavior}
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The quadratic entropy ${H_{2}}$ of a probability law $(\mu_1,...,\mu_s)$ is defined by the formula: [^6] $${H_{2}}(\mu_1,\cdots, \mu_s) := 1- \sum_{i=1}^s \mu_i^2.$$
\[prop:asymptotic\_behavior\] For each $n\in {\mathbb{N}}$, let $\{k_i(n)\}_{i=1}^s $ be a set of positive real numbers such that $\sum_{i=0}^s k_i= n$ (we write $k_i$ when $n$ is clear from context). Suppose that, for each $i\in\{1,...,s\}$, it is verified that $k_i(n)\to l_i \in [0,\infty]$ as $n\to \infty$. Then, $$\begin{gathered}
{ {n \brack k_1,...,k_s}_q } \sim \\(q^{-1};q^{-1})_\infty^{1- s}\prod_{i =1}^s (q^{-(l_i +1)};q^{-1})_\infty q^{n^2 {H_{2}}(\frac {k_1}{n},...,\frac{k_s}{n})/2}.\end{gathered}$$
Recall that $f_n\sim g_n$ means $f_n/g_n\to 1$ as $n\to \infty$. By convention, $(q^{-(\infty +1)};q^{-1})_\infty = 1$.
See Appendix \[proof\_prop\_asymptotic\_q\_multinomial\].
When $f_n$ and $g_n$ are positive, $f_n \sim g_n$ implies that $\lim_n \frac{1}{n}(\log_q f_n - \log_q g_n) = 0$. For instance, we can deduce that, for any fixed $\Delta \in {\mathbb{N}}$, $$\label{computation_symptotics_binomial}
\lim_n \frac 1n \log_q { {n \brack n-\Delta}_q } = \lim_n \frac n 2 {H_{2}}(\Delta/n) = \Delta,$$ where the last equality comes from a direct computation.
As an immediate application of Theorem \[prop:asymptotic\_behavior\], we obtain the equivalent of Proposition \[log-asymptotics-H1\].
For each $n\in {\mathbb{N}}$, let $\{k_i(n)\}_{i=1}^s $ be a set of positive real numbers such that $\sum_{i=0}^s k_i= n$ (we write $k_i$ when $n$ is clear from context). Suppose that $k_i/n \to \mu_i \in [0,1]$ as $n\to \infty$, for all $i$. Then $$\lim_{n\to \infty} \frac{2}{n^2} \log_q { {n \brack k_1, ...., k_s }_q } = {H_{2}}(\mu_1,\cdots, \mu_s).$$
If $f/g\to 1$, then $\log_q(f/g) \to 0$. Therefore, $$\begin{gathered}
\log_q{ {n \brack k_1,...,k_s}_q } - \log_q\left(\frac{(q^{-1};q^{-1})_\infty^{1- s}}{\prod_{i =1}^s (q^{-(l_i +1)};q^{-1})_\infty}\right) \\ - \frac{n^2}{2} {H_{2}}\left(\frac{k_1}{n},...,\frac{k_s}{n}\right) = o(1).\end{gathered}$$ Multiply this by $2/n^2$ and use the continuity of ${H_{2}}$ to conclude.
Combinatorial explanation for nonadditivity of Tsallis 2-entropy {#sec:combinatorial_explanation}
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Additivity corresponds to the following property of Shannon entropy: if $X$ is a random variable with law $P=\{p_x\}_{x\in S_X}$ and $Y$ another with law $Q=\{q_y\}_{y\in S_Y}$, independent of $X$, then the joint variable $(X,Y)$ has law $P\otimes Q=\{p_xq_y\}_{(x,y)\in S_X\times S_Y}$ and $$\label{additivity_H1}
{H_{1}}[(X,Y)](P\otimes Q) = {H_{1}}[X](P) + {H_{1}}[Y](Q).$$
For simplicity (the arguments work in general), we suppose that $X$, $Y$ are binary variables, i.e. $S_X=S_Y = \{0,1\}$. Consider the sequences counted by ${N\choose N_{00}, N_{01}, N_{10}, N_{11}}$; they are the possible results of $N$ independent trials of the variable $(X,Y)$, under the assumption that the result $(i,j)$ is obtained $N_{ij}$ times, for each $(i,j)\in \{0,1\}^2$. We treat the particular case $N_{ij} = p_iq_jN$, that correspond to the expected number of appearances of $(i,j)$. The independence between $Y$ and $X$ means that, given $N_0:=N_{00}+N_{01} = p_0 N$ occurrences of $X=0$ (resp. $N_1:=N_{10}+N_{11}=p_1N$ occurrences of $X=1$) in the sequences of length $N$ counted above, there are $q_0N_i$ occurrences of $Y=0$ and $q_1N_i$ occurrences of $Y=1$ in the corresponding subsequence defined by the condition $X=i$, irrespective of the value of $i$. In this case, specializes to $$\label{iterative_multinomial}
{N\choose N_{00}, N_{01}, N_{10}, N_{11}} = {N \choose N_0} {N_0 \choose q_0 N_0 }{N_1 \choose q_0N_1}.$$ Applying $\frac 1N \ln (-)$ to both sides and taking the limit $N\to \infty$, we recover . (This is just a particular case of the computations in Section \[sec:combinatorial\_shannon\].)
In the $q$-case, ${ {N \brack N_{00}, N_{01}, N_{10}, N_{11}}_q }$ counts the number of flags $V_{00}\subset V_{01}\subset V_{10}\subset V_{11} = {\mathbb{F}}_q^n$ of type $(N_{00}, N_{01}, N_{10}, N_{11})$. When $N_{ij} = p_iq_j N$, such a flag can be determined by an iterated choice of subspaces, whose dimensions are chosen independently: pick first a subspace $V_0 \subset {\mathbb{F}}_q^n$ of dimension $N_0=N_{00}+N_{01} = p_0 N$ (there are ${ {N \brack N_0}_q }$ of those) and then pick a subspace of dimension $ q_0N_0 \subset V_0$ and another subspace of dimension $q_0N_1$ in ${\mathbb{F}}_q^n/V_0$. This corresponds to the combinatorial identity $$\label{iterative_qmultinomial}
{ {N \brack N_{00}, N_{01}, N_{10}, N_{11}}_q } = { {N \brack N_0}_q } { {N_0 \brack q_0 N_0}_q } { {N_1 \brack q_0 N_1}_q }.$$ Applying $\frac 2{N^2} \log_q (-)$ to both sides and taking the limit $N\to \infty$, we obtain $$\begin{aligned}
{H_{2}} & (p_0q_0,p_0q_1,p_1q_0,p_1q_1) \\&= {H_{2}}(p_0,p_1) + p_0^2 {H_{2}}(q_0,q_1) + (1-p_0)^2 {H_{2}}(q_0,q_1) \\
&= {H_{2}}(p_0,p_1) + {H_{2}}(q_0,q_1) - {H_{2}}(p_0,p_1){H_{2}}(q_0,q_1).\end{aligned}$$ In both cases, the trees that represent the iterated counting are the same, see Fig. \[fig:tree\] (and compare this with Figure 6 in Shannon’s paper [@shannon1948]). The main difference lies in the exponential growth of the combinatorial quantity of interest and how the correspondent exponents are combined. In the $q$-case, even if you choose the dimensions in two independent steps, the exponents do not simply add; in fact, the counting of sequences is nongeneric in this respect. Remark also that the interpretation of probabilities as relative *frequencies* of symbols only make sense for the case of words; more generally they correspond to ratios or relative proportions.
=\[level distance=4cm, sibling distance=2cm\] =\[level distance=3cm, sibling distance=1.5cm\]
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child [ node\[end\] child [ node\[end\] edge from parent node\[below\] [$q_0$]{} ]{} child [ node\[end\] edge from parent node\[above\] [$q_1$]{} ]{} edge from parent node\[below\] [$p_0$]{} ]{} child [ node\[end\] child [ node\[end\] edge from parent node\[below\] [$q_0$]{} ]{} child [ node\[end\] edge from parent node\[above\] [$q_1$]{} ]{} edge from parent node\[above\] [$p_1$]{} ]{};
Maximum entropy principle {#sec:max_ent}
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In the simplest models of statistical mechanics, one assumes that the system is composed of $n$ particles, each one in certain state from a finite set $S=\{s_1,...,s_m\}$ (in certain contexts, the elements of $S$ are called *spins*). A configuration of the system is a feasible vector ${\mathbf {x}}\in S^n$; when all particles are independent, $S^n$ is the sets of all configurations.
We have in mind a new type of statistical mechanics, where a configuration of the $n$ particle system is represented by a flag of vector spaces $V_1 \subset V_2 \subset ... \subset V_m = {\mathbb{F}}_q^n$.
In the classical case of independent particles, the total energy of a configuration ${\mathbf {x}}$ just depends on its type $(k_i)_{1\leq i \leq m}$, where $k_i$ is the number of appearances of the symbol $s_i$ in ${\mathbf {x}}$. In fact, the mean (internal) energy is $\sum_{i=1}^m \frac{k_i}{n} E_i$, where $E_i \in {\mathbb{R}}$ is the energy associated to the spin $s_i$. Setting $E_{m+1}=0$, $\tilde E_i = E_i - E_{i+1}$ and $r_i = \sum_{j=1}^i k_j$, one can write $\sum_{i=1}^n \frac{r_i}{m} \tilde E_i$ instead of $\sum_{i=1}^m \frac{k_i}{n} E_i$.
Now we plan to move beyond independence, so it is convenient to see the energy as a “global” function that depends on the type of the sequence. We assume now that the energy associated to a flag of vector spaces $V_1 \subset V_2 \subset ... \subset V_m = {\mathbb{F}}_q^n$ just depends on its type $(k_1,...,k_m)$ and is of the form $$\sum_{i=1}^m \frac{k_i}{n} E_i=\sum_{i=1}^n \frac{r_i}{m} \tilde E_i = \sum_{i=1}^m \frac{(\dim V_i)}{n} \tilde E_i$$ where $r_i = \sum_{j=1}^i k_j$, as before.
In general, if $n>1$, the equations $$\begin{aligned}
\sum_{i=1}^m \frac{k_i}{n} E_i &= \langle E \rangle \label{ME_constraints1}\\
\sum_{i=1}^m k_i &= n, \label{ME_constraints2}\end{aligned}$$ where $\langle E \rangle\in {\mathbb{R}}$ is a prescribed mean energy, do not suffice to determine the type $(k_1,...,k_m)$ and an additional principle must be introduced to select the “best” estimate: the *principle of maximum entropy* [@jaynes]. This principle—attributed to Boltzmann and popularized by Jaynes—states that, between all the types that satisfy and , we should select the one that corresponds to the greatest number of configurations of the system. This means that we must maximize $$W(k_1,...,k_m):= { {n \brack k_1, k_2,...,k_m}_q }$$ under the constraints and . The maximization of $W(k_1,...,k_m)$ is equivalent to the maximization of $2\log_q W(k_1,...,k_m)/n^2$; as $n\to \infty$, the latter quantity approaches ${H_{2}}(g_1,...,g_m)$, with $g_i := \lim_n k_i/n$. The maximum entropy principle says that the best estimate to $(g_1,...,g_m)$ corresponds to the solution to the following problem $$\begin{aligned}
\max &\quad {H_{2}}(g_1,...,g_m) \\
\text{subject to} & \quad\sum_{i=1}^m g_i E_j = \langle E \rangle \\
& \quad\sum_{i=1}^m g_i = 1.\end{aligned}$$ This is different from usual presentations of the maximum entropy principle in the literature concerning nonextensive statistical mechanics. Usually the constraints are written in terms of escort distributions derived from $(g_1,...,g_m)$; these have proven useful in several domains, e.g. the analysis of multifractals. However, it is not clear for us how to derive them from combinatorial facts.
Dynamical model {#sec:dynamical_model}
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When $q$-is a prime power, the $q$-binomial coefficients count vector spaces. As explained in the introduction, this motivates a generalization of information theory where messages are vector spaces in correspondence with the usual information theory for memoryless Bernoulli sources. Table \[table\_correspondence\] outline the correspondence. Sections \[sec:q-binomial-distribution\] and \[sec:grassmannian\_proc\] justify the last row of this table. Section \[sec:q-binomial-distribution\] describes the $q$-deformed version of the binomial distribution, associated to the $q$-binomial coefficients. Section \[sec:grassmannian\_proc\] introduces an original stochastic model for the generation of generalized messages: a discrete-time stochastic process that gives at time $n$ a vector subspace of ${\mathbb{F}}_q^n$. We call it *Grassmannian process*. Finally, Section \[sec:asymptotics-grassmannian-process\] establishes some facts about the asymptotic behavior of this process.
The $q$-binomial distribution {#sec:q-binomial-distribution}
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Let $Z$ be a random variable that takes the value $1$ with probability $\xi \in[0,1]$ and the value $0$ with probability $1-\xi$ (Bernoulli distribution). Its characteristic function is $${\mathbb{E}}({\mathrm{e}}^{itZ}) = \xi {\mathrm{e}}^{it} + (1-\xi).$$ Let $W_n$ be a random variable with values in $\{0,...,n\}$, such that $k$ has probability ${\operatorname{Bin}}(k|n,\xi): = {n \choose k} \xi^k(1-\xi)^{n-k}$, where $\xi \in[0,1]$. The binomial theorem implies that ${\operatorname{Bin}}(\cdot|n,\xi)$ is a probability mass function, corresponding to the so-called binomial distribution. The theorem also implies that $$\begin{split}
({\mathbb{E}\left({\mathrm{e}}^{itZ}\right)})^n &{ }={ } (\xi {\mathrm{e}}^{it}+(1-\xi))^n \\
&{ }={ } \sum_{k=0}^n {n \choose k} {\mathrm{e}}^{it k} \xi^k(1-\xi)^{n-k} \\
&{ }={ } {\mathbb{E}\left({\mathrm{e}}^{itW_n}\right)},
\end{split}$$ which means that $W_n = Z_1 + ... + Z_n$ (in law), where $Z_1,...,Z_n$ are $n$ i.i.d. variables with the same distribution than $Z$ [@doob Ch. I, Sec. 11]. Given a collection $\{Z_i\}_{i\geq 1}$ of i.i.d. random variables such that $Z_i \sim {\operatorname{Ber}}(\xi)$, the process $\{W_n\}_{n\geq 1}$ defined by $W_1=Z_1$ and $W_n = W_{n-1}+Z_n$ when $n>1$ is an ${\mathbb{N}}$-valued markovian stochastic process.
There is a well known combinatorial interpretation for all this: if you generate binary sequences of length $n$ by tossing $n$ times a coin that gives $1$ with probability $\xi$ and $0$ with probability $1-\xi$, any sequence with exactly $k$ ones has probability $\xi^k(1-\xi)^{n-k}$ and there are ${n\choose k}$ of them. Therefore, if $Y$ is the sum of the outputs of all the coins (the number of ones in the generated sequence), the probability of observing $Y=k$ is ${n\choose k} \xi^k(1-\xi)^{n-k}$.
There is also a $q$-binomial theorem, known as the Gauss binomial formula [@kac2001quantum Ch. 5]: $$\label{gauss_binomial_theorem}
(x+y)(x+y q) \cdots (x+y q^{n-1}) = \sum_{k=0}^n { {n \brack k}_q } q^{{k\choose 2}}y^k x^{n-k}.$$ Let us write $(x+y)^n_q$ instead of $(x+y)(x+y q) \cdots (x+y q^{n-1})$: the $q$-analog of $(x+y)^n$. Then implies that $$\label{qbinomial-law-xy}
{\operatorname{Bin}_q}(k|n,x,y) := { {n \brack k}_q } \frac{q^{{k\choose 2}}y^k x^{n-k}}{(x+y)^n_q}$$ is a probability mass function for $k\in \{0,...,n\}$, with parameters $n\in {\mathbb{N}}$, $x\geq 0$ and $y \geq 0$. Moreover, the factorization $$\label{factorization_q_binomial_law}
\prod_{j=0}^{n-1} \frac{(x+y{\mathrm{e}}^{it}q^j)}{(x+yq^j)} = \sum_{k=0}^n { {n \brack k}_q } \frac{{\mathrm{e}}^{itk}y^k x^{n-k}q^{{k\choose 2}}}{(x+y)^n_q}$$ shows that a variable $Y_n$ with law ${\operatorname{Bin}_q}(n,x,y)$ can be written as the sum of $n$ independent variables $X_1,...,X_n$, such that $X_i$ takes the value $0$ or $1$ with probability $x/(x+yq^{i-1})$ and $yq^{i-1}/(x+yq^{i-1})$, respectively.
If we begin with a collection $\{X_i\}_{i\geq 1}$ of independent variables such that $X_i\sim {\operatorname{Ber}}\left(\frac{yq^{i-1}}{x+yq^{i-1}}\right)$, then the process $\{Y_n\}_{n\geq 1}$ defined by $Y_n = X_1 + \cdots + X_n$ is an ${\mathbb{N}}$-valued markovian stochastic process. When $q\to 1$, each $X_i$ becomes a Bernoulli variable with parameter $y/(x+y)$ and $Y$ has a ${\operatorname{Bin}}(n, \frac{y}{x+y})$ distribution. Equation also implies that $${\mathbb{E}}(Y) = \sum_{j=0}^{n-1} \frac{yq^j}{x+yq^j} = n - \sum_{j=0}^{n-1} \frac{x}{x+yq^j}.$$
Provided that $x\neq 0$, one can write the mass function of the $q$-binomial as follows: $$\label{qbinomial-law}
{\operatorname{Bin}_q}(k|n,\theta) := { {n \brack k}_q } \frac{q^{{k\choose 2}}\theta^k}{(-\theta;q)_n},$$ where $\theta = y/x \geq 0$. We adopt here the classical notation $(-\theta; q)_n$ instead of $(1+\theta)^n_q$.[^7] Strictly speaking, this is the $q$-binomial distribution found in the literature [@kemp]. The expectation and the variance of this simplified distribution are respectively $$\begin{aligned}
{\mathbb{E}}(Y) &= \sum_{j=0}^{n-1} \frac{\theta q^j}{1+\theta q^j} = n - \sum_{j=0}^{n-1} \frac{1}{1+\theta q^j},\label{mean} \\
{\mathbin{\mathbb{V}}}(Y) &= \sum_{j=0}^{n-1} \frac{\theta q^j}{(1+\theta q^j)^2}.\end{aligned}$$ The statistical estimation of $\theta$ is addressed in the Appendix \[app:maximum\_likelihood\].
Set $c_n(\theta):=\sum_{j=0}^{n-1} \frac{1}{1+\theta q^j}$; this sequence is monotonic in $n$ and convergent to certain $c(\theta)$. We do not include $q$ in the notation, since it is fixed from the beginning.
A vector-space valued stochastic process associated to the $q$-binomial distribution {#sec:grassmannian_proc}
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The vector $(Z_1,...,Z_n)$ is a random binary sequence, but its $q$-deformation $(X_1,...,X_n)$, obtained in the previous section, cannot be identified in an obvious way with a vector space. This motivates the introduction of an associated stochastic process $\{V_i\}_{i\in {\mathbb{N}}}$ such that, for each $n\in {\mathbb{N}}$, $V_n$ is vector subspace of ${\mathbb{F}}_q^n$ and the law of $\{X_i\}_{i\in {\mathbb{N}}^*}$ can be recovered from that of $\{V_i\}_{i\in {\mathbb{N}}}$.
Let ${\operatorname{Gr}}(k,n)$ be the set of $k$-dimensional vector subspaces of ${\mathbb{F}}_q^n$ and define the total $n$-th Grassmannian by $${\operatorname{Gr}}(n):= \bigcup_{i=0}^n {\operatorname{Gr}}(i,n).$$
Let $\langle 0\rangle={\mathbb{F}}_q^0 \hookrightarrow {\mathbb{F}}_q^1 \hookrightarrow {\mathbb{F}}_q^2 \hookrightarrow .... \hookrightarrow {\mathbb{F}}_q^n \hookrightarrow...$ be a sequence of linear embeddings; note that it induces embeddings at the level of Grassmannians, that will be implicit in what follows. Define $V_0 := {\mathbb{F}}_q^0$, the trivial vector space; for each $n\geq 0$, let $V_{n+1}$ be a random variable taking values in ${\operatorname{Gr}}(n+1)$ with law defined by $$\begin{aligned}
{\mathbb{P}\left(V_{n+1} =v | V_n = w, X_{n+1} = 0\right)} &= \delta_w(v) ,\\
{\mathbb{P}\left(V_{n+1} =v | V_n = w, X_{n+1} = 1\right)} &= \frac{[v\in {\operatorname{Dil}}_{n+1}(w)]}{|{\operatorname{Dil}}_{n+1}(w)|}.\end{aligned}$$ The $(n+1)$-dilations of $w$, ${\operatorname{Dil}}_{n+1}(w)$, are defined as $${\{\,v \in {\operatorname{Gr}}(n+1) \, : \, w\subset v, \: v\not\subset {\mathbb{F}}_q^n, \: \dim v - \dim w = 1\,\} }.$$ We shall refer to $\{V_n\}_{n\in {\mathbb{N}}}$ as the Grassmannian process associated to the $q$-binomial process.
\[law\_Vn\] Let $v$ be a subspace of ${\mathbb{F}}_q^n$ such that $\dim(v) = k$. Then, $$\label{pmf_vector_spaces}
{\mathbb{P}\left(V_n = v \right)}=\frac{\theta^kq^{k(k-1)/2}}{(-\theta;q)_n}.$$
See Appendix \[proof\_law\_Vn\]
$${\mathbb{P}\left(\dim V_n =k\right)} = { {n \brack k}_q } \frac{\theta^kq^{k(k-1)/2}}{(-\theta;q)_n}.$$
This is a consequence of Proposition \[law\_Vn\] and the fact that ${ {n \brack k}_q }$ counts the number of $k$ dimensional subspaces of ${\mathbb{F}}_q^n$.
\[prop:simplified\_law\_Vn\] Let $\{Y_n\}_{n\in {\mathbb{N}}^*}$ denote a $q$-binomial process, $Y_n \sim {\operatorname{Bin}}_q(n,\theta)$, and $\{V_n\}_{n\in {\mathbb{N}}}$ its associated Grassmannian process. Let $v$ be a subspace of ${\mathbb{F}}_q^n$ of dimension $k=n-d$, for $d \in \llbracket 0, n\rrbracket$. Then, $$\label{proba_Vn_for_d}
{\mathbb{P}\left(V_n = v\right)} = \frac{q^{-\frac 12 (d -(\frac 12 - \log_q\theta))^2 + \frac 12 (\frac 12 - \log_q\theta)^2 - \frac{n^2}{2} {H_{2}}(d/n) } }{(-\theta^{-1}; q^{-1})_n}.$$
We shall rewrite the various factors in . In the first place, $$(-\theta;q)_n = \prod_{i=0}^{n-1} \theta q^i (1+\frac{1}{\theta q^i}) = \theta^n q^{n(n-1)/2} (-\theta^{-1};q^{-1})_n.$$ Note also that $n^2 {H_{2}}(d/n) = n^2 - k^2 -d^2$, which implies $$\begin{aligned}
q^{{k\choose 2}} = q^{k^2/2}q^{-k/2}
&= q^{(n^2-n^2{H_{2}}(d/n) -d^2)/2 } q^{(d-n)/2}.\end{aligned}$$ Finally, $\theta^k = \theta^{n-d}$. Replace all this in and simplify to obtain $${\mathbb{P}\left(V_n = v\right)} = \frac{q^{-\frac{d^2}{2} + d (\frac 12 - \log_q\theta)} q^{-\frac{n^2}{2} {H_{2}}(d/n)}}{(-\theta^{-1}; q^{-1})_n}.$$ Complete the square in the exponent to conclude.
Asymptotics {#sec:asymptotics-grassmannian-process}
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Let us define a function $\mu:{\mathbb{N}}\to (0,\infty)$ by $$\mu(d) := \frac{q^{-\frac 12 (d -(\frac 12 - \log_q\theta))^2 + \frac 12 (\frac 12 - \log_q\theta)^2 } (q^{-(d+1)};q^{-1})_\infty}{(q^{-1};q^{-1})_\infty (-\theta^{-1}; q^{-1})_\infty},$$ and introduce the notation $\mu(\llbracket a, b\rrbracket):=\sum_{d=a}^b \mu(d)$.
The asymptotic formula in Theorem \[prop:asymptotic\_behavior\], combined with Proposition \[prop:simplified\_law\_Vn\], implies that $${\mathbb{P}\left(V_n \in {\operatorname{Gr}}(n-d,n)\right)}={ {n \brack n-d}_q } {\mathbb{P}\left(V_n={\mathbb{F}}_q^{n-d}\right)} \to \mu(d),$$ for each fixed $d\in {\mathbb{N}}$.
\[mu\_proba\] $$\sum_{d=0}^\infty \mu(d) = 1.$$
See Appendix \[proof\_mu\_proba\].
Therefore, there is a well defined function $ \Delta:[0,1)\to {\mathbb{N}}$ that associates to each $p\in [0,1)$ the smallest $d$ such that $\mu(\llbracket 0, d\rrbracket) \geq p$; explicitly $$\Delta(p) = \sum_{k=0}^\infty [p>\mu(\llbracket 0, k\rrbracket)].$$ The sum is finite for every $p\in [0,1)$. Note that $ \Delta$ is *left* continuous. This function plays an important role in the proof of Theorem \[them:AEP\].
Generalized information theory {#sec:generalized_info}
==============================
In this section, we prove a fundamental result on measure concentration for the Grassmannian process (Theorem \[them:AEP\]), that generalizes the asymptotic equipartition property to this setting. It justifies the definition of “typical subspaces”. Section \[sec:coding\] applies this result to source coding.
Remarks on measure concentration
--------------------------------
The following definition covers the different stochastic models discussed so far. We use it to clarify the correspondence between Shannon’s information theory for sequences and our version for vector subspaces from the probabilistic viewpoint.
Let $\pi:(A, \mathcal A) \to (B,\mathcal B)$ be a surjection of measurable spaces and $p$ a probability measure on $(B,\mathcal B)$. The law has a refinement with respect to $\pi$ (or $\pi$-refinement) whenever there exists a probability distribution $\tilde p$ on $(A, \mathcal A)$ such that $\pi_* \tilde p = p$, where $\pi_* \tilde p$ denotes the image law (the push-forward of $\tilde p$, its marginalization).
In applications, $p$ is the law of a $(B,\mathcal B)$-valued random variable $X$ and $\tilde p$, the law of a $(A, \mathcal A)$-valued random variable $Y$. When $B\subset {\mathbb{C}}$, $${\mathbb{E}}_{\tilde p}({\mathrm{e}}^{it\pi(Y)})={\mathbb{E}}_p({\mathrm{e}}^{itX}).$$
There are four fundamental examples:
1. \[exam\_refinement\_binomial\] The probability measure ${\operatorname{Ber}}(\xi)^{\times n}$ on $\{0,1\}^n$, that assigns to every sequence with $k$ ones the probability $\xi^k(1-\xi)^{n-k}$, is a refinement of the law ${\operatorname{Bin}}(n,\xi)$ with respect to the surjection $\pi_1:\{0,1\}^n \to \{0,1,...,n\}, (x_1,...,x_n)\mapsto \sum_i x_i$.
2. The previous example generalizes to the so-called multinomial distribution. Let $S=\{s_1,...,s_m\}$ be a finite set and $\mu$ any probability law on $S$; set $p_i:=\mu(\{s_i\})$. The law $\mu^{\otimes n}$ assigns to a sequence $x$ in $S^n$ the probability $\prod_{i=1}^m p_i^{a_i(x)}$, where $a_i(x)$ denotes the number of appearances of the symbol $s_i$ in the sequence $x$. Let $T={\{\,(k_1,...,k_m)\in {\mathbb{N}}^m \, : \, \sum_{i=1}^m k_i= n\,\} }$; there is a surjection $\pi_2:S^n\to T$ given by $x\mapsto (a_1(x),...,a_m(x))$. Denote by $\nu$ the marginalization of $\mu^{\otimes n}$ under this map, given explicitly by $\nu(\{(k_1,...,k_m)\}) = { n \choose k_1,...,k_n } \prod_{i=1}^m p_i^{k_i}$. Then $\mu^{\otimes n}$ is a $\pi_2$-refinement of $\nu$.
3. The probability measure $\prod_{i=1}^{n-1} {\operatorname{Ber}}(\frac{\theta q^i}{1+\theta q^{i}})$ on $\{0,1\}^n$ is a refinement of the law ${\operatorname{Bin}}_q(n,\theta)$ under the application $\pi_1$ introduced above, see .
4. The probability measure on ${\operatorname{Gr}}(n)$ defined by , that we denote ${\operatorname{Grass}}(n,\theta)$, is also a refinement of ${\operatorname{Bin}}_q(n,\theta)$ with respect to the surjection $\pi_3:{\operatorname{Gr}}(n)\to \{0,1,...,n\}, V\mapsto\dim V$.
Let us consider for a moment the binomial case \[exam\_refinement\_binomial\]. For $W_n\sim {\operatorname{Bin}}(n,p)$, Chebyshev’s inequality reads ${\mathbb{P}\left( |W_n-pn| >n^{\frac 12+\xi}\right)}\leq p(1-p)/n^{2\xi}$, which goes to $0$ as long as $\xi>0$. In other words, the measure ${\operatorname{Bin}}(n,p)$ concentrates on the interval $I_{n,\xi}=\llbracket np-n^{\frac 12 + \xi}, np+n^{\frac 12 + \xi}\rrbracket \cap \llbracket 0, n \rrbracket$, in the sense that ${\mathbb{P}\left(W_n\in I_n^c\right)} \to 0$ as $n\to \infty$, and therefore the measure ${\operatorname{Ber}}(\xi)^{\times n}$ concentrates on $\pi_1^{-1}(I_{n,\xi})$, that can be regarded as a set of “typical sequences”. Moreover, the different type classes $\pi^{-1}(t)$, for $t\in I_{n,\xi}$, have cardinality $\exp\{nH_1(p)+o(n)\}$. An analogous argument shows that the measure ${\operatorname{Bin}}_q (n,\theta)$ concentrates on the interval $J_{n,\xi}=\llbracket k_n^* - n^\xi, k_n^* + n^\xi \rrbracket \cap \llbracket 0, n \rrbracket$ around the mean $k_n^*$, for any $\xi > 0$, and hence ${\operatorname{Grass}}(n,\theta)$ concentrates on $\pi_3^{-1}(J_{n,\xi})$. However, there is a difference: while ${\operatorname{Bin}}(k|n,p)$ goes to $0$ for any value of $k$, and in fact on needs more than $\sqrt{n}$ different types $k$ to accumulate asymptotically a prescribed probability $p_{\varepsilon}:= 1-{\varepsilon}$, the values of ${\operatorname{Grass}}(k|n,\theta) = {\mathbb{P}\left(V_n\in {\operatorname{Gr}}(k.n)\right)}$ tend to the constant value $\mu(d)$, independent of $n$. In the limit, only a finite number of different types $k$ are necessary to accumulate probability $p_{\varepsilon}$, and the corresponding type classes differ in size (even asymptotically). Theorem \[them:AEP\] bellow reflects this particular situation.
Typical subspaces
-----------------
We are ready to prove the main result of this article, which extends Theorems 3 and 4 of Shannon’s seminal article [@shannon1948] to this setting.
\[them:AEP\] Let $\{Y_n\}_{n\in {\mathbb{N}}^*}$ denote a $q$-binomial process, $Y_n \sim {\operatorname{Bin}}_q(n,\theta)$; $\{V_n\}_{n\in {\mathbb{N}}}$ its associated Grassmannian process; and $\delta \in (0,1)$ an arbitrary number. Let ${\varepsilon}>0$ be such that $p_{\varepsilon}:=1-{\varepsilon}$ is a continuity point of $ \Delta$. Define $A_n=\bigcup_{k=0}^{a_n} {\operatorname{Gr}}(n-k,n)$ as the smallest set of the form $\bigcup_{k=0}^m {\operatorname{Gr}}(n-k,n)$ such that ${\mathbb{P}\left(V_n \in A_n^c\right)} \leq {\varepsilon}$. Then, there exists $n_0\in {\mathbb{N}}$ such that, for every $n\geq n_0$,
1. \[value\_Delta\] $A_n = \bigcup_{k=0}^{ \Delta(p_{\varepsilon})} {\operatorname{Gr}}(n-k,n)$;
2. \[approx\_proba\] for any $v\in A_n$ such that $\dim v = k$, $$\label{approx_proba_eq}
\left| \frac{\log_q ({\mathbb{P}\left(V_n = v\right)}^{-1})}{n} - \frac{n}{2}{H_{2}}(k/n) \right| \leq \delta.$$
The size of $A_n$ is optimal, up to the first order in the exponential: let $s(n,{\varepsilon})$ denote $\min{\{\,|B_n| \, : \, B_n\subset {\operatorname{Gr}}(n) \text{ and } {\mathbb{P}\left(V_n \in B_n\right)} \geq 1-{\varepsilon}\,\} }$; then $$\label{size_optimality}
\begin{split}
\lim_n \frac{1}{n} \log_q|A_n| & = \lim_n \frac{1}{n} \log_q s(n,{\varepsilon}) \\
& = \lim_n \frac{n}{2} {H_{2}}(\Delta(p_{\varepsilon})/n) \\
& = \Delta(p_{\varepsilon}).
\end{split}$$
The set $A_n$ correspond to the “typical subspaces”, in analogy with typical sequences.
To shorten the notation, let us write ${\mathbb{P}_{n}\left(A\right)}$ instead of ${\mathbb{P}\left(V_n\in A\right)}$, and $G_{k}^n$ instead of ${\operatorname{Gr}}(k,n)$.
Given any $\eta>0$, there exists $n(\eta)\in {\mathbb{N}}$ such that, for every $n\geq n(\eta)$ and every $d\in \llbracket 0, \Delta(p_{\varepsilon})\rrbracket$, $$|{\mathbb{P}_{n}\left( G_{n-d}^n\right)} - \mu(d)| < \frac{\eta}{ \Delta(p_{\varepsilon}) +1},$$ because ${\mathbb{P}_{n}\left(G_{n-d}^n\right)} \to \mu(d)$ for each $d$.
Since $p_{\varepsilon}$ is a continuity point of $\Delta$, a piece-wise constant function, there exists $\xi>0$ such that $$\Delta(1-{\varepsilon}- \xi) = \Delta(1-{\varepsilon}) = \Delta(1-{\varepsilon}+ \xi).$$ Remark now that, for every $n\geq n(\xi)$, $$\label{bound_proba_A_n_Delta_inf}
\sum_{d=0}^{\Delta(p_{\varepsilon})} {\mathbb{P}_{n}\left(G_{n-d}^n\right)} > \sum_{d=0}^{\Delta(p_{\varepsilon})} \mu(d) -\xi \geq 1-{\varepsilon},$$ because $\mu(\llbracket 0, \Delta(p_{\varepsilon})\rrbracket) = \sum_{d=0}^{\Delta(p_{\varepsilon})} \mu(d) \geq 1-{\varepsilon}+ \xi$. This is a direct consequence of $\Delta(p_{\varepsilon}) = \Delta(1-{\varepsilon}+ \xi)$.
Analogously, for each $n\geq n(\xi)$, $$\begin{aligned}
\sum_{d=0}^{\Delta(p_{\varepsilon})-1} {\mathbb{P}_{n}\left( G_{n-d}^n\right)} & < \sum_{d=0}^{\Delta(p_{\varepsilon})-1} \mu(d) + \frac{\Delta(p_{\varepsilon})}{\Delta(p_{\varepsilon}) + 1} \xi \nonumber\\
& < 1-{\varepsilon}- \frac{\xi}{\Delta(p_{\varepsilon}) + 1} \nonumber\\
& < 1-{\varepsilon},\label{bound_proba_A_n_Delta_sup}\end{aligned}$$ because $\mu(\llbracket 0, \Delta(p_{\varepsilon})-1\rrbracket) < 1-{\varepsilon}- \xi$: if this is not the case, $\Delta(1-{\varepsilon}- \xi)\leq \Delta({\varepsilon})-1$. The inequalities and imply the part \[value\_Delta\] of the theorem whenever $n \geq n(\xi)$.
We suppose now that $n>n(\xi)$. Let $v$ be an element of $A_n$ of dimension $k$, and set $d = n-k$. The formula in Proposition \[prop:simplified\_law\_Vn\] can be stated as $$- \frac{\log_q {\mathbb{P}\left(V_n = v\right)}}{n} = \frac{g(d,n)}{n} + \frac{n}{2} {H_{2}}(d/n),$$ where we have set $g(d,n)= \frac 12 (d -(\frac 12 - \log_q\theta))^2 - \frac 12 (\frac 12 - \log_q\theta)^2 + \log_q (-\theta^{-1}; q^{-1})_n$. Since $d$ belongs to the interval $\llbracket 0, \Delta(p_{\varepsilon})\rrbracket$, independent on $n$, and $(-\theta^{-1}; q^{-1})_n \to (-\theta^{-1}; q^{-1})_\infty$, there exists $n_0 \geq n(\xi)$ such that, for every $n\geq n_0$ and every $d\in \llbracket 0, \Delta(p_{\varepsilon})\rrbracket$, $g(d,n)/n <\delta,$ which proves part \[approx\_proba\] of the theorem.
For $n$ big enough, $\Delta(p_{\varepsilon})$ belongs to the interval $[n/2,n]$. The inequalities in Lemma \[lemma:unimodality\_binomial\] imply that $$\label{inequalities_size_An}
\begin{split}
{ {n \brack n-\Delta(p_{\varepsilon})}_q } &\leq |A_n| \\
&\leq \sum_{k=0}^{\Delta(p_{\varepsilon})} { {n \brack n-k}_q } \\
&\leq (\Delta(p_{\varepsilon})+1) { {n \brack n-\Delta(p_{\varepsilon})}_q }.
\end{split}$$ Therefore, $$\label{asymptotic_size_An}
\lim_n \frac 1n \log_q |A_n| = \lim_n \frac 1n \log_q { {n \brack n-\Delta(p_{\varepsilon})}_q } = \Delta(p_{\varepsilon}),$$ where the second equality comes from .
For any ${\varepsilon}$, we show now how to build iteratively a set $B_n$ of minimal cardinality such that ${\mathbb{P}_{n}\left(B_n^c\right)} \leq {\varepsilon}$: start with $B_n = \emptyset$ and then add vector subspaces of ${\mathbb{F}}_q^n$ one-by-one, picking at each time any of the vector subspaces of *highest dimension* in $B_n^c$, until you attain ${\mathbb{P}_{n}\left(B_n^c\right)} \leq {\varepsilon}$. Let $n-b_n$ be the dimension of the last space included in $B_n$. It is easy to prove that $b_n < 2\sqrt{n}$, as a consequence of Chebyshev’s inequality (the interval $[n-2\sqrt{n},n]$ accumulates probability $p_{\varepsilon}$ when $n$ is big enough). This construction gives in fact the smallest possible set, because the function $f_n:[0,n] \to {\mathbb{R}}, \: x \mapsto \theta^x q^{x(x-1)/2}/(-\theta,q)_n$ is strictly convex and attains its minimum at $x_0=\frac 12 - \log_q\theta$; therefore, all the subspaces are included in $B_n$ in decreasing order of probability, and the probability of the last space included is bounded bellow by $\theta^{n-2\sqrt n} q^{({n-2\sqrt n})({n-2\sqrt n}-1)/2}/(-\theta,q)_n$, which is much bigger that $(-\theta,q)_n^{-1}$, the maximum of $f_n$ on $[0,x_0]$, when $n$ is big enough.
Two versions of $B_n$ only differ in the particular subspaces of dimension $n-b_n$ they include, but they coincide on $\bigcup_{k=0}^{b_n-1} G_{n-k}^n$. In what follows, $B_n$ denotes any of the possible sets. Remark also that $B_n \subset A_n$; even more, $a_n=b_n$ (a strict inequality between the two contradicts the minimality of either $B_n$ or $a_n$). It is also true in general that $$\begin{aligned}
p_{\varepsilon}&\leq {\mathbb{P}_{n}\left( B_n\right)} \nonumber\\
& = \sum_{k=0}^{a_n} {\mathbb{P}_{n}\left( B_n\cap G_{n-k}^n\right)} \nonumber\\
& = {\mathbb{P}_{n}\left(B_n\cap G_{n-a_n}^n\right)} + \sum_{k=0}^{a_n-1} {\mathbb{P}_{n}\left(B_n\cap G_{n-k}^n\right)}.\label{inequality_probas_B_n}\end{aligned}$$
We restrict ourselves again to the case in which $p_{\varepsilon}$ is continuity point of $\Delta$, in such a way that $\Delta(p_{\varepsilon})= a_n=b_n$. Under these hypotheses, we are able to lower-bound uniformly the term ${\mathbb{P}_{n}\left(B_n\cap G_{n-\Delta(p_{\varepsilon})}^n\right)}$ by using , and deduce from this that $|B_n|$ grows like $|A_n|$, that in turn grows like $|G_{n-\Delta(p_{\varepsilon})}^n|$, as shown in . In fact, we have that $$\begin{aligned}
\sum_{k=0}^{\Delta(p_{\varepsilon})-1} {\mathbb{P}_{n}\left( B_n\cap G_{n-k}^n\right)} &\leq \sum_{k=0}^{\Delta(p_{\varepsilon})-1} {\mathbb{P}_{n}\left(G_{n-k}^n\right)} \nonumber \\
& { }<{ } 1 - {\varepsilon}- \frac{\xi}{\Delta(p_{\varepsilon})+1}, \label{bound_B_n_inter_grassmanian_2} \end{aligned}$$ where we have used again the bound in . Inequalities and imply that $$\frac{\xi}{\Delta(p_{\varepsilon})+1} < {\mathbb{P}_{n}\left(B_n\cap G_{n-\Delta(p_{\varepsilon})}^n\right)}.$$ When $n>n_0$, the part entails that ${\mathbb{P}_{n}\left(x\right)} \leq q^{-n^2 {H_{2}}(\Delta/n)/2 + n\delta}$ for every $x\in G_{n-\Delta(p_{\varepsilon})}^n$, or equivalently ${\mathbb{P}_{n}\left(x\right)} q^{n^2 {H_{2}}(\Delta/n)/2 - n\delta} \leq 1$. Then, $$\begin{aligned}
|B_n| & \geq |B_n \cap G_{n-\Delta(p_{\varepsilon})}^n| \nonumber\\
& \geq \sum_{x\in B_n \cap G_{n-\Delta(p_{\varepsilon})}^n} {\mathbb{P}_{n}\left(x\right)} q^{n^2 {H_{2}}(\Delta(p_{\varepsilon})/n)/2 - n\delta} \nonumber\\
& \geq q^{n^2 {H_{2}}(\Delta(p_{\varepsilon})/n)/2 - n\delta} {\mathbb{P}_{n}\left(B_n \cap G_{n-\Delta(p_{\varepsilon})}^n\right)} \nonumber\\
& > q^{n^2 {H_{2}}(\Delta(p_{\varepsilon})/n)/2 - n\delta} \frac{\xi}{\Delta(p_{\varepsilon})+1}.\label{lower_bound_Bn_with_probas}\end{aligned}$$ We deduce that $$\label{lim_inf_Bn}
\liminf_n \frac 1n \log_q |B_n| \geq \lim_n \frac{n}{2} {H_{2}}(\Delta(p_{\varepsilon})/n) - \delta.$$ On the other hand, since $B_n \subset A_n$, it is clear that $$\label{lim_sup_Bn}
\begin{split}
\limsup \frac 1n \log_q |B_n| &\leq \lim_n \frac 1n \log_q |A_n| \\
&{ }={ } \lim_n \frac{n}{2} {H_{2}}(\Delta(p_{\varepsilon})/n).
\end{split}$$ Since $\delta>0$ is arbitrarily small, and imply that $\lim_n \frac 1n \log_q |B_n|$ exists and equals $\Delta(p_{\varepsilon})$. The theorem is proved.
The definition of $A_n$ still makes sense when $p_{\varepsilon}$ is a discontinuity point of $ \Delta$. In this case, there exists $\xi>0$ such that $\Delta(p_{\varepsilon})+1 = \Delta(p_{\varepsilon}+ \xi)$ and $\Delta(p_{\varepsilon})= \Delta(p_{\varepsilon}-\xi)$ . Inequality can be easily adapted to show that $\sum_{k=0}^{\Delta(p_{\varepsilon})+1} {\operatorname{Gr}}(n-k,n) \geq 1-{\varepsilon}$, which implies that $a_n\leq \Delta(p_{\varepsilon})+1$; by , $a_n\geq \Delta(p_{\varepsilon})$. Of course, part \[approx\_proba\] in the Theorem still makes sense. We also have that $B_n\subset A_n$ and $a_n = b_n$. The problems appear in the comparison of $|B_n|$ and $|A_n|$; it is possible that ${\mathbb{P}_{n}\left(B_n\cap {\operatorname{Gr}}(n-\Delta(p_{\varepsilon}),n)\right)}$ goes to zero very fast when $n\to \infty$, and is not valid any more. However, we can still adapt the bounds in to prove $$\begin{split}
\liminf_n \frac 1n \log_q |A_n| & { }\geq{ } \liminf_n \frac 1n \log_q |B_n| \\
&{ }\geq{ } \lim_n \frac 1n \log_q { {n \brack n-(\Delta(p_{\varepsilon})-1)}_q } \\
&{ }={ } \Delta(p_{\varepsilon})-1,
\end{split}$$ because $b_n = a_n \geq \Delta(p_{\varepsilon})$ and therefore $ {\operatorname{Gr}}(n-(\Delta(p_{\varepsilon})-1),n)\subset B_n$. Analogously, $B_n\subset A_n$ and $a_n \leq \Delta(p_{\varepsilon})+1$ lead to $$\begin{split}
\limsup_n \frac 1n \log_q |B_n| & { }\leq{ } \limsup_n \frac 1n \log_q |A_n| \\
&{ }\leq{ } \lim_n \frac 1n \log_q { {n \brack n-(\Delta(p_{\varepsilon})+1)}_q } \\
&{ }={ } \Delta(p_{\varepsilon})+1,
\end{split}$$ where we have used again .
In the classical case of sequences, all the typical sequences tend to be equiprobable, in the sense of . This is not valid for the process $V_n$: a typical space $v\in A_n$ of dimension $n-d$ satisfy asymptotically the bounds $q^{-n(\frac n2 {H_{2}}(d/n)+\delta)} \leq {\mathbb{P}\left(V_n = v\right)}\leq q^{-n(\frac n2 {H_{2}}(d/n)-\delta)}$, for any $\delta>0$, and $\frac n2 {H_{2}}(d/n) = d + O(1/n)$.
Coding {#sec:coding}
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Inspired by [@csiszar1981information], we define a generalized $n$-to-$k$ $q$-ary block code as a pair of mappings $f:{\operatorname{Gr}}(n)\to \{1,...,q\}^k$ and $\phi:\{1,...,q\}^k\to {\operatorname{Gr}}(n)$. For a given stochastic process $W_n$, such that $W_n$ takes values in ${\operatorname{Gr}}(n)$, we define the probability of error of this code as $e(f,\phi)={\mathbb{P}\left(\phi(f(W_n))\neq W_n\right)}$. Small $k$ and small probability of error are good properties for codes, but there is a trade-off between the two. Let $k(n,{\varepsilon})$ be the smallest $k$ such that there exists a generalized $n$-to-$k$ $q$-ary block code $(f,\phi)$ that satisfies $e(f,\phi)\leq {\varepsilon}$.
\[minimum\_block\_code\] For the Grassmanian process $V_n$ introduced above and for all ${\varepsilon}>0$ such that $p_{\varepsilon}=1-{\varepsilon}$ is a continuity point of $\Delta$, one has $$\lim_n \frac{k(n,{\varepsilon})}{n}= \Delta(p_{\varepsilon}).$$
The existence of an $n$-to-$k$ $q$-ary block code $(f,\phi)$ such that $e(f,\phi)\leq {\varepsilon}$ is equivalent to the existence of a set $B_n\subset {\operatorname{Gr}}(n)$ such that ${\mathbb{P}\left(V_n\in B_n\right)}\geq 1-{\varepsilon}$ and $|B_n|\leq q^k$ (let $B_n$ be the set of sequences that are reproduced correctly...). As in the main theorem, let $s(n,{\varepsilon})$ denote the minimum cardinality of such a set. The statement in Proposition \[minimum\_block\_code\] is therefore equivalent to $\lim_n \frac 1n \log_q s(n,{\varepsilon}) = \Delta(p_{\varepsilon})$, which is already proved.
In simpler terms, it is always possible to code all the typical subspaces $A_n=\bigcup_{k=0}^{ \Delta(p_{\varepsilon})} {\operatorname{Gr}}(n-k,n)$ with different code-words if one disposes of $q^{n(\Delta(p_{\varepsilon}) +\xi)}$ such words, for $\xi$ positive and arbitrarily small, as long as $n$ is big enough. In contrast, it is asymptotically impossible if one disposes of $q^{n(\Delta(p_{\varepsilon}) -\xi')}$ different code-words, for any $\xi'>0$.
Further remarks
===============
A recent paper [@jensen2016statistical] proposes the study of “exploding” phase spaces: statistical systems such that the cardinality of the space of configurations grows faster than $k^n$, the combination of $n$ components that can occupy $k$ states. The total grassmannians ${\operatorname{Gr}}(n)={\operatorname{Gr}}(n, {\mathbb{F}}_q)$ are an example, since their cardinality grows like $q^{\frac{n^2}{4}+o(n^2)}$. This can be deduced from the unimodality of the $q$-binomial coefficients (Lemma \[lemma:unimodality\_binomial\]) and our asymptotic formulae, because $${ {n \brack \floor{n/2}}_q } \leq |{\operatorname{Gr}}(n)| \leq (n+1){ {n \brack \floor{n/2}}_q }$$ and therefore $$\begin{split}
\lim_n \frac{2}{n^2} \log_q |{\operatorname{Gr}}(n)| &{ }={ } \lim_n \frac{2}{n^2} \log_q { {n \brack \floor{n/2}}_q } \\
&{ }={ } {H_{2}}\left(\frac 12,\frac 12\right) = \frac 12.
\end{split}$$ In fact, the values of $\lim_{n\to\infty} |{\operatorname{Gr}}(2n+1)|q^{-(2n+1)^2/4}$ and $\lim_{n\to\infty} |{\operatorname{Gr}}(2n)|q^{-(2n)^2/4}$ depend only on $q$ and can be determined explicitly in terms of the Euler’s generating function for the partition numbers and the Jacobi theta functions $\vartheta_2$ and $\vartheta_3$, see [@kousidis Cor. 3.7]
A link between Tsallis entropy and the size of the *effective* phase space (the configurations whose probability is non-zero) is already suggested by Tsallis in [@tsallis-book Sec. 3.3.4]. There, $H_{(\rho-1)/\rho}$ appears naturally as a extensive quantity when the effective phase space grows like $N^\rho$, for $\rho>0$.
Finally, we conjecture the existence of other combinatorial quantities ${{n}\choose {k_1,...,k_s}}_{\mathrm{gen}}$ that satisfy the multiplicative relations , but such that $${{n}\choose {p_1n,...,p_sn}}_{\mathrm{gen}}\sim \exp(f(p_1,...,p_s)n^\beta + o(n^\beta)).$$ If this is the case, the function $f(p_1,...,p_s)$ would satisfy the functional equation for $\alpha=\beta$, and therefore be equal to $KH_\beta$, for an appropriate constant $K$.
Parameter estimation by the maximum likelihood method {#app:maximum_likelihood}
=====================================================
Let us suppose we make $n$ independent trials of a variable $Y$ with distribution ${\operatorname{Bin}_q}(n,\theta)$, obtaining results $y_1,...,y_m$. The probability of this outcome is $$P(y_1,...,y_m|\theta) = \prod_{i=1}^m { {n \brack y_i}_q } \frac{\theta^{y_i} q^{y_i(y_i-1)/2}}{(-\theta;q)_n}.$$ This implies that $${\frac{\partial \log P}{\partial \theta}} = \frac 1\theta \left(\sum_{i=1}^n y_i - m \sum_{j=0}^{n-1} \frac{\theta q^j}{(1+\theta q^j)}\right).$$ By the maximum likelihood method, the best estimate for $\theta$, say $\hat \theta$, should maximize $P$ and therefore satisfy $\left.{\frac{\partial \log P}{\partial \theta}}\right|_{\theta = \hat \theta}=0$; in turn, this equation implies that the empirical mean $$\bar y := \sum_{i=1}^m y_i$$ should coincide with the theoretical mean $$m_{q,n}(\theta) := \sum_{j=0}^{n-1} \frac{\theta q^j}{1+\theta q^j}.$$
The map $\theta \mapsto m_{q,n}(\theta) $ establishes a bijection between $[0,\infty)$ and $[0,n)$.
If this correspondence is extended by $m_{q,n}(\infty)=n$ |which corresponds to the case $x=0$| the value of $\hat \theta$ is uniquely determined by the equation $m_{q,n}(\hat \theta) = \bar y$.
Since $${\frac{\mathrm{d} }{\mathrm{d} \theta}}\left(\frac{\theta q^j}{1+\theta q^j}\right) = \frac{q^j}{(1+\theta q^j)^2} > 0,$$ $m_{q,n}(\theta)$ is strictly increasing. Moreover, $m_{q,n}(0)= 0$ and $\lim_{\theta \to \infty} m_{q,n}(\theta)=n$.
Proof of Proposition \[prop:asymptotic\_behavior\] {#proof_prop_asymptotic_q_multinomial}
==================================================
First, we substitute in (the powers of $(q-1)$ cancel): $$\begin{gathered}
\label{first_exp_binom}
{ {n \brack k_1,...,k_s}_q } = \\
(q^{-1};q^{-1})_\infty^{1-s} q^{n^2 {H_{2}}(\frac {k_1}{n},...,\frac{k_s}{n})/2} \frac{\prod_{i=1}^s (q^{-(k_i+1)};q^{-1})_\infty}{ (q^{-(n+1)};q^{-1})_\infty}. \end{gathered}$$ Theorem \[prop:asymptotic\_behavior\] is a direct consequence of this equality and the following fact: for any sequence $\{t_n\}_n$ of positive numbers, $$\label{eq:limit_pochhammer_1}
\lim_{n\to \infty} (q^{-(t_n +1)};q^{-1})_\infty = 1$$ if $ t_n \to \infty$, and $$\label{eq:limit_pochhammer_2}
\lim_{n\to \infty} (q^{-(t_n +1)};q^{-1})_\infty=(q^{-(t + 1)};q^{-1})_\infty$$ if $ t_n \to t \in [0,\infty)$. To establish and , remark first that $$(q^{-(t_n +1)};q^{-1})_\infty=\sum_{j=0}^\infty q^{-j(t_n +1)}/(q^{-1};q^{-1})_j$$ can be written as $\int_{{\mathbb{N}}} f_n(x) \nu(dx)$, where $\nu$ denotes the counting measure and $f_n:{\mathbb{N}}\to [0,\infty)$ is given by $$f_n(x) = \frac{q^{-x(t_n +1)}}{(q^{-1};q^{-1})_x}$$ Moreover, $|f_n(x)| \leq g(x):=q^{-x}/(q^{-1};q^{-1})_x$, because $t_n \geq 0$, and $g(x)$ is integrable, $\int_{\mathbb{N}}g(x) \nu(dx) \leq (q^{-1},q^{-1})_\infty^{-1} \frac{1}{1-q^{-1}}$. Therefore, in virtue of Lebesgue’s dominated convergence theorem, $$\begin{aligned}
\lim_{n\to \infty} \sum_{j=0}^\infty \frac{q^{-j(t_n +1)}}{(q^{-1};q^{-1})_j} &= \lim_n \int_{{\mathbb{N}}} f_n(x) \nu(dx) \\
& =\int_{{\mathbb{N}}} \lim_n f_n(x) \nu(dx) \end{aligned}$$ The point-wise limit $\lim_n f_n(x)$ is $[x=0]$ when $t_n\to \infty$ and $ \frac{q^{-x(t +1)}}{(q^{-1};q^{-1})_x}$ when $t_n\to t$.
Proof of Proposition \[law\_Vn\] {#proof_law_Vn}
=================================
To shorten notation, we write in this section ${\mathbb{P}_{X}\left(x\right)}$ instead of ${\mathbb{P}\left(X=x\right)}$, and ${\mathbb{P}_{X|Y}\left(x|y\right)}$ instead of ${\mathbb{P}\left(X=x|Y=y\right)}$.
Our proof is by recurrence. The case $n=1$ is straightforward; for instance, $$\begin{aligned}
{\mathbb{P}_{V_1}\left(\langle 0\rangle\right)} &= {\mathbb{P}_{V_1|V_0}\left(\langle 0 \rangle| \langle 0 \rangle\right)} \\
&= {\mathbb{P}_{V_1|V_0,X_1}\left(\langle 0 \rangle| \langle 0 \rangle, 0\right)}{\mathbb{P}_{X_1}\left(0\right)}, \\
&= {\mathbb{P}_{X_1}\left(0\right)}\end{aligned}$$ because $\langle 0 \rangle$ it is not a dilation of itself.
Suppose the formula is valid up to $n\geq 1$. Let $v'$ be a subspace of ${\mathbb{F}}_q^{n+1}$ of dimension $k$. When $v'$ is contained in ${\mathbb{F}}_q^n$, $$\begin{aligned}
{\mathbb{P}_{V_{n+1}}\left( v' \right)} &= {\mathbb{P}_{V_{n+1}|V_n,X_{n+1} }\left(v' | v', 0\right)} {\mathbb{P}_{X_{n+1}}\left(0\right)}{\mathbb{P}_{V_n}\left( v'\right)} \\
& = 1 \cdot \frac{1}{1+\theta q^n} \frac{\theta^kq^{k(k-1)/2}}{(-\theta;q)_n} = \frac{\theta^k q^{k(k-1)/2}}{(-\theta;q)_{n+1}}.
\end{aligned}$$ If $v'\not \subset {\mathbb{F}}_q^n$,
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= \_[w(n)]{} [\_[V\_[n+1]{}|V\_n,X\_[n+1]{}]{}(v’|w, 1)]{} [\_[Y\_n]{}( w)]{} [\_[X\_[n+1]{}]{}( 1)]{}\
= \_ ( )\
= .
The formula $F(U)+F(V)=F(U+V)+F(U\cap V)$ entails that $v\cap {\mathbb{F}}_q^n$ has dimension $k-1$. Any $w \in {\operatorname{Gr}}(k-1,n)$ such that $w\subset v$ must be contained in $v\cap {\mathbb{F}}_q^n$ and have the same dimension, implying that $w = v\cap {\mathbb{F}}_q^n$; this explain the last equality above.
Finally, let $w$ be a $k-1$ dimensional subspace in ${\mathbb{F}}_q^n$; to dilate it into a $v\in {\operatorname{Gr}}(k,n+1){\smallsetminus}{\operatorname{Gr}}(k,n)$, one must pick a vector $x$ outside ${\mathbb{F}}_q^n$: there are $q^{n+1}-q^n$ of those. However, since $w+\langle x\rangle$ has $q^k$ points and $w$ just $q^{k-1}$, there are $q^k-q^{k-1}$ choices of $x$ that give the same dilation $v$. Therefore, the number of different dilations is $$\frac{q^{n+1}-q^n}{q^k-q^{k-1}} = q^{n-(k-1)}.$$ In particular, the quantity $|{\operatorname{Dil}}_{n+1}(v\cap {\mathbb{F}}_q^n)|$ equals $q^{n-(k-1)}$.
Proof of Proposition \[mu\_proba\] {#proof_mu_proba}
==================================
We prove first a lemma that will be useful in the proof of Proposition \[mu\_proba\].
For every $n\in {\mathbb{N}}$ and every $d\in [0,n]$, $$\label{upper_bound_quotient}
\frac{(q^{-(n-d+1)};q^{-1})_\infty}{q^{-(n+1)};q^{-1})_\infty} \leq 1.$$ Moreover, for every $n\in {\mathbb{N}}$ and every $d\in\llbracket 0,2\sqrt{n}\rrbracket$, $$\label{lower_bound_quotient}
1-c(q)q^{-(\sqrt n +1)^2} \leq \frac{(q^{-(n-d+1)};q^{-1})_\infty}{q^{-(n+1)};q^{-1})_\infty},$$ where $c(q)=2(q^{-1};q^{-1})_\infty$.
In this proof we use repeatedly the $q$-binomial theorem . For any $k\in{\mathbb{N}}$, $q^{-k(n+1)}\leq q^{-k(n-d+1)}$, which in turn implies : $$\begin{aligned}
\frac{1}{(q^{-(n+1)};q^{-1})_\infty} &{ }={ } \sum_{k=0}^\infty \frac{q^{-k(n+1)}}{(q^{-1};q^{-1})_k}\\
& { }\leq{ } \sum_{k=0}^\infty \frac{q^{-k(n-d+1)}}{(q^{-1};q^{-1})_k} \\
& { }={ } \frac{1}{(q^{-(n-d+1)};q^{-1})_\infty}.\end{aligned}$$ To prove , first remark that $$\begin{aligned}
\frac{1}{(q^{-(n-d+1)};q^{-1})_\infty} & - \frac{1}{(q^{-(n+1)};q^{-1})_\infty} \\
& { }={ } \sum_{k=1}^\infty \frac{q^{-k(n+1)}(q^{kd}-1)}{(q^{-1};q^{-1})_k} \\
& { }\leq{ } (q^{-1};q^{-1})_\infty^{-1} \sum_{k=1}^\infty q^{-k(n+1)}q^{kd} \\
& { }\leq{ } (q^{-1};q^{-1})_\infty^{-1} \sum_{k=1}^\infty q^{-k(\sqrt n+1)^2}. \end{aligned}$$ Remark that we omit the term corresponding to $k=0$, since it vanishes. The first of these inequalities is implied by the trivial bound $x-1\leq x$ and the fact that $\{(q^{-1};q^{-1})_k\}_k$ decreases with $k$; the second, from $ d \leq 2\sqrt{n}$. The geometric series $\sum_{k=1}^\infty q^{-k(\sqrt n+1)^2}$ equals $q^{-(\sqrt n+1)^2} (1-q^{-(\sqrt n+1)^2})^{-1}$, that is upper-bounded by $2q^{-(\sqrt n+1)^2}$, because $q\geq 2$. Hence, we have $$\begin{aligned}
\frac{1}{(q^{-(n-d+1)};q^{-1})_\infty} &- \frac{1}{(q^{-(n+1)};q^{-1})_\infty} \\
& { }\leq{ } 2(q^{-1};q^{-1})_\infty^{-1} q^{-(\sqrt n+1)^2} \\
&{ }={ }c(q)q^{-(\sqrt n+1)^2} .
\end{aligned}$$ Finally, note that $\frac{1}{(q^{-(n-d+1)};q^{-1})_\infty} = 1 +$ (positive term)$\geq 1$, therefore it is also true that $$\begin{split}
\frac{1}{(q^{-(n-d+1)};q^{-1})_\infty} & - \frac{1}{(q^{-(n+1)};q^{-1})_\infty} \\
& { }\leq{ } \frac{c(q)q^{-(\sqrt n+1)^2}}{(q^{-(n-d+1)};q^{-1})_\infty}.
\end{split}$$
To simplify notation, set $$A(d):=-\frac 12 (d -(\frac 12 - \log_q\theta))^2 + \frac 12 (\frac 12 - \log_q\theta)^2.$$ and $B_n=(-\theta^{-1};q^{-1})_n^{-1}$. Recall from that $${ {n \brack n-d}_q } = \frac{q^{n^2 {H_{2}}(d/n)/2}(q^{-(d+1)};q^{-1})_\infty (q^{-(n-d+1)};q^{-1})_\infty}{(q^{-1};q^{-1})_\infty (q^{-(n+1)};q^{-1})_\infty }.$$ This and give $$\begin{aligned}
1&{ }={ } \sum_{d=0}^n {\mathbb{P}\left(V_n\in {\operatorname{Gr}}(n-d,n)\right)}\nonumber\\
& { }={ } B_n\sum_{d=0}^n \frac{q^{A(d)}(q^{-(d+1)};q^{-1})_\infty}{(q^{-1};q^{-1})_\infty} \frac{(q^{-(n-d+1)};q^{-1})_\infty}{(q^{-(n+1)};q^{-1})_\infty}\nonumber\\
& { }\leq{ } B_n \sum_{d=0}^n \frac{q^{A(d)}(q^{-(d+1)};q^{-1})_\infty}{(q^{-1};q^{-1})_\infty}.\label{lower_bound_sum_mu}\end{aligned}$$ At the end we have used the inequality . In turn, implies that $$(-\theta^{-1};q^{-1})_\infty \leq \sum_{d=0}^\infty \frac{q^{A(d)}(q^{-(d+1)};q^{-1})_\infty}{(q^{-1};q^{-1})_\infty}$$ We shall see that in fact this is an equality, as the proposition claims. Using this time , we obtain $$\begin{aligned}
1 & \geq \sum_{d=0}^{\floor{2 \sqrt n}} {\mathbb{P}\left(V_n\in {\operatorname{Gr}}(n-d,n)\right)} \\
& \geq B_n \sum_{d=0}^{\floor{2 \sqrt n}} \frac{q^{A(d)}(q^{-(d+1)};q^{-1})_\infty}{(q^{-1};q^{-1})_\infty} (1-c(q)q^{-(\sqrt n +1)^2}).\end{aligned}$$ which is equivalent to $$\sum_{d=0}^{\floor{2 \sqrt n}} \frac{q^{A(d)}(q^{-(d+1)};q^{-1})_\infty}{(q^{-1};q^{-1})_\infty} \leq \frac{(-\theta^{-1};q^{-1})_n}{1-c(q)q^{-(\sqrt n +1)^2}}.$$ In the limit, $$\sum_{d=0}^{\infty} \frac{q^{A(d)}(q^{-(d+1)};q^{-1})_\infty}{(q^{-1};q^{-1})_\infty} \leq (-\theta^{-1};q^{-1})_\infty.$$ and this finishes the proof.
Acknowledgements {#acknowledgements .unnumbered}
================
I am very grateful to Matilde Marcolli, who pointed out the combinatorial meaning of the $q$-multinomial coefficients during a conversation we had at CIRM. I also want to thank Daniel Bennequin for his constant encouragement and multiple suggestions.
[Juan Pablo Vigneaux Ariztía]{} received the B.Eng.Sc. and the Engineer’s degree in industrial engineering from the Pontifical Catholic University of Chile (PUC), Santiago, Chile, in 2014 and the master’s degree in fundamental mathematics from Pierre and Marie Curie University (Paris VI), France, in 2015. He is presently pursuing a Ph.D. in mathematics at Paris Diderot University (Paris VII), France, under the supervision of Prof. Daniel Bennequin.
His work focuses on algebraic characterizations of information functions and related objects in probability theory and combinatorics, along with the application of homological and homotopical techniques in these domains. He is also interested in the relation between information theory and statistical mechanics, particularly the principle of free energy minimization and related algorithms (e.g. generalized belief propagation), as well as the use of this principle in machine learning and neuroscience.
[^1]: Manuscript received August 28, 2018; revised February 8, 2019; accepted March 7, 2019. This paper was presented in part at the Latin American Week on Coding and Information 2018 (Campinas, Brazil).
Juan Pablo Vigneaux is with the [Institut de Mathématiques de Jussieu-Paris Rive Gauche]{}, attached administratively to Université Paris Diderot, F-75013 Paris, France; to Sorbonne Université, F-75005 Paris, France, and also to CNRS, F-75016 Paris, France (e-mail:[email protected]).
Communicated by P. Harremoës, Associate Editor for Probability and Statistics.
Digital Object Identifier 10.1109/TIT.2019.2907590
Copyright (c) 2019 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to [email protected].
[^2]: In this work, the range of every random variable is supposed to be a finite set.
[^3]: In the physics literature, it is customary to use the letter $q$ instead of $\alpha$, but we reserve $q$ for the ’quantum’ parameter that appears in the $q$-integers, $q$-multinomial coefficients, etc.
[^4]: Originally, this was called *non-extensivity*, which explains the name ‘nonextensive statistical mechanics’.
[^5]: “...the entropy to be used for thermostatistical purposes would be not universal but would depend on the system or, more precisely, on the nonadditive universality class to which the system belongs.”[@tsallis-book p. xii]
[^6]: We fix the constant $1$ in front of $1- \sum_{i=1}^s \mu_i^2$. In [@vigneaux2017generalized] we have characterized Tsallis $\alpha$-entropy ($\alpha>0$) with system of functional equations (as a $1$-cocycle in cohomology), whose general solution is $\frac{K}{2^{1-\alpha}-1} \left(1- \sum_{i=1}^s \mu_i^\alpha\right)$, for $K$ an arbitrary constant.
[^7]: The notation can be misleading, because the terms $1$ and $\theta$ do not commute inside $(1+\theta)^n_q$.
|
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---
abstract: |
In this paper we prove the continuity of all Lyapunov exponents, as well as the continuity of the Oseledets decomposition, for a class of irreducible cocycles over strongly mixing Markov shifts. Moreover, gaps in the Lyapunov spectrum lead to a Hölder modulus of continuity for these quantities. This result is an application of the abstract continuity theorems obtained in [@LEbook], and generalizes a theorem of E. Le Page on the Hölder continuity of the maximal LE for one-parameter families of strongly irreducible and contracting cocycles over a Bernoulli shift.
This is a draft of a chapter in our forthcoming research monograph [@LEbook].
address:
- |
Departamento de Matemática and CMAFIO\
Faculdade de Ciências\
Universidade de Lisboa\
Portugal
- |
Department of Mathematical Sciences\
Norwegian University of Science and Technology (NTNU)\
Trondheim, Norway\
and IMAR, Bucharest, Romania
author:
- Pedro Duarte
- Silvius Klein
title: Large deviation type estimates for random cocycles
---
Introduction and statements {#random_intro}
===========================
We define the class of random cocycles over Markov shifts and describe our assumptions on them. We then formulate the main statements, and sketch the argument for proving large deviation type estimates. Finally we relate our findings to other results for similar models.
Description of the model {#random_model}
------------------------
The spectral method {#random_sm}
-------------------
Literature review {#random_lit_review}
-----------------
An abstract setting {#random_as}
===================
In this section we specialize an abstract setting in [@HH], from which we derive an abstract theorem on the existence of uniform LDT estimates for Markov processes.
The assumptions {#random_setting}
---------------
An abstract theorem {#random_abs_theor}
-------------------
The proof of LDT estimates {#random_proof}
==========================
We prove here the base-LDT and uniform fiber-LDT estimates for irreducible cocycles over mixing Markov shifts. These results follow from the abstract Theorem \[ALDE:uniform\].
Base LDT estimates {#random_base_ldt}
------------------
Fiber LDT estimates {#random_fiber_ldt}
-------------------
Deriving continuity of the Lyapunov exponents {#random_continuity}
=============================================
In this last section we use the LDT estimates (theorems \[Base:LDT\] and \[Fiber:LDT\]) to derive the continuity of the Lyapunov exponents and of the Oseledets’s filtration / decomposition. We give some simple generalizations of the continuity results and explain the method’s limitations regarding the continuity of the LE in the reducible case.
Proof of the continuity {#random_cont_proof}
-----------------------
Some generalizations {#random_generalizations}
--------------------
Method limitations {#random_limitations}
------------------
Acknowledgments {#acknowledgments .unnumbered}
---------------
Both authors would like to acknowledge fruitful conversations they had with A. Baraviera, G. Del Magno and J. P. Gaivão about large deviation estimates for random cocycles.
The first author was supported by Fundação para a Ciência e a Tecnologia, UID/MAT/04561/2013.
The second author was supported by the Norwegian Research Council project no. 213638, “Discrete Models in Mathematical Analysis”. He is also grateful to the University of Lisbon and to the Institut Mittag-Leffler (through its "Research in Peace” program) for their hospitality during the summer of 2013, when this project began.
[10]{}
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Philippe Bougerol and Jean Lacroix, *Products of random matrices with applications to [S]{}chrödinger operators*, Progress in Probability and Statistics, vol. 8, Birkhäuser Boston, Inc., Boston, MA, 1985. [MR ]{}[886674 (88f:60013)]{}
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Pedro Duarte and Silvius Klein, *An abstract continuity theorem of the [L]{}yapunov exponents*, preprint (2015), 1–32.
[to3em]{}, *The [A]{}valanche [P]{}rinciple and other estimates on [G]{}rassmann manifolds*, preprint (2015), 1–49.
[to3em]{}, *Continuity of the [O]{}seledets decomposition*, preprint (2015), 1–32.
[to3em]{}, *Lyapunov exponents of linear cocycles, continuity via large deviations*, Atlantis Studies in Dynamical Systems, Atlantis Press, to appear in 2016.
H. Furstenberg and H. Kesten, *Products of random matrices*, Ann. Math. Statist. **31** (1960), 457–469. [MR ]{}[0121828 (22 \#12558)]{}
Harry Furstenberg, *Noncommuting random products*, Trans. Amer. Math. Soc. **108** (1963), 377–428. [MR ]{}[0163345 (29 \#648)]{}
Y. Guivarc’h and A. Raugi, *Frontière de [F]{}urstenberg, propriétés de contraction et théorèmes de convergence*, Z. Wahrsch. Verw. Gebiete **69** (1985), no. 2, 187–242. [MR ]{}[779457 (86h:60126)]{}
Hubert Hennion and Lo[ï]{}c Herv[é]{}, *Limit theorems for [M]{}arkov chains and stochastic properties of dynamical systems by quasi-compactness*, Lecture Notes in Mathematics, vol. 1766, Springer-Verlag, Berlin, 2001. [MR ]{}[1862393 (2002h:60146)]{}
Alexander S. Kechries, *Classical descriptive set theory*, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995.
S G Krein and Yu I Petunin, *Scales of banach spaces*, Russian Mathematical Surveys **21** (1966), no. 2, 85.
mile Le Page, *Théorèmes limites pour les produits de matrices aléatoires*, Probability measures on groups ([O]{}berwolfach, 1981), Lecture Notes in Math., vol. 928, Springer, Berlin-New York, 1982, pp. 258–303. [MR ]{}[669072 (84d:60012)]{}
Eugene Lukacs, *Characteristic functions*, Hafner Publishing Co., New York, 1970, Second edition, revised and enlarged. [MR ]{}[0346874 (49 \#11595)]{}
S. V. Nagaev, *Some limit theorems for stationary [M]{}arkov chains*, Teor. Veroyatnost. i Primenen. **2** (1957), 389–416. [MR ]{}[0094846 (20 \#1355)]{}
Karl Petersen, *Ergodic theory*, Cambridge Studies in Advanced Mathematics, vol. 2, Cambridge University Press, Cambridge, 1983. [MR ]{}[833286 (87i:28002)]{}
A. Raugi and J. Rosenberg, *Fonctions harmoniques et th[é]{}or[è]{}mes limites pour les marches al[é]{}atoires sur les groupes*, Bulletin de la Soci[é]{}t[é]{} math[é]{}matique de France. M[é]{}moire, no. no. 54, Soci[é]{}t[é]{} math[é]{}matique de France, 1977.
F. Riesz and B. Sz[ő]{}kefalvi-Nagy, *Functional analysis*, Ungar, 1955.
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---
bibliography:
- 'refs.bib'
---
[**[On classical de Sitter solutions\
in higher dimensions ]{}**]{}
\
\
[**Abstract**]{}
> We derive necessary criteria for the existence of classical meta-stable de Sitter solutions in flux compactifications of type II supergravity down to dimensions higher than four. We find that the possibilities in higher dimensions are much more restricted than in four dimensions. The only models that satisfy the criteria are derived from O6 compactifications to $D=5,6$ and O5 compactifications to $D=5$ and no meta-stable solutions can exist in dimensions higher than six. All these models have in common that the compact dimensions are negatively curved.
Introduction
============
An anti-de Sitter conspiracy theory?
------------------------------------
By now there is little doubt that string theory has an enormous (possibly infinite) amount of meta-stable solutions with four large and six small dimensions. What is less obvious is whether any of these solutions can describe our observed universe. A popular idea is that the high amount of solutions implies that many are close (and one is equal) to our observed world since values for various observable quantities are distributed around the landscape with some non-zero probability. This vision entails a possible danger as it ignores any subtle and non-obvious property of the landscape, if any.
An important observable property of a solution is the size of the cosmological constant. All *explicit* and fully trustworthy solutions that have ever been constructed in string theory have a non-positive cosmological constant. One reason for this is that de Sitter solutions necessarily break all susy and are therefore more “dirty”. However, non-susy and trustworthy vacua have been known for some time now (see e.g. the non-susy no-scale vacua of [@Giddings:2001yu]), so the lack of susy cannot be the only explanation. Maybe the true reason is that string theory is simply very constraining when it comes to the existence of meta-stable de Sitter solutions. The current state of affairs is that all de Sitter proposals necessarily invoke quantum corrections that cannot be computed exactly, or feature localised sources whose backreaction is potentially problematic (see e.g. [@McGuirk:2009xx; @Bena:2009xk; @Douglas:2010rt; @Blaback:2010sj; @Bena:2010gs; @Blaback:2011nz; @Giecold:2011gw; @Massai:2011vi; @Burgess:2011rv; @Blaback:2011pn]). From this viewpoint one can understand that all proposals for de Sitter solutions that are claimed to be successful (which started with [@Kachru:2003aw]) have a lack of explicitness, since the problems hide in those places where explicit computations are difficult. To fully appreciate this fact it is noteworthy that dS solutions, that are not even phenomenologically interesting, have not been constructed in a trustworthy and explicit way. Examples of this would be de Sitter solutions in dimensions different from four or de Sitter solutions without fine-tuned size for the cosmological constant, all of which would be very interesting play grounds for studying de Sitter space in string theory nonetheless. Hence, there seems room to believe in some “anti-de Sitter conspiracy”, by which we mean that the landscape of flux vacua does not contain any meta-stable dS solution at all or, in a weaker version, that the amount of dS vacua is dramatically less than assumed sofar.
Disproving the strongest version of this conspiracy theory requires only one explicit counter example that does not have to obey any special property but being meta-stable, de Sitter and trustworthy. It is allowed to be of any dimension, to have any susy-breaking scale, or have only planets populated by purple aliens.
Classical dS solutions in $D=4$?
--------------------------------
Perhaps the easiest place to find simple and trustworthy de Sitter solutions is at the classical level governed by the 10-dimensional supergravity approximation. In this setup, the Maldacena-Nunez nogo-theorem [@Maldacena:2000mw] (see also [@deWit:1986xg]) gives us a useful lead: a solution with a time-independent, regular, compact space without a boundary should involve brane-sources.
The idea that four-dimensional Sitter solutions could arise at the purely classical level was revived in [@Hertzberg:2007wc]. Shortly after this, some attempts were made to find explicit solutions from orientifolds of negatively curved extra dimensions [@Silverstein:2007ac; @Haque:2008jz]. However these solutions can be shown to explicitly fail to solve the 10-dimensional equations of motion. A safer approach is to use either directly the 10-dimensional equations of motion with smeared sources, or, equivalently, use a consistently truncated effective action in the lower dimension. Few de Sitter solutions have been constructed within the first [@Danielsson:2009ff; @Danielsson:2010bc] and second approach[@Flauger:2008ad; @Caviezel:2008tf; @Caviezel:2009tu] (see [@deCarlos:2009qm; @Dibitetto:2010rg] for more related work). A review and extension of these solutions has been presented in [@Danielsson:2011au][^1]. Those solutions that were obtained with purely geometrical ingredients (fluxes, branes and curved spaces) and for which a consistent truncation exists that allows a trustworthy computation of moduli masses, have all been shown to be perturbatively unstable. The known nogo theorems against stability [@Covi:2008ea; @GomezReino:2008bi; @Borghese:2010ei; @Shiu:2011zt] are evaded in the quoted examples and we therefore still lack a simple explanation for the presence of tachyons. On the other hand, whenever supersymmetry is broken there is no particular reason for a solution to be meta-stable. Even more, if one makes the naive assumption that the signs of moduli masses are randomly distributed then meta-stable solutions will be rare since four-dimensional models generically have many moduli[^2].
Even more, if a meta-stable solution will be found in the future, the approximation of smeared orientifolds, might be invalid[@Douglas:2010rt; @Blaback:2010sj; @Blaback:2011nz; @Blaback:2011pn]. In any case, at the moment there is no trustworthy, meta-stable, classical de Sitter solution known[^3]. For now we ignore the issues of smeared orientifolds and backreaction of localised sources and we will be satisfied with classical de Sitter solutions with smeared sources that are perturbatively stable.
Classical dS solutions in $D>4$?
--------------------------------
Our interest for flux vacua in dimensions higher than four is mainly coming from the increase of simplicity when one goes higher in dimension. This simplicity arises because there are much less moduli and the possibilities to wrap branes and fluxes in the compact dimensions is less. Because this paper will strictly be concerned with flux compactifications down to dimensions higher than four we list the various motivations for this:
- As explained above, to disprove a hypothetical anti-de Sitter conspiracy in string theory, it suffices to find one de Sitter solution, which should be easier in dimensions higher than four since there are less moduli.
- A simple classical de Sitter solution in string theory will be very useful as an example to study the hypothetical dS/CFT correspondence [@Strominger:2001pn]. For this purpose the dimension of the de Sitter solution does not have to be four.
- Meta-stable de Sitter solutions in dimensions higher than four can be dimensionally reduced to meta-stable quintessence solutions in four dimensions [@Townsend:2001ea; @Rosseel:2006fs]. Or in some cases meta-stable de Sitter solutions in five dimensions can be linked to meta-stable de Sitter solutions in four dimensions [@Ogetbil:2008tk].
- It has been shown that under certain circumstances the decay of higher-dimensional de Sitter solutions can give a dynamical mechanism for compactification [@Carroll:2009dn].
The rest of this paper is organised as follows. In the next section we review some basic properties of orientifold compactifications that will be needed. In section \[criteria\] we derive the necessary existence and meta-stability criteria for de Sitter solutions in dimensions higher than four. In section \[dSmodels\] we discuss in some details the models that fulfill our necessary criteria. We end with section \[discussion\] where we discuss the obtained results and the interesting directions for further research they imply.
Tree-level energy
=================
The energy sources
------------------
At the classical level there are three ingredients that enter the energy of the lower-dimensional theory: fluxes, branes and curvature of the extra dimensions. Experience with compactifications down to four-dimensions have demonstrated that typical ingredients required for de Sitter solutions are negative tension objects, like orientifold planes, and negatively curved compact dimensions [@Silverstein:2007ac; @Haque:2008jz]. Below we will find similar results for higher dimensions. Since orientifolds are crucial we briefly repeat the most essential properties that we need.
An obvious restriction on the possible $Op$ planes for compactifications down to $D$ dimensions is that $p>D-2$, otherwise the source can not fill the $D$-dimensional space. When it comes to models with orientifold solutions of different types one can use the intersection rules for branes in flat space to understand what kind of mixtures are possible[^4]. The following combined sources could be possible $$\begin{aligned}
& D=5 : O4/O8, O6/O8, O5/O9\,,\nonumber\\
& D=6 : O5/O9, O6/O8\,,\nonumber\end{aligned}$$ and no mixture of $Op$-plane types in higher dimensions. In what follows we discard all models with $O9$ planes since the O9 tadpole can only be canceled by introducing sixteen D9 branes that cancel the negative tension.
The standard way to cancel the tadpole of O$p$ planes with $p<7$, without having to cancel the negative tension as well, uses fluxes. The Bianchi identity for the RR $(8-p)$-form field strength, that is magnetically sourced by an $Op$-plane, is given by $${\textrm{d}}F_{8-p} = H\wedge F_{6-p} + J_{Op}\,,$$ where $J_{Op}$ symbolizes the orientifold brane source and corresponds to a singular (delta-function-like) form or to a regular form in the smeared approximation. The tadpole condition arises when the Bianchi identity is integrated over the compact cycle perpendicular to the source. Clearly the integrated flux combination $H\wedge F_{6-p}$ can cancel the charge (integrated source form) for suitable fluxes. However, this is not possible for $O7-$ and $O8$-planes[^5]. The only way to solve the tadpole condition (without loosing the negative tension) in these cases, would be by considering pairs of $Op$ and anti-$Op$ planes or to make the $Op$ planes wrap cycles for which the source form $J_{Op}$ is exact[^6].
Finally we recall the specific transformation properties of the fluxes under the $Op$ targetspace involutions. The position of the $Op$-plane is given by the surface that is invariant under the target space involution $\sigma$. For all $Op$ planes the $H$-flux must be odd $\sigma(H)= -H$. To discuss the parity of the RR form field strengths $F_i$ we follow [@Koerber:2007hd] and introduce the operator $\alpha$ that reverses the indices of a form. Explicitly this means that $$\begin{aligned}
&\alpha(F_i)= + F_i\,,\qquad i=0,1, 4, 5, 8, 9\,,\nonumber\\
&\alpha(F_i)= - F_i\,,\qquad i=2, 3, 6, 7\,.\nonumber\end{aligned}$$ We then have the following parity conditions $$\begin{aligned}
&\sigma(F_i)= +\alpha(F_i) \,,\qquad (O2, O3, O6, O7 )\,,\nonumber\\
&\sigma(F_i)= -\alpha(F_i) \,,\qquad (O0, O1, O4, O5, O8, O9 )
\,.\nonumber\end{aligned}$$ Given that the source form $J_p$ is even for odd $p$ and odd for even $p$ we find that the parity rules are nicely consistent with the Bianchi-identity, which ensures that we can always employ fluxes to cancel the charge tadpole[^7].
Coupling and volume dependence of the energy
--------------------------------------------
Our starting point is the 10-dimensional supergravity action of type II string theory in string frame $$S = \int \exp(-2\phi)\sqrt{-g}\Bigl( R + (4\partial\phi)^2
-\tfrac{1}{2}\tfrac{1}{3!}H^2\Bigr) - \int
\sqrt{-g}\tfrac{1}{2}\sum_q \tfrac{1}{q!}|F_q^{RR}|^2\,,$$ where $q$ runs over $0, 2, 4$ for type IIA sugra and over $1, 3, 5$ for type IIB supergravity, with the usual caveat for the self-dual 5-form field strength. We suppressed the Chern-Simons terms as we will not need them. The string coupling constant is given by $g_s=\exp(\phi)$. There are also localised sources whose action can be added to the bulk action. The only piece we need is the DBI part that couples to the metric. For $Op$ and $Dp$ sources this is given by $$S_{Dp/Op} = - T_p\int_p \exp(-\phi) \sqrt{g_{p+1}}\,,$$ where $g_{p+1}$ is the induced metric on the source worldvolume and $T_p$ is the tension, which is negative for $Op$ sources and positive for $Dp$ sources. We can also consider NS5 branes, whose action differs from the $D5$ brane by an extra power of $\exp(-\phi)$.
In the unwarped limit, the 10-dimensional metric, describing a compactification downto $D$ dimensions, can be written as follows $${\textrm{d}}s^2_{10} = \tau^{-2}{\textrm{d}}s_D^2 + \rho {\textrm{d}}s^2_{10-D}\,.$$ The modulus $\rho$ is the string frame volume and consequently we normalised $\int \sqrt{g_{10-D}}$ to one in string units. The modulus $\tau$ can be shown to be $$\tau^{D-2}=\exp(-2\phi)\rho^{\frac{10-D}{2}}\,,$$ in order to have $D$-dimensional Einstein frame. The variables $\rho$ and $\tau$ span a flat 2-dimensional subspace of the general modulispace. In this parametrisation $\rho$ and $\tau$ do not have standard kinetic terms, but this is not required for the analysis done in this paper. We have chosen this specific parametrisation in order to make contact with the previous literature on the topic.
Using the above we can derive the dependency of the various energy contributions on the string coupling and volume modulus. The total energy $V$ is the sum of various energies $V=\sum_i V_i$. These separate energies $V_i$ come from the curved extra dimensions, $V_i=V_R$, the $H$-flux, $V_i=V_H$, the RR q-form fluxes $V_i=
V_{RR}^q$, and the sources $V_i= V_{Dp/Op}$ and $V_i=V_{NS5}$. Specifically we find $$\begin{aligned}
& V_R \sim -\tilde{R}_{10-D}(\varphi) \rho^{-1}\tau^{-2}\,,\\
& V_H \sim |\tilde{H}|^2(\varphi) \rho^{-3}\tau^{-2}\,,\\
& V_{RR}^q \sim |\tilde{F}_q|^2(\varphi)
\rho^{\frac{10-D}{2}-q}\tau^{-D}\,,\\
& V_{Dp/Op} \sim
T_p(\varphi)\rho^{\frac{2p-D-8}{4}}\tau^{-\frac{D+2}{2}}\,,\\
&V_{NS5} \sim T(\varphi)\rho^{-2}\tau^{-2}\,.\end{aligned}$$ The notation is such that the tilde contractions $|\tilde{F}|^2$ are done using the internal metric with the volume modulus $\rho$ factored out. Furthermore we introduced the democratic notation for fluxes that are space-filling. As an example, an $F_4$ flux filling four non-compact dimensions (when $D=4$) will be considered through its Hodge dual $F_6$. We have symbolically introduced the hidden dependence on the non-universal moduli as $\varphi$. These non-universal moduli could be shape moduli of the internal dimensions, or gauge potential moduli. The way they appear in the flux contribution and source contribution depends on the details of the cycles thread and wrapped by fluxes and sources.
In this paper we do not consider any KK monopoles or fractional Wilson lines as in [@Silverstein:2007ac]. The backreaction of KK monopoles is worry some, as well is it unclear how to find the stable cycles for such branes. In our approach we stick to brane setups that lead to lower-dimensional supergravity theories. This requires the sources to be calibrated. The benefit of this restriction is that supersymmetry of the lower-dimensional action restricts the possible corrections, and secondly, that the branes are wrapped in a consistent manner.
Existence and stability criteria {#criteria}
================================
With the $\rho, \tau$ dependence at hand one can deduce necessary (but not sufficient) criteria for the existence of de Sitter critical point to the potential. This boils down to analysing the constraints coming from the following two equations and one inequality $$\label{constraints}
\partial_{\rho} V=0\,,\qquad \partial_{\tau} V=0\,, \qquad V>0\,.$$ For the case $D=4$ this has been initiated in [@Hertzberg:2007wc] and systematically worked out in [@Danielsson:2009ff; @Wrase:2010ew]. The equations (\[constraints\]) can easily shown to coincide with specific linear combinations of the 10-dimensional dilaton equation and the trace of the 10-dimensional Einstein equation over the compact dimensions [@Danielsson:2009ff].
The universal moduli $\rho$ and $\tau$ do not only allow us to find existence criteria for de Sitter solutions, they also allow us to find minimal requirements for meta-stability of the de Sitter solutions [@Shiu:2011zt]. This goes as follows: if the two by two Hessian $$\begin{pmatrix}
\partial^2_{\rho} V & \partial_{\rho}\partial_{\tau} V\\
\partial_{\rho}\partial_{\tau} V & \partial^2_{\tau} V
\end{pmatrix}\,,$$ has a negative eigenvalue it implies that the full mass matrix will also have negative eigenvalues, through Silvesters criterium. The eigenvalues of the above two by two Hessian will not be part of the spectrum of the full $N$ by $N$ mass matrix due to moduli mixing, but this is not of any importance when it comes to finding necessary conditions for stability. In what follows it is useful to remind that the eigenvalues $\lambda_{+}, \lambda_{-}$ of a symmetric two by two matrix $$\begin{pmatrix}
t_1 & s\\
s & t_2
\end{pmatrix}$$ are given by $$2\lambda_{\pm} = t_1 + t_2 \pm\sqrt{(t_1+t_2)^2 - 4( t_1t_2-s^2)}\,.$$ Furthermore if either $t_1$ or $t_2$ is negative so will be at least one of the eigenvalues due to Sylvester’s criterium. The smallest eigenvalue is always $\lambda_{-}$ and stability therefore requires $$\label{stability}
\lambda_{-}>0 \qquad \Leftrightarrow \qquad \boxed{t_1t_2 > s^2}\
\,,$$ with both $t_1$ and $t_2$ positive. If this condition is violated we cannot have a meta-stable de Sitter solution. Equation (\[stability\]) is easy to interpret. It simply states that a negative determinant implies negative eigenvalues since the determinant equals the product of eigenvalues.
We will now list all existence and stability criteria for de Sitter solutions that can be obtained from the universal moduli. We simplify the calculations using a rescaling of the moduli. Consider a critical point of the potential at the values $\rho_c,\tau_c$ then we redefine the variables $\rho$ and $\tau$ as follows[^8] $$\rho\rightarrow \frac{\rho}{\rho_c}\,,\qquad \tau\rightarrow
\frac{\tau}{\tau_c}\,.$$ In this notation the critical point is always at the values $\rho=1,\tau=1$. As a consequence the coefficients in the potential are not anymore $|H|^2, |F_p|^2$ or $T$, but are rescaled by some powers of $\rho_c$ and $\tau_c$.
Trivial example : $D=9$ with D8/O8 sources
------------------------------------------
Consider compactifications with only one compact direction. This restricts the possible brane sources to be $D8/O8$ and we are necessarily in IIA. Then the possible flux is $F_0$ flux. We write the potential as follows $$V = f_0^2\tau^{-9}\rho^{1/2} + T \tau^{-11/2}\rho^{-1/4}\,,$$ where, up to numerical factors and rescalings with $\rho_c$ and $\tau_c$, $f_0$ corresponds to the Romans mass and $T$ to the tension. Then we immediately find $$\begin{aligned}
& \partial_{\rho}V=0 \Rightarrow f_0^2 = \tfrac{1}{2}T\,,\\
& \partial_{\tau}V=0 \Rightarrow f_0^2 = -\tfrac{11}{18}T\,.\end{aligned}$$ Hence no solution is possible at all, whether dS, AdS or Minkowski. We have even been too mild here since, strictly speaking, the $F_0$ flux should be projected out by the $O8$ plane and we furthermore have no way to cancel the $O8$ tadpole without canceling the $O8$ tension. In what follows we are more careful in taking into account the orientifold involutions and tadpole conditions.
Less trivial example: $D=7$ with D6/O6 sources
----------------------------------------------
Things start to get more interesting in $D=7$ where one possibility is to have space-filling $D6/O6$ sources. The possible fluxes are $F_0, F_2$ and $H$. Note that $F_2$ is odd and given that it has two legs outside the $O6$-plane it will normally not survive the orientifold projection. We nonetheless keep it with us, in case there is no $O6$ plane or it can somehow survive[^9]
We write the scalar potential as $$V = f_0^2 \tau^{-7}\rho^{3/2} + f_2^2\tau^{-7}\rho^{1/2} +
h^2\tau^{-2}\rho^{-3} - R\tau^{-2}\rho^{-1} + T
\tau^{-9/2}\rho^{-3/4}\,.$$ We introduced a few new symbols: $R$ equals the curvature of the internal dimensions (up to a positive constant), $h^2$ equals the $H^2$ (again with some positive proportionality constant which we will not mention anymore) and $f_2^2$ equals $F_2^2$. We then find $$\begin{aligned}
& \partial_{\rho}V=0 \Rightarrow R = -\tfrac{3}{2}f_0^2 +\tfrac{1}{2}f_2^2 + 3 h^2 +\tfrac{3}{4}T \,,\\
& \partial_{\tau}V=0 \Rightarrow T = -\tfrac{10}{3}f_0^2 -2 f_2^2
+\tfrac{4}{3}h^2 \,.\end{aligned}$$ If we plug this into the on-shell value for $V$ we find $$V=\tfrac{5}{3}(f_0^2 -h^2)\,.$$ This can have any sign, so this model could allow Minkowski, AdS and dS solutions.
The Hessian is given by $$\partial_i\partial_j V = \begin{pmatrix} \tfrac{35}{8}f_0^2 +\tfrac{1}{8}f_2^2
+\tfrac{23}{4}h^2 & -\tfrac{55}{4}f_0^2
-\tfrac{5}{4}f_2^2 +\tfrac{5}{2}h^2 \\
-\tfrac{55}{4}f_0^2 -\tfrac{5}{4}f_2^2 +\tfrac{5}{2}h^2 &
-\tfrac{5}{2}f_0^2 +\tfrac{25}{2}f_2^2 +15h^2
\end{pmatrix}\,.$$ When $V=0$ (and $f_2^2=0$) we find that $R=0$ and that the Hessian has one positive and zero eigenvalue, consistent with the fact that the scalar potential can then be written as a square $$V = f_0^2(\tau^{-7/2}\rho^{3/4} - \tau^{-1}\rho^{-3/2} )^2\,.$$ These solutions are the no-scale Minkowski solutions constructed in [@Blaback:2010sj] and they exist in dimensions $D=2\,\ldots 7$ and were first studied in $D=4$ [@Giddings:2001yu].
In order to have de Sitter solutions we need $f_0^2 > h^2$ and in that case we can demonstrate that the tension is necessarily negative, which corresponds to having net $O6$ sources. Let us therefore take $f_2^2=0$ in what follows. The determinant of the Hessian is given by $$det(\partial_i\partial_j V) = 40f_0^4(-5 + 2\frac{h^4}{f_0^4} + 3
\frac{h^2}{f_0^2})\,.$$ De Sitter requires $h^2/f_0^2<1$ with the no-scale Minkowski solution at the turning point $h^2/f_0^2=1$. Hence we find an elegant structure: *exactly at the Minkowski turning point, a tachyon appears and the de Sitter critical points can never be a local minima of the potential.* These unstable de Sitter solutions could possible be engineered with pairs of $O6$ and anti-$O6$ planes as pointed out in [@Blaback:2011nz].
Summary of all possibilities
----------------------------
As shown in the two examples the technique is to use the two $\partial V=0$ equations to eliminate $R$ and $T$ in terms of the fluxes, which are strict positive. Then this is plugged into the on-shell value for $V$ to read of the sign of the cosmological constant. To determine stability we do the same for the Hessian.
Let us summarise the result of the computation:
1. $D>7$: No dS critical points are possible.
2. $D=7$: A dS critical point build from $O6$ planes and negative curvature is allowed but necessarily unstable as shown explicitly in the previous section.
3. $D=6$: $O5$ sources with negatively curved compact dimensions allow dS critical points, which are necessarily unstable. However $O6$ sources can have critical points which are perturbatively stable in the $\rho,\tau$ directions if $F_2$ is large enough. The curvature of the internal dimensions is required to be negative.
4. $D=5$: $O4$ models with negatively curved compact dimensions again only allow unstable dS critical points, whereas $O5$ and $O6$ models can evade unstable directions at a dS critical point. Also here the curvature of the internal dimensions is negative.
Most relevant to observe here is that meta-stable solutions in $D>6$ cannot exist and that for $D=5, 6$ there is only a small amount of models that potentially allow meta-stable de Sitter solutions.
There is a recurring pattern in each dimension $D$ that allows a no-scale Minkowski solution with space-filling $O(D-1)$-planes (which are the values $D=2,\ldots, 7$ [@Blaback:2010sj]). These models all allow de Sitter critical points by negatively curving the compact dimensions of the no-scale solution and changing the ratio between net tension and charge of the orientifolds (such that there is more net negative tension than net negative charge). These solutions always have an unstable direction, which coincides with the massless direction of the no-scale Minkowski solution. In other words, the results of section 3.2 for the case of $O6$ planes in $D=7$ extends to the other dimensions as well.
We have furthermore checked that none of the mixed orientifold plane combinations mentioned in section 2 fulfill the criteria. This is straightforward for the $O5/09$ combination since we discarded solutions with net $O9$ tension as the $O9$ tadpole cannot be canceled without canceling the $O9$ tension. The $O6/O8$ combinations naively fulfill the criteria in $D=5$ and $D=6$ since $O6$ planes separately do. However the presence of the $O8$ plane implies that the necessary $F_0$ flux is projected out and that tadpole constraint cannot be solved easily. The same tadpole constraint problem is there for the $O4/O8$ model in $D=5$. Upon neglecting tadpole constraints the $O4/O8$ combination does satisfy the criteria for de Sitters solutions that are stable in the $(\rho,\tau)$-directions.
dS building grounds {#dSmodels}
===================
In this section we provide some details of the models in $D=5$ and $D=6$ that potentially allow meta-stable dS solutions.
dS model I: $O6$ planes in $D=6$
--------------------------------
In this case the $O6$ planes wrap a one-cycle which projects out any $F_4$ flux, since the flux would have one leg along the 1-cycle wrapped by the $O6$, giving it odd parity. The remaining fluxes are $H$, $F_0$ and $F_2$. The $\partial V =0$ equations lead to $$\begin{aligned}
& R = \frac{11}{3}(h^2 - f_0^2) - f^2_2\,,\\
& 3T = -10 f_0^2 -6f_2^2 + 4 h^2\,.\end{aligned}$$ When plugged into the potential we find $$V = \frac{4}{3}(f_0^2 - h^2)\,,$$ which shows that dS solutions require $f^2_0>h^2$ and imply negative internal curvature $R<0$ and net $O6$-plane tension $T<0$.
The necessary stability condition (\[stability\]) requires us to compute the determinant of the Hessian. The Hessian is given by $$\partial_i\partial_j V =\begin{pmatrix}\frac{41}{6}f_0^2 + \frac{17}{3}h^2 + \frac{1}{2}f_2^2 & -\frac{34}{3}f_0^2 + \frac{4}{3}h^2 - 2f_2^2 \\
-\frac{34}{3}f_0^2 + \frac{4}{3}h^2 - 2f_2^2 & -\frac{8}{3}f_0^2 +
\frac{32}{3}h^2 + 8 f_2^2
\end{pmatrix}\,.$$ For the Minkowski solutions $f_0^2 = h^2$, the determinant of the Hessian simplifies to $$det(\partial_i\partial_j V) = 64 f_0^2f_2^2\,,$$ which demonstrates that the Minkowski solutions are stable in the $\rho,\tau$-directions, and that the solution is no-scale when either $f_0$ or $f_2$ vanish. For the case $f_0^2 = h^2 =0$ and $f_2\neq 0$ this no-scale solution has been explicitly constructed in [@Blaback:2010sj] from T-dualising GKP[@Giddings:2001yu] and this solution corresponds to an $O6$ wrapping a one-cycle in a nilmanifold.
We can verify that the Hessian can be positive definite for de Sitter solutions by slightly perturbing the Minkowski solutions to: $$f_0^2 = h^2 +\delta\,,\qquad \delta>0\,.$$ Where we think of $\delta$ as arbitrary tiny and positive. At first-order in $\delta$ the determinant then becomes $$det(\partial_i\partial_j V) = 64 h^2f_2^2 -\frac{616}{3}h^2\delta +
8f_2^2\delta \,.$$ Hence it is a simple consequence of continuity of the determinant of the Hessian that de Sitter solutions that are stable in the ($\rho,\tau$)-directions exist when perturbing the no-scale Minkowski vacua given by $f_0=h=0$ and $f_2\neq 0$. This is an interesting place to look for dS solutions since solutions could be tuned very close to a (susy) Minkowski solution.
dS model II: $O5$ planes in $D=5$
---------------------------------
This situation is as good as identical to the O6 planes in $D=6$, with the roles of the $F_0$ and $F_2$ flux now played by the $F_1$ and the $F_3$ fluxes. First one observes that the $F_5$-flux must be projected out (just like the $F_4$ before). The $\partial V=0$ equations lead to $$\begin{aligned}
&R = \frac{9}{2}h^2 - \frac{9}{2}f_1^2 - f_3^2\,,\\
&T = 2h^2 - 4f_1^2 - 2 f_3^2\,,\\
&V = \frac{3}{2}(f_1^2 - h^2)\,.\end{aligned}$$ Again this implies that de Sitter solutions have negatively curved internal dimensions and net orientifold tension. Minkowski solutions exist whenever $f_1 = h$. The Hessian is given by $$\partial_i\partial_j V =\begin{pmatrix}\frac{45}{8}h^2 +\frac{9}{2}f_1^2 +\tfrac{1}{8}f_3^2 & \frac{9}{4}h^2 - 9 f_1^2 -\frac{3}{4}f_3^2 \\
\frac{9}{4}h^2 - 9 f_1^2 -\frac{3}{4}f_3^2 & \frac{21}{2}h^2 - 6
f_1^2 +\frac{9}{2}f_3^2
\end{pmatrix}\,.$$ This Hessian has the same structure as the example above. When slightly perturbing the Minkowski solutions towards de Sitter solutions by $f_1^2 = h^2 +\delta$, where $\delta>0$, the Hessian simplifies and the determinant becomes $$det(\partial_i\partial_j V) = 36 h^2f_3^2 - 162h^2\delta +
6f_3^2\delta\,.$$ We notice that the perturbation of the no-scale solutions with with $f_1=h=0$ (and $f_3\neq 0$) can give de Sitter solutions which are stable in the $(\rho,\tau)$-directions.
dS model III: $O6$ planes in $D=5$
----------------------------------
The two $\partial V =0$ equations entail $$\begin{aligned}
& R = \frac{10}{3}h^2 -\frac{10}{3}f_0^2 - f_2^2 +\frac{4}{3}f_4^2\,,\\
& T = \frac{4}{3}h^2 -\frac{10}{3}f_0^2 -2 f_2^2
-\frac{2}{3}f_4^2\,,\\
&V = f_0^2 - h^2 - f_4^2\,.\end{aligned}$$ As before, we find that de Sitter solutions necessarily require negative curvature ($R<0$) and negative tension ($T<0$). The Hessian is given by $$\partial_i\partial_j V = \begin{pmatrix} \frac{23}{4}h^2 + \frac{75}{8}f_0^2 +\frac{9}{8}f_2^2 + \frac{7}{8}f_4^2 & \frac{1}{2}h^2 -\frac{35}{4}f_0^2 -\frac{9}{4}f_2^2 +\frac{17}{4}f_4^2\\
\frac{1}{2}h^2 -\frac{35}{4}f_0^2 -\frac{9}{4}f_2^2
+\frac{17}{4}f_4^2 & 7h^2 -\frac{5}{2}f^2_0 +\frac{9}{2}f_2^2
+\frac{23}{2}f_4^2\end{pmatrix}\,.$$ The Minkowski solutions are defined by $$f_0^2 = h^2 + f_4^2\,.$$ In this case the determinant of the Hessian is given by $$det(\partial_i\partial_j V) = 9 ( 8f_4^4 + 4h^2f_2^2 + 12 h^2f_4^2 +
4 f_4^2f_2^2) \,.$$ Hence dS points close by are stable in the $(\rho,
\tau)$-directions. Note that no-scale Minkowski solutions are defined by $h^2 = f_0^2 = f_4^2$ and only $f_2^2\neq 0$. Perturbing those no-scale solutions into the dS regime by $f_0^2 =\delta$ leads to an instability in the $(\rho, \tau)$-directions.
Discussion
==========
Obtained results
----------------
In this paper we derived necessary (but not sufficient) conditions for the existence of classical meta-stable de Sitter solutions in $D>4$ dimensions from orientifold compactifications of type II supergravity. The necessary conditions are derived from the universal dependence of the scalar potential on the dilaton and volume modulus. This is a continuation of the results known for compactifications to $D=4$, as derived in [@Hertzberg:2007wc; @Danielsson:2009ff; @Wrase:2010ew; @Shiu:2011zt], for which we have simplified the method significantly.
The results show that no meta-stable de Sitter solution in $D>6$ can exist and that only few possibilities in $D=5, 6$ can work. The three cases that fulfill the necessary criteria ($O6$ in $D=5, 6$ and $O5$ in $D=5$) have in common that the net source tension has to be negative and that the curvature of the internal dimensions has to be negative. This is in line with almost all examples in $D=4$[^10].
Some of the criteria can be weakened when wrapped NS5 branes are included. We have not explicitly listed the new possibilities that arise with NS5 branes, because NS5 branes are difficult to incorporate in simple models. This comes from the NS5 tadpole condition, which cannot be satisfied using fluxes. Therefore one necessarily has to wrap the NS5 branes on trivial cycles or allow anti-NS5 branes in homologous cycles without the branes annihilating. Such possibilities are not easy to construct in a trustworthy manner [@Conlon:2011qp].
An important drawback of our derivation is the smeared (unwarped) approximation. This approximation works for deriving BPS solutions [@Giddings:2001yu; @Blaback:2010sj], but could easily be problematic for non-BPS solutions [@Blaback:2011nz; @Blaback:2011pn]. This is especially easy to spot from the 10-dimensional Einstein equations for dS solutions from orientifolds of negatively curved compact dimensions [@Douglas:2010rt], which comprises all our examples.
Interesting problems for future research
----------------------------------------
The obtained results suggest many different directions for further research, which we list here:
- Both the $O6$ in $D=6$ model and the $O5$ in $D=5$ model have Minkowski vacua and the would-be de Sitter solutions close to those Minkowski vacua are stable in the coupling and volume directions. However, preliminary investigations similar to the ones in [@Blaback:2010sj] indicate that dS solutions might be excluded in these models[^11]. This would point to the very exciting possibility that only the $O6$ model in $D=5$ is left.
- Our criteria are derived from the volume modulus and dilaton. But clearly, for every scalar field that is added one obtains a new stability and existence criterium. It should be possible to add one more “universal” scalar field, without having to fix the geometry and topology of the model. Such a scalar field could be the volume of the cycle wrapped by the orientifold.
- An obvious method that presents itself is to classify 4 and 5 dimensional group manifolds that allow the proper $O5$ and $O6$ involutions, similar to the investigation in [@Danielsson:2011au; @Andriot:2010ju]. This allows to scan a large set of models for de Sitter solutions. We hope to report on this in the future.
- It would be useful to compare the existence and stability issues for de Sitter solutions from type II orientifolds with those from $\alpha'$-corrections in heterotic supergravity as these should be dual to each other[^12]. In this respect the results of the recent paper [@Green:2011cn] indicate that meta-stable de Sitter solutions can be ruled out in large regions of parameter space.
Acknowledgments {#acknowledgments .unnumbered}
===============
I have benefited from discussions with Ulf Danielsson and Timm Wrase and I am especially grateful to Timm Wrase for spotting errors and typos in the first draft. My work is supported by the ERC Starting Independent Researcher Grant 259133-ObservableString.
[^1]: See also references [@Saltman:2004jh; @Andriot:2010ju; @Dong:2010pm] for examples that break susy at the compactification scale.
[^2]: It is useful to consider the situation for extended gauged supergravities. The only models that have meta-stable de Sitter solutions occur for $\mathcal{N}\preceq 2$ supergravity [@Fre:2002pd; @Gunaydin:2000xk; @Cosemans:2005sj] (see also [@Roest:2009tt]). Unfortunately the higher-dimensional origin of these meta-stable de Sitter solutions is not known.
[^3]: There are some recent claims of simple classical warped de Sitter solutions [@Neupane:2010is; @Minamitsuji:2011xs]. If these examples originate from higher dimensional supergravity they are either necessarily singular solutions or they are to be regarded as non-compactifications, as in [@Gibbons:2001wy]. It has been explicitly shown that the de Sitter solutions of [@Neupane:2010is] are of the form of a curved brane with de Sitter worldvolume [@Chemissany:2011gr]. This could still be of relevance for a brane-world type scenario [@Neupane:2010ey].
[^4]: We assume that violating these restrictions make the orientifold wrap cycles that are not calibrated, which probably indicates an inconsistency or at least an instability of the model.
[^5]: The same problem occurs for NS5 branes.
[^6]: In the case of nilmanifolds there are Op solutions known that wrap a 1-cycle whose volume form is not closed and whose Hodge-dual form $J_{Op}$ then becomes exact, see e.g. [@Blaback:2010sj] .
[^7]: This corrects a wrong statement about $O4$ models in [@Danielsson:2009ff].
[^8]: I am grateful to Ulf Danielsson for pointing out this simplifying trick.
[^9]: Note that localised $D6/O6$ sources lead to non-trivial $F_2$ profiles, but this is not really counted as flux, it is rather a consequence of backreaction.
[^10]: We have noticed that the $D=4$ examples were negative curvature is not required probably require fluxes that are projected out by the orientifold or require difficult-to-satisfy tadpole constraints.
[^11]: This is similar to the way dS solutions can be excluded in IIB supergravity with BPS $D3/O3$ sources [@Giddings:2001yu]
[^12]: I would like to thank Callum Quigley for pointing this out.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The electron capture decay of the isotope $^{163}$Ho has been proposed since a long time as a candidate for measuring the electron neutrino mass and recently the interest on this idea has been renewed. In this letter we note that a direct observation of the cosmic antineutrino background could be made using a target made of this isotope. We further discuss the requirements for an experiment aiming to obtain this result.'
author:
- |
Maurizio Lusignoli and Marco Vignati\
\
[ Sapienza, Università di Roma, and INFN, Sezione di Roma]{}\
Piazza A. Moro 2, I-00185 Roma, Italy
title: 'Relic antineutrino capture on $^{163}$Ho decaying nuclei'
---
Introduction
============
For many years the possibility of direct observation of the cosmic neutrino background (C$\nu$B), namely of those neutrinos that are relics of the Big Bang, has been considered and its terrible difficulty stressed. We know that their average number density in the Universe should be $n_{\nu} \sim 55$ cm$^{-3}$ for neutrinos (or antineutrinos) of each flavour and that they decoupled at a temperature around 1 MeV. Since then, the Universe expanded by about 10 orders of magnitude and therefore the average C$\nu$B momenta are today $\sim 10^{-4}$ eV.
Deviations of the spectrum in beta decays near to the end point due to a possible neutrino chemical potential in C$\nu$B were first suggested many years ago by Weinberg [@Weinberg:1962zz], but present limits on the chemical potential from nucleosynthesis [@BBN] make this effect unobservable. It is the effect of the nonzero mass of the neutrino[^1] that could instead provide a hope, if its value is close to the present experimental bound. If this is the case, gravitational clustering could increase the number density $n_{\nu}$ by one or two orders of magnitude [@Ringwald:2004np]. We recall that present limits are $\aprle 2$ eV from tritium decay [@Lobashev:2003kt; @Kraus:2004zw] and $\aprle 0.5$ eV from cosmology [@Fogli:2008ig; @Thomas:2009ae].
It has been recently proposed [@Cocco:2007za] to try to observe the process of capture of a neutrino in the C$\nu$B by a $\beta$–decaying nucleus. In this case the electron in the final state has energy larger than the maximum energy of $\beta$–rays by twice the value of the neutrino mass, and could therefore be distinguished, with a *very* good resolution, if the neutrino mass is large enough. An obvious candidate for the target is tritium, due to its small $Q$–value (18.6 keV) and good sensitivity to neutrino mass effects.
The detection of antineutrinos in C$\nu$B could analogously be made using as target radioactive atoms that decay by electron capture (EC). This possibility has been examined in ref. [@Cocco:2009rh], but apparently discarded as much less promising. In this letter, we want to deepen the examination in order to show that this is not correct, and that in fact the capture of antineutrinos in the C$\nu$B by nuclei of $^{163}$Ho (the record element for low $Q$–value in EC decays) could be a valid alternative.
Electron capture in $^{163}$Ho {#theory1}
==============================
The energy spectrum of neutrinos produced in EC decays is given by a series of lines, each at an energy $Q-E_i$ (where $Q$ is the mass difference of the two atoms in their ground states and $E_i$ is the binding energy of the electron hole in the final atom). The decay process that we consider is: $$^{163}{\rm Ho} \to \; ^{163}{\rm Dy}_i ^*+ \nu_e \to \; ^{163}{\rm Dy} + E_i + \nu_e \;.
\label{decay}$$ The EC decay rate can be expressed, following [@Bambynek:1977zz], as a sum over the possible captured levels: $$\label{ECrate}
\lambda_{EC} = {G_{\beta}^2 \over {4 \pi^2}} \; \sum_i n_i \, C_i\, \beta_i^2 B_i\, (Q-E_i)[(Q-E_i)^2-m_{\nu}^2]^{1/2} \; .$$ In this equation $G_{\beta} = G_F \cos \theta_C$, $n_i$ is the fraction of occupancy of the i-th atomic shell, $C_i $ is the nuclear shape factor, $\beta_i$ is the Coulomb amplitude of the electron radial wave function (essentially, the modulus of the wave function at the origin) and $B_i$ is an atomic correction for electron exchange and overlap. The spin and parity of the nuclei involved in reaction (\[decay\]) obey the relations $\Delta J= 1,\;\Pi_f \Pi_i = +1$, and the transition is dubbed as allowed. The $Q$–value for this reaction is so small that only electrons from levels $M_1, M_2, N_1, N_2, O_1, O_2, P_1$ can be captured. In a very good approximation the nuclear shape factors $C_i$ are all equal in an allowed transition, as it has been discussed in ref. [@Bambynek:1977zz], and can be factored out from the sum in eq.(\[ECrate\]). Different determinations of the $Q$–value can be found in the literature [@NDS]. In this letter we will use values ranging from an optimistic 2.3 keV to a pessimistic 2.8 keV.
The low Q-value of this transition prompted many years ago several proposals to use $^{163}$Ho decays to search for a signal of nonzero neutrino mass (as opposed to the antineutrino mass measured in tritium decays). The neutrino mass in fact affects the capture rates from different levels [@Bennett:1981], as in eq.(\[ECrate\]), and it modifies the spectra of inner brehmstrahlung photons [@DeRujula:1981ti] and emitted electrons [@DeRujula:1982bq] near to their endpoints. Several experiments have been performed [@expHo], obtaining upper bounds on the neutrino mass larger than 200 eV. These measurements were mainly limited by the poor knowledge of complicated atomic corrections that can modify the emitted X–rays spectrum.
A more promising technique would be a calorimetric experiment, embedding the radioactive $^{163}$Ho source in a bolometer. This was suggested many years ago [@DeRujula:1982qt] and it is presently being developed [@Gatti:1997] thanks to the huge improvements of the technique in terms of energy resolution. The advantage with respect to other techniques is that all the de–excitation energy is measured and does not remain partly trapped in invisible channels. The atomic levels have a finite (albeit often small) natural width, and therefore the lines have a Breit–Wigner resonance form, so that the spectrum of “calorimetric” energy $E_c$ should be given by[^2]
$$\begin{aligned}
\label{E_c-distr}
{d \lambda_{EC}\over dE_c} &=& {G_{\beta}^2 \over {4 \pi^2}}(Q-E_c) \sqrt{(Q-E_c)^2-m_{\nu}^2} \;
\times \nonumber \\
&& \sum_i n_i C_i \beta_i^2 B_i {\Gamma_i \over 2\,\pi}{1 \over (E_c-E_i)^2+\Gamma_i^2/4} \;. \end{aligned}$$
A calculated de–excitation energy spectrum is presented in fig.[\[fig:fig1\]]{} and the effect of a nonzero neutrino mass near to the end point is shown in fig.[\[fig:fig2\]]{}.
![ Expected de–excitation energy spectrum of the EC decay of $^{163}$Ho with $Q=2.5$ keV. Detector resolution effects are not included. The parameters used in the calculation are discussed in Section \[numbers\].[]{data-label="fig:fig1"}](spectrum.pdf){width="70.00000%"}
![ Energy spectrum in EC decays for $^{163}$Ho with $Q=2.5$ keV near to the end point for neutrino of zero mass (solid black line) or with $m_{\nu}=0.5$ eV (dashed red line).[]{data-label="fig:fig2"}](endpoint.pdf){width="70.00000%"}
Bolometers, however, have the disadvantage of being slow, and therefore pile–up could be a problem. The way to tame it is to use a large number of smallish detectors [@Gatti:1997].
Relic antineutrino capture in $^{163}$Ho {#theory2}
========================================
Consider now the capture by the original nucleus of a very low energy $\bar \nu_e$ and an electron from the $i$–th atomic shell: $$\bar \nu_e + \, ^{163}{\rm Ho} \to \; ^{163}{\rm Dy}_i^* \; .
\label{reaction}$$ The procedure to evaluate the rate for this process has been presented in ref. [@Cocco:2009rh], following the formalism for EC decays introduced in ref. [@Bambynek:1977zz] and can be written as $$\label{reac-rate}
\lambda_{\bar \nu} = n_{\bar \nu} {G_{\beta}^2 \over 2} \; \sum_i n_i \,C_i \,
\beta_i^2 B_i\, \rho_i(E_{\bar \nu})$$ where $n_{\bar \nu}$ is the number density of incoming antineutrinos, $E_{\bar \nu}$ is their energy ($\simeq m_{\nu}$ for C$\nu$B) and $ \rho_i(E_{\bar \nu})$ is the density of final states. Again, the nuclear shape factors can be factored out of the sum.
The final states in reaction (\[reaction\]) are unstable and the final value of the de–excitation energy must be $Q + m_{\nu}$ for a zero energy incoming antineutrino. Even if this value does not coincide with the maximum of the Breit–Wigner curve it can be reached anyhow, although of course the rate will be suppressed. As a consequence, the number of available final states per unit energy $\rho_i(E_{\bar \nu})$ defined in ref. [@Cocco:2009rh] should be modified as follows: $$\label{BW}
\rho_i(E_{\bar \nu}) = \delta (E_{\bar \nu} + Q - E_i) \longrightarrow
{1 \over \pi} \cdot {\Gamma_i/2 \over {(E_{\bar \nu}+Q-E_i)^2+\Gamma_i^2/4}} \;.$$ The ratio of the rates of the two processes considered is approximately: $$\label{ratio}
{\lambda_{\bar \nu} \over \lambda_{EC}} \simeq 2\,\pi^2\,n_{\bar \nu}\;{\sum_i n_i \beta_i^2 B_i \;
\rho_i(E_{\bar \nu}) \over \sum_i n_i \beta_i^2 B_i (Q-E_i)^2} \;.$$
As we will show in the next section, for C$\nu$B antineutrinos this is a [*really*]{} small number. However, the EC events that can be a background are only those falling in a narrow energy interval before the end point, namely $$Q-\Delta-m_{\nu} \leq E_c \leq Q-m_{\nu}\;.$$ The fraction of EC events falling in this region is given by the so–called factor of merit: $$\label{fig_of_merit}
F(\Delta, m_{\nu},Q) = {1 \over \lambda_{EC}} \; \int_{Q-\Delta-m_{\nu}}^{Q-m_{\nu}}
{d \lambda_{EC}\over dE_c} dE_c\;.$$ Neglecting the variations of the different Breit-Wigner over the small scale $\Delta$ and neglecting the neutrino mass with respect to $Q-E_i$, one has: $$\begin{aligned}
\label{fig1_of_merit}
F(\Delta, m_{\nu},Q)& \simeq {\Delta^3 \over 3} \left(1+{2 m_{\nu} \over \Delta} \right)^{3/2} \times\nonumber \\
&{ \sum_i n_i \beta_i^2 B_i \; (\Gamma_i / 2\,\pi)\;\left[(Q-E_i)^2+\Gamma_i^2/4\right]^{-1}
\over \sum_i n_i \beta_i^2 B_i\; (Q-E_i)^2} \; .\end{aligned}$$
As a consequence, the ratio of the counting rates of the antineutrino capture and the EC decays near to the end point is: $$\begin{aligned}
\label{final}
R(\Delta, m_{\nu}, Q) &=& {1 \over F(\Delta, m_{\nu},Q)} {\lambda_{\bar \nu} \over \lambda_{EC}} \nonumber\\
&\simeq& 6\,\pi^2\, {n_{\bar \nu} \over \Delta^3} \left(1+ {2 m_{\nu} \over \Delta} \right)^{-{3 \over 2}}\;.\end{aligned}$$ The above expression of $R(\Delta, m_{\nu}, Q)$ does not depend on $Q$ and is equal to the analogous result for $\beta$ decay [@Cocco:2007za], showing that both types of radioactive decaying nuclei are in principle equally good as targets to detect C$\nu$B.
Numerical results {#numbers}
=================
We proceed to give numerical results, based on estimates found in the literature for the parameters appearing in the previous equations. The levels of the electrons that can be captured are fully occupied ($n_i=1$). Their binding energies and widths are reported in Table \[En-Wid\]. Note that the real values may slightly differ [@Bennett3] from these, obtained in dysprosium excitation, but they will be precisely determined in future calorimetric experiments, from widths and positions of the peaks in the measured $E_c$ distribution, see fig.(\[fig:fig1\]).
Level $E_i$ (eV) $\Gamma_i$ (eV)
------- ------------ -----------------
M$_1$ 2047 13.2
M$_2$ 1842 6.0
N$_1$ 414.2 5.4
N$_2$ 333.5 5.3
O$_1$ 49.9
O$_2$ 26.3
: Energy levels of the captured electrons, with their widths, for $^{163}$Dy [@param1].
\[En-Wid\]
In Table \[waveratio\] we report the relative values of the squared wave functions, namely the ratios of the parameters $\beta_i^2/\beta_{\rm M_1}^2$. The exchange and overlap corrections[^3] are neglected (i.e. $B_i\sim 1$). The validity of this approximation far from the peaks may be doubted, however the [*shape*]{} of the spectrum in an interval of O($\Delta$) near to the end–point is anyhow determined by the neutrino phase-space factor. The [*rate*]{} at the end–point can be modified: a phenomenological model has been proposed in [@Riisager:1988wy], where it was suggested that $F(\Delta, m_{\nu},Q)$ may be suppressed by a factor about 2. In this case, it is obvious that also the capture rate of C$\nu$B would be suppressed by the same amount, leaving the ratio $R(\Delta, m_{\nu}, Q)$ unchanged. In the following, we will derive results using our expressions, but keep in mind the possibility of a small further suppression in the counting rate. On the other hand, we are neglecting the overdensity due to gravitational clustering, that will certainly increase the rate.
Levels Ratio
------------- --------
M$_2$/M$_1$ 0.0526
N$_1$/M$_1$ 0.2329
N$_2$/M$_1$ 0.0119
O$_1$/M$_1$ 0.0345
O$_2$/M$_1$ 0.0015
P$_1$/M$_1$ 0.0021
: Electrons squared wave functions at the origin $\beta_i^2$ relative to $\beta_{\rm M_1}^2$ [@Band:1985gm].
\[waveratio\]
Assuming for the unknown (and not very relevant) parameters the values $\Gamma_i = (3,3,1)$ eV for the levels (O$_1$, O$_2$, P$_1$) and $E_i \sim 0$ for P$_1$, we find that the ratio of C$\nu$B antineutrino captures to the total EC events defined in eq.(\[ratio\]) is: $$\label{ratioval}
{\lambda_{\bar \nu} \over \lambda_{EC}} = (7.7\cdot10^{-22}, 5.8\cdot10^{-23}, 1.4\cdot10^{-23})$$ for $Q=(2.3, 2.5, 2.8)~{\rm keV}$, values higher than the analogous result for tritium $\beta$–decays [@Cocco:2007za]: $\lambda_{\nu} / \lambda_{\beta} = 6.6\cdot10^{-24}$. Assuming as an example a value of 0.5 eV for the neutrino mass and $\Delta = 0.2$ eV, we have $$\label{figmervall}
F(0.2~{\rm eV}, 0.5~{\rm eV},Q) =
(3.6\cdot10^{-12}, 2.7\cdot10^{-13}, 6.5\cdot10^{-14})$$ for $Q=(2.3, 2.5, 2.8)~{\rm keV}$, to be compared with the value $3\cdot10^{-14}$ for tritium. The half–lives are $T_{1/2} = 4570\;(12.32)$ y for $^{163}$Ho ($^3$H). Therefore we confirm that a calorimetric experiment with $^{163}$Ho, having a higher factor of merit, may be a competitor of $^3$H for hunting the neutrino mass effect.
For the detection of C$\nu$B the ratio of the counting rates of the antineutrino capture and the EC decays near to the end point, $R(\Delta, m_{\nu}, Q)$, is equal to the corresponding quantity for $\beta$–decaying nuclei. We made an analysis including the effect of the detector resolution to determine the discovery potential of a future experiment using a $^{163}$Ho target. Let us consider the total number of signal events: $$S = {\lambda_{\bar \nu} \over \lambda_{EC}} {\log 2 \over T_{1/2}} N_A n_{\rm mol} t\;,$$ where $N_A$ is Avogadro’s number, $n_{\rm mol}$ the number of mols, $t$ the exposure time, $T_{1/2}$ the half–life of $^{163}$Ho and assume that we require a minimum number of 10 events observed. Using the values in eq.(\[ratioval\]) this correspond to a minimum quantity of $^{163}$Ho of (23.2, 307, 1274) kg$\cdot$y for $Q = (2.3, 2.5, 2.8)$ keV.
The number of background events falling in an interval of amplitude $ \Delta_{\rm FWHM} = 2.35\, \Delta$ centered at $Q+m_{\nu}$ can be obtained by convoluting the energy distribution in EC events with a gaussian of variance $\Delta^2$. Defining $$b(\Delta, m_{\nu},Q) = {1 \over \lambda_{EC}}\,{1 \over \sqrt{2 \pi} \Delta} \; \int_{Q+m_{\nu}-\Delta_{\rm FWHM}/ 2}^{Q+m_{\nu}+\Delta_{\rm FWHM}/ 2} dE' \int_{0}^{Q-m_{\nu}} dE
{d \lambda_{EC}\over dE}(E) \; e^{-{(E-E')^2 \over 2 \Delta^2}}$$ the number of background events is: $$\label{bckg1}
B(\Delta, m_{\nu},Q) = b(\Delta, m_{\nu},Q) \;{\log 2 \over T_{1/2}} N_A n_{\rm mol} t \;.$$ Defining the statistical significance as $S / \sqrt {B}$, in fig.[\[fig:fig3\]]{} we present the boundary of the discovery region, where the statistical significance is larger than 5, in the plane ($m_{\nu}$, $\Delta_{\rm FWHM}$). The dependence on $\Delta$ is so sharp that the variation with $Q$ cannot be appreciated given the thickness of the line.
![Detector resolution needed as a function of the neutrino mass. The discovery region, where $S / \sqrt {B} \geq 5$ for $S=10$, falls below the line.[]{data-label="fig:fig3"}](sensitivity.pdf){width="70.00000%"}
The boundary of the discovery region presented in fig.[\[fig:fig3\]]{} also applies to an experiment using a tritium target (and looking for neutrinos in the C$\nu$B instead of antineutrinos). Due to the different half–life and mass number, in this case to have a minimum of 10 signal events one needs 137 g$\cdot$y of $^3$H. At present, the requirements for both EC and $\beta$ decaying nuclei seem very demanding and we do not know which of the two very different technologies may have more chances. Note that our estimates are pessimistic, since the inclusion of gravitational clustering effects would enhance the number of signal events.
Conclusions
===========
We have presented in this work an estimate of the requirements for an experiment aiming to detect antineutrinos in the C$\nu$B using a target of $^{163}$Ho. The request to have a reasonable number of events in the signal gives a constraint on the mass of the source and on the exposure time that is very sensitive to the $Q$–value, not yet well known. Assuming $Q=2.5$ keV one would need ten years of observation of a source of 30 kilograms to have 10 events of signal. Even more stringent maybe are the sensitivity requirements, that instead are practically independent on $Q$: for a neutrino mass of 0.5 eV, for instance, one would need a resolution FWHM better than 0.33 eV in order to attain a statistical significance of 5, the usual requirement for a discovery. Nonetheless, maybe the neutrino mass is higher, the $Q$–value is smaller and the experimental ingenuity may arrive at resolutions better than the present ones.
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[^1]: As usual, we mean by this the mass of the mass eigenstate more strongly coupled to the electron neutrino state.
[^2]: Some justifications concerning the neglect of interference terms and the absence of corrections due to final particles phase space have been given in [@DeRujula:1982qt].
[^3]: They are not given for all the levels needed in [@Bambynek:1977zz]. Those given are less than $\sim 10\%$.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We give new constructions of pair of functions $(f, g)$, analytic in the unit disc, with $g\in H^\infty $ and $f$ an unbounded Bloch function, such that the product $g\cdot f$ is not a Bloch function.'
address: |
Análisis Matemático\
Facultad de Ciencias\
Universidad de Málaga\
29071 Málaga\
Spain
author:
- Daniel Girela
date: 'October 25, 2019'
title: Products of unbounded Bloch functions
---
[^1]
Introduction and statements of the results {#intro}
==========================================
Let ${\mathbb D}=\{z\in{\mathbb C}:|z|<1\}$ denote the open unit disc in the complex plane $\mathbb C$. The space of all analytic functions in ${\mathbb D}$ will be denoted by ${\mathcal Hol}({\mathbb D})$.
For $0<p\le\infty $, the classical Hardy space $H\sp p $ is defined as the set of all $f\in{\mathcal Hol}({\mathbb D})$ for which $$\|f\|_{H\sp
p}{\stackrel{\text{def}}{=}}\sup_{0<r<1}M\sb p(r,f)<\infty ,$$ where, for $0<r<1$ and $f\in {\mathcal Hol}({\mathbb D})$, $$\begin{aligned}
M_p(r,f)\,=\,&\left (\frac{1}{2\pi}\int_0^{2\pi}
|f(re\sp{i\theta })|\sp p d\theta\right )\sp{1/p}, \,(0<p<\infty );
\\M_\infty (r,f)\,=\,&\sup _{\theta \in \mathbb R}|f(re\sp{i\theta
})|.\end{aligned}$$ We mention [@Du:Hp] as a general reference for the theory of Hardy spaces.
A function $f\in{\mathcal Hol}({\mathbb D})$ is said to be a Bloch function if $$\Vert f\Vert _{\mathcal B}{\stackrel{\text{def}}{=}}\vert f(0)\vert +
\sup _ {z\in {\mathbb D}}\,(1-\vert z\vert\sp 2)\vert f\sp\prime (z)\vert
<\infty .$$ The space of all Bloch functions is denoted by $\mathcal B$, it is a Banach space with the just defined norm $\Vert \cdot \Vert
_{\mathcal B}$. It is well known that $$H\sp\infty\subsetneq\mathcal B.$$ A typical example of an unbounded Bloch function is the function $f$ defined by $$f(z)\,=\,\log \frac{1}{1-z},\quad z\in\mathbb D.$$ We mention [@ACP] as a general reference for the theory of Bloch functions.
A function $f$ which is meromorphic in ${\mathbb D}$ is said to be a normal function in the sense of Lehto and Virtanen [@LV] if $$\sup _{z\in {\mathbb D}}(1-\vert z\vert \sp 2)\frac{\vert f\sp\prime (z)\vert }
{1+\vert f(z)\vert \sp 2}<\infty .$$ For simplicity, we shall let $\mathcal N $ denote the set of all holomorphic normal functions in ${\mathbb D}$. It is clear that any Bloch function is a normal function, that is, we have $\mathcal B\subset \mathcal N$. We refer to [@ACP], [@LV] and [@Po:UF] for the theory of normal functions. In particular, we remark here that if $f\in \mathcal N$, $\xi \in
\partial {\mathbb D}$ and $f$ has the asymptotic value $L$ at $\xi$, (that is, there exists a curve $\gamma $ in ${\mathbb D}$ ending at $\xi $ such that $f(z)\to L$, as $z\to \xi $ along $\gamma $) then $f$ has the non-tangential limit $L$ at $\xi $.
Let us recall that if a sequence of points $\{a_n\} $ in the unit disc satisfies the *Blaschke condition*: $\sum\sb{n=1}\sp{\infty} (1-|a\sb n|)<\infty $, the corresponding Blaschke product $B $ is defined as $$B(z)=\prod\sb{n=1}\sp{\infty} \frac{|a\sb{n}|}{a\sb{n}}
\frac{a\sb{n}-z}{1-\overline{a\sb{n}}z} \,.$$ Such a product is analytic in ${\mathbb D}$. In fact, it is an inner function, that is, an $H^\infty $-function with radial limit of absolute value $1$ at almost every point of $\partial \mathbb D$ (cf. [@Du:Hp Chapter 2]).
If $\{a_n\} $ is a Blaschke sequence and there exists $\delta >0$ such that $$\prod_{m\neq n} \left\vert \frac{a_n-a_m}{1-\overline
a_na_m}\right \vert\ge \delta ,\quad\text{for all $n$,}$$ we say that the sequence $\{a_n\} $ is *uniformly separated* and that $B$ is an *interpolating Blaschke product*. Equivalently, $$\label{inter}
B \,\, {\it is \, an \, interpolating\, Blaschke\, product} \,\,\,
\Leftrightarrow\,\,\, \inf _{n\ge 1}(1-\vert a_n\vert \sp 2)\vert
B'(a_n)\vert >0.$$ We refer to [@Du:Hp Chapter 9] and [@Gar Chapter VII] for the basic properties of interpolating Blaschke products. In particular, we recall that an exponential sequence is uniformly separated and that the converse holds if all the $a_k$’s are positive.
Lappan [@La Theorem 3] proved that if $B$ is an interpolating Blaschke product and $f$ is a normal analytic function in ${\mathbb D}$, the product $B\cdot f$ need not be normal. Lappan used this to show that $\mathcal N$ is not a vector space.
Lappan’s result is a consequence of the following easy fact: if $B$ is an interpolating Blaschke product whose sequence of zeros is $\{
a_n\} $ and $G$ is an analytic function in $\mathbb D$ with $G(a_n)\to\infty$, then $f=B\cdot G$ is not a normal function (and hence it is not a Bloch function either). This result has been used by several authors (see [@Cam; @Ya1; @Ya2; @Gi:Pri; @Gi:Ill; @BGM]) to construct distinct classes of non-normal functions.
The author and Suárez proved in [@GS] a result of this kind dealing with Blaschke products with zeros in a Stolz angle but not necessarily interpolating, improving a result of [@GGP]. Namely, Theorem1 of [@GS] is the following.
\[Th-GGP\] Let $B$ be an infinite Blaschke product whose sequence of zeros $\{
a_n\} $ is contained in a Stolz angle with vertex at $1$ and let $G$ be analytic in $\mathbb D$ with $G(z)\to\infty$, as $z\to 1$. Then the function $f=B\cdot G$ is not a normal function.
It is natural to ask whether it is possible to prove results similar to those described, substituting Blaschke products by some other classes of $H^\infty
$-functions. Our first result in this paper deals with the atomic singular inner function.
\[Bloch-inner\] Let $S$ be the atomic singular inner function defined by $$S(z)=\exp\left (-\frac{1+z}{1-z}\right ),\quad
z\in \mathbb D,$$ and let $f$ be a Bloch function with $$\lim_{z\to 1}\vert f(z)\vert =\infty .$$ Then the function $F$ defined by $F(z)=S(z)f(z)$ is not a normal function (hence, it is not a Bloch function).
In particular, the function $f$ defined by $f(z)=S(z)\cdot \log
\frac{1}{1-z}$ ($z\in \mathbb D$) is not normal. Since a Bloch function satisfies $M_\infty (r,f)\,=\,\operatorname{O}\left
(\log\frac{1}{1-r}\right )$, Theorem\[Bloch-inner\] follows from the following result.
\[unbound-inner\] Let $S$ be the singular inner function defined by $S(z)=\exp\left
(-\frac{1+z}{1-z}\right )$ ($z\in \mathbb D$) and let $f$ be an analytic function in $\mathbb D$ satisfying:
- $\lim_{z\to 1}\vert f(z)\vert =\infty
$.
- $\vert f(r)\vert \,=\,\operatorname{o}\left
(\exp\frac{1+r}{1-r}\right )$,as $r\to 1^-$.
Then the function $F$ defined by $F(z)=S(z)f(z)$ is not a normal function (hence, it is not a Bloch function).
For $g\in{\mathcal Hol}({\mathbb D})$, the multiplication operator $M_g$ is defined by $$M_g(f)(z){\stackrel{\text{def}}{=}}g(z)f(z),\quad f\in {\mathcal Hol}({\mathbb D}),\,\, z\in {\mathbb D}.$$ Let us recall that if $X$ and $Y$ are two spaces of analytic function in ${\mathbb D}$ and $g\in{\mathcal Hol}({\mathbb D})$ then $g$ is said to be a multiplier from $X$ to $Y$ if $M_g(X)\subset Y$. The space of all multipliers from $X$ to $Y$ will be denoted by $M(X,Y)$ and $M(X)$ will stand for $M(X,X)$. Brown and Shields [@BS] characterized the space of multipliers of the Bloch space $M\mathcal (B)$ as follows.
\[mult-Bloch\]A function $g\in{\mathcal Hol}(\mathbb D)$ is a multiplier of the Bloch space if and only if $g\in H^\infty \cap
\mathcal B_{\log }$, where $\mathcal B_{\log }$ is the Banach space of those functions $f\in{\mathcal Hol}({\mathbb D})$ which satisfy $$\label{Blog}\Vert f\Vert _{B_{\log }}{\stackrel{\text{def}}{=}}\vert f(0)\vert +\sup_{z\in \mathbb D}(1-\vert z\vert ^2)\left (\log
\frac{2}{1-\vert z\vert ^2}\right )\vert f^\prime (z)\vert <\infty
.$$
Thus, if $g\in H^\infty \setminus \mathcal B_{\log }$ there exists a function $f\in \mathcal B\setminus H^\infty $ such that $g\cdot
f\notin \mathcal B$. It is easy to see that the analytic Lipschitz spaces $\Lambda _\alpha $ ($0<\alpha \le 1$) and the mean Lipschitz spaces $\Lambda ^p_{\alpha }$ ($1<p<\infty $, $1/p<\alpha \le 1$) are contained in $M(\mathcal B)$ We refer to [@Du:Hp Chapter5] for the definitions of these spaces, let us simply recall here that $$\Lambda ^1_1=\{ f\in {\mathcal Hol}(\mathbb D) : f^\prime \in H^1\} .$$ On the other hand, Theorem1 of [@GGH] shows the existence of a Jordan domain $\Omega $ with rectifiable boundary and $0\in \Omega
$, and such that the conformal mapping $g$ from $\mathbb D$ onto $\Omega $ with $g(0)=0$ and $g^\prime (0)>0$ does not belong to $\mathcal B_{\log }$. For this function $g$ we have that $g\in
\Lambda ^1_1$ but $g$ is not a multiplier of $\mathcal B$. Thus we have: $$\Lambda ^1_1\,\not\subset \,M(\mathcal B).$$
In view of this and the results involving Blaschke products that we have mentioned above, it is natural to ask the following question:
\[question\] Is it true that for any given $f\in
\mathcal B\setminus H^\infty $ there exists a function $g\in \Lambda
^1_1$ such that $g\cdot f\notin \mathcal B$?
We shall show that the answer to this question is affirmative. Actually we shall prove a stronger result.
We let $B^1$ denote the minimal Besov space which consists of those functions $f\in {\mathcal Hol}(\mathbb D)$ such that $$\int_{\mathbb D}\vert
f^{\prime \prime }(z)\vert \,dA(z)\,<\,\infty .$$ Here $dA$ denotes the area measure on $\mathbb D$. Alternatively, the space $B^1$ can be characterized as follows (see [@AFP]):
For $f\in{\mathcal Hol}({\mathbb D})$, we have that $f\in B^1$ if and only there exist a sequence of points $\{ a_k\} _{k=1}^\infty \subset {\mathbb D}$ and a sequence $\{
\lambda_k\} _{k=0}^\infty \in \ell ^1$ such that $$\label{B1def}
f(z)=\lambda _0+\sum_{k=1}^\infty\lambda_k\varphi_{a_k}(z),\quad
z\in{\mathbb D}.$$ Here, for $a\in \mathbb D$, $\varphi
_a:\mathbb D\rightarrow \mathbb D$ denotes the Möbius transformation defined by $$\label{phia}\varphi_a(z)=\frac{a-z}{1-\overline az},\quad z\in \mathbb D
.$$ is is well known that $B^1\subset \Lambda^1_1$ (see [@AFP; @DGV1]) and then our next result implies that the answer to question\[question\] is affirmative.
\[not-mul-B1\] If $f\in \mathcal B\setminus H^\infty $ then there exists $g\in B^1$ such that $g\cdot f\notin \mathcal B$.
The proofs of Theorem\[unbound-inner\] and Theorem\[not-mul-B1\] will be presented in section\[proofs\]. We close this section noticing that throughout the paper we shall be using the convention that $C=C(p,
\alpha ,q,\beta , \dots )$ will denote a positive constant which depends only upon the displayed parameters $p, \alpha , q, \beta
\dots $ (which often will be omitted) but not necessarily the same at different occurrences. Moreover, for two real-valued functions $E_1, E_2$ we write $E_1\lesssim E_2$, or $E_1\gtrsim E_2$, if there exists a positive constant $C$ independent of the arguments such that $E_1\leq C E_2$, respectively $E_1\ge C E_2$. If we have $E_1\lesssim E_2$ and $E_1\gtrsim E_2$ simultaneously then we say that $E_1$ and $E_2$ are equivalent and we write $E_1\asymp E_2$.
The proofs {#proofs}
==========
[*Theorem\[unbound-inner\].*]{} For $0<a<1$, set $\Gamma _a=\{ z\in \mathbb D : \vert z-a\vert
={1-a}\} $. If $z\in \Gamma _a$ then $\operatorname{Re}\frac{1+z}{1-z}\,=\,\frac{a}{1-a}$ and, hence, $$\label{modSGammaa}\vert S(z)\vert \,=\,\exp\left
(\frac{-a}{1-a}\right ),\quad z\in \Gamma _a.$$ This, together with (i), implies that $$F(z)\to \infty ,\quad \text{as $z\to 1$ along $\Gamma _a$}.$$ Hence $F$ has the asymptotic value $\infty $ at $1$. On the other hand, (ii) implies that $F$ has the radial limit $0$ at $1$. Then it follows that $F$ is not normal.
[*Theorem\[not-mul-B1\].*]{} Take $f\in \mathcal B\setminus H^\infty $. Set $$\varphi (r)\,=\,M_\infty (r,g),\quad 0<r<1.$$ Clearly, $\varphi (r)\to\infty $, as $r\to 1$ and it is well known that $$\phi (r)\,=\,\operatorname{O}\left (\log \frac{1}{1-r}\right ).$$ This implies that $$\label{op-square}(1-r)^2\varphi (r)\to
0,\quad \text{as $r\to 1$}.$$ Choose a sequence of numbers $\{ r_n\} \subset (0,1)$ satisfying the following properties:
- $\{ r_n\} $ is increasing.
- $(1-r_n)^2\varphi (r_n)\,=\,\operatorname{o}\left (\left (
\frac{1-r_{n-1}}{n}\right )^2\right ),\quad \text{as $n\to \infty
$.}$
- $\varphi (r_n)\,\ge\,2\varphi (r_{n-1})$, for all $n$.
- $\frac{1-r_{n+1}}{1-r_n}\to 0$, as $n\to \infty $.
The existence of such a sequence is clear, bearing in mind (\[op-square\]) and the the fact that $\varphi (r)\to\infty $, as $r\to 1$.
Set $$\lambda _k\,=\,\varphi (r_k)^{-1/2},\quad k=1, 2, \dots .$$ For each $k$, take $a_k\in \mathbb D$ with $\vert a_k\vert
\,=\,r_k$ and $\vert f(a_k)\vert \,=\,\varphi (r_k)$. Using (iii), it follows that $$\label{sumlam}\sum_{k=1}^\infty
\lambda _k\,<\,\infty .$$ Define $$\label{def-f}g(z)\,=\,\sum_{k=1}^\infty \lambda
_k\varphi _(z),\quad z\in \mathbb D.$$ Using (\[sumlam\]) we see that the sum in (\[def-f\]) defines an analytic function in $\mathbb D$ which belongs to $B^1$. Set $$F(z)=g(z)f(z),\quad z\in \mathbb D.$$ Since $g\in H^\infty $ and $f\in\mathcal B$ we see that $$\label{gfprime}\vert g(a_n)f^\prime (a_n)\vert
\,=\,\operatorname{O}\left (\frac{1}{1-\vert a_n\vert }\right ).$$ On the other hand, $$\label{gprime}\vert g^\prime (a_n)f(a_n)\vert
\,\gtrsim \,I\,-\,II\,-\,III,$$ where $$\begin{aligned}
I\,=&\,\vert f(a_n)\vert \lambda_n \vert
\varphi_{a_n}(a_n)\vert,\,\,\,II\,\lesssim \,\vert f(a_n)\vert
\sum_{k=1}^{n-1}\lambda _k\frac{1-\vert a_k\vert ^2}{\vert
1-\overline {a_k}\,a_n\vert ^2},
\\ III\,\lesssim &\,\vert
f(a_n)\vert \sum_{k=n+1}^{\infty }\lambda _k \frac{1-\vert a_k\vert
^2}{\vert 1-\overline {a_k}\,a_n\vert ^2}.\end{aligned}$$ Clearly, $$\label{I}
I=\,\vert f(a_n)\vert \lambda_n \vert \varphi_{a_n}(a_n)\vert
\,\asymp \,\frac{\varphi (r_n)^{1/2}}{1-r_n}.$$ Using the definitions, the facts that $\varphi $ and the sequence $\{
r_n\} $ are increasing, and (ii), we obtain $$\begin{aligned}
\label{II}II\,\lesssim \,&\vert f(a_n)\vert
\sum_{k=1}^{n-1}\lambda _k\frac{1-\vert a_k\vert ^2}{\vert
1-\overline {a_k}\,a_n\vert ^2}\\ \,\lesssim \,& \varphi
(r_n)\sum_{k=1}^{n-1}\varphi (r_k)^{-1/2}\frac{1-\vert a_k\vert
}{[(1-\vert a_k\vert )+(1-\vert a_n)]^2} \nonumber
\\ \,\lesssim \,&\varphi
(r_n)\sum_{k=1}^{n-1}\frac{1}{\varphi (r_k)^{1/2}(1-r_k)} \nonumber
\\
\,\lesssim \,&\frac{n\varphi (r_n)}{1-r_{n-1}} \nonumber
\\
\,=\,& \frac{\varphi (r_n)^{1/2}}{1-r_n}\,\varphi
(r_n)^{1/2}\,\frac{n(1-r_n)}{1-r_{n-1}}\nonumber
\\
\,=\,&\operatorname{o}\left (\frac{\varphi (r_n)^{1/2}}{1-r_n}\right )\nonumber.\end{aligned}$$ Likewise, using the definitions, the facts that $\varphi $ and the sequence $\{ r_n\} $ are increasing, (iii), and (iv), we obtain $$\begin{aligned}
\label{III}
III\,\lesssim \,&\varphi (r_n)\sum_{k=n+1}^\infty \frac{\varphi
(r_k)^{-1/2}(1-r_k)}{[(1-r_k)+(1-r_n)]^2}
\\ \,\lesssim \,&\varphi
(r_n)\sum_{k=n+1}^\infty \varphi
(r_k)^{-1/2}\frac{1-r_k}{(1-r_n)^2}\nonumber
\\ \,\lesssim \,&\varphi
(r_n)\frac{1-r_{n+1}}{(1-r_n)^2}\sum_{k=n+1}^\infty \varphi
(r_k)^{-1/2}\nonumber
\\ \,\lesssim \,&
\frac{\varphi (r_n)^{1/2}}{1-r_n}\cdot
\frac{1-r_{n+1}}{1-r_n}\nonumber
\\
\,=\,&\operatorname{o}\left (\frac{\varphi (r_n)^{1/2}}{1-r_n}\right ).\nonumber\end{aligned}$$ Using (\[I\]), (\[II\]), (\[III\]), and the fact that $\lim\varphi (r_n)=\infty $, we deduce that $$(1-\vert a_n\vert )\vert g^\prime (a_n)f(a_n)\vert \,\rightarrow
\,\infty ,\quad\text{as $n\to\infty$}.$$ This and (\[gfprime\]) imply that $F$ is not a Bloch function.
[10]{}
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[^1]: This research is supported in part by a grant from El Ministerio de Economía y Competitividad, Spain (PGC2018-096166-B-I00) and by grants from la Junta de Andalucía (FQM-210 and UMA18-FEDERJA-002).
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We present a spectroscopic catalog of 70,841 visually inspected M dwarfs from the seventh data release (DR7) of the Sloan Digital Sky Survey (SDSS). For each spectrum, we provide measurements of the spectral type, a number of molecular bandheads, and the H$\alpha$, H$\beta$, H$\gamma$, H$\delta$ and Ca II K emission lines. In addition, we calculate the metallicity-sensitive parameter $\zeta$ and 3D space motions for most of the stars in the sample. Our catalog is cross-matched to Two Micron All Sky Survey (2MASS) infrared data, and contains photometric distances for each star. Future studies will use these data to thoroughly examine magnetic activity and kinematics in late-type M dwarfs and examine the chemical and dynamical history of the local Milky Way.'
author:
- 'Andrew A. West$^{1}$, Dylan P. Morgan$^{1}$, John J. Bochanski$^{2,3}$, Jan Marie Andersen$^{1}$, Keaton J. Bell$^{4}$, Adam F. Kowalski$^{4}$, James R. A. Davenport$^{4}$, Suzanne L. Hawley$^{4}$, Sarah J. Schmidt$^{4}$, David Bernat$^{5}$, Eric J. Hilton$^{4}$, Philip Muirhead$^{5}$, Kevin R. Covey$^{1,5}$, B[á]{}rbara Rojas-Ayala$^{5}$, Everett Schlawin$^{5}$, Mary Gooding$^{6}$, Kyle Schluns$^{1}$, Saurav Dhital$^{7}$, J. Sebastian Pineda$^{3}$, David O. Jones$^{1}$'
title: The Sloan Digital Sky Survey Data Release 7 M Dwarf Spectroscopic Catalog
---
Description of the Catalog
==========================
We provide a brief description of some of the measured quantities in our Sloan Digital Sky Survey [SDSS; @york00] Data Release 7 [DR7; @dr7] value-added catalog below. A much more thorough description of the catalog selection and its bulk characteristics can be found in @west10. The catalog will eventually be available on the Vizier[^1] site, but can also be obtained immediately by contacting the primary author (AAW).
We visually inspected 116,161 M dwarf candidates (color selected from the SDSS database) and manually assigned spectral types. The sample was divided among 17 individuals[^2] who used the manual “eyecheck” mode of the Hammer [v. 1\_2\_5; @covey07] to assign spectral types and remove non-M dwarf interlopers, resulting in 70,841 M dwarfs (see Figure \[fig:bam\]). We also matched our catalog to the 2MASS point source catalog [@2mass], matching only to unique 2MASS counterparts within 5$^{\prime\prime}$ of the SDSS position that do not fall within the boundaries of an extended source ([gal\_contam]{} $=$0).
Radial velocities (RVs) were measured by cross-correlating each spectrum with the appropriate @bootem M dwarf template. This method has been shown to produce uncertainties ranging from 7-10 kms$^{-1}$ [@bootem]. All of the DR7 objects were cross-matched to the USNO-B/SDSS proper motion catalog [@munn04; @munn08], identifying 39,151 M dwarfs with good proper motions. Distances to each star were calculated using the $M_r$ vs. $r-z$ color-magnitude relation given in @boo10. The proper motions and distances were combined with the RVs to produce 3-dimensional space motions for the DR7 M dwarfs.
As part of our analysis, we measured a number of spectral lines and molecular features in each M dwarf spectrum. All of the spectral measurements were made using the RV corrected spectra. The TiO1, TiO2, TiO3, TiO4, TiO5, TiO8, CaOH, CaH1, CaH2, and CaH3 molecular bandhead indices and their formal uncertainties were measured using the Hammer with the molecular bandheads as defined by @pmsu1 and @gizis97. We also measured the chromospheric hydrogen Balmer and Ca II lines that are associated with magnetic activity. We expanded the H$\alpha$ analysis of @west04 [@west08] to include H$\beta$, H$\gamma$, H$\delta$ and Ca II K (H$\epsilon$ and Ca II H are blended in SDSS data and were not included in our sample). All of the line measurements were made by integrating over the specific line region (8 Å wide centered on the line) and subtracting off the mean flux calculated from two adjacent continuum regions. Equivalent widths (EW) were computed for each line by dividing the integrated line flux by the mean continuum value.
For all of the active stars in the sample [see @west10 for definition of activity] we computed the ratio of luminosity in the emission line as compared to the bolometric luminosity (L$_{\rm{line}}$/L$_{\rm{bol}}$). We followed the methods of @hall96, @walkowicz04 and @westhawley08 who derived $\chi$ factors for the Balmer and Ca II chromospheric lines as a function of M dwarf spectral type. The L$_{\rm{line}}$/L$_{\rm{bol}}$ values were computed by multiplying the EW of each active star by the appropriate $\chi$ value. Formal uncertainties were computed for each L$_{\rm{line}}$/L$_{\rm{bol}}$ value and are included in the final database.
We also computed the metal sensitive parameter $\zeta$, defined by @lepine07, which uses a combination of the TiO5, CaH2, and CaH3 molecular band indices to separate the sample into different metallicity classes. This is similar to the @gizis97 classification system but was re-calibrated using wide common proper motion pairs that were assumed to be at the same metallicity. Stars with solar metallicity (\[Fe/H\]=0) have $\zeta$ values $\sim$1 and stars with \[Fe/H\]=-1 have $\zeta$ $\sim$0.4 [@woolf09]. Although there is considerable scatter in the \[Fe/H\] versus $\zeta$ relation at high-metallicities, this parameter is very useful for finding and classifying low-metallicity stars that are likely members of the Galactic halo.
As with previous SDSS spectroscopic catalogs of low-mass stars, we remind the community that these data do not represent a complete sample and that the complicated SDSS spectral targeting introduces a variety of selection effects. However, our new sample covers a large range of values for many of the physical attributes of the M dwarfs, including parameters that are sensitive to activity, metallicity, and Galactic motion, making accurate activity, kinematic, and chemical analyses possible. In addition, because some of the derived quantities are computed by automatic routines, values for a small percentage of individual stars may be incorrect; this should not affect large statistical results. Users are nevertheless cautioned to understand the origin of specific data products before using them indiscriminately.
natexlab\#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\]\[\][[\#2](#2)]{}
, K. N., et al. 2009, , 182, 543.
, J. J., [Hawley]{}, S. L., [Covey]{}, K. R., [West]{}, A. A., [Reid]{}, I. N., [Golimowski]{}, D. A., & [Ivezi[ć]{}]{}, [Ž]{}. 2010, , 139, 2679.
, J. J., [West]{}, A. A., [Hawley]{}, S. L., & [Covey]{}, K. R. 2007, , 133, 531.
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, J. A., [Monet]{}, D. G., [Levine]{}, S. E., [Canzian]{}, B., [Pier]{}, J. R., [Harris]{}, H. C., [Lupton]{}, R. H., [Ivezi[ć]{}]{}, [Ž]{}., [Hindsley]{}, R. B., [Hennessy]{}, G. S., [Schneider]{}, D. P., & [Brinkmann]{}, J. 2004, , 127, 3034
— 2008, , 136, 895
, I. N., [Hawley]{}, S. L., & [Gizis]{}, J. E. 1995, , 110, 1838
, L. M., [Hawley]{}, S. L., & [West]{}, A. A. 2004, , 116, 1105.
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, A. A., [Hawley]{}, S. L., [Bochanski]{}, J. J., [Covey]{}, K. R., [Reid]{}, I. N., [Dhital]{}, S., [Hilton]{}, E. J., & [Masuda]{}, M. 2008, , 135, 785.
, A. A., [Hawley]{}, S. L., [Walkowicz]{}, L. M., [Covey]{}, K. R., [Silvestri]{}, N. M., [Raymond]{}, S. N., [Harris]{}, H. C., [Munn]{}, J. A., [McGehee]{}, P. M., [Ivezi[ć]{}]{}, [Ž]{}., & [Brinkmann]{}, J. 2004, , 128, 426.
, A. A., et al. 2010, , submitted
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, D. G., et al. 2000, , 120, 1579.
[^1]: http://vizier.u-strasbg.fr/cgi-bin/VizieR
[^2]: The order of the co-authors was based on the number of spectra examined.
|
{
"pile_set_name": "ArXiv"
}
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---
abstract: 'We study the simple-stellar-population-equivalent (SSP-equivalent) age and chemical composition measured from the Lick/IDS line-strength indices of composite stellar populations (CSP). We build two sets of $\sim$30000 CSP models using stellar populations synthesis models, combining an old population and a young population with a range of ages and chemical compositions representative of early-type galaxies. We investigate how the SSP-equivalent stellar parameters of the CSP’s depend on the stellar parameters of the two input populations; how they depend on $V$-band luminosity-weighted stellar parameters; and how SSP-equivalent parameters derived from different Balmer-line indices can be used to reveal the presence of a young population on top of an old one. We find that the SSP-equivalent age depends primarily on the age of the young population and on the mass fraction of the two populations, and that the SSP-equivalent chemical composition depends mainly on the chemical composition of the old population. Furthermore, while the SSP-equivalent chemical composition tracks quite closely the $V$-band luminosity weighted one, the SSP-equivalent age does not and is strongly biased towards the age of the young population. In this bias the age of the young population and the mass fraction between old and young population are degenerate. Finally, assuming typical error bars, we find that a discrepancy between the SSP-equivalent parameters determined with different Balmer-line indices can reveal the presence of a young stellar population on top of an old one as long as the age of the young population is less than $\sim$2.5 Gyr and the mass fraction of young to old population is between 1% and 10%. Such disrepancy is larger at supersolar metallicities.'
author:
- |
P. Serra[^1] and S.C. Trager\
Kapteyn Astronomical Institute, RuG, Landleven 12, 9747 AD, Groningen, NL
date: 6 September 2006
title: 'On the interpretation of age and chemical composition of composite stellar populations determined with line-strength indices.'
---
\[firstpage\]
galaxies: stellar content
Introduction {#intro}
============
The knowledge of age and chemical composition of stars in early-type galaxies is a fundamental piece in the puzzle of galaxy formation and evolution. For a long time, optical-wavelength studies in this direction have been hampered by the age-metallicity degeneracy: an age variation of a factor of $\sim$3 mimics a metallicity variation of a factor of $\sim$2 in the spectra of old stellar populations (e.g., Faber 1973; O’Connell 1986; Worthey 1994). The effort of various authors during the past two decades culminated in the work of Worthey (1994), who showed that age and metallicity can be disentangled by the joint use of pairs of line-strength indices, one metal-line and one Balmer-line index, measured from the optical spectra of galaxies. A system of line-strength indices was defined (the Lick/IDS system, see Burstein et al. 1984; Worthey et al. 1994 and references therein; Worthey & Ottaviani 1997) and is now widely used in order to determine the age [$t$]{}, metallicity [\[$Z$/H\]]{} and abundance ratio [\[E/Fe\]]{} of stars in galaxies ([\[E/Fe\]]{} is defined in Trager et al. 2000a as a way of parameterising deviations from the solar abundance pattern).
In practise, one compares the indices measured from the optical spectrum of a galaxy to their values predicted by stellar populations models (provided for example by Worthey 1994; Vazdekis 1999; Bruzual & Charlot 2003; Thomas, Maraston & Bender 2003). The stellar [$t$]{}, [\[$Z$/H\]]{} and [\[E/Fe\]]{} of the galaxy are the ones of the model whose indices best agree with the measured ones. Because each model consists of a single-burst stellar population (SSP) whose stars, unlike in real galaxies, all have the same [$t$]{}, [\[$Z$/H\]]{} and [\[E/Fe\]]{}, the derived stellar parameters are labelled as *SSP-equivalent*. We will refer to them as [$t_{\rm SSP}$]{}, [\[$Z$/H\]$_{\rm SSP}$]{} and [\[E/Fe\]$_{\rm SSP}$]{}.
Early-type galaxies are the most massive stellar systems for which the SSP approximation seems to hold. Therefore, many authors have measured their line-strength indices in order to determine their stellar content (see for example Trager et al. 2000b; Caldwell, Rose & Concannon 2003; Denicoló et al. 2005; Thomas et al. 2005; Clemens et al. 2006). However, many results suggest that recent star formation occurred in these galaxies (Trager et al. 2000b; Yi et al. 2005), so that a small fraction of their current stellar mass formed few Gyr ago. It is natural to wonder about the meaning of the line-strength indices analysis under such circumstances (i.e., in the presence of more than one SSP in the same galaxy). How do the SSP-equivalent parameters relate to the average properties of a galaxy and to the ones of the many SSP’s that it hosts? And what do we actually learn from SSP-equivalent parameters?
Previous authors have already looked into this problem. Using a limited number of models of composite stellar populations (CSP), Trager et al. (2000b) found that the SSP-equivalent age is heavily biased towards the age of the young stars present in a galaxy. In this paper we address the same questions in a more systematic and extensive way from the point of view of the stellar population models.
We build two sets of CSP’s by using the models of Bruzual & Charlot (2003, hereafter BC03) and Worthey (1994, hereafter W94). Each dataset contains $\sim$3$\times10^4$ CSP models composed of one old and one young SSP. Different CSP’s correspond to different stellar parameters of the parent SSP’s. The old SSP (SSP$_1$) is always chosen to be more massive than the young one (SSP$_2$), as inferred in many early-type galaxies (e.g., Trager et al. 2000b; Leonardi & Worthey 2000; Jeong et al. 2006). We analyse the line-strength indices of the CSP’s and derive the corresponding [$t_{\rm SSP}$]{}, [\[$Z$/H\]$_{\rm SSP}$]{} and [\[E/Fe\]$_{\rm SSP}$]{} as is usually done for observed galaxies. We then analyse how the result depends on the input parameters ([$t$]{}$_1$, [\[$Z$/H\]]{}$_1$, [\[E/Fe\]]{}$_1$, [$t$]{}$_2$, [\[$Z$/H\]]{}$_2$,[\[E/Fe\]]{}$_2$ and $\mu$=$M_2$/$M_1$ where $M$ is the stellar mass) and on the luminosity-weighted properties of the CSP’s. We restrict our study to systems composed of only two SSP’s because this case is still relatively easy to treat and might be a reasonable first-order approximation for early-type galaxies. In Sect.\[models\] we explain in some details the construction of the two datasets, we present the results in Sect.\[results\], and finally draw some conclusions.
Models of composite stellar populations {#models}
=======================================
[l|l]{} parameter & values\
[$t$]{}$_1$ (Gyr) & 10.0,13.0\
[\[$Z$/H\]]{}$_1$ & $-$0.15,0.0,0.15,0.3\
[\[E/Fe\]]{}$_1$ & 0.0,0.15,0.3,0.45\
[$t$]{}$_2$ (Gyr) & 1,1.4,1.8,2.5,3.4,4.7\
[\[$Z$/H\]]{}$_2$ & 0.0,0.15,0.3,0.45\
[\[E/Fe\]]{}$_2$ & $-$0.15,0,0.15,0.3\
$\mu$=$M_2$/$M_1$ & 0.0,0.001,0.005,0.01,0.025,0.05,0.1,0.2,0.35,0.5\
Stellar parameters are labelled as “1” and “2” corresponding to the old population (SSP$_1$) and the young population (SSP$_2$) respectively. The last row lists the values adopted for the mass fraction between the two populations.
We define a set of old and young SSP’s in Table \[input\]. All the possible combinations of one old and one young SSP for each of the values of the mass fraction $\mu$ give the 30720 CSP models that form each of the datasets. A dataset consists of a 7-dimensional grid in the parameters ([$t$]{}$_1$, [\[$Z$/H\]]{}$_1$, [\[E/Fe\]]{}$_1$, [$t$]{}$_2$, [\[$Z$/H\]]{}$_2$,[\[E/Fe\]]{}$_2$, $\mu$). At each position in the grid we store a vector that contains the line-strength indices of the two input SSP’s and of the resulting CSP, the luminosity-weighted stellar parameters ([$t_{\rm LW}$]{}, [\[$Z$/H\]$_{\rm LW}$]{}, [\[E/Fe\]$_{\rm LW}$]{}) and the SSP-equivalent stellar parameters ([$t_{\rm SSP}$]{}, [\[$Z$/H\]$_{\rm SSP}$]{},[\[E/Fe\]$_{\rm SSP}$]{}).
The luminosity-weighted stellar parameters are defined in terms of the $V$-luminosity of the parent populations at the respective ages (i.e., taking into account the change in mass-to-light ratio as a population ages).
![*Top panel*: BC03 parent SSP’s of solar [\[E/Fe\]]{} plotted on top of BC03 model grid; filled triangles represent the young SSP’s (SSP$_2$); empty circles represent the old SSP’s (SSP$_1$) which dominate the mass of the CSP’s (see Table \[input\]). *Bottom panel*: distribution of the BC03 CSP’s built from the solar-[\[E/Fe\]]{} SSP’s.[]{data-label="solarsample"}](BC03_solarsample_in.eps){width="7cm"}
![*Top panel*: BC03 parent SSP’s of solar [\[E/Fe\]]{} plotted on top of BC03 model grid; filled triangles represent the young SSP’s (SSP$_2$); empty circles represent the old SSP’s (SSP$_1$) which dominate the mass of the CSP’s (see Table \[input\]). *Bottom panel*: distribution of the BC03 CSP’s built from the solar-[\[E/Fe\]]{} SSP’s.[]{data-label="solarsample"}](BC03_solarsample_out.eps){width="7cm"}
The SSP-equivalent parameters are determined by comparing the Lick/IDS line-strength indices Mg$b$, Fe5270 and Fe5335 and a Balmer-line index of the CSP to their values according to models (BC03 or W94 depending on the dataset). As Balmer-line index we use alternatively H$\beta$, H$\gamma_A$, H$\gamma_F$, H$\delta_A$ or H$\delta_F$, obtaining SSP-equivalent parameters for each of them separately. These will then be labelled according to the Balmer-line index used in deriving them (e.g., [$t$]{}$_{H\beta}$ is the SSP-equivalent age derived using Mg$b$, Fe5270 and Fe5335 and H$\beta$). The use of different Balmer lines is important because they respond differently to the presence of a young stellar component on top of the old one (Schiavon et al. 2004).
[cccccccc]{} & $\log$ [$t$]{}$_1$ & $\log$ [$t$]{}$_2$ & [\[$Z$/H\]]{}$_1$ & [\[$Z$/H\]]{}$_2$ & [\[E/Fe\]]{}$_1$ & [\[E/Fe\]]{}$_2$ & $\mu$\
$\log$ [$t_{\rm SSP}$]{}& 0.05 , 0.10 & *0.51 , 0.56* & $-$0.14 , $-$0.07 & 0.09 , 0.12 & $-$0.01 , 0.01 & $-$0.01 , 0.00 & *$-$0.57 , $-$0.65*\
[\[$Z$/H\]$_{\rm SSP}$]{}& 0.02 , 0.06 & $-$0.19, $-$0.32 & *0.78 , 0.85* & 0.09 , 0.16 & $-$0.01 , 0.01 & 0.00 , 0.01 & 0.25\
[\[E/Fe\]$_{\rm SSP}$]{}& 0.00 , 0.01 & $-$0.11, $-$0.01 & 0.02 , 0.07 & $-$0.06 , $-$0.04 & *0.85 , 0.87* & 0.23 & $-$0.19 , $-$0.17\
Each entry is the range within which the covariance coefficient varies when changing Balmer-line index (or the value of the coefficient if this does not vary). The largest coefficients for each SSP-equivalent parameter are given in italics.
![H$\beta$-based [$t_{\rm SSP}$]{}, [\[$Z$/H\]$_{\rm SSP}$]{} and [\[E/Fe\]$_{\rm SSP}$]{} plotted versus the respective $V$-band luminosity-weighted quantities using BC03 dataset. The dashed line of each plot is the identity line. The colour codes [$t$]{}$_2$.[]{data-label="z&e"}](tpaper2.eps){width="6.3cm"}
![H$\beta$-based [$t_{\rm SSP}$]{}, [\[$Z$/H\]$_{\rm SSP}$]{} and [\[E/Fe\]$_{\rm SSP}$]{} plotted versus the respective $V$-band luminosity-weighted quantities using BC03 dataset. The dashed line of each plot is the identity line. The colour codes [$t$]{}$_2$.[]{data-label="z&e"}](zpaper2.eps){width="6.3cm"}
![H$\beta$-based [$t_{\rm SSP}$]{}, [\[$Z$/H\]$_{\rm SSP}$]{} and [\[E/Fe\]$_{\rm SSP}$]{} plotted versus the respective $V$-band luminosity-weighted quantities using BC03 dataset. The dashed line of each plot is the identity line. The colour codes [$t$]{}$_2$.[]{data-label="z&e"}](epaper2.eps){width="6.3cm"}
Although W94 and BC03 models are available only with [\[E/Fe\]]{}=0, Table \[input\] contains also parent SSP’s of non-solar [\[E/Fe\]]{}. In these cases we correct the line-strength indices given by the models according to the [\[E/Fe\]]{} variations. We then use the corrected values when both building the CSP models and measuring their SSP-equivalent parameters. The correction scheme is the one described in Trager et al. (2000a) but improved by the use of new response functions computed and kindly provided by G. Worthey. For details see Trager, Faber & Dressler (2006).
Fig.\[solarsample\] shows the distribution of the solar-[\[E/Fe\]]{} parent SSP’s drawn from the BC03 models and of the resulting CSP’s on the plane \[\[MgFe\],H$\beta$\], where age and metallicity are efficiently decoupled. The points are plotted on top of the BC03 solar-[\[E/Fe\]]{} model grid. The CSP models cover most of the area where early-type galaxies have been observed to lie (e.g., Trager et al. 2000a; Denicoló et al. 2005; Thomas et al. 2005).
Results
=======
We calculate the covariance coefficients between the SSP-equivalent stellar parameters and the input parameters in order to understand which input parameters are mostly driving the variations in [$t_{\rm SSP}$]{}, [\[$Z$/H\]$_{\rm SSP}$]{} and [\[E/Fe\]$_{\rm SSP}$]{}. Table \[covariance\] shows the result of this calculation performed on the BC03 dataset. The covariance coefficients do not change much when changing Balmer-line index. In the table we give the range within which each coefficient varies when changing Balmer-line index. Furthermore, we have verified that the W94 dataset gives the same result. The following comments apply therefore to all Balmer-line indices and to both datasets.
- The variations in [$t_{\rm SSP}$]{} are mostly driven by variations in [$t$]{}$_2$, the age of the young population, and $\mu$, the mass fraction, while other parameters play a secondary role. The sign and absolute value of these two covariance coefficients clearly show the strong degeneracy between [$t$]{}$_2$ and $\mu$: the same [$t_{\rm SSP}$]{} can result from a small mass of young stars or a sufficiently large mass of older stars.
- The variations in [\[$Z$/H\]$_{\rm SSP}$]{} are by far dominated by variations in [\[$Z$/H\]]{}$_1$, the metallicity of the old population. The mass fraction $\mu$ and the age of the young population [$t$]{}$_2$ also play a relevant role with the usual degeneracy. The latter correlations must be (at least partially) due to the fact that in the dataset [\[$Z$/H\]]{}$_2$ is on average larger than [\[$Z$/H\]]{}$_1$. However, we have verified that the covariance coefficients between [\[$Z$/H\]$_{\rm SSP}$]{} and $\mu$ and between [\[$Z$/H\]$_{\rm SSP}$]{} and [$t$]{}$_2$ remain significantly larger than zero when considering a subset of models with [\[$Z$/H\]]{}$_1$ and [\[$Z$/H\]]{}$_2$ sampled in identical ways (in particular, the covariance coefficients drop by a factor of $\sim$2 and $\sim$1.3 respectively).
- [\[E/Fe\]$_{\rm SSP}$]{} seems to be varying mostly because of variations in the abundance ratios of the two parent populations, with the old, massive population being dominant. As for [\[$Z$/H\]$_{\rm SSP}$]{}, the correlation with $\mu$ and [$t$]{}$_2$ is only in part due to the different sampling of [\[E/Fe\]]{}$_1$ and [\[E/Fe\]]{}$_2$. Using a subset of models with identical sampling of [\[E/Fe\]]{}$_1$ and [\[E/Fe\]]{}$_2$ reduces the covariance coefficient with $\mu$ and increases the one with [$t$]{}$_2$ by a factor of $\sim$3.
![Difference between H$\beta$- and H$\gamma_A$-based SSP-equivalent parameters as a function of $\mu$. The plots are obtained using the BC03 dataset and solar [\[$Z$/H\]]{} and [\[E/Fe\]]{} for both SSP$_1$ and SSP$_2$. The age of SSP$_1$ is fixed at 13 Gyr.[]{data-label="differences"}](ballpaper2.eps){width="8cm"}
Covariance coefficients highlight which of the input parameters play the dominant role in determining the variation of the SSP-equivalent ones. It is also interesting to see how the latter relate to the average properties of the model CSP’s. Fig.\[z&e\] shows the comparison between the H$\beta$-based SSP-equivalent parameters and the $V$-band luminosity-weighted ones. The behaviour is substantially the same when using different Balmer-line indices.
[\[$Z$/H\]$_{\rm SSP}$]{} and [\[E/Fe\]$_{\rm SSP}$]{} seem to track quite closely [\[$Z$/H\]$_{\rm LW}$]{} and [\[E/Fe\]$_{\rm LW}$]{} respectively; strong deviations are observed only for the youngest [$t$]{}$_2$. On the other hand [$t_{\rm SSP}$]{} is always much smaller than [$t_{\rm LW}$]{} and lies somewhere between the latter and [$t$]{}$_2$. This effect was already known (Trager et al. 2000b). Its explanation is that the determination of [$t_{\rm SSP}$]{} relies primarily on Balmer-line indices (see the model grid in Fig.\[input\]). These are dominated by young stars and therefore [$t_{\rm SSP}$]{} is strongly biased towards the age of the young stellar component. As Fig.\[z&e\] illustrates, the younger SSP$_2$ the stronger this bias is.
Fig.\[z&e\] demonstrates that it is not correct to use [$t_{\rm SSP}$]{} as an estimate of when a galaxy formed its stars (yet, this is often done; see for example Clemens et al. 2006). A fair statement would be that *[$t_{\rm SSP}$]{} is a Balmer-line-weighted age* and it should always be kept in mind that such age is strongly biased towards the age of young stellar components. Furthermore, as highlighted by the covariance coefficient and confirmed by Fig.\[z&e\], the effect of [$t$]{}$_2$ and $\mu$ on [$t_{\rm SSP}$]{} is degenerate. An increasingly older SSP$_2$ can produce the same [$t_{\rm SSP}$]{} as long as $\mu$ is properly increased (in Fig.\[z&e\] $\mu$ increases towards decreasing [$t_{\rm SSP}$]{} and [$t_{\rm LW}$]{}).
As mentioned, SSP-equivalent parameters derived from different Balmer-line indices behave substantially in the same way (i.e., Fig.\[z&e\] looks roughly the same for all of them). However, different Balmer-line indices are sensitive to the presence of young stars at different levels (Schiavon et al. 2004). Because of this [$t_{\rm SSP}$]{}, [\[$Z$/H\]$_{\rm SSP}$]{} and [\[E/Fe\]$_{\rm SSP}$]{} (of a CSP) computed with different Balmer-line indices will not be in agreement. Fig.\[differences\] illustrates this concept for a subset of the CSP models chosen to have solar chemical composition and [$t$]{}$_1$=13 Gyr. It can also be seen that the difference between H$\beta$- and H$\gamma_A$-based SSP-equivalent parameters goes to zero for $\mu$ approaching 0 and 1 and peaks between these two extremes at a position dependent on [$t$]{}$_2$. Furthermore, Fig.\[differences\] shows once more the degeneracy between [$t$]{}$_2$ and $\mu$. The same difference between, for example, [$t$]{}$_{H\beta}$ and [$t$]{}$_{H\gamma_A}$ can be caused by increasingly older SSP$_2$’s as long as the mass fraction $\mu$ is sufficiently increased.
![Difference between H$\beta$- and H$\gamma_A$-based [\[$Z$/H\]$_{\rm SSP}$]{} (top) and [\[E/Fe\]$_{\rm SSP}$]{} (bottom) plotted versus the difference in [$t_{\rm SSP}$]{}. Each point corresponds to a BC03 CSP model. The presence of a young stellar population on top of an old one causes SSP-equivalent parameters based on different Balmer-line indices to disagree. This effect must be and indeed is observable simultaneously in [$t_{\rm SSP}$]{}, [\[$Z$/H\]$_{\rm SSP}$]{} and [\[E/Fe\]$_{\rm SSP}$]{}. In particular, points seem to be distributed along a very tight relation in the age-metallicity plane.[]{data-label="totaldeltas"}](deltaallpaper.eps){width="8cm"}
For clarity Fig.\[differences\] shows only a subset of the CSP models. A similar trend is anyway observed in the whole sample (and in W94 dataset), showing that it is possible to detect the presence of a young stellar component on the basis of the disagreement between SSP-equivalent parameters obtained with different Balmer-line indices. We actually expect that more dramatic disagreements in, for example, [$t_{\rm SSP}$]{} are accompanied by larger differences in [\[$Z$/H\]$_{\rm SSP}$]{} and [\[E/Fe\]$_{\rm SSP}$]{}. This is indeed observed and showed in Fig.\[totaldeltas\], where the difference between H$\beta$- and H$\gamma_A$-based [\[$Z$/H\]$_{\rm SSP}$]{} and [\[E/Fe\]$_{\rm SSP}$]{} is plotted versus the difference in [$t_{\rm SSP}$]{}. In particular the age-metallicity plot shows a very tight relation. This could be used in order to test the correctness of one’s results.
Different Balmer lines can be used as a diagnostic for the presence of a young stellar component only as long as the differences in the SSP-equivalent parameters are larger than the observational errors. These are typically of 0.1 on the logarithm of [$t_{\rm SSP}$]{} and on [\[$Z$/H\]$_{\rm SSP}$]{} and of 0.05 on [\[E/Fe\]$_{\rm SSP}$]{} (see for example Trager et al. 2000a; Thomas et al. 2005). Fig.\[totaldeltas\] shows that with these errors [$t_{\rm SSP}$]{} measurements are the most efficient in revealing a young component, allowing the detection of SSP$_2$’s younger than $\sim$2.5 Gyr. As suggested by Fig.\[differences\], this is however possible only within a certain range of $\mu$. The actual range depends on [$t$]{}$_2$ but we find it to be roughly between 1% and 10%. [\[$Z$/H\]$_{\rm SSP}$]{} and [\[E/Fe\]$_{\rm SSP}$]{} are only sensitive to SSP$_2$’s younger than $\sim$1.5 Gyr with $\mu$ between 2% and 10%. It is important to stress that for $\mu\geq$10% there is no detectable difference between the SSP-equivalent parameters derived from different Balmer-line indices. At these values of $\mu$ the Balmer-line indices are so heavily dominated by the younger populations that they all “see” the same age, which is very close to the age of the young population.
![Difference between H$\beta$- and H$\gamma_A$-based [$t_{\rm SSP}$]{} plotted versus the H$\gamma_A$-based [\[$Z$/H\]]{}. There is a clear trend that gives larger [$t_{\rm SSP}$]{} discrepancies at larger [\[$Z$/H\]$_{\rm SSP}$]{}.[]{data-label="newfind"}](roleofz.eps){width="6.5cm"}
Fig.\[differences\] and Fig.\[totaldeltas\] show another interesting feature: the largest difference between [$t$]{}$\rm_{H\beta}$ and [$t$]{}$\rm_{H\gamma_A}$ in Fig.\[differences\] is small compared to the one in Fig.\[totaldeltas\]. Recall that Fig.\[differences\] is relative to CSP’s where both SSP$_1$ and SSP$_2$ have solar chemical composition, while Fig.\[totaldeltas\] represents the whole BC03 sample, with [\[$Z$/H\]]{} growing significantly above solar. The two figures suggest that [\[$Z$/H\]]{} plays an important role with respect to the difference [$t$]{}$\rm_{H\beta}$–[$t$]{}$\rm_{H\gamma_A}$. Fig.\[newfind\] shows that indeed high (and therefore more easily detectable) differences in [$t_{\rm SSP}$]{} occurr only at high metallicities. Similar plots hold for the difference in [\[$Z$/H\]]{} and [\[E/Fe\]]{}. We therefore find that the use of more than one Balmer-line index can reveal the presence of a young stellar populations, but that this is possible only for a small range of [$t$]{}$_2$ and $\mu$ and depends also on the metallicity of the populations.
We would like to remind the reader that a disagreement between Balmer-line-based SSP-equivalent parameters, in principle revealing the presence of a young stellar component, could also result from the approach used when analysing the data. In particular, it is important to remember that different Balmer-line indices respond differently to variations in [\[E/Fe\]]{}. Thomas et al. (2004) and Thomas & Davies (2006) pointed out that this causes a discrepancy between SSP-equivalent parameters determined from different Balmer-line indices when using as a comparison models with solar [\[E/Fe\]]{} only. This effect could mimic the presence of a young stellar component. However, no such problem should occur when using models that account properly for [\[E/Fe\]]{} variations, as was done here.
Another delicate point when using Balmer-line indices is their increase caused by hot star populations like blue horizontal branch stars or blue stragglers (Maraston & Thomas 2000; Trager et al. 2005). Although this is not an issue for the present study, where we do not explore the metal-poor regime at which these stars are expected to be found, this problem should always be kept into consideration when dealing with real data.
Conclusions
===========
We have built two datasets of composite stellar populations (CSP) using Bruzual & Charlot (2003) and Worthey (1994) models. Each CSP model in the datasets consists of an old single-burst stellar population (SSP$_1$) and a younger, less massive one (SSP$_2$). We have investigated how the SSP-equivalent parameters determined by measuring the Lick/IDS line-strength indices of the CSP’s depend on the stellar parameters of SSP$_1$ and SSP$_2$. By means of covariance coefficients we have found that, regardless of the particular stellar populations models used in building the CSP’s and of the Balmer-line index used for the analysis: [$t_{\rm SSP}$]{}, the SSP-equivalent age, depends primarily on [$t$]{}$_2$, the age of the young population, and $\mu$, the mass fraction between the two populations; variations in [\[$Z$/H\]$_{\rm SSP}$]{}, the SSP-equivalent metallicity, are mostly driven by variations in [\[$Z$/H\]]{}$_1$, the metallicity of the old population; and [\[E/Fe\]$_{\rm SSP}$]{} the SSP-equivalent abundance ratio, depends mainly on [\[E/Fe\]]{}$_1$, the abundance ratio of the old population.
Furthermore, we have found that [\[$Z$/H\]$_{\rm SSP}$]{} and [\[E/Fe\]$_{\rm SSP}$]{} track quite closely the $V$-band luminosity-weighted metallicity and abundance ratio ([\[$Z$/H\]$_{\rm LW}$]{} and [\[E/Fe\]$_{\rm LW}$]{}) except in case of very young (and significantly massive) SSP$_2$. On the other hand, [$t_{\rm SSP}$]{} does not follow [$t_{\rm LW}$]{}, being strongly biased towards the the age [$t$]{}$_2$ of the young population. The SSP-equivalent age [$t_{\rm SSP}$]{} is simply a Balmer-line-weighted age *and should not be interpreted as the time passed since the formation of most of the stars in a galaxy.*
Finally, as found by Schiavon et al. (2004), using more than one Balmer-line index can reveal the presence of a young stellar component on top of an old one. In this case, SSP-equivalent parameters derived from different Balmer-line index give discrepant results. This is true however only for values of $\mu$ between 1% and 10% and for [$t$]{}$_2\leq$2.5 Gyr assuming typical errors on [$t_{\rm SSP}$]{}, [\[$Z$/H\]$_{\rm SSP}$]{} and [\[E/Fe\]$_{\rm SSP}$]{}. Furthermore, these discrepancies are higher at supersolar [\[$Z$/H\]$_{\rm SSP}$]{}. Finally, this method does not appear to break the degeneracy between the age and the mass fraction of the young population, especially when considering the size of the typical error bars. In this respect, what is really needed is an age-sensitive index dependent on the age of the old stellar population (i.e., RGB stars), to be used in combination with Balmer-line indices.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors would like to thank the referee, Daniel Thomas, for useful comments that helped clarify important points of the discussion. We also thank Guy Worthey for providing new index responses to abundance ratio variations in advance of publication.
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[^1]: E-mail: [email protected]
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{
"pile_set_name": "ArXiv"
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abstract: 'Inferring the information of 3D layout from a single equirectangular panorama is crucial for numerous applications of virtual reality or robotics (e.g., scene understanding and navigation). To achieve this, several datasets are collected for the task of 360$^\circ$ layout estimation. To facilitate the learning algorithms for autonomous systems in indoor scenarios, we consider the Matterport3D dataset with their originally provided depth map ground truths and further release our annotations for layout ground truths from a subset of Matterport3D. As Matterport3D contains accurate depth ground truths from time-of-flight (ToF) sensors, our dataset provides both the layout and depth information, which enables the opportunity to explore the environment by integrating both cues. Our annotations are made available to the public at <https://github.com/fuenwang/LayoutMP3D>.'
author:
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Fu-En Wang$^{* 1}$\
[[email protected]]{}
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Yu-Hsuan Yeh$^{* 2}$\
[[email protected]]{}
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Min Sun$^{1}$\
[[email protected]]{}
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Wei-Chen Chiu$^{2}$\
[[email protected]]{}
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Yi-Hsuan Tsai$^{3}$\
[[email protected]]{}
title: 'LayoutMP3D: Layout Annotation of Matterport3D'
---
[[ ]{}]{} [[ ]{}]{} [[ ]{}]{} [[ ]{}]{}
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{
"pile_set_name": "ArXiv"
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abstract: 'Dispersive representations of the $\pi\pi$ scattering amplitudes and pion form factors, valid at two-loop accuracy in the low-energy expansion, are constructed in the presence of isospin-breaking effects induced by the difference between the charged and neutral pion masses. Analytical expressions for the corresponding phases of the scalar and vector pion form factors are computed. It is shown that each of these phases consists of the sum of a “universal” part and a form-factor dependent contribution. The first one is entirely determined in terms of the $\pi\pi$ scattering amplitudes alone, and reduces to the phase satisfying Watson’s theorem in the isospin limit. The second one can be sizeable, although it vanishes in the same limit. The dependence of these isospin corrections with respect to the parameters of the subthreshold expansion of the $\pi\pi$ amplitude is studied, and an equivalent representation in terms of the $S$-wave scattering lengths is also briefly presented and discussed. In addition, partially analytical expressions for the two-loop form factors and $\pi\pi$ scattering amplitudes in the presence of isospin breaking are provided.'
author:
- 'S. Descotes-Genon'
- 'M. Knecht'
title: 'Two-loop representations of low-energy pion form factors and $\pi\pi$ scattering phases in the presence of isospin breaking'
---
INTRODUCTION {#intro}
============
In recent years, our knowledge of low-energy pion-pion scattering has improved in a very significant way and in several respects. Firstly, the high precision $K^+_{e4}$ experiments performed at the BNL AGS by the E865 experiment [@Pislak:2001bf; @Pislak:2003sv] and, more recently, by the NA48/2 collaboration [@Batley:2007zz; @Batley:2010zza] at the CERN SPS, have provided very accurate determinations of the difference $\delta_0^0 - \delta_1^1$ of the pion-pion phase shifts in the $S$ and $P$ waves in the energy range between threshold and the kaon mass. Next, one should mention the measurement of the invariant mass distribution in $K^{\pm} \to \pi^\pm \pi^0 \pi^0$ decays [@Batley:2005ax; @Batley:2000zz], that gives information on the $S$-wave $\pi\pi$ scattering lengths [@Cabibbo:2004gq] (see also [@Cabibbo:2005ez; @Gamiz:2006km; @Gasser:2011ju]). Finally, forthcoming analyses of the data collected by the NA48/2 experiment on the $K^+_{e4}$ decay channel into a pair of neutral pions (for preliminary reports, see [@Masetti:2007de; @NA48-2_Ke4_cusp]), or on the $K^0_L \to \pi^0 \pi^0 \pi^0$ decay mode [@Madigozhin:2009zz], together with the measurement of the pionium lifetime by the DIRAC collaboration [@Yazkov:2009zz], should provide additional information, and might sharpen the picture even more. In the meantime, the accuracy obtained on $K_{\ell 4}$ decays from NA48/2 implies that these data clearly drive the current determination of the difference between the $S$ and $P$ phase shifts at low energies, and in particular of the two scattering lengths $a_0^0$ and $a_0^2$, for which very accurate predictions are available [@Colangelo:2001df]. This provides a particularly stringent test of two-flavour chiral perturbation theory [@Gasser:1983yg], and its underlying assumptions [@DescotesGenon:2001tn].
In order to extract relevant information on low-energy pion-pion scattering from the above processes, it has become mandatory to take isospin violations into account. This is certainly quite easy to understand in the case of the $K^{\pm} \to \pi^\pm \pi^0 \pi^0$ decay, where one exploits the presence of a unitarity cusp in the invariant $\pi^0\pi^0$ mass distribution, which occurs only if the masses of the charged and neutral pions differ [@cusp61; @Cabibbo:2004gq]. Perhaps somewhat more unexpectedly, isospin-violating corrections proved also of importance [@Gasser:2007de] in the analysis of the $K^+_{e4}$ data, in order to account for the high precision reached by the recent NA48/2 experiment, and to make comparison with theory meaningful [@Colangelo:2007df; @Colangelo:2008sm]. Actually, once isospin corrections are applied also to the E865 data, there remains a disagreement with NA48/2 [@Colangelo:2007df], whose origin seems to lie in the original analysis performed by the E865 collaboration (for details, see the errata quoted under refs. [@Pislak:2001bf; @Pislak:2003sv]). Anyway, the analysis of the full data set collected by NA48/2 has by now completely superseded the E865 results, and one should focus on the former to study pion-pion scattering from $K^+_{e4}$ decays.
In the present paper, we propose to address the issue of isospin-violating effects in low-energy pion-pion interactions using an approach based on a dispersive construction of the various $\pi\pi$ scattering amplitudes and pion form factors in the presence of isospin breaking. Ultimately, we wish to extend this program [@wip] to the $K_{\ell 4}$ form factors analysed in the NA48/2 experiment. Before undertaking this enterprise and hitting the full complexity of this four-body decay, we want to demonstrate its feasibility and exhibit the general features of such a method by considering the somewhat simpler setting provided by the scalar and vector form factors of the pion.
As far as the amplitude for elastic $\pi\pi$ scattering in the isospin limit is concerned, the general framework has been laid down in ref. [@FSS93], and the explicit construction of the two-loop amplitude has subsequently been performed along these lines in detail in ref. [@KMSF95]. Concerning the pion form factors, the corresponding dispersive representations in the framework of the chiral expansion have been studied in ref. [@GasserMeissner91] in the isospin limit, but only one-loop expressions were given in analytical form. Full two-loop expressions of the vector form factors have been obtained by integrating the corresponding dispersive integrals in ref. [@Colangelo96]. In ref. [@Bijnens98] the two-loop expressions of the vector and scalar form-factors have also been obtained in the absence of isospin violation by the direct evaluation of Feynman graphs generated from the effective chiral Lagrangian at next-to-next-to-leading order. A similar calculation for the pion-pion scattering amplitude in the isospin limit has been achieved in ref. [@pi-pi2loops]. Finally, let us also mention that the reconstruction theorem for elastic $\pi\pi$ scattering in the isospin limit of ref. [@FSS93] was extended by the authors of ref. [@NovZdra08] to the whole set of scattering amplitudes involving the mesons of the lightest pseudoscalar octet. Applications of this framework to the decay modes $P \rightarrow \pi\pi\pi$, with $P = K , \eta$, have also been considered [@P_to_3pi; @eta_to_3pi].
These dispersive constructions generate subtraction polynomials with unspecified coefficients. The latter are in one-to-one correspondence with the appropriate combinations of low-energy constants and chiral logarithms that would be encountered in a calculation of the corresponding Feynman diagrams generated by the chiral lagrangian. In the case of the form factors, these coefficients may be identified with their slopes and curvatures. In the case of the $\pi\pi$ scattering amplitudes, they can be expressed in terms of the subthreshold parameters occurring in the expansions of these amplitudes as Taylor series around the center of the Mandelstam triangle. This was the option considered in the isospin-symmetric case in ref. [@KMSF95]. By no means, however, is this choice a necessity. It has, for instance, become customary to rather let the scattering lengths play a prominent role. They have a more direct physical interpretation than the subthreshold parameters, and are thus considered as more “experimentalist friendly”. We will therefore also provide expressions where the subtraction polynomials are given in terms of the two $S$-wave $I=0$ and $I=2$ scattering lengths, $a_0^0$ and $a_0^2$, in the isospin limit. In the isospin-symmetric situation, this provides an alternative to the choice made in ref. [@KMSF95]. Of course, taking the expressions at two-loop order provided in the latter reference, one could convert the expressions for the $\pi\pi$ scattering amplitude given there to the one presented here in terms of the scattering lengths. The two formulations are equivalent, up to corrections that are of higher order. In the situation where isospin is broken, this allows us to discuss the size of the corresponding corrections to the phases of the form factors in terms of $a_0^0$ and $a_0^2$. This second option is of course the most interesting in the present context, where these scattering lengths are the quantities one would eventually like to determine from the data. It is thus important that the corrections due to isospin breaking are not studied for a fixed [*a priori*]{} value for them. Indeed, given the precision reached by the latest experiments, one ought to perform a quantitative evaluation of the possible bias introduced if isospin corrections are evaluated for fixed values of these scattering lengths. This provides another motivation for the present work.
Here we will mainly concentrate on the phases of the pion form factors. Full two-loop expressions for the scattering amplitudes and form factors themselves require the evaluation of dispersion integrals corresponding to specific topologies of two-loop three-point Feynman diagrams of the non-factorizing type (“acnode” of “fish” diagrams, cf. fig. \[fig3\]). Explicit analytical expressions for them do not seem to be available in the literature in the cases where several distinct masses are present. We therefore present only partially analytical expressions for the scattering amplitudes and form factors. Note that a similar situation arises in the evaluation of the $SU(3)$ vector [@Bijnens:2002hp] and scalar [@Bijnens:2003xg] form factors at two loops without isospin breaking, but where the difference between the pion and kaon masses has to be dealt with. These difficulties do not show up in the computation of the phases of the two-loop form factors, where the technically most demanding step is the computation of the projections of the one-loop amplitudes on the $S$ and $P$ partial waves, which can be done analytically.
Coming now to the outline of this paper, our first purpose will be to extend the frameworks of refs. [@FSS93; @GasserMeissner91] to the situation where the difference between the masses of charged and neutral pions is taken into account. The general framework leading to two-loop representations for form factors and scattering amplitudes when isospin symmetry no longer holds is thus described in section \[general\]. In section \[1stIteration\], we implement the program of constructing the corresponding form factors and scattering amplitudes at the one-loop level and provide explicit expressions for them. The second iteration, leading to two-loop representations of the form factors and the scattering amplitudes, is discussed in section \[2ndIteration\]. The issue of isospin breaking in the phases of the two-loop form factors is addressed in section \[IB\_in\_phases\]. Section \[numerics\] is devoted to the numerical analysis of the isospin-breaking contributions in the phases of the form factors. Finally, a summary of this study and our conclusions are to be found in section \[conclusion\]. This main body of the text is supplemented with six appendices, where details concerning more computational or technical aspects have been gathered.
GENERAL FRAMEWORK AND PRELIMINARY REMARKS {#general}
=========================================
The objects of our study are the scalar and vector form factors of the pion, defined through the following matrix elements $$\begin{aligned}
\!\!\!
\langle \pi^0(p_1) \pi^0(p_2) \vert {\widehat m}({\overline u}u + {\overline d}d)(0)\vert\Omega\rangle
&=&
+ F_S^{\pi^0}(s)
\quad { }
\nonumber
\\
\!\!\!\langle \pi^+(p_{\mbox{\tiny{$ +$}}}) \pi^-(p_{\mbox{\tiny{$ -$}}}) \vert {\widehat m}({\overline u}u + {\overline d}d) (0)\vert\Omega\rangle
&=&
- F_S^{\pi}(s) ,
\quad { }
\label{eq:F_S}\end{aligned}$$ with ${\widehat m}\equiv (m_u + m_d)/2$, and $$\begin{aligned}
\!\!\!\!\!\!\!\!\!\!
\frac{1}{2}\langle \pi^+ \pi^- \vert
({\overline u}\gamma_\mu u -
{\overline d}\gamma_\mu d)(0)\vert\Omega\rangle
\!
&=&
\!
(p_{\mbox{\tiny{$ -$}}} - p_{\mbox{\tiny{$ +$}}})_\mu F_V^{\pi}(s) ,
\nonumber\\
&&
\label{eq:F_V}\end{aligned}$$ respectively, in the presence of isospin-breaking corrections induced by the difference in the masses of the charged and neutral pions. In each case, $s$ denotes the squared invariant mass of the dipion system, $s=(p_1 + p_2)^2$ or $s=(p_{\mbox{\tiny{$ -$}}} + p_{\mbox{\tiny{$ +$}}})^2$, with $p_{1,2}^2= M_{\pi^0}^2$, $p_{\mbox{\tiny{$\pm$}}}^2 = M_{\pi}^2$, and $\vert\Omega\rangle$ stands for the QCD vacuum state. The mass of the neutral pion is denoted by $M_{\pi^0}$, while $M_{\pi}$ stands for the mass of the charged pion. We will define the isospin limit as the case when the neutral pion mass tends to the charged pion mass, $M_{\pi^0} \to M_{\pi}$, while keeping the latter fixed. This explains the convention that we follow in this paper, namely that all quantities without superscript refer to the charged pion case (default case), and that we refer to quantities involving neutral pions by an explicit $0$ superscript. The minus sign in the definition of $F_S^{\pi}(s)$ reflects a choice of phase for the charged-pion states. In addition, we choose the crossing phases to be $-1$ for charged pions and $+1$ for neutral pions. These choices are compatible with the Condon and Shortley phase convention in the isospin-symmetric situation. We further assume throughout that symmetry under charge conjugation holds.
These form factors, while being perfectly well-defined observable quantities in QCD, are however not observables from a strictly experimental point of view: they can only be measured indirectly, and should thus at best be considered as pseudo-observables. For instance, in the Standard Model, the vector form factor $F_V^{\pi}(s)$ appears in the physical process $e^+ e^- \to \pi^+ \pi^-$ through the exchange of a single neutral spin-one gauge boson, which in practice reduces to only photon exchange at low energies. As far as the scalar form-factors $F_S^{\pi^0}(s)$ and $F_S^{\pi}(s)$ are concerned, a similar statement can in principle also be made, but is of little use in practice, since the Standard Model contributions to the processes $e^+ e^- \to \pi^+ \pi^-\,,\ \pi^0 \pi^0$, arising from the exchange of a Higgs particle, are well below the level of sensitivity that one could expect for any experiment of this type in the foreseeable future. Despite these limitations on the experimental side, these form factors prove useful as a theoretical laboratory. They allow us to discuss and to illustrate several issues related to the structure of isospin-breaking contributions within a rather simple context. The full complexity of experimentally more interesting situations, like the $K_{\ell 4}$ form factors or the decay amplitudes of light pseudoscalar mesons (eta or kaons) into three pions, can then be addressed on the basis of these considerations and the general framework developed here, see [@wip] and the forthcoming publication [@P_to_3pi] in the former or the latter case, respectively.
In the present section, we aim at tackling two issues, namely discussing precisely the isospin contributions that we intend to deal with, and describing our general theoretical framework. Then we can focus on the specific pion form factors that we use as an illustration.
Electromagnetic corrections {#sec:em_corr}
---------------------------
At the fundamental level, isospin violations have two origins within the Standard Model: the electroweak interactions mediated by the gauge bosons, and the quark mass difference $m_u - m_d$, arising through the coupling of the light $u$ and $d$ quark flavours to the Higgs boson. Both effects contribute to the mass difference between charged and neutral pions, although the second one turns out to be marginal: the pion mass difference is mainly an electromagnetic effect [@Weinberg77; @GasserLeut82].
Chiral perturbation theory [@Wein79; @Gasser:1983yg] including electromagnetism [@Urech:1994hd; @Neufeld:1994eg; @Neufeld:1995mu; @Knecht:1997jw; @Meissner:1997fa; @Schweizer:2002ft] provides in principle a suitable framework to deal with these isospin-breaking contributions in the low-energy domain. It has been applied to the computation of several quantities at the one-loop level, including the $\pi\pi$ scattering amplitudes [@Knecht:1997jw; @Meissner:1997fa; @KnechtNehme02] and pion form factors [@Kubis:1999db] in the two-flavour case. Unfortunately, from a practical point of view, this is not a level of accuracy able to match the experimental one in several cases of interest (low-energy pion-pion scattering or $K_{l4}$ decay, for instance). One might of course contemplate the extension of this effective Lagrangian framework to next-to-next-to-leading order, but this is a more ambitious program, the interest of which might be limited eventually by the proliferation of low-energy constants. We will therefore not pursue this issue here.
Instead, we will rather consider the point of view described in refs. [@Gasser:2007de; @Colangelo:2008sm]: we thus assume a situation, as is, actually, often the case in the analyses of experimental data (such as for E865 and NA48/2), where part of the radiative corrections due to real and virtual corrections have already been dealt with in some manner, while those that may remain are supposed to be negligible. In this kind of procedure, radiative corrections of the type shown on the left-hand side of fig. \[fig1\], for instance, together with emission of soft photons, are subtracted away, but other photonic effects, like the one shown on the right-hand side of the same figure, are not taken into account, but are considered to be negligible. Notice that the latter would be included in a genuine two-loop calculation in the framework of the QCD+QED effective theory. One might also think of considering the possibility of treating them in the dispersive framework that we are using here, for instance upon including also photons among the possible intermediate states in the unitarity conditions for the relevant partial waves. While this remains an interesting issue, it would however lead us beyond the purposes of the present work. Within the framework assumed here, this leaves the difference in the pion masses as the only remaining source of isospin breaking that we have to consider. In practice, it means that the charged and neutral pion masses will be kept at their experimental values, but the origin of the difference in their masses will not be addressed. In other words, we assume that general properties like analyticity, unitarity, and crossing, together with chiral counting, can be used to describe a world where the charged and neutral pion masses differ, even though the interaction at the origin of this difference is not explicitly accounted for.
Dispersive construction of the form factors
-------------------------------------------
The starting point of our study is provided by the dispersion relations satisfied by the pion form factors and scattering amplitudes. Here we will only be interested in the structure of these quantities in the low-energy region. In order to obtain dispersive representations that agree with their analytic structures up to two loops in the low-energy expansion, it is convenient to consider thrice subtracted dispersion relations (for a discussion of this issue in the isospin limit, see e.g. ref. [@GasserMeissner91] for the pion form factors, and ref. [@FSS93] for the $\pi\pi$ scattering amplitude). Let us start with the form factors, for which these dispersive representations read $$\begin{aligned}
F_S^{\pi^0}\!(s) &=& F_S^{\pi^0}\!(0) \! \left[
1 + \frac{1}{6}\langle r^2\rangle_S^{\pi^0}s + c_S^{\pi^0} \! s^2
+ U_S^{\pi^0}\!(s)
\right]
\nonumber\\
F_S^{\pi}(s) &=& F_S^{\pi}(0) \! \left[
1 + \frac{1}{6}\langle r^2\rangle_S^{\pi}\,s + c_S^{\pi} \, s^2
+ U_S^{\pi}(s)
\right]
\nonumber\\
F_V^{\pi}(s) &=&
1 + \frac{1}{6}\langle r^2\rangle_V^{\pi}\,s + c_V^{\pi} s^2
+ U_V^{\pi}(s)
.
\label{eq:dispersiveff}\end{aligned}$$ In the last of these relations, the condition $F_V^{\pi}(0) = 1$, due to the conservation of the electromagnetic current and thus valid to all orders, has been used. Through crossing, $F_S^{\pi^0}(0)$ and $F_S^{\pi}(0)$ are equal to the corresponding sigma-term type form-factors, $\langle \pi^0(p) \vert {\widehat m}({\overline u}u + {\overline d}d)(0)\vert \pi^0(p)\rangle$ and $\langle \pi^\pm(p) \vert {\widehat m}({\overline u}u + {\overline d}d)(0)\vert \pi^\pm(p)\rangle$, respectively, for which there also exist relations [@GasserZepeda] valid to all orders, that follow from the Feynman-Hellmann theorem [@FeynHell], $$F_S^{\pi^0}\!(0)\,=\,{\widehat m}\,\frac{\partial M_{\pi^0}^2}{\partial {\widehat m}} ,
\ F_S^{\pi}(0)\,=\,{\widehat m}\,\frac{\partial M_{\pi}^2}{\partial {\widehat m}} .$$ Since the dominant contribution to the pion mass difference is purely of electromagnetic origin and independent of the quark masses [@Eckeretal89], one has $$\frac{F_S^{\pi}(0)}{F_S^{\pi^0}\!(0)} \,=\, 1 \,+\,\dots ,
\label{ratioF_S}$$ where the ellipsis denotes higher order terms. The unitarity parts are given in terms of dispersion integrals, $$\begin{aligned}
U_S^{\pi^0}(s) &=&\frac{s^3}{\pi}\,\int\frac{dx}{x^{ 3}}\,
\frac{{\mbox{Im}}F_S^{\pi^0}\!(x)/F_S^{\pi^0}\!(0)}{x - s -i0}
\nonumber\\
U_S^{\pi}(s) &=& \frac{s^3}{\pi}\,\int\frac{dx}{x^{ 3}}\,
\frac{{\mbox{Im}}F_S^{\pi}(x)/F_S^{\pi}(0)}{x - s -i0}
\nonumber\\
U_V^{\pi}(s) &=& \frac{s^3}{\pi}\,\int\frac{dx}{x^{ 3}}\,
\frac{{\mbox{Im}}F_V^{\pi}(x)}{x - s -i0}\,.\end{aligned}$$ In the low-energy region, the form factors are analytical functions in the complex $s$-plane, except for cut singularities on the positive real axis, starting at $s=4 M_{\pi}^2$ in the case of $F_V^{\pi}(s)$, and at $s=4 M_{\pi^0}^2$ in the cases of $F_S^{\pi}(s)$ and $F_S^{\pi^0}(s)$. For $s$ real and below these cuts, the form factors are real. In the chiral expansion, the form factors behave dominantly as $$\begin{aligned}
&{\mbox{Re}}F_S^{\pi(\pi^0)}(s) \sim {\cal O}(E^2),\quad
&{\mbox{Im}}F_S^{\pi(\pi^0)}(s) \sim {\cal O}(E^4),
\nonumber\\
&{\mbox{Re}}F_V^{\pi}(s) \sim {\cal O}(E^0),\quad
&{\mbox{Im}}F_V^{\pi}(s) \sim {\cal O}(E^2) ,
\label{countingFF}\end{aligned}$$ where $E$ denotes either a pion momentum or a pion mass. Furthermore, intermediate states with more than two pions contribute only from the three-loop level onwards. Therefore, below the thresholds involving other states than the pions, and up to and including two loops in the two-flavour chiral expansion, only discontinuities arising from two-pion intermediate states need to be retained [@GasserMeissner91], as illustrated in fig. \[unitaritydiag\]. In order to distinguish among the different $\pi \pi$ scattering channels, we use the following superscripts: $00$ for $\pi^0 \pi^0 \to \pi^0 \pi^0$, $++$ for $\pi^+ \pi^+ \to \pi^+ \pi^+$, $+-$ for $\pi^+ \pi^- \to \pi^+ \pi^-$, $+0$ for $\pi^+ \pi^0 \to \pi^+ \pi^0$, and $x$ for the inelastic channels $\pi^0 \pi^0 \to \pi^+ \pi^-$ and $\pi^+ \pi^- \to \pi^0 \pi^0$. We have $$\begin{aligned}
{\mbox{Im}}F_S^{\pi^0}(s) &=& {\mbox{Re}}\bigg\{
\frac{1}{2}\,\sigma_{0}(s)f_0^{00}(s) F_S^{\pi^0\! *}(s) \theta(s-4M_{\pi^0}^2)
-
\sigma (s)f_0^{x}(s)F_S^{\pi *}(s) \theta(s-4M_{\pi}^2)
\bigg\} + {\cal O}(E^8)
\nonumber\\
{\mbox{Im}}F_S^{\pi}(s) &=& {\mbox{Re}}\bigg\{
\sigma (s)f_0^{\mbox{\tiny{$ +-$}}}(s)F_S^{\pi *}(s)\theta(s-4M_{\pi}^2)
-
\frac{1}{2}\,\sigma_{0}(s)f_0^{x}(s)F_S^{\pi^0\! *}(s)\theta(s-4M_{\pi^0}^2)
\bigg\} + {\cal O}(E^8)
\nonumber\\
{\mbox{Im}}F_V^{\pi}(s) &=& {\mbox{Re}}\bigg\{
\sigma (s)f_1^{\mbox{\tiny{$ +-$}}}(s)F_V^{\pi *}(s)\theta(s-4M_{\pi}^2) \bigg\}
+ {\cal O}(E^6) ,
\label{discFF}\end{aligned}$$ where we define the phase-space functions $$\sigma_{0}(s)\,=\,\sqrt{1 - \frac{4M_{\pi^0}^2}{s}}\,,\ \sigma(s)\,=\,\sqrt{1 - \frac{4M_{\pi}^2}{s}} .
\label{def_sigma}$$
In these expressions, $f_0(s)$ and $f_1(s)$ denote the $S$ and $P$ partial waves, respectively, of the $\pi\pi$ scattering amplitudes $A(s,t)$ in the corresponding channels. These partial waves have been defined as usual by projections of the corresponding amplitudes, $$f_\ell (s) \,=\, \frac{1}{32\pi} \int_{-1}^{+1} dz A(s,t) P_\ell (z) ,
\label{PWproj}$$ with $P_0(z)=1$ and $P_1(z)=z$ the appropriate Legendre polynomials, and $z=\cos \theta$, where $\theta$ denotes the scattering angle in the center-of-mass frame. The relation with the Mandelstam variables $s,t,u$ (summing up to the squared masses of the incoming and outgoing particles) depends on the process under consideration. For processes involving four particles with the same mass, it is simply given by $$t \,=\, - \frac{s - 4M^2}{2}\,(1-z) .$$ This is, in particular, the case for $A^{00}(s,t)$ (with $M=M_{\pi^0}$), as well as for $A^{\mbox{\tiny{$++$}}}(s,t)$ and for $A^{\mbox{\tiny{$+-$}}}(s,t)$ (with now $M=M_\pi$). In the reactions involving both charged and neutral pions, it becomes $$t \,=\, - \frac{1}{2}\,(s - 2 M_{\pi^0}^2 - 2 M_\pi^2) + \frac{z}{2}
\sqrt{(s - 4 M_{\pi^0}^2)(s - 4 M_\pi^2)}$$ for $A^x(s,t)$, and $$t \,=\, - \frac{\lambda(s)}{2s}\,(1-z) \,,\quad \lambda(s) = [s - (M_{\pi} + M_{\pi^0})^2][s - (M_{\pi} - M_{\pi^0})^2] .$$ for $A^{\mbox{\tiny{$+$}} 0}(s,t)$. These expressions hold above the kinematical threshold, $s\ge 4 M_{\pi}^2$ for $A^x(s,t)$, and $s\ge (M_{\pi} + M_{\pi^0})^2$ for $A^{\mbox{\tiny{$+$}} 0}(s,t)$, for instance. Let us point out that the normalization in (\[PWproj\]) differs from the usual definition of the $\pi\pi$ partial-wave amplitudes by a factor of $2$: it would correspond to a decomposition of the scattering amplitudes given by $$A(s,t) \,=\, 16\pi \sum_{\ell\ge 0} (2\ell +1) P_{\ell}(\cos\theta) f_{\ell}(s) ,
\label{PWdecomp}$$ i.e. with the normalization factor $16\pi$ instead of the usual $32\pi$. This modification is motivated by the fact that, in the presence of isospin breaking, the two-pion states do no longer obey (generalized) Bose symmetry, except in the cases of two neutral pions, or of two identically charged pions. In these cases, the appropriate symmetry factor has been included instead in the expressions of the corresponding phase spaces, cf. eq. (\[discFF\]). In the chiral expansion, the $\pi\pi$ partial waves behave as $$\begin{aligned}
&&
{\mbox{Re}}f_{\ell}(s) \sim {\cal O}(E^2),
\, {\mbox{Im}}f_{\ell}(s) \sim {\cal O}(E^4),
\, \ell = 0,1 ,
\nonumber\\
&&
{\mbox{Re}}f_{\ell}(s) \sim {\cal O}(E^4),
\, {\mbox{Im}}f_{\ell}(s) \sim {\cal O}(E^8),
\, \ell \ge 2 .
\label{countingPW}\end{aligned}$$ This is in perfect agreement with the chiral counting (\[countingFF\]) of the form factors, together with the expressions (\[discFF\]) of their low-energy discontinuities.
Dispersive construction of $\pi\pi$ scattering amplitudes {#sec:dispconstpipi}
---------------------------------------------------------
It is actually possible to obtain dispersive representations for the two-loop $\pi\pi$ scattering amplitudes themselves, following the same procedure as for the form factors. As in the isospin-symmetric case [@FSS93], they follow from fixed-$t$ dispersion relations, combined with very general properties like relativistic invariance, unitarity, analyticity, crossing, and from the chiral counting properties that have just been recalled. Isospin breaking is not expected to modify the asymptotic high-energy behaviour of the amplitudes, so that two subtractions should be enough in order to obtain convergent dispersion relations. We start with three subtractions in order to construct low-energy expressions of the scattering amplitudes that are valid up to and including two loops in the chiral expansion.
Whereas the various channels can all be described in terms of a single amplitude $A(s\vert t,u)$ as long as isospin symmetry holds, several independent amplitudes, not related by crossing, are necessary in order to deal with all the different available channels once isospin is broken. Otherwise, the derivation proceeds as in the isospin-symmetric case [@Knecht97], up to the kinematical peculiarities due to the presence of particles with different masses. The relevant features can be inferred from the discussion of ref. [@NovZdra08], devoted to the extension of the results of ref. [@FSS93] to the scattering amplitudes of the mesons belonging to the octet of lightest pseudoscalar states, pions, kaons, and eta. We will therefore directly write down the resulting expressions, and then provide a few additional comments on their structure.
*i) Elastic scattering involving only neutral pions* remains the simplest case, with a single, fully crossing invariant amplitude, which has the following two-loop structure (for convenience, we display, from now on, the dependence on the three Mandelstam variables $s,t,u$, although they are not independent) $$\begin{aligned}
A^{00}(s,t,u) &=&
P^{00}(s,t,u) \,+\, W^{00}_0(s) \,+\, W^{00}_0(t) \,+\, W^{00}_0(u) \,+\, {\cal O}(E^8) .
\label{A00}\end{aligned}$$ It involves a polynomial $P^{00}(s,t,u)$ of third order in $s,t,u$, symmetric under any permutation of its variables, together with a dispersive integral $W^{00}(s)$. This function has a discontinuity on the positive real $s$-axis starting at $s=4M_{\pi^0}^2$, and specified by the $\ell = 0$ partial-wave amplitude $f^{00}_0(s)$, $${\mbox{Im}} W^{00}_0(s) \,=\, 16\pi\, {\mbox{Im}}f^{00}_0 (s) \,\theta (s-4M_{\pi^0}^2) .$$ Again, at the order under consideration, this discontinuity is provided by the unitarity condition, in terms of the $\ell = 0$ partial waves in the relevant channels, $f^{00}_0(s)$ and $f^{x}_0(s)$, $$\frac{1}{16\pi}\,{\mbox{Im}}W^{00}_0(s) \,=\,
\frac{1}{2}\,\sigma_0 (s)\,
\left\vert f^{00}_0(s) \right\vert ^2
\theta (s-4M_{\pi^0}^2)
\,+\,
\sigma (s)\,
\left\vert f^{x}_0(s) \right\vert ^2
\theta (s-4M_{\pi}^2) + {\cal O}(E^8).\ \ \quad{ }
\label{Im_W00}$$
*ii) The processes involving exactly two neutral pions,* i.e. $\pi^\pm \pi^0 \to \pi^\pm \pi^0$ and $\pi^+ \pi^- \to \pi^0 \pi^0$, provide the next family of amplitudes that are related under crossing. They display the following structure at two loops in the chiral expansion: $$\begin{aligned}
A^{x}(s,t,u) &=&
- P^{x}(s,t,u) - W^{x}_{0}(s)
- \left[W^{\mbox{\tiny{$ +$}}0}_{0}(t) +3(s-u)W^{\mbox{\tiny{$ +$}}0}_{1}(t)\right]
- \left[W^{\mbox{\tiny{$ +$}}0}_{0}(u) +3(s-t)W^{\mbox{\tiny{$ +$}}0}_{1}(u)\right]
+ {\cal O}(E^8) ,
\nonumber\\
&& \qquad{ }
\label{Ax}\end{aligned}$$ whereas $A^{\mbox{\tiny{$ +$}}0}(s,t,u) = - A^x(t,s,u)$ through crossing. In the above expression, $P^{x}(s,t,u)$ represents a polynomial of third order in the Mandelstam variables, symmetric under exchange of $t$ and $u$ (Bose symmetry). The functions $W^{\mbox{\tiny{$ +$}}0}_{0,1}(s)$ and $W^{x}_{0}(s)$ have discontinuities on the positive real $s$-axis, starting at $s=(M_{\pi} + M_{\pi^0})^2$ and at $s=4M_{\pi^0}^2$, respectively. These discontinuities are again given in terms of the appropriate lowest ($S$ and $P$) $\pi\pi$ partial waves $$\begin{aligned}
{\mbox{Im}}W^{\mbox{\tiny{$ +$}}0}_{0}(s)
&=&
16\pi
\bigg[{\mbox{Im}}f^{\mbox{\tiny{$ +$}}0}_0(s) + \frac{3\left(M_{\pi}^2 - M_{\pi^0}^2\right)^2}{\lambda (s)}
\,{\mbox{Im}}f^{\mbox{\tiny{$ +$}}0}_1(s)
\bigg]
\theta \!\left(s-(M_{\pi} + M_{\pi^0})^2\right)
\nonumber\\
{\mbox{Im}}W^{\mbox{\tiny{$ +$}}0}_{1}(s)
&=&
16\pi
\frac{s}{\lambda (s)} \, {\mbox{Im}}f^{\mbox{\tiny{$ +$}}0}_1(s)
\theta \!\left(s-(M_{\pi} + M_{\pi^0})^2\right)
\nonumber\\
{\mbox{Im}}W^{x}_{0}(s) &=&
- 16\pi \,{\mbox{Im}}f^{x}_0(s) \,\theta (s-4M_{\pi^0}^2)
.
\label{Im_Wx}\end{aligned}$$ Up to higher-order contributions, the unitarity condition allows to rewrite these expressions in terms of the same lowest partial waves. In the elastic channel, there is only one contribution, arising from the $\pi^+ \pi^0$ intermediate state, whereas the inelastic channel involves two contributions: $$\begin{aligned}
\frac{1}{16\pi}\,{\mbox{Im}}W^{\mbox{\tiny{$ +$}}0}_{0}(s)
&=&
\bigg[
\frac{\lambda^{1/2}(s)}{s}
\left\vert f^{\mbox{\tiny{$ +$}}0}_0(s) \right\vert ^2 +
\frac{3\left(M_{\pi}^2 - M_{\pi^0}^2\right)^2}{s \lambda^{1/2} (s)}
\, \left\vert f^{\mbox{\tiny{$ +$}}0}_1(s) \right\vert ^2
\bigg]
\theta \!\left(s-(M_{\pi} + M_{\pi^0})^2\right)
\,+\, {\cal O}(E^8)
\nonumber\\
\frac{1}{16\pi}\,{\mbox{Im}}W^{\mbox{\tiny{$ +$}}0}_{1}(s)
&=&
\frac{1}{\lambda^{1/2} (s)} \, \left\vert f^{\mbox{\tiny{$ +$}}0}_1(s) \right\vert ^2
\,\theta \!\left(s-(M_{\pi} + M_{\pi^0})^2\right)
\,+\, {\cal O}(E^8)
\nonumber\\
\frac{1}{16\pi}\,{\mbox{Im}}W^{x}_{0}(s) &=&
- \frac{1}{2}\,\sigma_0 (s)\,
f^{x}_0(s) \left[ f^{00}_0(s) \right] ^\star
\theta (s-4M_{\pi^0}^2) -
\sigma (s)\,
f^{\mbox{\tiny{$ +-$}}}_0(s)
\left[ f^{x}_0(s) \right] ^\star
\theta (s-4M_{\pi}^2)
\,+\, {\cal O}(E^8) . \qquad{ }\end{aligned}$$
*iii) Finally, the subset of elastic scattering processes involving only charged pions* remains to be considered. Their amplitudes being all related by crossing, it is enough to display explicitly one of them, for instance, $$\begin{aligned}
A^{\mbox{\tiny{$ +-$}}}(s,t,u) &=&
P^{\mbox{\tiny{$ +-$}}}(s,t,u) +
\left[ W^{\mbox{\tiny{$ +-$}}}_{0}(s) + 3 (t-u)W^{\mbox{\tiny{$ +-$}}}_{1}(s) \right] +
\left[ W^{\mbox{\tiny{$ +-$}}}_{0}(t) + 3 (s-u)W^{\mbox{\tiny{$+-$}}}_{1}(t) \right] +
W^{\mbox{\tiny{$ ++$}}}_{0}(u) + {\cal O}(E^8) ,
\nonumber\\
&& \ \quad{ }
\label{A+-}\end{aligned}$$ where the third order polynomial $P^{\mbox{\tiny{$ +-$}}}(s,t,u)$ is symmetric under exchange of $s$ and $t$ (Bose symmetry in the crossed $u$-channel). The three functions $W^{\mbox{\tiny{$ +-$}}}_{0,1}(s)$ and $W^{\mbox{\tiny{$ ++$}}}_{0}(s)$ have cut singularities along the real $s$ axis, starting at $s=4M_{\pi^0}^2$ or at $s=4 M_{\pi}^2$. The corresponding discontinuities read $$\begin{aligned}
{\mbox{Im}} W^{\mbox{\tiny{$ +-$}}}_{0}(s) &=& 16\pi {\mbox{Im}}f^{\mbox{\tiny{$ +-$}}}_0(s) \theta (s-4M_{\pi^0}^2)
\nonumber\\
{\mbox{Im}} W^{\mbox{\tiny{$ +-$}}}_{1}(s) &=& 16\pi {\mbox{Im}}f^{\mbox{\tiny{$ +-$}}}_1(s) \theta (s-4M_{\pi}^2)
\nonumber\\
{\mbox{Im}} W^{\mbox{\tiny{$ ++$}}}_{0}(s) &=& 16\pi {\mbox{Im}}f^{\mbox{\tiny{$ ++$}}}_0(s) \theta (s-4M_{\pi}^2) .\end{aligned}$$ The unitarity condition for the three $\pi\pi$ partial waves involved then leads to $$\begin{aligned}
\frac{1}{16\pi}\,{\mbox{Im}}
W^{\mbox{\tiny{$ +-$}}}_{0}(s) &=& \sigma (s)\,
\left\vert f^{\mbox{\tiny{$ +-$}}}_0(s) \right\vert ^2
\theta (s-4M_{\pi}^2)
\,+\,
\frac{1}{2}\,
\sigma_0 (s)\,
\left\vert f^{x}_0(s) \right\vert ^2
\theta (s-4M_{\pi^0}^2)
\,+\, {\cal O}(E^8)
\nonumber\\
\frac{1}{16\pi}\,{\mbox{Im}}
W^{\mbox{\tiny{$ +-$}}}_{1}(s) &=& \sigma (s)\,
\left\vert f^{\mbox{\tiny{$ +-$}}}_1(s) \right\vert ^2
\theta (s-4M_{\pi}^2)
\,+\, {\cal O}(E^8)
\nonumber\\
\frac{1}{16\pi}\,{\mbox{Im}}
W^{\mbox{\tiny{$ ++$}}}_{0}(s) &=&
\frac{1}{2}\,\sigma (s)\,
\left\vert f^{\mbox{\tiny{$ ++$}}}_0(s) \right\vert ^2
\theta (s-4M_{\pi}^2)
\,+\, {\cal O}(E^8) .
\label{Im_Wpm}\end{aligned}$$
The reason why only the lowest $S$ and $P$ partial waves play a role in these expressions follows again from the chiral counting (\[countingPW\]) for the partial waves. In the following, we will make use of the chiral expansion for the real parts of the $\ell=0,1$ partial waves, for values of $s$ corresponding to the cut along the positive real axis, $$\label{partialwavesamplitude}
{\mbox{Re}} f_\ell (s) \,=\, \varphi_\ell (s) + \psi_\ell (s) + {\cal O}(E^6) ,$$ with $\varphi_\ell (s) \sim {\cal O}(E^2)$ and $\psi_\ell (s) \sim {\cal O}(E^4)$, so that $$\left\vert f_\ell (s) \right\vert ^2 \,=\, \left[ {\mbox{Re}} f_\ell (s) \right]^2
\,+\, {\cal O}(E^8)\,=\, \left[ \varphi_\ell (s) \right]^2 + 2 \varphi_\ell (s) \psi_\ell(s)
\,+\, {\cal O}(E^8) , \ \ell=0,1 .$$ Let us also emphasize that the functions $W(s)$ only have a right-hand cut, that coincides with the right-hand cut of the corresponding $S$ and $P$ $\pi\pi$ partial-wave projections [@KMSF95]. This structure is in agreement with the analyticity properties of the $\pi\pi$ scattering amplitudes $A(s,t,u)$ required by unitarity and crossing. The decompositions (\[A00\]), (\[Ax\]), and (\[A+-\]) satisfy these constraints, to the given order in the low-energy expansion. Of course, the partial-wave amplitudes have a more complicated analytical structure, coming from the projection in eq. (\[PWproj\]), with also a left-hand cut, and even a circular cut in the case of the $\pi^\pm \pi^0 \to \pi^\pm \pi^0$ channel (for a description of the analytic structure of partial-wave amplitudes in a general context, see e.g. [@PWanalyticity]). At this stage we should also stress that a full partial-wave decomposition (\[PWdecomp\]) of the $\pi\pi$ amplitudes is actually not required. For our purposes, it is sufficient to know that the discontinuity of the latter in the complex $s$-plane can be written, in the low-energy region of interest here, as $${\mbox{Im}} A(s,t) \,=\, 16\pi \left[ {\mbox{Im}} f_0(s) + 3 z {\mbox{Im}} f_{1}(s) \right] + \Phi_{\ell\ge 2}(s,t) ,
\label{PWdecomp2}$$ with $\Phi_{\ell\ge 2}(s,t) \sim {\cal O}(E^8)$ as its dominant chiral behaviour.
(375,130)(0,130)
(0,200)(70,20) (0,200)\[\][$A$ at order [$E^{2k}$]{}]{}
(35,200)(90,200)
(62.5,240)\[\][projection]{} (62.5,230)\[\][over partial waves]{}
(125,200)(70,20) (125,200)\[\][$f$ at order [$E^{2k}$]{}]{}
(160,200)(202.5,200)
(180,230)\[\][unitarity]{}
(250,200)(95,20) (250,200)\[\][Im $f$ at order [$E^{2k+2}$]{}]{}
(297.5,200)(335,200)
(317.5,230)\[\][dispersion relation]{}
(375,200)(80,20) (375,200)\[\][$A$ at order [$E^{2k+2}$]{}]{}
(375,190)(375,150) (375,150)(0,150) (0,150)(0,190)
The very general features and the results that have just been presented are at the basis of the construction of two-loop representations of the pion form factors and scattering amplitudes in the low-energy regime. This construction is achieved trough a two-step recursive process of which we now give a short outline, summarised also in fig. \[iterconst\]. Chiral counting provides the initial information, namely that at lowest order the form factors reduce to real constants, to be identified with their values at $s=0$, while the $\pi\pi$ scattering amplitudes consist of ${\cal O}(E^2)$ polynomials of at most first order in the Mandelstam variables. This initial input, together with unitarity, fixes the discontinuities of the form factors and of the amplitudes, through the expressions of the functions $\varphi_\ell (s)$ at next-to-leading order. The complete one-loop expressions are then recovered up to a subtraction polynomial of at most first order (second order) in $s$ (in the Mandelstam variables) in the case of the form factors (of the scattering amplitudes). In turn, these one-loop expressions then provide the discontinuities at next-to-next-to-leading order, and thus the form factors (and amplitudes) themselves at order ${\cal O}(E^6)$, up to a polynomial ambiguity of second order in $s$ (third order in the Mandelstam variables). In the case of the $\pi\pi$ scattering amplitudes, crossing imposes further restrictions on the possible terms that may appear in these polynomials. Notice that the presence of these polynomials reflect the fact that the functions $W^{00}_{0}(s)$, $W^{+0}_{0,1}(s)$, etc. are only specified by their analytical properties, in particular as far as their discontinuities are concerned. This leaves room for polynomial ambiguities in the expressions of these functions, and the maximal degree of the polynomials is then limited by chiral power counting.
This second iteration relies on the possibility to obtain analytical expressions for the ${\cal O}(E^4)$ pieces $\psi_\ell(s)$ of the real parts of the lowest partial waves from the one-loop $\pi\pi$ amplitudes, whose structures are no longer polynomial. This represents the technically most demanding step. The following sections are therefore devoted to the detailed implementation of this program. Since our main interest lies in discussing the effects of isospin breaking on the phases of the pion form factors, we will provide explicit two-loop expressions for the latter only. This means that we will stop in the middle of the second iteration of the recursive procedure. Completing this second iteration would provide full two-loop expressions for the form factors, and not only that of their imaginary part as needed for the phase shifts. This in turn would require one to obtain analytical expressions for the corresponding dispersion integrals, a daunting task as already explained in the introduction. We therefore defer this remaining step to future work.
The subtraction polynomials: subthreshold parameters vs. scattering lengths {#sub_poly}
---------------------------------------------------------------------------
The two-loop dispersive construction provides representations of the $\pi\pi$ scattering amplitudes that involve subtraction polynomials of at most third order in the Mandelstam variables. These polynomials depend on a certain number of parameters that are not fixed by the general properties, listed at the beginning of subsection \[sec:dispconstpipi\], on which the representations for the $\pi\pi$ scattering amplitudes rest. Furthermore, beyond general constraints coming, for instance, from the crossing property, there is, of course, a certain degree of arbitrariness in the form of these polynomials, and thus on the physical meaning of the corresponding coefficients.
In the isospin-symmetric case treated in refs. [@FSS93; @KMSF95], the form of the subtraction polynomial $P(s \vert t,u)$ was chosen so that some of its coefficients were identified with the subthreshold parameters of the amplitudes (the coefficients of its Taylor expansion around the center of the Mandelstam triangle). Among other reasons, this was motivated by the fact that the chiral expansions of these quantities show better convergence properties than, for instance, the scattering lengths. The latter can then be obtained from their expressions in terms of these subthreshold parameters (see, e.g., the corresponding two-loop expressions in Appendix B of ref. [@KMSF95] and the discussion in ref. [@DescotesGenon:2001tn]). Along the same line of thought, in the presence of isospin breaking the most general subtraction polynomials of third order in the Mandelstam variables, and compatible with the symmetries of the amplitudes under crossing, can then be written as $$\begin{aligned}
P^{00}(s,t,u) &=& \frac{\alpha_{00} M_{\pi^0}^2}{F_{\pi}^2}
\nonumber\\
&& +\,
\frac{3\lambda_{00}^{(1)}}{F_{\pi}^4}\left[
(s-2M_{\pi^0}^2)^2 + (t-2M_{\pi^0}^2)^2 + (u-2M_{\pi^0}^2)^2
\right]
\nonumber\\
&& +\,
\frac{3\lambda_{00}^{(2)}}{F_{\pi}^6}\left[
(s-2M_{\pi^0}^2)^3 + (t-2M_{\pi^0}^2)^3 + (u-2M_{\pi^0}^2)^3
\right]
\nonumber\\
P^x(s,t,u) &=& \frac{\beta_{x}}{F_{\pi}^2}\,
\left( s - \frac{2}{3}M_{\pi}^2 - \frac{2}{3}M_{\pi^0}^2\right)\,+\,
\frac{\alpha_{x} M_{\pi^0}^2}{3F_{\pi}^2}
\nonumber\\
&& +\,
\frac{\lambda_{x}^{(1)}}{F_{\pi}^4}(s-2M_{\pi^0}^2)(s-2M_{\pi}^2)
\,+\,
\frac{\lambda_{x}^{(2)}}{F_{\pi}^4}\left[
(t - M_{\pi}^2 - M_{\pi^0}^2)^2 + (u - M_{\pi}^2 - M_{\pi^0}^2)^2
\right]
\nonumber\\
&& +\,
\frac{\lambda_{x}^{(3)}}{F_{\pi}^6}\left[
(s-2M_{\pi^0}^2)(s-2M_{\pi}^2)^2 + (s-2M_{\pi^0}^2)^2(s-2M_{\pi}^2)
\right]
\nonumber\\
&& +\,
\frac{\lambda_{x}^{(4)}}{F_{\pi}^6}\left[
(t - M_{\pi}^2 - M_{\pi^0}^2)^3 + (u - M_{\pi}^2 - M_{\pi^0}^2)^3
\right]
\nonumber\\
P^{\mbox{\tiny{$ +-$}}}(s,t,u) &=&
\frac{\beta_{\mbox{\tiny{$ +-$}}}}{F_{\pi}^2}\,
\left( s + t - \frac{8}{3}M_{\pi}^2 \right)\,+\,
\frac{2\alpha_{\mbox{\tiny{$ +-$}}} M_{\pi^0}^2}{3F_{\pi}^2}
\nonumber\\
&& +\,
\frac{\lambda_{\mbox{\tiny{$ +-$}}}^{(1)} + \lambda_{\mbox{\tiny{$ +-$}}}^{(2)}}{F_{\pi}^4}\left[
(s- 2M_{\pi}^2 )^2 + (t - 2M_{\pi}^2)^2
\right]
\,+\,
\frac{2 \lambda_{\mbox{\tiny{$ +-$}}}^{(2)}}{F_{\pi}^4}
(u- 2M_{\pi}^2 )^2
\nonumber\\
&& +\,
\frac{\lambda_{\mbox{\tiny{$ +-$}}}^{(3)} + \lambda_{\mbox{\tiny{$ +-$}}}^{(4)}}{F_{\pi}^6}\left[
(s- 2M_{\pi}^2 )^3 + (t - 2M_{\pi}^2)^3
\right]
\,+\,
\frac{2 \lambda_{\mbox{\tiny{$ +-$}}}^{(4)}}{F_{\pi}^6}
(u- 2M_{\pi}^2 )^3 .
\label{polynomials_P}\end{aligned}$$ In each of these polynomials, the first line is of at most first order in $s$, $t$, and $u$, and corresponds to the tree-level amplitudes. The second line corresponds to the subtraction terms at one-loop level: to construct the scattering amplitudes at one-loop precision, it is enough to consider only twice-subtracted dispersion relations [@FSS93; @KMSF95].
On the other hand, it has become customary to rather let the scattering lengths play a prominent role, since they are usually considered to have a more direct physical interpretation than subthreshold parameters. The point we would like to stress here is that the framework developed in refs. [@FSS93; @KMSF95], and that we extend in the present work to the isospin-violating situation, is rather flexible from this point of view, and can accommodate several choices of parameters, according to one’s needs or purposes. It is simply a matter of appropriately choosing the forms of the lowest-order amplitudes and of the subtraction polynomials introduced at each of the two iterations. Different sets of parameters can be related, order by order in the chiral expansion, and the corresponding two-loop amplitudes differ only by higher-order terms. In the present article, we will use the representation in terms of subthreshold parameters, with the subtraction polynomials given in eq. (\[polynomials\_P\]) above. In appendix \[app:scatt\_lengths\], we provide the corresponding expressions in terms of the scattering lengths, and give an outline of how the present analysis can be reformulated in terms of the latter quantities. For more details, we refer the reader to our forthcoming work [@wip].
Whatever choice is eventually considered, these polynomials altogether depend on fifteen independent subtraction constants. In the isospin limit, only six independent subtraction constants are required: isospin symmetry induces nine linear relations among these constants. These relations can be summarised by the statements that, as $M_{\pi^0}\rightarrow M_\pi$, one has $$\begin{aligned}
P^{00}(s,t,u) & \rightarrow & P(s \vert t,u) + P(t \vert s,u) + P(u \vert s,t)
\nonumber\\
P^x(s,t,u) & \rightarrow & P(s \vert t,u)
\nonumber\\
P^{\mbox{\tiny{$ +-$}}}(s,t,u) & \rightarrow & P(s \vert t,u) + P(t \vert s,u)
,\end{aligned}$$ where $$\begin{aligned}
P(s \vert t,u) &=& \frac{\beta}{F_{\pi}^2}\,
\left( s - \frac{4}{3}M_{\pi}^2 \right)\,+\,
\frac{\alpha M_{\pi}^2}{3F_{\pi}^2}
\nonumber\\
&& +\,
\frac{\lambda_{1}}{F_{\pi}^4} (s-2M_{\pi}^2)^2
\,+\,
\frac{\lambda_{2}}{F_{\pi}^4}\left[
(t - 2 M_{\pi}^2 )^2 + (u - 2 M_{\pi}^2 )^2
\right]
\nonumber\\
&& +\,
\frac{\lambda_{3}}{F_{\pi}^6} (s-2 M_{\pi}^2)^3
\,+\,
\frac{\lambda_{4}}{F_{\pi}^6} \left[
(t - 2 M_{\pi}^2 )^3 + (u - 2 M_{\pi}^2 )^3
\right] \end{aligned}$$ is the subtraction polynomial for the isospin-symmetric scattering amplitude $A(s \vert t,u)$, cf. reference [@KMSF95]. Some relations between these subtraction constants are given explicitly below \[see, for instance, section \[FF\_and\_Amp\_tree\] and the end of section \[Amp\_1loop\]\].
FIRST ITERATION: ONE-LOOP EXPRESSIONS {#1stIteration}
=====================================
In this section, we discuss the pion form factors and the $\pi\pi$ scattering amplitudes at leading order, and then proceed with the construction of the corresponding one-loop expressions along the lines described above.
Leading-order form factors and $\pi\pi$ amplitudes {#FF_and_Amp_tree}
--------------------------------------------------
At lowest order in the chiral expansion, the form factors are constants, that may be identified with their values at $s=0$, $F_S^{\pi}(0)$, $F_S^{\pi^0}(0)$, and $F_V^{\pi}(0) = 1$.
At the same order, the $\pi\pi$ scattering amplitudes in the relevant channels read \[cf. eq. (\[polynomials\_P\])\] $$\begin{aligned}
A^{x}(s,t) &=& -\frac{\beta_{x}}{F_{\pi}^2}\!
\left( s - \frac{2}{3}M_{\pi}^2 - \frac{2}{3}M_{\pi^0}^2\right)\! -
\frac{\alpha_{x} M_{\pi^0}^2}{3F_{\pi}^2}
\nonumber\\
A^{\mbox{\tiny{$ +-$}}}(s,t) &=& \frac{\beta_{\mbox{\tiny{$\! +-$}}}}{F_{\pi}^2}\!
\left( s + t - \frac{8}{3}M_{\pi}^2 \right)\! +
\frac{2\alpha_{\mbox{\tiny{$\! +-$}}} M_{\pi^0}^2}{3F_{\pi}^2}
\nonumber\\
A^{00}(s,t) &=& \frac{\alpha_{00} M_{\pi^0}^2}{F_{\pi}^2} .
\label{AmpTree1}\end{aligned}$$ From these amplitudes, the partial wave projections are obtained as ${\mbox{Re}}f_{\ell}(s) = \varphi_{\ell}(s) + {\cal O}(E^4)$, $\ell = 0,1$, with $$\begin{aligned}
\varphi_0^{x}(s) &=& -\frac{\beta_{x}}{16\pi F_{\pi}^2}
\left( s - \frac{2}{3}M_{\pi}^2 - \frac{2}{3}M_{\pi^0}^2\right) -
\frac{\alpha_{x} M_{\pi^0}^2}{48\pi F_{\pi}^2}
\nonumber\\
\varphi_0^{\mbox{\tiny{$ +-$}}}(s) &=& \frac{\beta_{\mbox{\tiny{$\! +-$}}}}{32\pi F_{\pi}^2}
\left( s - \frac{4}{3}M_{\pi}^2 \right) +
\frac{\alpha_{\mbox{\tiny{$\! +-$}}} M_{\pi^0}^2}{24\pi F_{\pi}^2}
\nonumber\\
\varphi_1^{\mbox{\tiny{$ +-$}}}(s) &=& \frac{\beta_{\mbox{\tiny{$\! +-$}}}}{96\pi F_{\pi}^2}
\left( s - {4}M_{\pi}^2 \right)
\nonumber\\
\varphi_0^{00}(s) &=& \frac{\alpha_{00} M_{\pi^0}^2}{16\pi F_{\pi}^2} .
\label{PWTree1}\end{aligned}$$ The various parameters, like $\beta_{x}$ or $\alpha_{00}$, that occur in these expressions are free, in the sense that they are not fixed by the general principles (analyticity, unitarity, crossing, and chiral symmetry). The occurrence, in these expressions, of $M_{\pi^0}^2$ rather than $M_{\pi}^2$ in the terms proportional to $\alpha_{x}$, $\alpha_{\mbox{\tiny{$\! +-$}}}$, or $\alpha_{00}$ is a pure matter of choice, and can be considered as part of the definition of these parameters. The presence, in the denominator, of $F_\pi$, the pion decay constant in the isospin limit, is likewise a matter of convention.
As discussed in subsection \[sub\_poly\], in the isospin limit $\alpha_{x}$, $\alpha_{\mbox{\tiny{$\! +-$}}}$, and $\alpha_{00}$ take a common value $\alpha$. Similarly, the parameters $\beta_{x}$ and $\beta_{\mbox{\tiny{$\! +-$}}}$ become equal to the same quantity $\beta$ in this limit. Let us notice that there is no analogous quantity $\beta_{00}$ in the case of elastic $\pi^0\pi^0$ scattering, due to the Bose symmetry and the identity $s+t+u = 4 M_{\pi^0}^2$. Both $\alpha$ and $\beta$ were introduced in ref. [@FSS93]. They remain finite in the chiral limit, and describe the $I=0$ and $I=2$ $S$-wave scattering lengths $a_0^I$ in the isospin limit [@FSS93; @KMSF95] at lowest order, $$a_0^0 \,=\, \frac{M_{\pi}^2}{96 \pi F_\pi^2}\,(5\alpha + 16 \beta)\ ,\quad
a_0^2 \,=\, \frac{M_{\pi}^2}{48 \pi F_\pi^2}\,(\alpha - 4 \beta) .
\label{alpha_beta_a00_a02}$$ On the other hand, the lowest-order $S$-wave scattering lengths corresponding to the amplitudes $A^{x}(s,t)$, $A^{\mbox{\tiny{$ +-$}}}(s,t)$, and $A^{00}(s,t)$ were computed in ref. [@Knecht:1997jw] and read $$\begin{aligned}
a_0^{x} &=& \frac{2}{3} \left(- a_0^0 + a_0^2 \right)
- (4 \beta - \alpha) \,\frac{M_{\pi}^2 - M_{\pi^0}^2}{48 \pi F_\pi^2}
\nonumber\\
a_0^{\mbox{\tiny{$ +-$}}} &=& \frac{1}{3} \left(2 a_0^0 + a_0^2 \right)
+ (4 \beta - \alpha) \,\frac{M_{\pi}^2 - M_{\pi^0}^2}{24 \pi F_\pi^2}
\nonumber\\
a_0^{00} &=& \frac{2}{3} \left( a_0^0 + 2 a_0^2 \right) - \alpha \,\frac{M_{\pi}^2 - M_{\pi^0}^2}{16 \pi F_\pi^2} ,\end{aligned}$$ As compared to ref. [@Knecht:1997jw], where $\beta = 1$ was taken at lowest order, we have kept the dependence with respect to $\beta$ in the isospin-violating correction terms. These expressions also account for the difference in the normalization of the partial-wave amplitudes as compared to [@Knecht:1997jw], see eq. (\[PWdecomp\]). Upon comparing these formulae with the expressions of the scattering lengths computed directly from the amplitudes displayed in (\[AmpTree1\]), we obtain the following identifications: $$\begin{aligned}
\beta_{x} &=& \beta_{\mbox{\tiny{$\! +-$}}} \ =\ \beta
\nonumber\end{aligned}$$ $$\begin{aligned}
\alpha_{x} &=& \alpha \,+\, 2\beta\,\frac{M_{\pi}^2 - M_{\pi^0}^2}{M_{\pi^0}^2}\,, \quad
\alpha_{\mbox{\tiny{$\! +-$}}} \ =\ \alpha \,+\, 4\beta\,\frac{M_{\pi}^2 - M_{\pi^0}^2}{M_{\pi^0}^2}
\,,\quad \alpha_{00} \ =\ \alpha \,.
\label{alphabetaLO}\end{aligned}$$ At higher orders, and in the absence of isospin symmetry, all these coefficients become independent, and these simple expressions receive additional contributions. The computation of the corresponding isospin-breaking corrections at next-to-leading order will be addressed below, see subsection \[sub\_csts\_at\_NLO\] and appendix \[app:subtraction\].
Finally, let us recall, from ref. [@Knecht:1997jw], that the lowest-order $\pi \pi$ amplitudes (\[AmpTree1\]) take the form $$\begin{aligned}
A^{x}(s,t) &=& - A(s\vert t,u)\qquad\qquad\qquad\ \qquad [s+t+u = 2 M_{\pi^0}^2 + 2 M_{\pi}^2 ]
\nonumber\\
A^{\mbox{\tiny{$ +-$}}}(s,t) &=& A(s\vert t,u) \,+\, A(t\vert s,u)\qquad\qquad\qquad\,\ \quad [s+t+u = 4 M_{\pi}^2 ]
\nonumber\\
A^{00}(s,t) &=& A(s\vert t,u) \,+\, A(t\vert u,s) \,+\, A(u\vert s,t)\qquad [s+t+u = 4 M_{\pi^0}^2 ] ,\end{aligned}$$ with $$A(s\vert t,u) \,=\, \frac{s - 2{\widehat m} B}{F^2}
,
\label{pipiAmp_LO}$$ and one has to be aware that the variable $u$ that appears in $A(s\vert t,u)$ takes a different meaning in each case, as indicated between brackets. Identifying these expressions with the ones in eq. (\[AmpTree1\]) then gives $$\alpha \,=\, \frac{F_\pi^2 }{F^2} \left(4 - 3 \, \frac{2{\widehat m} B}{ M_{\pi^0}^2} \right)
, \ \beta \,=\, \frac{F_\pi^2 }{F^2}
\label{alpha_beta_LO}$$ at this order. Beyond leading order, the expressions (\[alpha\_beta\_LO\]) involve the low-energy constants ${\bar \ell}_3$ and ${\bar \ell}_4$ of ref. [@Gasser:1983yg], the appropriate formulae can be found in ref. [@KMSF95]. At this point, it may be useful to make briefly contact with the discussion towards the end of subsection \[sub\_poly\], after eq. (\[polynomials\_P\]). Indeed, one might actually consider three sets of independent quantities, $(\alpha, \beta )$, $(a_0^0, a_0^2)$, $({\bar \ell}_3 , {\bar \ell}_4)$ as the unknowns of the problem. From a theoretical point of view, they are, to some extent, interchangeable. The last set naturally arises in the quark mass expansion that is implemented through the calculation of Feynman graphs generated by the effective chiral lagrangian. The two first sets are better suited for addressing phenomenological issues related to the analysis of experimental data. Here, we choose to organize the discussion in terms of the set $(\alpha, \beta )$. The transcription in terms of the two $S$-wave scattering lengths can be found, as already mentioned, in appendix \[app:scatt\_lengths\] and in a forthcoming article [@wip].
Pion form factors at one loop {#FF_1loop}
-----------------------------
We can now start the procedure described in figure \[iterconst\]. At this stage, the unitarity conditions take then the following form: $$\begin{aligned}
\label{unitaroneloop1}
{\mbox{Im}}F_S^{\pi^0}(s) &=&
\frac{1}{2}\,\sigma_{0}(s)\varphi_0^{00}(s)F_S^{\pi^0}(0)\theta(s-4M_{\pi^0}^2)
-
\sigma (s)\varphi_0^{x}(s)F_S^{\pi}(0)\theta(s-4M_{\pi}^2)
\,+\,{\cal O}(E^6)
\nonumber\\
{\mbox{Im}}F_S^{\pi}(s) &=&
\sigma (s)\varphi_0^{\mbox{\tiny{$ +-$}}}(s)F_S^{\pi}(0)\theta(s-4M_{\pi}^2)
-
\frac{1}{2}\,\sigma_{0}(s)\varphi_0^{x}(s)F_S^{\pi^0}(0)\theta(s-4M_{\pi^0}^2)
\,+\,{\cal O}(E^6)
\nonumber\\
{\mbox{Im}}F_V^{\pi}(s) &=&
\sigma (s)\varphi_1^{\mbox{\tiny{$ +-$}}}(s) \theta(s-4M_{\pi}^2) \,+\,{\cal O}(E^4) .
\label{unitaroneloop2}\end{aligned}$$ We have now to determine the full form factors, exploiting the fact that we know their analytic structure, i.e., a cut along the positive real axis, and the value of the discontinuity along this cut. We introduce the well-known functions ${\bar J}_0(s)$ and ${\bar J}(s)$ defined by the following dispersive integrals $$\begin{aligned}
{\bar J}_0 (s) &=& \frac{s}{16\pi^2}\,\int_{4M_{\pi^0}^2}^{\infty}\,\frac{dx}{x}\,\frac{1}{x-s-i0}\,\sigma_0 (x)
\nonumber\\
{\bar J} (s) &=& \frac{s}{16\pi^2}\,\int_{4M_{\pi}^2}^{\infty}\,\frac{dx}{x}\,\frac{1}{x-s-i0}\,\sigma (x)
.
\label{JbarDisp}\end{aligned}$$ These functions correspond to the standard one-loop integrals subtracted at $s=0$, and through a change of variable the integrals can be brought into the more familiar form $${\bar J} (s) \,=\, \frac{-1}{16\pi^2}\int_0^1 dx \ln\left[1 - x(1-x)\frac{s}{M_\pi^2}\right] ,
\label{Jbar1loop}$$ and a similar expression for ${\bar J}_0 (s)$, with $M_\pi$ replaced by $M_{\pi^0}$. In the latter form, the integration is easy to perform for, say, $s<0$. The expression of ${\bar J}(s)$ for the remaining values of $s$ is obtained through analytic continuation, with the $s+i0$ prescription on the cut, as made explicit in the representation (\[JbarDisp\]). The result is well known and reads $${\bar J} (s) \,=\, \displaystyle{\frac{1}{16\pi^2}}\,\left[2\,+\,\sigma(s)\,L (s) + i\pi\sigma(s) \theta(s - 4M_{\pi}^2)\right]$$ Here, the function $L(s)$ is defined by $$\begin{aligned}
L (s) \ =\
\left\{
\begin{array}{l}
\ln\left(
{\displaystyle \frac{1 - \sigma(s) }{1 + \sigma(s) } }\right) \quad [s\ge 4 M_{\pi}^2]\\
\\
\ln\left(
{\displaystyle \frac{\sigma(s) - 1}{\sigma(s) + 1} }\right) \quad [s\le 4 M_{\pi}^2]
\end{array}
\right.
\quad {\mbox{with}}
\quad
\sigma (s) &=&
\left\{
\begin{array}{l}
\sqrt{1 - {\displaystyle\frac{4 M_{\pi}^2}{s}}} \quad [s\le 0\ {\mbox{or}}\ s \ge 4 M_{\pi}^2 ]\\
\\
i \sqrt{\displaystyle{\frac{4 M_{\pi}^2}{s}} - 1} \quad [0 \le s\le 4 M_{\pi}^2] \, ,\\
\end{array}
\right.\end{aligned}$$ where we have performed a similar analytical continuation of the phase-space function $\sigma(s)$, whose cut extends over the interval $0 \le s\le 4 M_{\pi}^2$. The function $\ln (s)$ is defined as usual with its cut on the negative real axis. Analogous functions $L_0(s)$ and $\sigma_0(s)$ are defined upon replacing $M_{\pi}$ by $M_{\pi^0}$ in the above expressions. Then one can easily find functions with the appropriate discontinuities (\[unitaroneloop2\]) to represent the form factors following the dispersive representation eq. (\[eq:dispersiveff\]): $$\begin{aligned}
U_S^{\pi^0}(s) &=& P_S^{\pi^0}(s) + 16\pi\,\frac{1}{2}\,\varphi_0^{00}(s) {\bar J}_0 (s)
- 16\pi \varphi_0^{x}(s)\,\frac{F_S^{\pi}(0)}{ F_S^{\pi^0}(0)}\, {\bar J} (s)
\,+\, {\cal O}(E^6)
\nonumber\\
U_S^{\pi}(s) &=& P_S^{\pi}(s) + 16\pi \varphi_0^{\mbox{\tiny{$ +-$}}}(s) {\bar J} (s)
- 16\pi\,\frac{1}{2}\,\varphi_0^{x}(s)\,
\frac{F_S^{\pi^0}(0)}{ F_S^{\pi}(0)}\,{\bar J}_0 (s)
\,+\, {\cal O}(E^6)
\nonumber\\
U_V^{\pi}(s) &=& P_V^{\pi}(s) + 16\pi \varphi_1^{\mbox{\tiny{$ +-$}}}(s) {\bar J} (s)
\,+\, {\cal O}(E^6) . \quad{ }\end{aligned}$$ $P_S^{\pi^0}(s)$, $P_S^{\pi}(s)$, and $P_V^{\pi}(s)$ represent calculable polynomials at most quadratic in $s$, which are determined by the property that the functions $U_{S,V}^{\pi}(s)$ and $U_S^{\pi^0}(s)$ have vanishing first and second derivatives at $s=0$. These polynomials can be reabsorbed into the subtraction constants such as to build up the (one-loop) radii and curvatures. At one loop, only one subtraction constant is required for each form factor. The corresponding expressions can therefore be rewritten as $$\begin{aligned}
F_S^{\pi^0}(s) &=& F_S^{\pi^0}\! (0)\! \Bigg[
1 + a_S^{\pi^0} \! s +
16\pi \frac{\varphi_0^{00}(s)}{2} {\bar J}_0 (s) \! \Bigg]
\,-\,16\pi {F_S^{\pi}(0)}\, \varphi_0^{x}(s)\,{\bar J} (s)
\nonumber\\
F_S^{\pi}(s) &=& F_S^{\pi}(0) \!\Bigg[
1 + a_S^{\pi}\,s +
16\pi \varphi_0^{\mbox{\tiny{$ +-$}}}(s) {\bar J} (s)
\Bigg]
\,-\,16\pi {F_S^{\pi^0}(0)} \,\frac{1}{2}\,\varphi_0^{x}(s)\,
{\bar J}_0 (s)
\nonumber\\
F_V^{\pi}(s) &=&
1 + a_V^{\pi}\,s
\,+\,16\pi \varphi_1^{\mbox{\tiny{$ +-$}}}(s) {\bar J} (s)
.\label{1loopFF}\end{aligned}$$ At this order, the subtraction constants $a_S^{\pi^0}$, $a_S^{\pi}$, and $a_V^{\pi}$ are then related to the radii through $$\begin{aligned}
\langle r^2\rangle_S^{\pi^0} &=& 6\, a_S^{\pi^0} \,-\,
\frac{1}{48\pi^2 F_{\pi}^2}\,\left[\,
\frac{F_S^{\pi}(0)}{ F_S^{\pi^0}(0)}\,\left(
2\beta_{x}\,\frac{M_{\pi}^2 + M_{\pi^0}^2}{M_{\pi}^2}\,
-\,\alpha_{x}\,\frac{M_{\pi^0}^2}{M_{\pi}^2} \right) \,-\,
\frac{3}{2}\,\alpha_{00} \right]
\nonumber\\
\langle r^2\rangle_S^{\pi} &=& 6\, a_S^{\pi} \,-\,
\frac{1}{96\pi^2 F_{\pi}^2}\,\left[\,
\frac{F_S^{\pi^0}(0)}{ F_S^{\pi}(0)}\,\left(
2\beta_{x}\,\frac{M_{\pi}^2 + M_{\pi^0}^2}{M_{\pi^0}^2}\,
-\,\alpha_{x}\right) \,+\,
4\beta_{\mbox{\tiny{$\! +-$}}} \,-\,4\alpha_{\mbox{\tiny{$\! +-$}}}
\,\frac{M_{\pi^0}^2}{M_{\pi}^2} \right]
\nonumber\\
\langle r^2\rangle_V^{\pi} &=& 6\, a_V^{\pi} \,-\, \frac{1}{24\pi^2 F_{\pi}^2}\,\beta_{\mbox{\tiny{$\! +-$}}} ,
\label{1loop_radii}\end{aligned}$$ while the curvatures are given by $$\begin{aligned}
c_S^{\pi^0} &=&
\frac{1}{2880\pi^2 F_{\pi}^2 M_{\pi}^2}\,\left[\,
\frac{F_S^{\pi}(0)}{ F_S^{\pi^0}(0)}\,\left(
28\beta_{x}\,-\,
2\beta_{x}\,\frac{M_{\pi^0}^2}{M_{\pi}^2}\,
+ \alpha_{x}\,\frac{M_{\pi^0}^2}{M_{\pi}^2} \right) \,+\,
\frac{3}{2}\,\alpha_{00}\,\frac{M_{\pi}^2}{M_{\pi^0}^2} \right]
\nonumber\\
c_S^{\pi} &=&
\frac{1}{5760\pi^2 F_{\pi}^2 M_{\pi}^2}\,\Bigg[\,
\frac{F_S^{\pi^0}(0)}{ F_S^{\pi}(0)}\,\frac{M_{\pi}^2}{M_{\pi^0}^2}\,\left(
28\beta_{x} \,-\,
2\beta_{x}\,\frac{M_{\pi}^2}{M_{\pi^0}^2}\,
+\,\alpha_{x}\right)
\,+\,
26\beta_{\mbox{\tiny{$\! +-$}}} \,+\,4\alpha_{\mbox{\tiny{$\! +-$}}}
\,\frac{M_{\pi^0}^2}{M_{\pi}^2} \Bigg]
\nonumber\\
c_V^{\pi} &=&
\frac{1}{960\pi^2 F_{\pi}^2 M_{\pi}^2}\,\beta_{\mbox{\tiny{$\! +-$}}} \, .
\label{1loop_curvatures} \end{aligned}$$
One-loop representation of $\pi\pi$ scattering amplitudes {#Amp_1loop}
---------------------------------------------------------
The form factors at two loops are obtained once we know the real parts of the one-loop $S$ and $P$ $\pi\pi$ partial wave projections, as shown in eq. (\[unitaroneloop1\]). With this aim in mind, we now undertake the construction of the $\pi\pi$ scattering amplitudes to one-loop in the presence of isospin breaking. The starting point is provided by the lowest-order expressions (\[AmpTree1\]) of these amplitudes, supplemented with two amplitudes that are deduced from the former ones by crossing, and that are needed to express unitarity in the crossed channels, $$\begin{aligned}
A^{\mbox{\tiny{$ +$}}0}(s,t) &=& \frac{\beta_{x} }{F_{\pi}^2}\,
\left( t - \frac{2}{3}M_{\pi}^2 - \frac{2}{3}M_{\pi^0}^2\right)\,+\,
\frac{\alpha_{x} M_{\pi^0}^2}{3F_{\pi}^2}
\nonumber\\
A^{\mbox{\tiny{$ ++$}}}(s,t) &=& - \frac{\beta_{\mbox{\tiny{$\! +-$}}}}{F_{\pi}^2}\,
\left( s - \frac{4}{3}M_{\pi}^2 \right)\,+\,
\frac{2\alpha_{\mbox{\tiny{$\! +-$}}} M_{\pi^0}^2}{3F_{\pi}^2} ,
\label{AmpTree2}\end{aligned}$$ together with the corresponding lowest-order partial wave projections, $$\begin{aligned}
\varphi_0^{\mbox{\tiny{$ +$}}0}(s) &=& \! -\frac{\beta_{x}}{16\pi F_{\pi}^2}\!
\left[ \frac{\lambda (s)}{2s} + \frac{2}{3} M_{\pi}^2 + \frac{2}{3} M_{\pi^0}^2 \! \right] \! +
\frac{\alpha_{x} M_{\pi^0}^2}{48\pi F_{\pi}^2}
\nonumber\\
\varphi_1^{\mbox{\tiny{$ +$}}0}(s) &=& \frac{\beta_{x}}{48\pi F_{\pi}^2}\,
\frac{\lambda (s)}{2s}
\nonumber\\
\varphi_0^{\mbox{\tiny{$ ++$}}}(s) &=& - \frac{\beta_{\mbox{\tiny{$\! +-$}}}}{16\pi F_{\pi}^2}\,
\left( s - \frac{4}{3}M_{\pi}^2 \right)\,+\,
\frac{\alpha_{\mbox{\tiny{$\! +-$}}} M_{\pi^0}^2}{24\pi F_{\pi}^2} .
\label{PWTree2}\end{aligned}$$ Applying the formulae given in sec. \[sec:dispconstpipi\], and recalling that at the one-loop order $\left\vert f_\ell (s) \right\vert ^2 = [\varphi_\ell(s)]^2 + {\cal O}(E^6)$, one easily obtains the expressions for the unitarity parts of the various amplitudes, up to a polynomial ambiguity that can be reabsorbed into the corresponding subtraction polynomials $P(s,t,u)$. For the amplitudes corresponding to the elastic channels, with either only neutral (\[Im\_W00\]) or only charged pions (\[Im\_Wpm\]), these expressions read $$\begin{aligned}
W_0^{00}(s) &=&
\frac{1}{2}\left[16\pi \varphi^{00}_0(s) \right] ^2 {\bar J}_0 (s)
\,+\,
\left[16\pi \varphi^{x}_0(s) \right] ^2\,{\bar J} (s)
\nonumber\\
W^{\mbox{\tiny{$+-$}}}_{0}(s) &=&
\left[ 16\pi \varphi^{\mbox{\tiny{$+-$}}}_0(s) \right] ^2
{\bar J} (s)
\,+\,
\frac{1}{2}\,
\left[ 16\pi \varphi^{x}_0(s) \right] ^2 {\bar J}_0 (s)
\nonumber\\
W^{\mbox{\tiny{$+-$}}}_{1}(s) &=&
\frac{\beta_{\mbox{\tiny{$\! +-$}}}^2}{36 F_{\pi}^4}
\,\left( s - {4}M_{\pi}^2 \right){\bar J} (s)
\nonumber\\
W^{\mbox{\tiny{$++$}}}_{0}(s) &=&
\frac{1}{2}\,
\left[ 16\pi \varphi^{\mbox{\tiny{$++$}}}_0(s) \right] ^2 {\bar J} (s) .\end{aligned}$$ For the amplitudes corresponding to the processes involving both charged and neutral pions (\[Im\_Wx\]), one obtains $$\begin{aligned}
W^{x}_{0}(s) &=& - \frac{1}{2}\,(16\pi)^2
\varphi^{x}_0(s)\,
\varphi^{00}_0(s) \,{\bar J}_0 (s)
\,-\,(16\pi)^2
\varphi^{\mbox{\tiny{$+-$}}}_0(s)\,
\varphi^{x}_0(s)\,{\bar J} (s)
\nonumber\\
W^{\mbox{\tiny{$+$}} 0}_{0}(s)&=& \bigg\{
\frac{\beta_{x}^2}{12 F_{\pi}^4}\,\frac{(M_{\pi}^2 - M_{\pi^0}^2)^2}{s}\,
\left[
s \,-\, 6(M_{\pi}^2 + M_{\pi^0}^2)
\right]
\nonumber\\
&&
+\,\frac{\beta_{x}^2}{F_{\pi}^4}\left[
\frac{s^2}{4} - \frac{s}{3}(M_{\pi}^2 + M_{\pi^0}^2) +
\frac{11 M_{\pi}^4 - 14 M_{\pi}^2 M_{\pi^0}^2 + 11 M_{\pi^0}^4}{18}
\right]
\nonumber\\
&&
-\,\frac{\beta_{x} \alpha_{x} M_{\pi^0}^2}{3 F_{\pi}^4}
\left[
s - \frac{2}{3} (M_{\pi}^2 + M_{\pi^0}^2) + \frac{(M_{\pi}^2 - M_{\pi^0}^2)^2}{s}
\right]
\,+\, \frac{\alpha_{x}^2 M_{\pi^0}^4}{9 F_{\pi}^4}\bigg\}\,
{\bar J}_{\mbox{\tiny{$\! + $}} 0} (s)
\nonumber\\
&&
+\,\frac{\beta_{x}^2}{3 F_{\pi}^4}\,\frac{(M_{\pi}^2 - M_{\pi^0}^2)^4}{s^2}\,\,
{\bar{\!\!{\bar J}}}_{\mbox{\tiny{$+ $}} 0} (s)
\nonumber\\
W^{\mbox{\tiny{$+$}} 0}_{1}(s)&=& \frac{\beta_{x}^2}{36 F_{\pi}^4}\,
\left[
s \,-\, 2(M_{\pi}^2 + M_{\pi^0}^2)\,+\,\frac{(M_{\pi}^2 - M_{\pi^0}^2)^2}{s}
\right]\,{\bar J}_{\mbox{\tiny{$+ $}} 0}(s) .\end{aligned}$$ In these last expressions, we have introduced another one-loop integral \[$s_{\mbox{\tiny{$\! + $}} 0} \equiv (M_{\pi} + M_{\pi^0})^2$\], $${\bar J}_{\mbox{\tiny{$\!+ $}} 0} (s) =
\frac{s}{16\pi^2} \! \int_{s_{\mbox{\tiny{$\!+ $}} 0}}^{\infty}
\!\frac{dx}{x}\,\frac{1}{x-s-i0}\,\frac{\lambda^{1/2} (x)}{x} ,
\label{Jbar_+0_Disp}$$ together with the subtracted integral $\ {\bar{\!\!{\bar J}}}_{\mbox{\tiny{$\! + $}} 0} (s) = {\bar J}_{\mbox{\tiny{$\! + $}} 0} (s)
- s {\bar J}_{\mbox{\tiny{$\! + $}} 0}^\prime (0)$, i.e. $${\bar{\!\!{\bar J}}}_{\mbox{\tiny{$\! +$}} 0} (s) =
\frac{s^2}{16\pi^2} \! \int_{s_{\mbox{\tiny{$\!\pm $}} 0}}^{\infty}
\!\frac{dx}{x^2}\,\frac{1}{x-s-i0}\,\frac{\lambda^{1/2} (x)}{x} .$$ The expression for ${\bar J}_{\mbox{\tiny{$\! + $}} 0} (s)$ can again be brought into the more familiar form of an integral over a Feynman parameter, $$\begin{aligned}
\!\!\!\!\!
{\bar J}_{\mbox{\tiny{$\! + $}} 0} (s) = \frac{-1}{16\pi^2}\!
\int_0^1 \!\! dx \ln\left[1 - \frac{x(1-x)s}{M_{\pi}^2 - x (M_{\pi}^2 - M_{\pi^0}^2)}\right] \!,
\label{Jbar_+0_Loop} \end{aligned}$$ through an appropriate change of variable.
The functions $W_0^{00}(s)$, $W^{\mbox{\tiny{$+-$}}}_{0}(s)$, etc. are defined by their discontinuities up to polynomial ambiguities. At one-loop order, these polynomials need only be of at most second order in the variables $s,t,u$. Taking into account the symmetry properties of the corresponding amplitudes, they may therefore be written as \[see also eq. (\[polynomials\_P\]) and the discussion following it\] $$\begin{aligned}
P^{00}(s,t,u) &=& \frac{\alpha_{00} M_{\pi^0}^2}{F_{\pi}^2}
\,+\,
\frac{3\lambda_{00}^{(1)}}{F_{\pi}^4}\left[
(s-2M_{\pi^0}^2)^2 + (t-2M_{\pi^0}^2)^2 + (u-2M_{\pi^0}^2)^2
\right]
\nonumber\\
P^{x}(s,t,u) &=& \frac{\beta_{x}}{F_{\pi}^2}\,
\left( s - \frac{2}{3}M_{\pi}^2 - \frac{2}{3}M_{\pi^0}^2\right)\,+\,
\frac{\alpha_{x} M_{\pi^0}^2}{3F_{\pi}^2}
\nonumber\\
&& +\,
\frac{\lambda_{x}^{(1)}}{F_{\pi}^4}(s-2M_{\pi^0}^2)(s-2M_{\pi}^2)
\,+\,
\frac{\lambda_{x}^{(2)}}{F_{\pi}^4}\left[
(t - M_{\pi}^2 - M_{\pi^0}^2)^2 + (u - M_{\pi}^2 - M_{\pi^0}^2)^2
\right]
\nonumber\\
P^{\mbox{\tiny{$ +-$}}}(s,t,u) &=& \frac{\beta_{\mbox{\tiny{$ +-$}}}}{F_{\pi}^2}\,
\left( s + t - \frac{8}{3}M_{\pi}^2 \right)\,+\,
\frac{2\alpha_{\mbox{\tiny{$ +-$}}} M_{\pi^0}^2}{3F_{\pi}^2}
\nonumber\\
&& +\,
\frac{\lambda_{\mbox{\tiny{$ +-$}}}^{(1)} + \lambda_{\mbox{\tiny{$ +-$}}}^{(2)}}{F_{\pi}^4}\left[
(s- 2M_{\pi}^2 )^2 + (t - 2M_{\pi}^2)^2
\right]
\,+\,
\frac{2 \lambda_{\mbox{\tiny{$ +-$}}}^{(2)}}{F_{\pi}^4}
(u- 2M_{\pi}^2 )^2
\label{poly_1loop_amp}\end{aligned}$$ The five subtraction constants $\lambda_{00}^{(1)}$, $\lambda_{x}^{(i)}$, $\lambda_{\mbox{\tiny{$ +-$}}}^{(i)}$, $i = 1,2$, are new free parameters. In the isospin limit, they are given by $$\begin{aligned}
\lambda_{x}^{(i)} \ \to\ \lambda_{i}\,,\quad \lambda_{\mbox{\tiny{$ +-$}}}^{(i)}\ \to\ \lambda_{i}\,,\quad
\lambda_{00}^{(1)} \ \to\ \frac{\lambda_{1} + 2\lambda_{2}}{3} ,
\label{lambdas_iso}\end{aligned}$$ where, at this order, $\lambda_{1,2}$ can be expressed in terms of the low-energy constants ${\bar \ell}_1$ and ${\bar \ell}_2$ of [@Gasser:1983yg], cf. [@KMSF95] and equation (\[lambdas\_l1\_l2\]) below.
The expressions for the $\pi\pi$ scattering amplitudes that follow from the above results agree with those derived in refs. [@Knecht:1997jw; @Meissner:1997fa; @KnechtNehme02] from a Feynman graph calculation based on the low-energy effective lagrangian for QCD+QED, provided that one drops the contributions coming from virtual photons, while keeping the difference between charged and neutral pion masses \[this procedure is discussed in greater detail in subsection \[sub\_csts\_at\_NLO\] and in appendix \[app:subtraction\]\].
Infrared behaviour of one-loop amplitudes and form factors {#Mto0_1loop}
----------------------------------------------------------
In the isospin-symmetric situation, the $\pi\pi$ scattering amplitude $A(s\vert t,u)$, the vector form factor $F_V^\pi(s)$ and the scalar form factor $F_S^\pi (s)$ are well behaved in the limit where the pion mass $M_\pi$ vanishes. For the pion form factors, this has been shown explicitly at the two loop level in [@GasserMeissner91]. This good behaviour in the chiral limit requires that some of the parameters that appear in these quantities develop themselves logarithmically singular terms in the limit whare the pion mass vanishes. This is necessary in order to compensate for the singularities coming from the unitarity part, for, as $M_\pi \rightarrow 0$, $${\bar J}(s)\, _{ \widetilde{_{ M_\pi \rightarrow\, 0\,\, }}}
\,\frac{1}{16\pi^2}\,\ln\left(\frac{M_{\pi}^2}{-s}\right)
\,+\, \frac{1}{8 \pi^2} .
\label{Jbar_IR_limit}$$ Taking into account the pion mass difference offers a wider range of possibilities from this point of view. Indeed, besides the path just described, reaching the isospin limit first, then letting the common pion mass vanish, two additional options might be considered, where one lets, say, the neutral pion mass tend to zero, keeping the charged pion mass fixed, or the other way around. In the first case, singular contributions in the unitarity part come from the function ${\bar J}_0(s)$, which behaves as in (\[Jbar\_IR\_limit\]), but with $M_\pi$ replaced by $M_{\pi^0}$. In the second case, eq. (\[Jbar\_IR\_limit\]) applies directly. Notice that the function ${\bar J}_{\mbox{\tiny{$\! + $}} 0} (s)$ remains finite as either of these two limits is taken. It requires that both pion masses vanish for it to develop an infrared singular behaviour.
Let us first consider the $\pi\pi$ scattering amplitudes obtained in the preceding sub-section. In order for the one-loop amplitudes $A^{00}$, $A^x$, and $A^{\mbox{\tiny{$ +-$}}}$ to remain finite as $M_{\pi^0} \rightarrow 0$, with $M_\pi$ fixed, we must have $$\begin{aligned}
\alpha_{00} M_{\pi^0} &\rightarrow & 0
\, ,\quad \alpha_x M_{\pi^0}^2 \ \rightarrow \ {\widehat\alpha}_x M_{\pi}^2
\, ,\ \beta_x \ \rightarrow \ {\widehat\beta}_x \,,\end{aligned}$$ and \[for the sake of simplicity, we keep the notation $F_\pi$ for the pion decay constant, which has a regular behaviour in either limit\] $$\begin{aligned}
\alpha_{\mbox{\tiny{$ +-$}}} M_{\pi^0}^2 &\sim & - \frac{M_\pi^4}{96\pi^2 { F}_\pi^2}
\,{\widehat\alpha}_x ( {\widehat\alpha}_x + 4 {\widehat\beta}_x ) \ln M_{\pi^0}^2\,+\,{\mbox{finite}}
\nonumber\\
\beta_{\mbox{\tiny{$ +-$}}} &\sim & - \frac{M_\pi^2}{48\pi^2 { F}_\pi^2}
\,{\widehat\beta}_x ( {\widehat\alpha}_x + 4 {\widehat\beta}_x) \ln M_{\pi^0}^2\,+\,{\mbox{finite}}
\nonumber\\
\lambda_{\mbox{\tiny{$ +-$}}}^{(1)} &\sim & -\frac{1}{32\pi^2}\,{\widehat\beta}_x ^2 \ln M_{\pi^0}^2\,+\,{\mbox{finite}}\,,\end{aligned}$$ whereas the remaining coefficients, $\lambda_{00}^{(1)}$, $\lambda_{\mbox{\tiny{$ +-$}}}^{(2)}$, $\lambda_x^{(1)}$, and $\lambda_x^{(2)}$ remain finite in this limit. Likewise, the coefficients ${\widehat\alpha}_x$ and ${\widehat\beta}_x$ are free of infrared singularities. In order to avoid any possible confusion, we remind the reader that the coefficients $\alpha_{00}$, $\alpha_x$, $\alpha_{\mbox{\tiny{$ +-$}}}$ appear in the amplitudes multiplied by $M_{\pi^0}$ as a pure matter of convention \[see the remark after eq. (\[AmpTree1\])\]. Therefore, $\alpha_x M_{\pi^0}^2$ and $\alpha_{\mbox{\tiny{$ +-$}}} M_{\pi^0}^2$ need not vanish as $M_{\pi^0}$ tends to zero with $M_\pi$ fixed. This feature is actually exhibited already by the lowest-order expressions given in eq. (\[alphabetaLO\]). Furthermore, the quantities which appear on the right-hand sides of the above equations have to be understood as taking their lowest-order values. We have not distinguished them from their values at next-to-leading order, which appear on the left-hand sides, in order not to overburden the notation. The appearance of infrared singular behaviours is a loop effect, and higher-order corrections will induce new singularities. At the next order, these may involve log-squared terms [@GasserMeissner91], with an additional $1/(4\pi F_\pi)^2$ loop suppression factor. Concretely, from eq. (\[alphabetaLO\]) one obtains the lowest-order values $${\widehat\alpha}_x \,=\, 2 \beta \, , \ {\widehat\beta}_x \,=\, \beta .
\label{alpha_beta_hat}$$ Taking now the second limit, $M_{\pi} \rightarrow 0$ with $M_{\pi^0}$ fixed, we find that the finiteness of the one-loop amplitudes $A^{00}$, $A^x$, and $A^{\mbox{\tiny{$ +-$}}}$ makes the various coefficients \[we denote their lowest-order, infrared finite, limiting values with a tilde on top of them, except for $F_\pi$\] behave as follows: $$\begin{aligned}
\alpha_{00} &\sim & - \frac{M_{\pi^0}^2}{48\pi^2 { F}_\pi^2}
\,{\widetilde\alpha}_x ( {\widetilde\alpha}_x + 4 {\widetilde\beta}_x) \ln M_{\pi}^2\,+\,{\mbox{finite}}
\nonumber\\
\lambda_{00}^{(1)} &\sim & -\frac{1}{48\pi^2}\,{\widetilde\beta}_x ^2 \ln M_{\pi}^2\,+\,{\mbox{finite}}
\nonumber\\
\alpha_x &\sim & - \frac{1}{48\pi^2}\, \frac{M_{\pi^0}^2}{{ F}_\pi^2}
\left[
2 {\widetilde\alpha}_{\mbox{\tiny{$ +-$}}} {\widetilde\alpha}_x + {\widetilde\beta}_{\mbox{\tiny{$ +-$}}}
({\widetilde\alpha}_x + 4 {\widetilde\beta}_x ) \right] \ln M_{\pi}^2\,+\,{\mbox{finite}}
\nonumber\\
\beta_x &\sim & - \frac{1}{96\pi^2}\, \frac{M_{\pi^0}^2}{{ F}_\pi^2}
\left[
4 {\widetilde\alpha}_{\mbox{\tiny{$ +-$}}} {\widetilde\beta}_x + {\widetilde\beta}_{\mbox{\tiny{$ +-$}}}
({\widetilde\alpha}_x + 4 {\widetilde\beta}_x ) \right] \ln M_{\pi}^2\,+\,{\mbox{finite}}
\nonumber\\
\lambda_x^{(1)} &\sim & - \frac{1}{32\pi^2}\, {\widetilde\beta}_x {\widetilde\beta}_{\mbox{\tiny{$ +-$}}}
\ln M_{\pi}^2\,+\,{\mbox{finite}}
\nonumber\\
\alpha_{\mbox{\tiny{$ +-$}}} &\sim & - \frac{5}{48\pi^2} \frac{M_{\pi^0}^2}{{ F}_\pi^2}
\,{\widetilde\beta}_{\mbox{\tiny{$ +-$}}}^2 \ln M_{\pi}^2\,+\,{\mbox{finite}}
\nonumber\\
\beta_{\mbox{\tiny{$ +-$}}} &\sim & - \frac{1}{12\pi^2} \frac{M_{\pi^0}^2}{{ F}_\pi^2}
\,{\widetilde\alpha}_{\mbox{\tiny{$ +-$}}} {\widetilde\beta}_{\mbox{\tiny{$ +-$}}} \ln M_{\pi}^2\,+\,{\mbox{finite}}
\nonumber\\
\lambda_{\mbox{\tiny{$ +-$}}}^{(1)} &\sim & \frac{1}{96\pi^2}
\,{\widetilde\beta}_{\mbox{\tiny{$ +-$}}}^2\ln M_{\pi}^2\,+\,{\mbox{finite}}
\nonumber\\
\lambda_{\mbox{\tiny{$ +-$}}}^{(2)} &\sim & - \frac{1}{48\pi^2}
\,{\widetilde\beta}_{\mbox{\tiny{$ +-$}}}^2\ln M_{\pi}^2\,+\,{\mbox{finite}}
\,.\end{aligned}$$ Now the lowest-order values inferred from eq. (\[alphabetaLO\]) read: $${\widetilde\alpha}_x \,=\, \alpha - 2 \beta \, , \ {\widetilde\alpha}_{\mbox{\tiny{$ +-$}}} \,=\, \alpha - 4 \beta \, ,
\ {\widetilde\beta}_x \,=\, {\widetilde\beta}_{\mbox{\tiny{$ +-$}}} \,=\, \beta .$$ In appendix \[app:subtraction\] we determine the various subtraction constants that appear in the expressions of the $\pi\pi$ scattering amplitudes obtained after the first iteration from the corresponding expressions obtained from a one-loop calculation. One may check that the formulae given in appendix \[app:subtraction\] indeed exhibit the expected infrared behaviour.
Let us now turn towards the form factors, and consider the same two limits. As $M_{\pi^0}$ vanishes while $M_\pi$ is kept fixed, the form factors remain free of infrared singularities provided $$\begin{aligned}
a_S^{\pi} &\sim & - \frac{1}{32\pi^2 F_\pi^2}
\, \frac{{\widehat F}_S^{\pi^0}(0)}{{\widehat F}_S^{\pi}(0)}
\,{\widehat\beta}_x \ln M_{\pi^0}^2\,+\,{\mbox{finite}}
\nonumber\\
F_S^{\pi}(0) &\sim & - \frac{1}{96\pi^2} \frac{M_{\pi}^2}{{ F}_\pi^2}
\,{\widehat F}_S^{\pi^0}(0) ( {\widehat\alpha}_x - 2 {\widehat\beta}_x)\ln M_{\pi^0}^2\,+\,{\mbox{finite}}\, ,\end{aligned}$$ while $a_S^{\pi^0}$ and $F_S^{\pi^0}(0)$ remain finite \[at this order\]. Actually, in view of (\[alpha\_beta\_hat\]), this is also true for $F_S^{\pi}(0)$. In the case of the second limit, $M_\pi \rightarrow 0$ and $M_{\pi^0}$ fixed, infrared finite form factors require that $$\begin{aligned}
a_V^\pi &\sim & - \frac{1}{96\pi^2 F_\pi^2}
\,{\widetilde\beta}_{\mbox{\tiny{$ +-$}}} \ln M_{\pi}^2\,+\,{\mbox{finite}}
\nonumber\\
a_S^{\pi^0} &\sim & - \frac{1}{16\pi^2 F_\pi^2}
\, \frac{{\widetilde F}_S^{\pi}(0)}{{\widetilde F}_S^{\pi^0}(0)}
\,{\widetilde\beta}_x \ln M_{\pi}^2\,+\,{\mbox{finite}}
\nonumber\\
F_S^{\pi^0}(0) &\sim & - \frac{1}{48\pi^2} \frac{M_{\pi^0}^2}{{ F}_\pi^2}
\,{\widetilde F}_S^\pi (0) ( {\widetilde\alpha}_x - 2 {\widetilde\beta}_x) \ln M_{\pi}^2\,+\,{\mbox{finite}}
\nonumber\\
a_S^{\pi} &\sim & - \frac{1}{32\pi^2 F_\pi^2}
\,{\widetilde\beta}_{\mbox{\tiny{$ +-$}}} \ln M_{\pi}^2\,+\,{\mbox{finite}}
\nonumber\\
F_S^{\pi}(0) &\sim & - \frac{1}{24\pi^2} \frac{M_{\pi^0}^2}{{ F}_\pi^2}
\, {\widetilde F}_S^\pi (0) {\widetilde\alpha}_{\mbox{\tiny{$ +-$}}} \ln M_{\pi}^2\,+\,{\mbox{finite}}\,.\end{aligned}$$ Again, one may check that the explicit one-loop expressions given in appendix \[app:subtraction\] reproduce the infrared behaviour obtained here from quite general arguments.
To conclude this short discussion, we may observe that many of these infrared singularities disappear once the second mass also tends to zero, thus restoring the infrared features of the isospin-symmetric chiral limit. Upon studying the expressions (\[1loop\_radii\]) and (\[1loop\_curvatures\]) in the light of the present discussion, a similar worsening of the infrared behaviour, as compared to the isospin-symmetric limit, can also be brought forward in the radii and curvatures of the scalar form factors. We thus can conclude that, to a certain extent, isospin symmetry tames the infrared behaviour of the soft pion clouds.
SECOND ITERATION: TWO-LOOP EXPRESSIONS {#2ndIteration}
======================================
So far, we have completed the first cycle of the procedure sketched in figure \[iterconst\]. The one-loop expressions of the form factors and scattering amplitudes obtained in the preceding section can now be used in order to construct the two-loop representations of the form factors. At next-to-leading order in the chiral expansion, the structure of the form factors is now as follows $$\begin{aligned}
{\mbox{Re}} F_S^{\pi}(s) &=& F_S^{\pi}(0) \left[
1 \,+\,\Gamma_S^{\pi}(s) \right] \,+\,{\cal O}(E^6)
\nonumber\\
{\mbox{Re}} F_S^{\pi^0}(s) &=& F_S^{\pi^0}(0) \left[
1 \,+\,\Gamma_S^{\pi^0}(s) \right] \,+\,{\cal O}(E^6) , \quad{ }\end{aligned}$$ and $${\mbox{Re}} F_V^{\pi}(s) \,=\, 1 + \Gamma_V^{\pi}(s) \,+\,{\cal O}(E^4).$$ The one-loop corrections $\Gamma_S^{\pi^0}(s)$, $\Gamma_S^{\pi}(s)$, and $\Gamma_V^{\pi}(s)$ are easy to extract from the expressions of the form factors obtained in the preceding section, $$\begin{aligned}
\Gamma_S^{\pi^0}(s) &=&
a_S^{\pi^0}\!s \,+\, 16\pi\,\frac{1}{2}\,\varphi_0^{00}(s)\,{\mbox{Re}}\, {\bar J}_0 (s)
\, -\,16\pi \varphi_0^{x}(s)\,\frac{F_S^{\pi}(0)}{F_S^{\pi^0}(0)}\,{\mbox{Re}}\, {\bar J} (s)
\nonumber\\
\Gamma_S^{\pi}(s) &=&
a_S^{\pi} s \,+\, 16\pi \varphi_0^{\mbox{\tiny{$+-$}}}(s) \,{\mbox{Re}}\, {\bar J} (s)
\, -\,16\pi\,\frac{1}{2}\,\varphi_0^{x}(s)\,
\frac{F_S^{\pi^0}(0)}{F_S^{\pi}(0)} \,{\mbox{Re}}\, {\bar J}_0 (s)
\nonumber\\
\Gamma_V^{\pi}(s) &=&
a_V^{\pi} s \,+\, 16\pi \varphi_1^{\mbox{\tiny{$+-$}}}(s) \,{\mbox{Re}}\, {\bar J} (s)
.
\label{eq:gammaoneloop}\end{aligned}$$
As far as the discontinuities are concerned, they start at ${\cal O}(E^4)$ for $F_S^{\pi}(s)$ and $F_S^{\pi^0}(s)$, and at ${\cal O}(E^2)$ for $F_V^{\pi}(s)$. Using this power counting and eq. (\[partialwavesamplitude\]), one obtains the discontinuities of the form factors at next-to-next-to-leading order $$\begin{aligned}
{\mbox{Im}}F_S^{\pi^0}(s) &=&
\frac{1}{2}\,\sigma_{0}(s)F_S^{\pi^0}(0) \left\{
\varphi_0^{00}(s)\left[
1 \,+\,\Gamma_S^{\pi^0}(s) \right] \,+\,\psi_0^{00}(s)
\right\}
\theta(s-4M_{\pi^0}^2)
\nonumber\\
&& \, -\,
\sigma(s)F_S^{\pi}(0)\left\{
\varphi_0^{x}(s)\left[
1 \,+\,\Gamma_S^{\pi}(s) \right] \,+\,\psi_0^{x}(s)
\right\}\theta(s-4M_{\pi}^2)
\,+\,{\cal O}(E^6)
\nonumber\\
{\mbox{Im}}F_S^{\pi}(s) &=&
\sigma(s)F_S^{\pi}(0)\left\{
\varphi_0^{\mbox{\tiny{$+-$}}}(s)\left[
1 \,+\,\Gamma_S^{\pi}(s) \right] \,+\,\psi_0^{\mbox{\tiny{$+-$}}}(s)
\right\}\theta(s-4M_{\pi}^2)
\nonumber\\
&& \, -\,
\frac{1}{2}\,\sigma_{0}(s)F_S^{\pi^0}(0)\left\{
\varphi_0^{x}(s)\left[
1 \,+\,\Gamma_S^{\pi^0}(s) \right] \,+\,\psi_0^{x}(s)
\right\}\theta(s-4M_{\pi^0}^2)
\,+\,{\cal O}(E^6)
\nonumber\\
{\mbox{Im}}F_V^{\pi}(s) &=&
\sigma (s)\left\{
\varphi_1^{\mbox{\tiny{$+-$}}}(s) \left[
1 \,+\,\Gamma_V^{\pi}(s) \right] \,+\,\psi_1^{\mbox{\tiny{$+-$}}}(s)
\right\}\theta(s-4M_{\pi}^2) \,+\,{\cal O}(E^4) .
\label{ImF_NLO}\end{aligned}$$
Partial-wave projections from the one-loop amplitudes {#PW_1_loop}
-----------------------------------------------------
The computation of the one-loop corrections $\psi_0^{00}(s)$, $\psi_0^{x}(s)$, $\psi_0^{\mbox{\tiny{$+-$}}}(s)$, $\psi_1^{\mbox{\tiny{$+-$}}}(s)$ to the $\pi\pi$ $S$ and $P$ partial wave projections from the one-loop scattering amplitudes is less straightforward, and represents the next issue to be addressed. In order to illustrate how this difficulty can be handled, let us start with the elastic scattering of neutral pions, i.e. the quantity $\psi^{00}_0(s)$ describing the next-to-leading-order correction to the real part of the $S$-wave projection for $A^{00}(s,t)$ in the range $s\ge 4M_{\pi^0}^2$. Trading the integration over the scattering angle for an integration over the variable $t$, with $t_{\mbox{\tiny{$-$}}}(s)\equiv -(s - 4 M_{\pi^0}^2) \le t \le 0$, we obtain (the functions ${\bar J}(t)$ and ${\bar J}_0(t)$ are real for $t\le 0$) $$\begin{aligned}
\psi^{00}_0(s) &=&
\frac{\lambda_{0 0}^{(1)} }{16\pi F_{\pi}^4}
(5 s^2 - 16 s M_{\pi^0}^2 + 20 M_{\pi^0}^4)
+\,
\frac{1}{32\pi}\left[ 16\pi \varphi_0^{00}(s)\right]^2\,
{\rm{Re}}\,{\bar J}_0(s)
\,+\,
\frac{1}{16\pi}\left[ 16\pi \varphi_0^{x}(s)\right]^2\,
{\rm{Re}}\,{\bar J} (s)
\nonumber\\
&&\!\!\!\!\!
+\,
\frac{1}{16\pi}\,\frac{1}{s - 4 M_{\pi^0}^2}\,\frac{\alpha_{0 0}^2 M_{\pi^0}^4}{F_{\pi}^4}\!
\int_{t_{-}(s)}^{0}\!\! dt\,{\bar J}_0(t)
+\,
\frac{1}{8\pi}\,\frac{1}{s - 4 M_{\pi^0}^2}\!
\int_{t_{-}(s)}^{0} \! dt \left[ 16\pi \varphi_0^{x}(t)\right]^2
\!\!{\bar J} (t) . \quad\ \,{ }
\label{psi_00_proj}\end{aligned}$$ It turns out that the remaining integrals can be performed analytically. The relevant formulae can be found in appendix \[app:integJbar\]. The resulting expression can be written as $$\begin{aligned}
16\pi \psi^{00}_0(s) &=& 2\,\frac{M_{\pi}^4}{F_\pi^4}\,
\sqrt{\frac{s}{s - 4M_{\pi^0}^2}}\,\Bigg\{
\xi_{00}^{(0)}(s) \sigma_{0}(s) \,+\, 2\xi^{(1;0)}_{00}(s) L_{0}(s)
\,+\, 2\xi^{(1;{\mbox{\tiny$\nabla$}})}_{00}(s)\,\frac{\sigma_{0}(s)}{\sigma_{\mbox{\tiny$\nabla$}}(s)}
\, L_{\mbox{\tiny$\nabla$}}(s)
\nonumber\\
&&
+\,2\xi^{(2;\pm)}_{00}(s) \sigma (s) \sigma_0(s) L(s) \,+\,
2\xi^{(2;0)}_{00}(s) \left(1 - \frac{4 M_{\pi^0}^2}{s}\right) L_0(s)
\nonumber\\
&&
+\, 3 \xi^{(3;0)}_{00}(s)\,\sigma_0 (s)\,
\frac{M_{\pi^0}^2}{s - 4M_{\pi^0}^2}\,L_{0}^2(s)
\,+\,
3 \xi^{(3;{\mbox{\tiny$\nabla$}})}_{00}(s)\,\sigma_{0}(s)
\frac{M_{\pi}^2}{s - 4M_{\pi^0}^2}
\,L_{\mbox{\tiny$\nabla$}}^2(s)
\Bigg\} ,
\label{psi_00}\end{aligned}$$ with $\sigma_{\mbox{\tiny $\nabla $}}(s) = \sigma (s-4M_{\pi^0}^2+4M_{\pi}^2)$ and $L_{\mbox{\tiny $\nabla $}}(s) = L(s-4M_{\pi^0}^2+4M_{\pi}^2)$. The various functions $\xi_{00}^{(0)}(s)$, $\xi^{(1;0)}_{00}(s)$, etc. that enter this expression of $\psi^{00}_0(s)$ are polynomials in $s$ and in the subthreshold parameters, which are given in appendix \[app:polynomials\]. We have written the result in a way that allows for a straightforward connection with the similar expressions for the isospin-symmetric case, as displayed in ref. [@KMSF95]. Indeed in the limit $M_{\pi^0} \to M_{\pi}$ (and $\alpha_{00}, \alpha_x, \alpha_{\mbox{\tiny{$+-$}}} \to \alpha$, $\beta_x, \beta_{\mbox{\tiny{$+-$}}} \to \beta$) one obtains the expected combination of $I=0$ and $I=2$ contributions, weighted by the corresponding $SU(2)$ Clebsch-Gordan coefficients, $$\psi^{00}_0(s)\rightarrow \ \stackrel{{ }_{\mbox{\scriptsize{$o$}}}}{\psi} \stackrel{00}{_{\!\! 0}} \!\! (s)
\,\equiv\, 2\,\frac{M_\pi^4}{F_\pi^4}\,
\sqrt{\frac{s}{s-4M_\pi^2}}\,
\sum_{n=0}^3\left[\frac{2}{3}\,\xi^{(n)}_2(s)\,+\,\frac{1}{3}\,\xi^{(n)}_0(s)\right]
k_n(s) ,
\nonumber
\label{lim_psi_00}$$ where, for the reader’s convenience, we reproduce the expressions of the functions $k_n(s)$ of ref. [@KMSF95] (the function $k_4(s)$ appears only in the $P$ wave component $\psi^{\mbox{\tiny{$+-$}}}_1(s)$, to be discussed below), $$\begin{aligned}
&& k_0(s) \,=\, \frac{1}{16\pi}\,\sqrt{\frac{s - 4 M_\pi^2}{s}}\,,
\qquad
k_1(s) \,=\, \frac{1}{8\pi}\,L(s)\,,
\nonumber\\
&& k_2(s) \,=\, \frac{1}{8\pi}\,\left(1\,-\,\frac{4 M_\pi^2}{s}\right)\,L(s)\,,
\qquad
k_3(s) \,=\, \frac{3}{16\pi}\,\frac{M_\pi^2}{\sqrt{s(s - 4 M_\pi^2)}}\,L^2(s)\,,
\nonumber\\
&& k_4(s) \,=\,\frac{1}{16\pi}\,\frac{M_\pi^2}{\sqrt{s(s - 4 M_\pi^2)}}\,
\left\{
1\,+\,\sqrt{\frac{s}{s - 4 M_\pi^2}}\,L(s)\,+\,\frac{M_\pi^2}{s - 4 M_\pi^2}\,L^2(s)
\right\}
.
\label{k_n}\end{aligned}$$ For the remaining notation we refer the reader to [@KMSF95] (see in particular eqs. (3.36), (3.37) and the Appendix B therein). Let us, however, point out that the factor of 2 in (\[lim\_psi\_00\]) takes care of the difference in normalization in the partial waves as compared to that reference, see eq. (\[PWdecomp\]) and the comment preceding it.
For the elastic scattering of charged pions, the computation of $\psi^{\mbox{\tiny{$+-$}}}_0 (s)$ and of $\psi^{\mbox{\tiny{$+-$}}}_1 (s)$, now in the range $s\ge M_{\pi}^2$, proceeds along similar lines. The starting point is provided by the following formulae, $$\begin{aligned}
\psi^{\mbox{\tiny{$+-$}}}_0(s) &=&
\frac{\lambda_{\mbox{\tiny{$+-$}}}^{(1)} + \lambda_{\mbox{\tiny{$+-$}}}^{(2)}}{ F_{\pi}^4}\,\left(s-2 M_{\pi}^2\right)^2
\,+\,
\frac{\lambda_{\mbox{\tiny{$+-$}}}^{(1)} + 3\lambda_{\mbox{\tiny{$+-$}}}^{(2)} }{3 F_{\pi}^4}\,
\left(s^2 - 2 s M_{\pi} + 4 M_{\pi}^4
\right)
\,+\,
\frac{1}{32\pi}\,\left[ 16\pi \varphi_0^{x}(s)\right]^2 {\rm{Re}}\,{\bar J}_0(s)
\nonumber\\
&&
+\,
\frac{1}{16\pi}\,\left[ 16\pi \varphi_0^{\mbox{\tiny{$+-$}}}(s)\right]^2 {\rm{Re}}\,{\bar J}(s)
\,+\,
\frac{1}{32\pi}\,\frac{1}{s - 4 M_{\pi}^2}\!
\int_{t_-(s)}^{0} \! dt \left[ 16\pi \varphi_0^{x}(t)\right]^2{\bar J}_0(t)
\nonumber\\
&&
+\,
\frac{1}{32\pi}\,\frac{1}{s - 4 M_{\pi}^2}\,
\int_{t_-(s)}^{0} dt\,\left\{
2\left[ 16\pi \varphi_0^{\mbox{\tiny{$+-$}}}(t)\right]^2 + \left[ 16\pi \varphi_0^{\mbox{\tiny{$++$}}}(t)\right]^2
\right\}
{\bar J} (t)
\nonumber\\
&&
+\,
\frac{1}{16\pi}\,\frac{1}{s - 4 M_{\pi}^2}\,
\int_{t_-(s)}^{0} dt\,
\frac{\beta_{\mbox{\tiny{$\! +-$}}}^2}{12F_\pi^4}\,(t - 4 M_{\pi}^2)(2s + t - 4 M_{\pi}^2)
{\bar J} (t)
,
\nonumber\end{aligned}$$ and $$\begin{aligned}
\psi^{\mbox{\tiny{$+-$}}}_1(s) &=&
-\,\frac{\lambda_{\mbox{\tiny{$+-$}}}^{(1)} - \lambda_{\mbox{\tiny{$+-$}}}^{(2)}}{96\pi F_{\pi}^4}\,s(s - 4 M_{\pi}^2)
\,+\,
\frac{1}{16\pi}\,\frac{\beta_{\mbox{\tiny{$\! +-$}}}^2}{36 F_{\pi}^4}\,(s - 4 M_{\pi}^2)^2
\,{\rm{Re}}\,{\bar J}(s)
\nonumber\\
&&
+\,
\frac{1}{32\pi}\,\frac{1}{s - 4 M_{\pi}^2}\!
\int_{t_-(s)}^{0} \! dt
\left[ 16\pi \varphi_0^{x}(t)\right]^2 \left( 1 + \frac{2t}{s - 4 M_{\pi}^2}\right) {\bar J}_0(t)
\nonumber\\
&&
+\,
\frac{1}{32\pi}\,\frac{1}{s - 4 M_{\pi}^2}\!
\int_{t_-(s)}^{0} \! dt
\left\{
2\left[ 16\pi \varphi_0^{\mbox{\tiny{$+-$}}}(t)\right]^2 - \left[ 16\pi \varphi_0^{\mbox{\tiny{$++$}}}(t)\right]^2
\right\}
\left( 1 + \frac{2t}{s - 4 M_{\pi}^2}\right){\bar J}(t)
\nonumber\\
&&
+\,
\frac{1}{16\pi}\,\frac{1}{s - 4 M_{\pi}^2}\,
\int_{t_-(s)}^{0} dt\,
\frac{\beta_{\mbox{\tiny{$\! +-$}}}^2}{12F_\pi^4}\,(t - 4 M_{\pi}^2)(2s + t - 4 M_{\pi}^2)
\left( 1 + \frac{2t}{s - 4 M_{\pi}^2}\right){\bar J} (t)
,\end{aligned}$$ with now $t_{\mbox{\tiny{$-$}}}(s) = -(s - 4 M_{\pi}^2)$. Performing the remaining integrations with the help of the formulae given in appendix \[app:integJbar\] leads then to $$\begin{aligned}
16\pi \psi^{\mbox{\tiny{$+-$}}}_0(s) &=& 2\,\frac{M_{\pi}^4}{F_\pi^4}\,
\sqrt{\frac{s}{s - 4M_{\pi}^2}}\,\Bigg\{
\xi_{{\mbox{\tiny{$+-$}}};S}^{(0)}(s) \sigma (s) \,+\, 2\xi^{(1;{\mbox{\tiny{$\pm$}}})}_{{\mbox{\tiny{$+-$}}};S}(s) L (s)
\,+\, 2\xi^{(1;{\mbox{\tiny$\Delta$}})}_{{\mbox{\tiny{$+-$}}}; S}(s)\,\frac{\sigma (s)}{\sigma_{\mbox{\tiny$\Delta$}}(s)}
\, L_{\mbox{\tiny$\Delta$}}(s)
\nonumber\\
&&
+\,2\xi^{(2;{\mbox{\tiny{$\pm$}}})}_{{\mbox{\tiny{$+-$}}};S}(s) \left(1 - \frac{4 M_{\pi}^2}{s}\right) L (s) \,+\,
2\xi^{(2;0)}_{{\mbox{\tiny{$+-$}}};S}(s) \sigma (s) \sigma_0(s) L_0(s)
\nonumber\\
&&
+\, 3 \xi^{(3;{\mbox{\tiny{$\pm$}}})}_{{\mbox{\tiny{$+-$}}};S}(s)\,\frac{M_{\pi}^2}{\sqrt{s(s - 4M_{\pi}^2)}}\,L^2(s) \,+\,
3 \xi^{(3;{\mbox{\tiny$\Delta$}})}_{{\mbox{\tiny{$+-$}}};S}(s)\,\frac{M_{\pi^0}^2}{\sqrt{s(s - 4M_{\pi}^2)}}
\,L_{\mbox{\tiny$\Delta$}}^2(s)
\Bigg\}
,
\label{psi_+-_0} \end{aligned}$$ $$\begin{aligned}
16\pi \psi^{\mbox{\tiny{$+-$}}}_1(s) &=& 2\,\frac{M_{\pi}^4}{F_\pi^4}\,
\sqrt{\frac{s}{s - 4M_{\pi}^2}}\,\Bigg\{
\xi_{{\mbox{\tiny{$+-$}}};P}^{(0)}(s) \sigma (s) \,+\, 2\xi^{(1;{\mbox{\tiny{$\pm$}}})}_{{\mbox{\tiny{$+-$}}};P}(s) L (s)
\,+\, 2\xi^{(1;{\mbox{\tiny$\Delta$}})}_{{\mbox{\tiny{$+-$}}}; P}(s)\,\frac{\sigma (s)}{\sigma_{\mbox{\tiny$\Delta$}}(s)}
\, L_{\mbox{\tiny$\Delta$}}(s)
\nonumber\\
&&
+\,
2\xi^{(2;{\mbox{\tiny{$\pm$}}})}_{{\mbox{\tiny{$+-$}}};P}(s) \left(1 - \frac{4 M_{\pi}^2}{s}\right) L (s)
\nonumber\\
&&
+\, 3 \xi^{(3;{\mbox{\tiny{$\pm$}}})}_{{\mbox{\tiny{$+-$}}};P}(s)\,\frac{M_{\pi}^2}{\sqrt{s(s - 4M_{\pi}^2)}}\,L^2(s) \,+\,
3 \xi^{(3;{\mbox{\tiny$\Delta$}})}_{{\mbox{\tiny{$+-$}}};P}(s)\,\frac{M_{\pi^0}^2}{\sqrt{s(s - 4M_{\pi}^2)}}
\,L_{\mbox{\tiny$\Delta$}}^2(s)
\nonumber\\
&&
+\,\xi^{(4;{\mbox{\tiny{$\pm$}}})}_{{\mbox{\tiny{$+-$}}};P}(s)\,\frac{M_{\pi}^2}{\sqrt{s(s-4 M_{\pi}^2)}}\,
\left[
1\,+\,\frac{1}{\sigma (s)}L(s)\,+\,\frac{M_{\pi}^2}{s-4 M_{\pi}^2}\,L^2(s)
\right]
\nonumber\\
&&
+\,\xi^{(4;{\mbox{\tiny$\Delta$}})}_{{\mbox{\tiny{$+-$}}};P}(s)\,\frac{M_{\pi^0}^2}{\sqrt{s(s-4 M_{\pi}^2)}}\,
\left[
1\,+\,\frac{1}{\sigma_{\mbox{\tiny$\Delta$}}(s)}\,L_{\mbox{\tiny$\Delta$}}(s)
\,+\,\frac{M_{\pi^0}^2}{s-4 M_{\pi}^2}\,L_{\mbox{\tiny$\Delta$}}^2(s)
\right]
\Bigg\}
,
\label{psi_+-_1} \end{aligned}$$ with $\sigma_{\mbox{\tiny$\Delta $}}(s) = \sigma_0 (s+4M_{\pi^0}^2-4M_{\pi}^2)$ and $L_{\mbox{\tiny$\Delta $}}(s) = L_0(s+4M_{\pi^0}^2-4M_{\pi}^2)$. The various polynomials that enter the expression of $\psi^{\mbox{\tiny{$+-$}}}_0(s)$ and $\psi^{\mbox{\tiny{$+-$}}}_1(s)$ have been gathered in appendix \[app:polynomials\]. Taking the isospin limit, as described above in the case of $\psi^{00}_0(s)$, one recovers, for the $S$ wave, the expected combination of $I=0$ and $I=2$ contributions, and, for the $P$ wave, the corresponding $I=1$ component, $$\begin{aligned}
&&\psi^{\mbox{\tiny{$+-$}}}_0(s)\rightarrow
\ \stackrel{{ }_{\mbox{\scriptsize{$o$}}}}{\psi} \stackrel{{\mbox{\tiny{$+-$}}}}{_{\!\!\!\! 0}} \! \! (s)
\,\equiv\, 2\,\frac{M_\pi^4}{F_\pi^4}\,
\sqrt{\frac{s}{s-4M_\pi^2}}\,
\sum_{n=0}^3\left[\frac{1}{6}\,\xi^{(n)}_2(s)\,+\,\frac{1}{3}\,\xi^{(n)}_0(s)\right]
k_n(s) ,
\nonumber\\
&&\psi^{\mbox{\tiny{$+-$}}}_1(s)\rightarrow
\ \stackrel{{ }_{\mbox{\scriptsize{$o$}}}}{\psi} \stackrel{{\mbox{\tiny{$+-$}}}}{_{\!\!\!\! 1}} \! \! (s)
\,\equiv\, 2\,\frac{M_\pi^4}{F_\pi^4}\,
\sqrt{\frac{s}{s-4M_\pi^2}}\,
\sum_{n=0}^4 \frac{1}{2}\,\xi^{(n)}_1(s) k_n(s) ,
\label{lim_psi_+-}\end{aligned}$$ in full agreement with the results of [@KMSF95].
Turning eventually towards the inelastic scattering $\pi^+ \pi^- \rightarrow \pi^0 \pi^0$, i.e. $\psi^{x}_0(s)$, the range of integration corresponding to $-1\le z\equiv\cos\theta\le +1$ is $t_{\mbox{\tiny{$-$}}}(s) \le t \le t_{\mbox{\tiny{$+$}}}(s)$, with $$t_{\mbox{\tiny{$\pm$}}}(s) \,=\, -\frac{1}{2} (s - 2 M_{\pi}^2 - 2 M_{\pi^0}^2)
\,\pm\,
\frac{1}{2}\,\sqrt{(s- 4 M_{\pi}^2)(s - 4 M_{\pi^0}^2)} .$$ For $s\ge 4 M_{\pi}^2$, one has $t\le 0$ and $u\le 0$. In terms of an integration over $t$, one thus obtains $$\begin{aligned}
\psi^{x}_0(s)
&=&-\,\frac{\lambda_{x}^{(1)}}{16\pi F_{\pi}^4}(s-2M_{\pi^0}^2)(s-2M_{\pi}^2)
\,-\,
\frac{\lambda_{x}^{(2)}}{24\pi F_{\pi}^4}\left[
s^2 - s (M_{\pi}^2 + M_{\pi^0}^2) + 4 M_{\pi}^2 M_{\pi^0}^2
\right]
\nonumber\\
&&
+\,
\varphi_0^{x}(s) \varphi_0^{00}(s) \,\,
\frac{1}{2\pi}\,\left[2\,+\,\sigma_0(s)\,L_0(s)\right]
+\,
\varphi_0^{\mbox{\tiny{$+-$}}}(s) \varphi_0^{x}(s)\, \,
\frac{1}{\pi}\,\left[2\,+\,\sigma (s)\,L (s)\right]
\nonumber\\
&&
-\,
\frac{1}{8\pi F_\pi^4}\,\frac{1}{\sqrt{(s- 4 M_{\pi}^2)(s - 4 M_{\pi^0}^2)}}\,
\int_{t_-(s)}^{t_+(s)} dt\,\left[
\frac{\beta_{x}}{2}\,\left(t - \frac{2}{3}M_{\pi}^2 - \frac{2}{3}M_{\pi^0}^2\right)\,-\,
\frac{\alpha_{x} M_{\pi^0}^2}{3}
\right]^2 {\bar J}_{\mbox{\tiny{$\! + $}} 0}(t)
\nonumber\\
&&
-\,
\frac{1}{24\pi F_{\pi}^4}\,
\frac{\beta_{x}(M_{\pi}^2 - M_{\pi^0}^2)^2}{\sqrt{(s- 4 M_{\pi}^2)(s - 4 M_{\pi^0}^2)}}\,
\int_{t_-(s)}^{t_+(s)} \frac{dt}{t}\,
\left[
\frac{\beta_{x}}{4}\,\left(7t - 6 M_{\pi}^2 - 6 M_{\pi^0}^2\right)\,-\,
{\alpha_{x} M_{\pi^0}^2}
\right] {\bar J}_{\mbox{\tiny{$\! + $}} 0}(t)
\nonumber\\
&&
-\,
\frac{1}{24\pi F_{\pi}^4}\,
\frac{\beta_{x}^2(M_{\pi}^2 - M_{\pi^0}^2)^4}{\sqrt{(s- 4 M_{\pi}^2)(s - 4 M_{\pi^0}^2)}}\,
\int_{t_-(s)}^{t_+(s)} \frac{dt}{t^2}\,
\ {\bar{\!\!{\bar J}}}_{\mbox{\tiny{$\! + $}} 0}(t)
\nonumber\\
&&
-\,
\frac{1}{96\pi F_{\pi}^4}\,
\frac{\beta_{x}^2}{\sqrt{(s- 4 M_{\pi}^2)(s - 4 M_{\pi^0}^2)}}\,
\int_{t_-(s)}^{t_+(s)} dt (2s + t - 2M_{\pi}^2 - 2M_{\pi^0}^2)
\nonumber\\
&&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\times
\left[
t - 2 M_{\pi}^2 - 2 M_{\pi^0}^2 \,+\,\frac{(M_{\pi}^2 - M_{\pi^0}^2)^2}{t}\,
\right] {\bar J}_{\mbox{\tiny{$\! + $}} 0}(t)
.
\label{psi_x_proj}\end{aligned}$$ With the help of the formulae displayed in appendix \[app:integJbar\], one finds $$\begin{aligned}
16\pi \psi^{x}_0(s) &=& 2\,\frac{M_{\pi}^4}{F_\pi^4}\,
\sqrt{\frac{s}{s - 4M_{\pi}^2}}\,\Bigg\{
\xi_{x}^{(0)}(s) \sigma (s) \,+\, 2\xi^{(1)}_{x} (s) \,\frac{1}{\sqrt{s(s - 4M_{\pi^0}^2)}}
\left[
\lambda^{1/2}(t_{\mbox{\tiny{$-$}}}(s)){\cal L}_{\mbox{\tiny{$-$}}} (s)
-
\lambda^{1/2}(t_{\mbox{\tiny{$+$}}}(s)){\cal L}_{\mbox{\tiny{$+$}}} (s)
\right]
\nonumber\\
&&
+\, 2\xi^{(2;{\mbox{\tiny{$\pm$}}})}_{x}(s) \left(1 - \frac{4 M_{\pi}^2}{s}\right) L(s)
+\, 2\xi^{(2;0)}_{x}(s) \sigma (s) \sigma_0(s) L_0(s)
\nonumber\\
&&
+\,
3 \xi^{(3)}_{x}(s)\,\frac{M_{\pi}^2}{\sqrt{s(s - 4M_{\pi^0}^2)}}
\left[
{\cal L}^2_{\mbox{\tiny{$-$}}} (s) - {\cal L}^2_{\mbox{\tiny{$+$}}} (s)
\right]
\Bigg\}
\,+\, 16\pi \Delta_1 \psi^{x}_0(s) \,+\, 16\pi \Delta_2 \psi^{x}_0(s) .
\label{psi_x_0} \end{aligned}$$
Here $\Delta_1 \psi^{x}_0(s)$ and $\Delta_2 \psi^{x}_0(s)$ represent two contributions that behave as ${\cal O}(M_{\pi}^2 - M_{\pi^0}^2)$ and as ${\cal O}\left( (M_{\pi}^2 - M_{\pi^0}^2)^2\right)$, respectively, as one approaches the isospin limit. Their expressions, together with those of the remaining polynomials $\xi_{x}^{(0)}(s)$, etc. are given in appendix \[app:polynomials\]. This expression of $\psi^{x}_0(s)$ involves the function ${\cal L}_{\mbox{\tiny{$\pm $}}}(s)$, defined as $${\cal L}_{\mbox{\tiny{$\pm $}}} (s) \,=\, \ln\left[ \chi(t_{\mbox{\tiny{$\pm $}}}(s)) \right] ,$$ with $$\chi(t) \,=\,
\frac{\sqrt{(M_{\pi} + M_{\pi^0})^2 - t} - \sqrt{(M_{\pi} - M_{\pi^0})^2 - t}}{\sqrt{(M_{\pi} + M_{\pi^0})^2 - t} + \sqrt{(M_{\pi} - M_{\pi^0})^2 - t}}\, .
\label{chi_def}$$ Let us point out that the expression (\[psi\_x\_0\]) for $\psi^{x}_0(s)$ holds in the range $s\ge 4 M_{\pi}^2$, where the functions $t_{\mbox{\tiny{$\pm $}}}(s)$ are real. An analytical continuation is necessary in order to describe $\psi^{x}_0(s)$ in, say, the range $ 4 M_{\pi^0}^2 \le s \le 4 M_{\pi}^2$, as required, for instance, for ${\mbox{Im}}F_S^{\pi}(s)$, cf. eq. (\[ImF\_NLO\]). For the applications that will be discussed in the following sections, we need not deal with this aspect, and the expression (\[psi\_x\_0\]) is sufficient. It is also useful to notice that in the isospin limit $t_{\mbox{\tiny{$+$}}} (s)$, $\lambda^{1/2}(t_{\mbox{\tiny{$+$}}}(s)){\cal L}_{\mbox{\tiny{$+$}}} (s)$, and ${\cal L}_{\mbox{\tiny{$\pm $}}}^2 (s)$ all behave as ${\cal O}\left( (M_{\pi}^2 - M_{\pi^0}^2)^2\right)$. We keep however the contributions involving ${\cal L}_{\mbox{\tiny{$+$}}} (s)$ as indicated in equation (\[psi\_x\_0\]), so that each of the three pieces, when taken separately, displays a regular behaviour as $s$ approaches $4 M_\pi^2$ (from above). Finally, in the limit where the value of the mass $M_{\pi^0}$ tends to $M_\pi$, one recovers the result of [@KMSF95], $$\begin{aligned}
&&\psi^{x}_0(s)\rightarrow \ \stackrel{{ }_{\mbox{\scriptsize{$o$}}}}{\psi} \stackrel{x}{_{0}} \! \! (s)
\,\equiv\, 2\,\frac{M_\pi^4}{F_\pi^4}\,
\sqrt{\frac{s}{s-4M_\pi^2}}\,
\sum_{n=0}^3\left[\frac{1}{3}\,\xi^{(n)}_2(s)\,-\,\frac{1}{3}\,\xi^{(n)}_0(s)\right]
k_n(s) .
\label{lim_psi_x}\end{aligned}$$
Two-loop representation of the form factors and scattering amplitudes
---------------------------------------------------------------------
Having obtained the partial wave projections in the $S$ and $P$ waves from the relevant one-loop $\pi\pi$ scattering amplitudes, we may now proceed towards obtaining the two-loop expressions of the form factors and scattering amplitudes. This requires one to evaluate the dispersive integrals in terms of which they are expressed. As mentioned previously, we will not be able to work out analytical expressions for all the integrals involved. Closed expressions will be obtained only for the contributions corresponding to so-called factorizing two-loop diagrams, see fig. \[fig3\]. They will involve the functions ${\bar K}_n (s)$, defined as [@KMSF95] $${\bar K}_n(s) \,=\, \frac{s}{\pi} \int_{4 M_\pi^2}^\infty \frac{dx}{x}\,\frac{k_n(s)}{x - s - i0} ,$$ with the functions $k_n(s)$ given in eq. (\[k\_n\]), and the understanding that $K_0(s)$ remains denoted by ${\bar J}(s)$. Explicit expressions of the functions ${\bar K}_n(s)$ in terms of ${\bar J}(s)$ can be found in ref. [@KMSF95]. There will also appear functions ${\bar K}_n^0(s)$, which are defined in the same way in terms of functions $k_n^0(s)$, identical to the functions (\[k\_n\]), but with the charged pion mass $M_\pi$ replaced by $M_{\pi^0}$. Similarly, we will keep the notation ${\bar J}_0(s)$ for ${\bar K}^0_0(s)$.
Starting with the form factors, we obtain the two-loop representations $$\begin{aligned}
F_S^{\pi^0}(s) &=& F_S^{\pi^0}\! (0)\! \left(
1 + a_S^{\pi^0} \! s + b_S^{\pi^0} \! s^2
\right)
\nonumber\\
&&
+\, 8\pi F_S^{\pi^0}\! (0) \varphi_0^{00}(s)
\left[
1 + a_S^{\pi^0} \! s + \frac{1}{\pi} \varphi_0^{00}(s) - \frac{2}{\pi} \frac{F_S^{\pi}(0)}{F_S^{\pi^0}(0)}\, \varphi_0^{x}(s)
\right] \! {\bar J}_0 (s)
\nonumber\\
&&
-\, 16\pi {F_S^{\pi}(0)}\, \varphi_0^{x}(s)
\left[
1 + a_S^{\pi}\,s + \frac{2}{\pi} \varphi_0^{\mbox{\tiny{$ +-$}}}(s)
- \frac{1}{\pi} \frac{F_S^{\pi^0}\! (0)}{F_S^{\pi}(0)}\, \varphi_0^{x}(s)
\right] \! {\bar J} (s)
\nonumber\\
&&
+\, \frac{M_{\pi}^4}{F_{\pi}^4}\,F_S^{\pi^0}\! (0)\! \left [
\xi^{(0)}_{00}(s) {\bar J}_0 (s) +
\xi^{(1;0)}_{00} (s) {\bar K}_1^0(s) +
2 \xi^{(2;0)}_{00} (s) {\bar K}_2^0(s) +
\xi^{(3;0)}_{00} (s) {\bar K}_3^0(s)
\right]
\nonumber\\
&&
-\, \,\frac{M_{\pi}^4}{F_{\pi}^4}\,F_S^{\pi}(0) \left[
2 \xi^{(0)}_{x} (s) {\bar J} (s) + 4 \xi^{(2;{\mbox{\tiny{$\!\pm $}}})}_{x} (s) {\bar K}_2(s)
\right]
\nonumber\\
&&
+\, 2 \,\frac{M_{\pi}^4}{F_{\pi}^4}
\left[ F_S^{\pi^0}\! (0)\xi^{(2;{\mbox{\tiny{$\!\pm $}}})}_{00} (s) - 2 F_S^{\pi}(0) \xi^{(2;0)}_{x} (s) \right]
\left[ 16 \pi^2 {\bar J} (s) {\bar J}_0 (s) - 2 {\bar J} (s) - 2 {\bar J}_0 (s) \right]
\nonumber\\
&&
+\, \Delta_{\mbox{\tiny NF}} F_S^{\pi^0}(s)
\,+\, {\cal O}(E^8) ,\end{aligned}$$ and $$\begin{aligned}
F_S^{\pi}(s) &=& F_S^{\pi} (0)\! \left(
1 + a_S^{\pi} s + b_S^{\pi} s^2
\right)
\nonumber\\
&&
+\, 16\pi F_S^{\pi} (0) \varphi_0^{\mbox{\tiny{$ +-$}}}(s)
\left[
1 + a_S^{\pi} s + \frac{2}{\pi} \varphi_0^{\mbox{\tiny{$ +-$}}}(s)
- \frac{1}{\pi} \frac{F_S^{\pi^0}\!(0)}{F_S^{\pi}(0)}\,\varphi_0^{x}(s)
\right] \! {\bar J} (s)
\nonumber\\
&&
-\, 8\pi {F_S^{\pi^0}\! (0)}\, \varphi_0^{x}(s)
\left[
1 + a_S^{\pi^0}\! s + \frac{1}{\pi} \varphi_0^{00}(s)
- \frac{2}{\pi} \frac{F_S^{\pi} (0)}{F_S^{\pi^0}\! (0)}\, \varphi_0^{x}(s)
\right] \! {\bar J}_0 (s)
\nonumber\\
&&
+\, 2\,\frac{M_{\pi}^4}{F_{\pi}^4}\,F_S^{\pi} (0) \left [
\xi_{{\mbox{\tiny{$+-$}}};S}^{(0)}(s) {\bar J} (s) +
\xi_{{\mbox{\tiny{$+-$}}};S}^{(1;{\mbox{\tiny{$\!\pm $}}})}(s) {\bar K}_1(s) +
2 \xi_{{\mbox{\tiny{$+-$}}};S}^{(2;{\mbox{\tiny{$\!\pm $}}})}(s) {\bar K}_2(s) +
\xi_{{\mbox{\tiny{$+-$}}};S}^{(3;{\mbox{\tiny{$\!\pm $}}})}(s) {\bar K}_3(s)
\right]
\nonumber\\
&&
-\, \frac{M_{\pi}^4}{F_{\pi}^4}\,F_S^{\pi^0}\! (0) \left[
\xi^{(0)}_{x} (s) {\bar J}_0 (s) + 2\xi^{(2;0)}_{x} (s) {\bar K}_2^0(s)
\right]
\nonumber\\
&&
+\, 2 \,\frac{M_{\pi}^4}{F_{\pi}^4}
\left[2 F_S^{\pi} (0) \xi_{{\mbox{\tiny{$+-$}}};S}^{(2;0)} (s) - F_S^{\pi^0}\! (0) \xi^{(2;{\mbox{\tiny{$\!\pm $}}})}_{x} (s) \right]
\left[ 16 \pi^2 {\bar J} (s) {\bar J}_0 (s) - 2 {\bar J} (s) - 2 {\bar J}_0 (s) \right]
\nonumber\\
&&
+\, \Delta_{\mbox{\tiny NF}} F_S^{\pi}(s)
\,+\, {\cal O}(E^8) , $$ for the scalar form factors, whereas the vector form factor reads $$\begin{aligned}
F_V^{\pi}(s) &=&
1 + a_V^{\pi} \! s + b_V^{\pi} \! s^2
\nonumber\\
&&
+\, 16 \pi \varphi_1^{\mbox{\tiny{$+-$}}} (s)
\left[
1 + a_V^{\pi} \! s + \frac{2}{\pi} \varphi_1^{\mbox{\tiny{$+-$}}}(s)
\right] \! {\bar J} (s)
\nonumber\\
&&
+\, 2 \frac{M_{\pi}^4}{F_{\pi}^4} \left [
\xi^{(0)}_{{\mbox{\tiny{$+-$}}};P} (s) {\bar J} (s) +
\xi^{(1;{\mbox{\tiny{$\!\pm $}}})}_{{\mbox{\tiny{$+-$}}};P} (s) {\bar K}_1(s) +
2 \xi^{(2)}_{{\mbox{\tiny{$+-$}}};P} (s) {\bar K}_2(s) +
\xi^{(3;{\mbox{\tiny{$\!\pm $}}})}_{{\mbox{\tiny{$+-$}}};P} (s) {\bar K}_3(s) +
\xi^{(4;{\mbox{\tiny{$\!\pm $}}})}_{{\mbox{\tiny{$+-$}}};P} (s) {\bar K}_4(s)
\right]
\nonumber\\
&&
+\, \Delta_{\mbox{\tiny NF}} F_V^{\pi}(s)
\,+\, {\cal O}(E^6) . \end{aligned}$$ The contributions $\Delta_{\mbox{\tiny NF}} F_S^{\pi^0}(s)$, $\Delta_{\mbox{\tiny NF}} F_S^{\pi}(s)$, and $\Delta_{\mbox{\tiny NF}} F_V^{\pi}(s)$ stemming from non-factorizing two-loop graphs, see fig. \[fig3\], are expressed as dispersive integrals, $$\begin{aligned}
\Delta_{\mbox{\tiny NF}} F_S^{\pi^0}(s) &=&
\frac{M_{\pi}^4}{F_{\pi}^4}\,F_S^{\pi^0}\! (0)
\frac{s}{\pi} \int_{4 M_{\pi^0}^2}^{\infty}
\frac{dx}{x}\,\frac{1}{x - s - i0} \!
\left[
\xi^{(1;{\mbox{\tiny$\nabla$}})}_{00}(s)\,\frac{1}{8 \pi}\,\frac{\sigma_{0}(x)}{\sigma_{\mbox{\tiny$\nabla$}}(x)}
\, L_{\mbox{\tiny$\nabla$}}(x) \,+\,
\xi^{(3;{\mbox{\tiny$\nabla$}})}_{00}(s)\,\frac{3}{16 \pi}\,\frac{M_{\pi}^2}{x\sigma_{0}(x)}
\, L_{\mbox{\tiny$\nabla$}}^2(x)
\right]
\nonumber\\
&&\!\!\!\!\!
-\,2 \frac{M_{\pi}^4}{F_{\pi}^4}\,F_S^{\pi} (0)
\frac{s}{\pi} \int_{4 M_{\pi}^2}^{\infty}
\frac{dx}{x}\,\frac{1}{x - s - i0}
\left[
\xi^{(1)}_{x} (s) \,\frac{1}{8 \pi}\,\frac{\lambda^{1/2}(t_{\mbox{\tiny $-$}}(x))}{x \sigma_0 (x)}\, {\cal L} (x) \,+\,
\xi^{(3)}_{x}(s)\,\frac{3}{16 \pi}\,\frac{M_{\pi}^2}{x \sigma_0 (x)}
\,{\cal L}^2(x)
\right]
\nonumber\\
&&\!\!\!\!\!
-\, F_S^{\pi} (0)
\frac{s}{\pi} \int_{4 M_{\pi}^2}^{\infty}
\frac{dx}{x}\,\frac{1}{x - s - i0}\,\sigma(x)
\left[
\Delta_1 \psi^{x}_0(x) \,+\, \Delta_2 \psi^{x}_0(x)
\right] $$ $$\begin{aligned}
\Delta_{\mbox{\tiny NF}} F_S^{\pi}(s) &=&
2 \frac{M_{\pi}^4}{F_{\pi}^4}\,F_S^{\pi} (0)
\frac{s}{\pi} \int_{4 M_{\pi}^2}^{\infty}
\frac{dx}{x}\,\frac{1}{x - s - i0}
\left[
\xi^{(1;{\mbox{\tiny$\Delta$}})}_{{\mbox{\tiny{$+-$}}}; S}(s) \,\frac{1}{8 \pi}\,\frac{\sigma (x)}{\sigma_{\mbox{\tiny$\Delta$}}(x)}
\, L_{\mbox{\tiny$\Delta$}}(x)
\,+\,
\xi^{(3;{\mbox{\tiny$\Delta$}})}_{{\mbox{\tiny{$+-$}}};S}(s) \,\frac{3}{16 \pi}\,
\frac{M_{\pi^0}^2}{x\sigma(x)}\,L_{\mbox{\tiny$\Delta$}}^2(x)
\right]
\nonumber\\
&&\!\!\!\!\!
-\, \frac{M_{\pi}^4}{F_{\pi}^4}\,F_S^{\pi^0}\! (0)
\frac{s}{\pi} \int_{4 M_{\pi^0}^2}^{\infty}
\frac{dx}{x}\,\frac{1}{x - s - i0}
\left[
\xi^{(1)}_{x} (s) \,\frac{1}{8 \pi}\,\frac{\lambda^{1/2}(t_{\mbox{\tiny $-$}}(x))}{x \sigma (x)}\, {\cal L} (x) \,+\,
\xi^{(3)}_{x}(s)\,\frac{3}{16 \pi}\,\frac{M_{\pi}^2}{x \sigma (x)}
\,{\cal L}^2(x)
\right]
\nonumber\\
&&\!\!\!\!\!
-\,\frac{1}{2}\, F_S^{\pi^0} (0)
\frac{s}{\pi} \int_{4 M_{\pi^0}^2}^{\infty}
\frac{dx}{x}\,\frac{1}{x - s - i0}\,\sigma_0(x)
\left[
\Delta_1 \psi^{x}_0(x) \,+\, \Delta_2 \psi^{x}_0(x)
\right] \end{aligned}$$ $$\begin{aligned}
\Delta_{\mbox{\tiny NF}} F_V^{\pi}(s) &=&
2 \frac{M_{\pi}^4}{F_{\pi}^4}
\frac{s}{\pi} \int_{4 M_{\pi}^2}^{\infty}
\frac{dx}{x}\,\frac{1}{x - s - i0}
\left\{
\xi^{(1;{\mbox{\tiny$\Delta$}})}_{{\mbox{\tiny{$+-$}}}; P}(s)
\,\frac{1}{8 \pi}\,\frac{\sigma (x)}{\sigma_{\mbox{\tiny$\Delta$}}(x)}
\, L_{\mbox{\tiny$\Delta$}}(x) \,+\,
\xi^{(3;{\mbox{\tiny$\Delta$}})}_{{\mbox{\tiny{$+-$}}};P}(s) \,\frac{3}{16 \pi}\,\frac{M_{\pi^0}^2}{x\sigma (x)}
\,L_{\mbox{\tiny$\Delta$}}^2(x)
\right.
\nonumber\\
&&
\left.
+\,\xi^{(4;{\mbox{\tiny$\Delta$}})}_{{\mbox{\tiny{$+-$}}};P}(s)\,\frac{1}{16 \pi}\,
\frac{M_{\pi^0}^2}{\sqrt{x(x-4 M_{\pi}^2)}}\,
\left[
1\,+\,\frac{1}{\sigma_{\mbox{\tiny$\Delta$}}(x)}\,L_{\mbox{\tiny$\Delta$}}(x)
\,+\,\frac{M_{\pi^0}^2}{x-4 M_{\pi}^2}\,L_{\mbox{\tiny$\Delta$}}^2(x)
\right]
\right\} .\end{aligned}$$ Their evaluation has to be performed numerically. Notice, however, that these representations are not necessarily best suited for this purpose, due to possible numerical stability problems. These functions are actually often expressed as two-dimensional integrals, which can be computed more efficiently. Examples of such representations can be found in the articles quoted under [@2loops], and we refer the reader to them and to the papers quoted therein for further discussions on these aspects.
We could proceed in a similar way in order to write down two-loop representations for the $\pi\pi$ amplitudes, but this is not very useful for the applications of interest here. For the sake of illustration, let us consider the function $W_0^{00}(s)$, involved in the amplitude for elastic scattering of two neutral pions, as an example. With the results already at our disposal, we obtain $$\begin{aligned}
W_0^{00}(s) &=&
\frac{1}{2}\left[16\pi \varphi^{00}_0(s) \right] ^2 {\bar J}_0 (s)
\,+\,
\left[16\pi \varphi^{x}_0(s) \right] ^2\,{\bar J} (s)
\nonumber\\
&&
+ 32 \pi \,\frac{M_{\pi}^4}{F_{\pi}^4}\,\varphi_0^{00}(s)\left [
\xi^{(0)}_{00}(s) {\bar J}_0 (s) +
\xi^{(1;0)}_{00} (s) {\bar K}_1^0(s) +
\xi^{(2;0)}_{00} (s) {\bar K}_2^0(s) +
\xi^{(3;0)}_{00} (s) {\bar K}_3^0(s)
\right]
\nonumber\\
&&
+ 64 \pi \frac{M_{\pi}^4}{F_{\pi}^4}\,\varphi^{x}_0(s) \left[
\xi^{(0)}_{x} (s) {\bar J} (s) + \xi^{(2;{\mbox{\tiny{$\!\pm $}}})}_{x} (s) {\bar K}_2(s)
\right]
\nonumber\\
&&
+ 64 \pi \,\frac{M_{\pi}^4}{F_{\pi}^4}
\varphi_0^{00}(s) \xi^{(2;{\mbox{\tiny{$\!\pm $}}})}_{00} (s)
\left[ 16 \pi^2 {\bar J} (s) {\bar J}_0 (s) - 2 {\bar J} (s) - 2 {\bar J}_0 (s) \right]
\nonumber\\
&&
+\,
\Delta_{\mbox{\tiny NF}} W_0^{00}(s) ,\end{aligned}$$ with $$\begin{aligned}
\Delta_{\mbox{\tiny NF}} W_0^{00}(s) &=&
32 \pi \,\frac{M_{\pi}^4}{F_{\pi}^4}\,\varphi_0^{00}(s)
\frac{s}{\pi} \int_{4 M_{\pi^0}^2}^{\infty}\!\!
\frac{dx}{x}\,\frac{1}{x - s - i0}
\left[
\xi^{(1;{\mbox{\tiny$\nabla$}})}_{00}(s)\,\frac{1}{8 \pi}\,\frac{\sigma_{0}(x)}{\sigma_{\mbox{\tiny$\nabla$}}(x)}
\, L_{\mbox{\tiny$\nabla$}}(x) +
\xi^{(3;{\mbox{\tiny$\nabla$}})}_{00}(s)\,\frac{3}{16 \pi}\,\frac{M_{\pi}^2}{x\sigma_{0}(x)}
\, L_{\mbox{\tiny$\nabla$}}^2(x)
\right]
\nonumber\\
&&\!\!\!\!\!
+\, 64 \pi \frac{M_{\pi}^4}{F_{\pi}^4}\,\varphi^{x}_0(s)
\frac{s}{\pi} \int_{4 M_{\pi}^2}^{\infty}
\frac{dx}{x}\,\frac{1}{x - s - i0}
\left[
\xi^{(1)}_{x} (s) \,\frac{1}{8 \pi}\,\frac{\lambda^{1/2}(t_{\mbox{\tiny $-$}}(x))}{x \sigma_0 (x)}\, {\cal L} (x)
\,+\,
\xi^{(3)}_{x}(s)\,\frac{3}{16 \pi}\,\frac{M_{\pi}^2}{x \sigma_0 (x)}
\,{\cal L}^2(x)
\right]
\nonumber\\
&&\!\!\!\!\!
+\, 32 \pi \frac{M_{\pi}^4}{F_{\pi}^4}\,\varphi^{x}_0(s)
\frac{s}{\pi} \int_{4 M_{\pi}^2}^{\infty}
\frac{dx}{x}\,\frac{1}{x - s - i0}\,\sigma (x)
\left[
\Delta_1 \psi^{x}_0(x) \,+\, \Delta_2 \psi^{x}_0(x)
\right] .\end{aligned}$$ The remaining $W$-functions can be handled in a similar way, but we will not pursue the matter further here. As for the corresponding subtraction polynomials, they were already given in eq. (\[polynomials\_P\]).
Isospin breaking in phase shifts {#IB_in_phases}
================================
In this section we turn to the issue of isospin breaking in the phases of the form factors, making use of the results obtained so far. We begin with a discussion of some general aspects of the phases of the form factors in the low-energy regime, and consider the lowest-order isospin-breaking corrections. The corrections at next order are then discussed in a framework where only contributions of first order in the difference of the pion masses are kept.
General discussion and leading-order results {#IB_in_phases_LO}
--------------------------------------------
The phases of the form factors are defined generically as $$F(s+i0) = e^{2i\delta (s)} F(s-i0)
,$$ where the phases will be denoted $\delta_0^{\pi^0}(s)$, $\delta_0^{\pi}(s)$, and $\delta_1^{\pi}(s)$ for the form factors $F_S^{\pi^0}(s)$, $F_S^{\pi}(s)$, and $F_V^{\pi}(s)$, respectively. For a discussion of the analyticity properties of the form factors, we refer the reader to ref. [@Colangelo:2008sm]. Each of these phases $\delta_{\ell}(s)$ has itself a low-energy expansion, $\delta_{\ell}(s) = \delta_{\ell,2}(s) + \delta_{\ell,4}(s)
+ {\cal O}(E^6)$. Our aim is to address the issue of isospin-breaking corrections in the phases order by order in this expansion, i.e. our interest lies in the differences $$\Delta \delta_\ell (s) \equiv \delta_\ell (s) - \stackrel{o}{\delta} _\ell \!\!(s)
= \Delta_{2} \delta_\ell (s) + \Delta_{4} \delta_\ell (s) + {\cal O}(E^6)$$ between the phases $\delta_\ell (s)$ in the presence of isospin breaking and the phases $\stackrel{o}{\delta} _\ell \!\!(s)$ in the isospin limit, with $\Delta_n \delta_\ell (s) \equiv \delta_{\ell , n} (s) - \stackrel{o}{\delta} _{\ell ,n} \!\!(s)$. For the cases under consideration, we have $$\begin{aligned}
\delta_{0}^{\pi^0}(s) &=&
\frac{1}{2}\,\sigma_{0}(s)\left[
\varphi_0^{00}(s) \,+\, \psi_0^{00}(s) \right]\theta(s-4M_{\pi^0}^2)
\nonumber\\
&&\!\!\!\!\!
-\,
\sigma (s) \,\frac{F_S^{\pi}(0)}{F_S^{\pi^0}(0)}\left[
\varphi_0^{x}(s)
\frac{1+\Gamma^{\pi}_S(s)}{1+\Gamma^{\pi^0}_S (s)}\,+\,
\psi_0^{x}(s)
\right]\theta(s-4M_{\pi}^2)
\,+\,{\cal O}(E^6)
\nonumber\\
\delta_{0}^{\pi}(s) &=&
\sigma (s)\left[ \varphi_0^{\mbox{\tiny{$+-$}}}(s) \,+\,\psi_0^{\mbox{\tiny{$+-$}}}(s)
\right]\theta(s-4M_{\pi}^2)
\nonumber\\
&&\!\!\!\!\!
-\,\frac{1}{2}\,\sigma_{0}(s)\,\frac{F_S^{\pi^0}(0)}{F_S^{\pi}(0)}
\left[ \varphi_0^{x}(s)
\frac{1+\Gamma^{\pi^0}_S(s)}{1+\Gamma^{\pi}_S (s)}\,+\,
\psi_0^{x}(s)
\right] \theta(s-4M_{\pi^0}^2)
\,+\,{\cal O}(E^6)
\nonumber\\
\delta_{1}^{\pi}(s) &=&
\sigma (s)\left[ \varphi_1^{\mbox{\tiny{$+-$}}} (s) \,+\,\psi_1^{\mbox{\tiny{$+-$}}} (s)
\right]\theta(s-4M_{\pi}^2)
\,+\,{\cal O}(E^6) .\end{aligned}$$ We thus deduce that $$\begin{aligned}
\delta_{0,2}^{\pi^0}(s) &=&
\frac{1}{2}\,\sigma_{0}(s) \varphi_0^{00}(s) \theta(s-4M_{\pi^0}^2)
\, -\,
\sigma (s) \varphi_0^{x}(s) \theta(s-4M_{\pi}^2)
\nonumber\\
\delta_{0,2}^{\pi}(s) &=&
\sigma (s) \varphi_0^{\mbox{\tiny{$+-$}}}(s) \theta(s-4M_{\pi}^2)
\, -\,
\frac{1}{2}\,\sigma_{0}(s) \varphi_0^{x}(s) \theta(s-4M_{\pi^0}^2)
\nonumber\\
\delta_{1,2}^{\pi}(s) &=&
\sigma (s)
\varphi_1^{\mbox{\tiny{$+-$}}}(s) \theta(s-4M_{\pi}^2) ,
\label{phases_LO}\end{aligned}$$ at leading order, while, at next-to-leading order, $$\begin{aligned}
\delta_{0,4}^{\pi^0}(s) &=&
\frac{1}{2}\,\sigma_{0}(s) \psi_0^{00}(s) \theta(s-4M_{\pi^0}^2)
\, -\,\sigma (s) \psi_0^{x}(s) \theta(s-4M_{\pi}^2)
\nonumber\\
&& \,-\,
\sigma (s)\varphi_0^{x}(s)
\left[\left(\frac{F_S^{\pi}(0)}{F_S^{\pi^0}(0)} - 1\right)
+\left(\Gamma_S^{\pi}(s) - \Gamma_S^{\pi^0}(s)\right)
\right] \theta(s-4M_{\pi}^2)
\nonumber\\
\delta_{0,4}^{\pi}(s) &=&
\sigma (s) \psi_0^{\mbox{\tiny{$+-$}}}(s) \theta(s-4M_{\pi}^2)
\, -\,
\frac{1}{2}\,\sigma_{0}(s) \psi_0^{x}(s) \theta(s-4M_{\pi^0}^2)
\nonumber\\
&& \, -\,
\frac{1}{2}\,
\sigma_{0}(s)\varphi_0^{x}(s)
\left[\left(\frac{F_S^{\pi^0}(0)}{F_S^{\pi}(0)} - 1\right)
-\left(\Gamma_S^{\pi}(s) - \Gamma_S^{\pi^0}(s)\right)
\right] \theta(s-4M_{\pi^0}^2)
\nonumber\\
\delta_{1,4}^{\pi}(s) &=&
\sigma (s)
\psi_1^{\mbox{\tiny{$+-$}}}(s) \theta(s-4M_{\pi}^2) .
\label{phases_NLO}\end{aligned}$$
Here we have used the property that the quantities $F_S^{\pi^0}(0)/F_S^{\pi}(0) - 1$, $\Gamma_S^{\pi}(s)$, and $\Gamma_S^{\pi^0}(s)$ are all of order ${\cal O}(E^2)$. At order ${\cal O}(E^n)$, the isospin-symmetric phases $\stackrel{o}{\delta} _{\ell ,n} \!\!(s)$ are obtained from the expressions (\[phases\_LO\]) and (\[phases\_NLO\]) by setting $M_{\pi^0}$ equal to $M_\pi$, and by replacing the lowest order expressions of the $\pi\pi$ $S$ and $P$ waves, $\varphi_\ell (s)$ and $\psi_\ell (s)$, by their counterparts in the isospin limit, $\stackrel{o}{\varphi} _\ell \!(s)$ and $\stackrel{{ }_{\mbox{\scriptsize{$o$}}}}{\psi} _\ell \!(s)$, respectively. In this limit, the differences $F_S^{\pi^0}(0) / F_S^{\pi}(0) - 1$ and $\Gamma_S^{\pi}(s) - \Gamma_S^{\pi^0}(s)$ vanish.
Before proceeding with the actual calculation, let us make a few comments. First, we should point out an important aspect that emerges from the above expressions, and that has already been observed in ref. [@Colangelo:2008sm]. At order ${\cal O}(E^2)$, the phases of the form factors are entirely determined by the $\pi\pi$ scattering data. In the case of the vector form factor where, due to Bose symmetry, the $\pi^+\pi^-$ channel alone contributes, Watson’s theorem is still operative: the phase of $F_V^\pi(s)$ coincides with the phase of the $P$-wave projection of the corresponding scattering amplitude $A^{\mbox{\tiny{$+-$}}}(s,t,u)$. For the scalar form factors, the situation is different, due to the mixing between the two channels that contribute to the unitarity sum in the $S$ wave. Nevertheless, the phase has still a “universal” character, in the sense that its expression involves only the partial waves of the $\pi\pi$ scattering amplitudes, and makes no explicit reference to the form factors themselves. This property, however, rests entirely on the fact that $F_S^{\pi^0}(0) / F_S^{\pi}(0) = 1$ at this order. Hence, this situation does no longer survive at the next order in the case of the scalar form factors. In addition to the universal parts, ${\Delta}_4^U \delta_0^{\pi^0}(s)$ and ${\Delta}_4^U \delta_0^{\pi}(s)$, provided by the ${\cal O}(E^4)$ partial waves of the $\pi\pi$ scattering amplitudes, there now appear contributions ${\Delta}_4^F \delta_0^{\pi^0}(s)$ and ${\Delta}_4^F \delta_0^{\pi^0}(s)$ that depends explicitly on the form factors considered: $$\begin{aligned}
{\Delta}_4 \delta_0^{\pi^0}(s) &=& {\Delta}_4^U \delta_0^{\pi^0}(s) \,+\, {\Delta}_4^F \delta_0^{\pi^0}(s)
\nonumber\\
{\Delta}_4 \delta_0^{\pi}(s) &=& {\Delta}_4^U \delta_0^{\pi}(s) \,+\, {\Delta}_4^F \delta_0^{\pi}(s)
.
\label{Delta_U_F}\end{aligned}$$ The universal parts ${\Delta}_4^U \delta_0^{\pi^0}(s)$ and ${\Delta}_4^U \delta_0^{\pi}(s)$ correspond to the first lines of the expressions of $\delta_{0,4}^{\pi^0}(s)$ and of $\delta_{0,4}^{\pi}(s)$ given in eq. (\[phases\_NLO\]), respectively, from which the isospin-symmetric contributions are subtracted. The second lines in these same expressions correspond to the form-factor dependent contributions ${\Delta}_4^F \delta_0^{\pi^0}(s)$ and ${\Delta}_4^F \delta_0^{\pi}(s)$. Finally, we also note that for the scalar form factors some contributions in the expressions (\[phases\_LO\]) and (\[phases\_NLO\]) start at $s=M_{\pi^0}^2$, while others appear only for $s\ge M_\pi^2$. This is of course the manifestation of the unitarity cusp in the phases themselves.
At order ${\cal O}(E^2)$, the isospin-symmetric phases $\stackrel{o}{\delta} _{\ell ,2} \!\!(s)$ are obtained from the expressions (\[phases\_LO\]), putting $M_{\pi^0}$ equal to $M_\pi$, and replacing the lowest order expressions of the $\pi\pi$ $S$ and $P$ waves $\varphi_\ell (s)$ by their counterparts in the isospin limit, $\stackrel{o}{\varphi} _\ell \!(s)$, which read $$\begin{aligned}
\stackrel{o}{\varphi} \stackrel{00}{_{\!\! 0}} \!\! (s)
&=& \frac{\alpha M_{\pi}^2}{16\pi F_{\pi}^2}
\nonumber\\
\stackrel{o}{\varphi} \stackrel{x}{_{0}} \! \! (s)
&=&
-\frac{\beta}{16\pi F_{\pi}^2}\,
\left( s - \frac{4}{3}M_{\pi}^2\right)\,-\,
\frac{\alpha M_{\pi}^2}{48\pi F_{\pi}^2}
\nonumber\\
\stackrel{o}{\varphi} \stackrel{{\mbox{\tiny{$+-$}}}}{_{\!\!\!\! 0}} \! \! (s)
&=& \frac{\beta}{32\pi F_{\pi}^2}\,
\left( s - \frac{4}{3}M_{\pi}^2 \right)\,+\,
\frac{\alpha M_{\pi}^2}{24\pi F_{\pi}^2}
\nonumber\\
\stackrel{o}{\varphi} \stackrel{{\mbox{\tiny{$+-$}}}}{_{\!\!\!\! 1}} \! \! (s)
&=& \frac{\beta}{96\pi F_{\pi}^2}\,
\left( s - {4}M_{\pi}^2 \right) .\end{aligned}$$ We then obtain $$\begin{aligned}
\Delta_2 \delta^{\pi^0}_0 (s) &=&
\frac{\alpha_{00} M_{\pi^0}^2}{32 \pi F_\pi^2}
\left[ \sigma_0 (s) \theta (s-4 M_{\pi^0}^2) - \sigma (s) \theta (s-4 M_{\pi}^2) \right]
+\,
\bigg[
\frac{4 \beta_x - 2 \alpha_x - 3 \alpha_{00}}{96 \pi} \frac{\Delta_\pi}{F_\pi^2}
\nonumber\\
&&
+\,
\frac{\beta_{x} - \beta}{16 \pi F_\pi^2} \left( s - \frac{4}{3} M_\pi^2 \right) \,+\,
\frac{\alpha_{x} - \alpha}{48 \pi} \frac{M_{\pi}^2}{F_\pi^2} \,+\,
\frac{\alpha_{00} - \alpha}{32 \pi} \frac{M_{\pi}^2}{F_\pi^2}
\bigg]
\sigma (s) \theta (s-4 M_{\pi}^2)
\nonumber\\
\Delta_2 \delta^{\pi}_0 (s) &=&
\frac{1}{32\pi F_{\pi}^2}
\left[
\beta_x \left( s - \frac{2}{3}M_{\pi^0}^2 - \frac{2}{3}M_{\pi}^2 \right) +\,
\frac{\alpha_x M_{\pi^0}^2}{3}
\right]
\left[ \sigma_0 (s) \theta (s-4 M_{\pi^0}^2) - \sigma (s) \theta (s-4 M_{\pi}^2) \right]
\nonumber\\
&&
+\,
\bigg[
\frac{2 \beta_x - \alpha_x - 4 \alpha_{\mbox{\tiny{$+-$}}}}{96 \pi} \,\frac{\Delta_\pi}{F_\pi^2}
\,+\,
\frac{(\beta_{x} - \beta) + (\beta_{\mbox{\tiny{$+-$}}} - \beta )}{32 \pi F_\pi^2} \left( s - \frac{4}{3} M_\pi^2 \right)
\nonumber\\
&&
+\,
\frac{\alpha_{x} - \alpha}{96 \pi} \frac{M_{\pi}^2}{F_\pi^2} \,+\,
\frac{\alpha_{\mbox{\tiny{$+-$}}} - \alpha}{24 \pi} \frac{M_{\pi}^2}{F_\pi^2}
\bigg]
\sigma (s) \theta (s-4 M_{\pi}^2)
\nonumber\\
\Delta_2 \delta_1^{\pi}(s) &=&
\frac{1}{96\pi F_{\pi}^2}\,
(\beta_{\mbox{\tiny{$+-$}}} - \beta)\left(s - 4 M_{\pi}^2\right)
\sigma (s) .
\label{Delta_2}\end{aligned}$$ If we make use of eq. (\[alphabetaLO\]), these expressions become $$\begin{aligned}
\Delta_2^{\mbox{\scriptsize{LO}}} \delta^{\pi^0}_0 (s) &=&
\frac{\alpha M_{\pi^0}^2}{32 \pi F_\pi^2}
\left[ \sigma_0 (s) \theta (s-4 M_{\pi^0}^2) - \sigma (s) \theta (s-4 M_{\pi}^2) \right]
+\,
\frac{8 \beta - 5\alpha}{96 \pi} \, \frac{\Delta_\pi}{F_\pi^2}\, \sigma (s) \theta (s-4 M_{\pi}^2)
\nonumber\\
\Delta_2^{\mbox{\scriptsize{LO}}} \delta^{\pi}_0 (s) &=&
\frac{1}{32\pi F_{\pi}^2}
\left[
\beta \left( s - \frac{4}{3}M_{\pi^0}^2 \right) +\,
\frac{\alpha M_{\pi^0}^2}{3}
\right]
\left[ \sigma_0 (s) \theta (s-4 M_{\pi^0}^2) - \sigma (s) \theta (s-4 M_{\pi}^2) \right]
\nonumber\\
&&
+\,
\frac{5(4 \beta - \alpha)}{96 \pi} \, \frac{\Delta_\pi}{F_\pi^2}\, \sigma (s) \theta (s-4 M_{\pi}^2) ,
\label{Delta_2_LO}\end{aligned}$$ and $\Delta_2^{\mbox{\scriptsize{LO}}} \delta^{\pi}_1 (s) = 0$. Here we have introduced the notation $$\Delta_\pi \,\equiv\, M_{\pi}^2 - M_{\pi^0}^2 \,.$$ Furthermore, $\Delta_2^{\mbox{\scriptsize{LO}}} \delta_\ell (s)$ represents the leading-order isospin-breaking correction when the lowest-order relations (\[alphabetaLO\]) for the subthreshold parameters $\alpha_{00}$, $\alpha_x\ldots$ have been used. However, the latter also receive higher-order corrections, that will be discussed below. This means that at next-to-leading order we also need to include a contribution $$\Delta_2^{\mbox{\scriptsize{NLO}}} \delta_\ell (s) \,=\,
\Delta_2 \delta_\ell (s) \,-\,
\Delta_2^{\mbox{\scriptsize{LO}}} \delta_\ell (s)
\label{Delta_2_NLO}$$ which takes them into account. We show, on fig. \[fig5\], a plot of $\Delta_2^{\mbox{\scriptsize{LO}}} \delta^{\pi}_0 (s)$ as a function of energy \[the numerical input values we use are given in table \[tab:LECs\] and in equation (\[inputs\])\]. The curve exhibits the characteristic cusp-type behaviour [@cusp61; @Cabibbo:2004gq] at the $\sqrt{s} = 2 M_\pi$ threshold, where $\Delta_2^{\mbox{\scriptsize{LO}}} \delta^{\pi}_0 (s)$ takes its maximal value, close to 20 milliradians. For $\sqrt{s}$ greater than $\sim 300$ MeV, the value of $\Delta_2^{\mbox{\scriptsize{LO}}} \delta^{\pi}_0 (s)$ stays practically constant, around 10 milliradians. The higher-order contributions mentioned in eq. (\[Delta\_2\_NLO\]) become sizeable only above the cusp. The curve for $\Delta_2^{\mbox{\scriptsize{LO}}} \delta^{\pi^0}_0 (s)$ follows a similar shape, see figure \[fig6\], but its magnitude is reduced by roughly a factor of three as compared to $\Delta_2^{\mbox{\scriptsize{LO}}} \delta^{\pi}_0 (s)$. In both cases, the difference $\Delta_2^{\mbox{\scriptsize{NLO}}} \delta_\ell (s)$ defined in equation (\[Delta\_2\_NLO\]) above is negligible in the cusp region, but becomes more and more important as the energy increases.
It is also interesting to investigate the sensitivity of the isospin-breaking corrections with respect to the (unknown) parameters $\alpha$ and $\beta$. This is done in figures \[fig6\] and \[fig7\]. We notice that the dependence is strongest in the vicinity of the cusp, and is mainly driven by $\alpha$. Indeed, at $s=4 M_\pi^2$, one has $$\begin{aligned}
\Delta_2^{\mbox{\scriptsize{LO}}} \delta^{\pi^0}_0 (4 M_\pi^2) &=&
\frac{\alpha M_{\pi^0}^2}{32 \pi F_\pi^2} \, \sigma_0 (4 M_\pi^2)
\nonumber\\
\Delta_2^{\mbox{\scriptsize{LO}}} \delta^{\pi}_0 (4 M_\pi^2) &=&
\frac{1}{32\pi F_{\pi}^2}
\left[
\beta \left( 4 M_\pi^2 - \frac{4}{3}M_{\pi^0}^2 \right) +\,
\frac{\alpha M_{\pi^0}^2}{3} \right] \sigma_0 (4 M_\pi^2) \, .
\label{Delta_2_LO_cusp}\end{aligned}$$ The value at the cusp of $\Delta_2^{\mbox{\scriptsize{LO}}} \delta^{\pi^0}_0$ is directly proportional to $\alpha$. In the case of $\Delta_2^{\mbox{\scriptsize{LO}}} \delta^{\pi}_0$, the value is driven by the contribution proportional to $\beta$, the contribution proportional to $\alpha$ being suppressed by a factor of three as compared to $\Delta_2^{\mbox{\scriptsize{LO}}} \delta^{\pi^0}_0$. However, the relative variation in $\beta$ only covers a restricted range, approximatively $\pm 5\%$, so that the variations in $\alpha$ account for the largest part of the effect. As one leaves the cusp region towards larger values of $s$, the contribution proportional to $\alpha$ looses weight, and the variations are less ample. This behaviour is conveyed in a simple manner by the high-energy asymptotic expressions of $\Delta_2^{\mbox{\scriptsize{LO}}} \delta^{\pi^0}_0 (s)$, and, even more strongly, of $ \Delta_2^{\mbox{\scriptsize{LO}}} \delta^{\pi}_0 (s)$: $$\begin{aligned}
\Delta_2^{\mbox{\scriptsize{LO}}} \delta^{\pi^0}_0 (s) &\sim&
\frac{8 \beta - 5\alpha}{96 \pi} \,\frac{\Delta_\pi}{F_\pi^2}\,
+\,\frac{\alpha - \beta }{6 \pi} \,\frac{\Delta_\pi}{F_\pi^2}\,\frac{M_\pi^2}{s} \,+ \dots
\nonumber\\
\Delta_2^{\mbox{\scriptsize{LO}}} \delta^{\pi}_0 (s) &\sim&
\frac{26 \beta - 5\alpha}{96 \pi} \,\frac{\Delta_\pi}{F_\pi^2}\,
+\,\frac{\alpha - 4 \beta }{8 \pi} \,\frac{\Delta_\pi}{F_\pi^2}\,\frac{M_\pi^2}{s} \,+ \dots
\label{Delta_2_LO_asym}\end{aligned}$$
Isospin-breaking corrections at next-to-leading order {#IB_in_phases_NLO}
-----------------------------------------------------
The evaluation of $\Delta_4 \delta_\ell (s)$, the isospin-breaking effects in the phases at next-to-leading order, relies on the results obtained in the preceding sections. The corresponding numerical analysis will be the subject of section \[numerics\] below. Here, we wish to proceed for a while at the analytical level, but for simplicity, and since the next-to-leading order isospin-breaking corrections are expected to be small, we will restrict ourselves to the first order in $\Delta_\pi$. For this purpose, we expand the various quantities of interest with respect to $\Delta_\pi$, and neglect contributions beyond the linear terms. In the case of the phase-space factor for the neutral two-pion state, an expansion like: $$\sigma_0 (s) \,=\, \sigma (s) \left[ 1 \,+\, \frac{2}{s - 4 M_\pi^2}\,\Delta_\pi \,+\, {\cal O}(\Delta_\pi^2) \right] ,
\label{sigma_0_expand}$$ will not make sense when $s$ remains close to $4 M_\pi^2$. This means that our expansion to first order in isospin breaking will only provide an adequate description in regions of phase space sufficiently away from the $\pi^0 \pi^0$ and $\pi^+ \pi^-$ thresholds. From an experimental point of view, this needs not constitute a serious drawback, since the vicinity of the two-pion thresholds is usually part of the regions of phase space where the acceptance is low, as can be seen, for instance, from [@Batley:2007zz; @Batley:2010zza] in the case of the $K^+_{e4}$ decay (see, however, the discussion on the $K^\pm \rightarrow \pi^0 \pi^0 e^\pm \nu_e$ decay mode in [@NA48-2_Ke4_cusp]). From a practical point of view, we gain the advantage of having to deal with expressions which remain tractable. In the rest of this section, we will therefore remain within the framework set up by these two conditions – staying away from the two-pion thresholds, and considering only first-order isospin-violating effects. For illustration, let us consider the lowest-order corrections in the $S$-wave phases, $\Delta_2^{\mbox{\scriptsize{LO}}} \delta_0^{\pi^0}(s)$ and $\Delta_2^{\mbox{\scriptsize{LO}}} \delta_0^{\pi^0}(s)$ that we have discussed in the preceding subsection. Applying the procedure that we have just described to the expressions (\[Delta\_2\_LO\]) yields, for $s > 4 M_\pi^2$, $$\begin{aligned}
\Delta_2^{\mbox{\scriptsize{LO}}} \delta_0^{\pi^0}(s) &=& \sigma (s)\left[
\frac{\alpha}{16\pi}\,\frac{M_{\pi}^2}{s - 4 M_{\pi}^2}
\,+\,\frac{8 \beta - 5 \alpha}{96\pi}
\right]
\frac{\Delta_\pi}{F_{\pi}^2}
\,+\,{\cal O}(\Delta_\pi^2)
\nonumber\\
\Delta_2^{\mbox{\scriptsize{LO}}} \delta_0^{\pi}(s) &=&\sigma (s)\left[
\frac{8 \beta + \alpha}{48\pi}\,\frac{M_{\pi}^2}{s - 4 M_{\pi}^2}
\,+\,\frac{26 \beta - 5 \alpha}{96\pi}
\right]
\frac{\Delta_\pi}{F_{\pi}^2}
\,+\,{\cal O}(\Delta_\pi^2) .
\label{Delta_2_LO_approx}\end{aligned}$$ On figures \[fig5\] and \[fig6\], respectively, these expressions for $\Delta_2^{\mbox{\scriptsize{LO}}} \delta_0^{\pi}(s)$ and $\Delta_2^{\mbox{\scriptsize{LO}}} \delta_0^{\pi^0}(s)$ are shown together with the exact expressions given in equations (\[Delta\_2\]) and (\[Delta\_2\_LO\]). We see that the curve corresponding to the approximate expression overshoots the exact formula by about 25% at $\sqrt{s} = 285$ MeV, while the difference drops to a level around 5% for $\sqrt{s}$ above $\sim325$ MeV.
The expressions of the next-to-leading partial waves $\stackrel{{ }_{\mbox{\scriptsize{$o$}}}}{\psi} \stackrel{00}{_{\!\! 0}} \!\! (s)$, $\stackrel{{ }_{\mbox{\scriptsize{$o$}}}}{\psi} \stackrel{{\mbox{\tiny{$+-$}}}}{_{ 0,1}} \! \! (s)$, and $\stackrel{{ }_{\mbox{\scriptsize{$o$}}}}{\psi} \stackrel{x}{_{0}} \! \! (s)$ in the isospin limit have been given in eqs. (\[lim\_psi\_00\]), (\[lim\_psi\_+-\]), and (\[lim\_psi\_x\]), respectively. In order to evaluate $\Delta_4 \delta_\ell (s)$, we will proceed as follows. First, we replace, in the polynomials $\xi^{(n)}_{00}(s)$, $\xi^{(n)}_{{\mbox{\tiny{$+-$}}};S,P}$, and $\xi^{(n)}_{x}(s)$, each occurrence of $M_{\pi^0}^2$ by $M_{\pi}^2 - \Delta_\pi$. Keeping only the terms of first order in $\Delta_\pi$, we thus obtain $$\xi^{(n)}_{00}(s)\,=\,{\overline\xi}^{(n)}_{00}(s)\,+\,
\frac{\Delta_\pi}{M_\pi^2}\,\delta\xi^{(n)}_{00}(s)\,+\,
{\cal O}(\Delta_\pi^2) \, , \label{delta_xi}$$ and so on. Next, we have to expand the functions that multiply the polynomials $\xi^{(n)}_{00}(s)$, $\xi^{(n)}_{{\mbox{\tiny{$+-$}}};S}$, $\xi^{(n)}_{{\mbox{\tiny{$+-$}}};P}$ and $\xi^{(n)}_{x}(s)$, see eqs. (\[psi\_00\]), (\[psi\_+-\_0\]), (\[psi\_+-\_1\]), and (\[psi\_x\_0\]), to first order in $\Delta_\pi$. Finally, when subtracting from the functions ${\overline\xi}^{(n)}_{00}(s)$, ${\overline\xi}^{(n)}_{{\mbox{\tiny{$+-$}}};S}$, ${\overline\xi}^{(n)}_{{\mbox{\tiny{$+-$}}};P}$, ${\overline\xi}^{(n)}_{x}(s)$ the corresponding combinations of polynomials $\xi^{(n)}_\ell (s)$ arising in the isospin limit, as given in (\[lim\_psi\_00\]), (\[lim\_psi\_+-\]), and (\[lim\_psi\_x\]), isospin breaking only occurs through differences like $\alpha_x - \alpha$, or $\beta_{\mbox{\tiny{$+-$}}} - \beta$, for instance. Collecting these three contributions provides us with an expression for $\Delta_4 \delta_\ell (s)$ accurate at first order in $\Delta_\pi$.
As an illustration, consider the case of $\psi^{00}_0(s)$, cf. eq. (\[psi\_00\]). The first step is a straightforward algebraic exercise. It produces functions ${\overline\xi}^{(n)}_{00}(s)$, $n=0,1,2,3$, with $${\overline\xi}^{\,(1)}_{00}(s) \,=\, {\overline\xi}^{\,(1;0)}_{00}(s) \,+\,
{\overline\xi}^{\,(1;{\mbox{\tiny$\nabla$}})}_{00}(s)\,,
\ {\overline\xi}^{\,(2)}_{00}(s) \,=\, {\overline\xi}^{\,(2;0)}_{00}(s) \,+\,
{\overline\xi}^{\,(2;\pm)}_{00}(s)\,,
\ {\overline\xi}^{\,(3)}_{00}(s) \,=\, {\overline\xi}^{\,(3;0)}_{00}(s) \,+\,
{\overline\xi}^{\,(3;{\mbox{\tiny$\nabla$}})}_{00}(s) .$$ For the second step, we expand the various functions that appear in the expression for $\psi^{00}_0(s)$ to first order in $\Delta_\pi$ \[the functions $k_n(s)$ are defined in equation (\[k\_n\])\]: $$\begin{aligned}
2 \frac{\sigma(s)}{\sigma_{0}(s)}\,L_{0}(s) &=& 16\pi k_1(s)\,-\,
32\pi\,\frac{\Delta_\pi}{M_{\pi}^2}\left\{
k_0(s)\,+\,
\frac{M_{\pi}^2}{s-4 M_{\pi}^2}\,
\left[4 k_0(s) + k_1(s)\right]\right\}
\,+\, {\cal O}(\Delta_\pi^2)
\nonumber\\
2 \frac{\sigma (s)}{\sigma_{\mbox{\tiny$\nabla$}}(s)}\, L_{\mbox{\tiny$\nabla$}}(s) &=&
16\pi k_1(s) + 8\pi\,\frac{\Delta_\pi}{M_{\pi}^2}\left\{
k_1(s) - k_2(s)\,-\,
\frac{4 M_{\pi}^2}{s-4 M_{\pi}^2}\,
\left[4 k_0(s) + k_1(s)\right]\right\}\,+\,{\cal O}(\Delta_\pi^2)
\nonumber\\
2 \sigma (s) \sigma_0(s) L_0(s) &=& 16\pi k_2(s) \,+\,8\pi\frac{\Delta_\pi}{M_{\pi}^2}\,
\left[k_1(s) - k_2(s) - 4 k_0(s)\right]\,+\,
{\cal O}(\Delta_\pi^2)
\nonumber\\
\frac{3 M_{\pi^0}^2 \sigma (s)}{s - 4M_{\pi^0}^2}\,L_{0}^2(s) &=& 16\pi k_3(s) \,-\,16\pi\,
\frac{\Delta_\pi}{M_{\pi}^2}\left\{
k_3(s) + \frac{M_{\pi}^2}{s-4 M_{\pi}^2}\,
\left[3 k_1(s) + 4 k_3(s)\right]\right\}\,+\,
{\cal O}(\Delta_\pi^2)
\nonumber\\
\frac{3 M_{\pi}^2 \sigma (s)}{s - 4M_{\pi^0}^2}\,L_{\mbox{\tiny$\nabla$}}^2(s) &=& 16\pi k_3(s) \,+\,
16\pi\,
\frac{\Delta_\pi}{M_{\pi}^2}\left\{\frac{3}{4}\,
\left[k_1(s) - k_2(s)\right] \,-\, \frac{M_{\pi}^2}{s-4 M_{\pi}^2}\,
\left[3 k_1(s) + 4 k_3(s) \right] \right\}
+\,
{\cal O}(\Delta_\pi^2) .\ \qquad{ }
.
\label{expansion_k_n}\end{aligned}$$ Collecting the two contributions allows us to rewrite eq. (\[psi\_00\]) as $$\psi^{00}_0(s)
\,=\, 2\,\frac{M_\pi^4}{F_\pi^4}\,
\sqrt{\frac{s}{s-4M_\pi^2}}\,
\sum_{n=0}^3\left[{\overline\xi}^{(n)}_{00}(s) \,+\, \frac{\Delta_\pi}{M_\pi^2}\,\Delta\xi^{(n)}_{00}(s) \right]
k_n(s) \,+\,
{\cal O}(\Delta_\pi^2) ,
\label{psi_00_exp}$$ where $\Delta\xi^{(n)}_{00}(s)$ is the sum of $\delta\xi^{(n)}_{00}(s)$ in eq. (\[delta\_xi\]) and of the contribution generated by the expansions (\[expansion\_k\_n\]) to first order in $\Delta_\pi$. The expressions of the functions $\Delta\xi^{(n)}_{00}(s)$ are displayed in appendix \[app:delta\_xi\]. Let us briefly comment on the appearance of contributions involving the factor $M_\pi^2/(s - 4 M_\pi^2)$ in equation (\[expansion\_k\_n\]). Upon closer inspection, one finds that the combinations $\left[4 k_0(s) + k_1(s)\right]/(s - 4 M_\pi^2)$, $\left[3 k_1(s) + 4 k_3(s)\right]/(s - 4 M_\pi^2)$, and $k_2(s)/(s - 4 M_\pi^2)$, become actually proportional to $\sigma(s)$ as $s$ approaches $4 M_\pi^2$ from above.
The extraction of the first-order isospin-breaking contributions from the remaining one-loop partial waves proceeds along similar lines, and we merely quote the resulting formulae: $$\begin{aligned}
\psi^{\mbox{\tiny{$+-$}}}_0(s) &=& 2\,\frac{M_{\pi}^4}{F_\pi^4}\,
\sqrt{\frac{s}{s-4M_{\pi}^2}}\,
\sum_{n=0}^3\left[{\overline\xi}^{(n)}_{{\mbox{\tiny{$+-$}}};S} (s)\,+\,
\frac{\Delta_\pi}{M_{\pi}^2}\,\Delta\xi^{(n)}_{{\mbox{\tiny{$+-$}}};S} (s)\right]
k_n(s) \,+\,
{\cal O}(\Delta_\pi^2)
\nonumber\\
\psi^{\mbox{\tiny{$+-$}}}_1(s) &=& 2\,\frac{M_{\pi}^4}{F_\pi^4}\,
\sqrt{\frac{s}{s-4M_{\pi}^2}}\,
\sum_{n=0}^4\left[{\overline\xi}^{(n)}_{{\mbox{\tiny{$+-$}}};P} (s)\,+\,
\frac{\Delta_\pi}{M_{\pi}^2}\,\Delta\xi^{(n)}_{{\mbox{\tiny{$+-$}}};P} (s)\right]
k_n(s) \,+\,
{\cal O}(\Delta_\pi^2)
\nonumber\\
\psi^{x}_0(s) &=& 2\,\frac{M_{\pi}^4}{F_\pi^4}\,
\sqrt{\frac{s}{s-4M_{\pi}^2}}\,
\sum_{n=0}^3\left[{\overline\xi}^{(n)}_{x}(s)\,+\,
\frac{\Delta_\pi}{M_{\pi}^2}\,\Delta\xi_{x}^{(n)}(s)\right]
k_n(s) \,+\,
{\cal O}(\Delta_\pi^2) .
\label{psi_+-_exp_and_psi_x_exp}\end{aligned}$$ More details, as well as expressions of the functions $\Delta\xi^{(n)}_{{\mbox{\tiny{$+-$}}};S} (s)$, $\Delta\xi^{(n)}_{{\mbox{\tiny{$+-$}}};P} (s)$, and $\Delta\xi_{x}^{(n)}(s)$ are given in appendix \[app:delta\_xi\]. Working at first order in $\Delta_\pi$ has allowed us to cast the functions $\psi^{00}_0(s)$, $\psi^{\mbox{\tiny{$+-$}}}_{0,1}(s)$, and $\psi^{x}_0(s)$ into a form that makes the comparison with their expressions in the isospin limit straightforward.
At next-to-leading order, the isospin-breaking contributions to the $P$-wave phase are thus simply proportional to the difference $\psi^{\mbox{\tiny{$+-$}}}_1 (s)-
\stackrel{{ }_{\mbox{\scriptsize{$o$}}}}{\psi} \stackrel{{\mbox{\tiny{$+-$}}}}{_{\!\!\!\! 1}} \! \! (s) $: $$\begin{aligned}
{\Delta}_4 \delta_1^{\pi}(s) &=&
2\,\frac{M_\pi^4}{F_\pi^4}\,
\sum_{n=0}^4\left[
{\overline\xi}^{(n)}_{{\mbox{\tiny{$+-$}}};P} (s)
\,-\,\frac{1}{2}\xi_1^{(n)}(s)
\,+\,\frac{\Delta_\pi}{M_{\pi}^2}\,\Delta\xi^{(n)}_{{\mbox{\tiny{$+-$}}};P} (s)
\right]
k_n(s) \,+\,
{\cal O}(\Delta_\pi^2) .\end{aligned}$$ In the case of the $S$-wave phases, the corresponding corrections are naturally split into a universal contribution ${\Delta}_4^U \delta_0(s)$ and a form-factor dependent piece ${\Delta}_4^F \delta_0(s)$, cf. eq. (\[Delta\_U\_F\]). Keeping only the first-order isospin-breaking contributions, the two universal pieces read, again for $s > 4 M_\pi^2$, $$\begin{aligned}
{\Delta}_4^U \delta_0^{\pi^0}(s) &=&
\sigma(s)\left\{
\frac{1}{2} \psi_0^{00}(s) - \frac{1}{2} \stackrel{{ }_{\mbox{\scriptsize{$o$}}}}{\psi} \stackrel{00}{_{\!\! 0}} \!\! (s)
\, -\, \psi_0^{x}(s) \theta(s-4M_{\pi}^2) + \stackrel{{ }_{\mbox{\scriptsize{$o$}}}}{\psi} \stackrel{x}{_{0}} \! \! (s)
\,+\,\frac{1}{2}\left( \frac{\sigma_{0}(s)}{\sigma(s)} - 1\right) \psi_0^{00}(s)
\right\}
\nonumber\\
&=&
2\,\frac{M_\pi^4}{F_\pi^4}\,
\sum_{n=0}^3\left\{ \frac{1}{2}{\overline\xi}^{\,(n)}_{00}(s)\,-\,
{\overline\xi}^{\,(n)}_{x}(s)
\,-\,\frac{1}{2}\xi_0^{(n)}(s)
\right.
\nonumber\\
&& \left. \!\!\!\!\!\!\!\!
+\,\frac{\Delta_\pi}{M_\pi^2}\,\left[
\frac{1}{2} \Delta\xi_{00}^{(n)}(s)\,-\,\Delta\xi_{x}^{(n)}(s)
\,+\,\frac{1}{3}\,\frac{M_\pi^2}{s - M_\pi^4}
\left( 2 \xi_2^{(n)}(s) + \xi_0^{(n)}(s) \right)\right]
\right\}
k_n(s) \,+\,
{\cal O}(\Delta_\pi^2) ,
\nonumber\\
\nonumber\\
{\Delta}_4^U \delta_0^{\pi}(s) &=&
\sigma(s)\left\{
\psi^{\mbox{\tiny{$+-$}}}_0 (s)-
\stackrel{{ }_{\mbox{\scriptsize{$o$}}}}{\psi} \stackrel{{\mbox{\tiny{$+-$}}}}{_{\!\!\!\! 0}} \! \! (s)
\,-\,
\frac{1}{2} \psi^{x}_0(s)\,+\,
\frac{1}{2} \stackrel{{ }_{\mbox{\scriptsize{$o$}}}}{\psi} \stackrel{x}{_{0}} \! \! (s)
\,-\,\frac{1}{2}\left( \frac{\sigma_{0}(s)}{\sigma(s)} - 1\right) \psi^{x}_0(s)
\right\}
\nonumber\\
&=&
2\,\frac{M_\pi^4}{F_\pi^4}\,
\sum_{n=0}^3\left\{ {\overline\xi}^{\,(n)}_{{\mbox{\tiny{$+-$}}};S}(s)\,-\,\frac{1}{2}
{\overline\xi}^{\,(n)}_{x}(s)
\,-\,\frac{1}{2}\xi_0^{(n)}(s)
\right.
\nonumber\\
&& \left. \!\!\!\!\!\!\!\!
+\,\frac{\Delta_\pi}{M_\pi^2}\,\left[
\Delta\xi_{{\mbox{\tiny{$+-$}}};S}^{(n)}(s)\,-\,\frac{1}{2} \Delta\xi_{x}^{(n)}(s)
\,-\,\frac{1}{3}\,\frac{M_\pi^2}{s - M_\pi^4}
\left(\xi_2^{(n)}(s) - \xi_0^{(n)}(s) \right)\right]
\right\}
k_n(s) \,+\,
{\cal O}(\Delta_\pi^2)
.\end{aligned}$$ Concerning the form-factor dependent parts, the one-loop result (\[eq:gammaoneloop\]) gives, at the same level of accuracy, $$\begin{aligned}
\Gamma_S^{\pi}(s) \,-\,\Gamma_S^{\pi^0}(s) &=&
s(a_S^{\pi} - a_S^{\pi^0})\,+\,
\frac{\Delta_\pi}{32\pi^2F_\pi^2}\left[
-\beta\frac{s}{M_\pi^2}\,+\,\frac{28 \beta + 2 \alpha}{3}\right]
\,+\,
\frac{\Delta_\pi}{96\pi^2F_\pi^2} (4 \beta - \alpha)\,\frac{L(s)}{\sigma(s)}
\nonumber\\
&&\!\!\!\!\!
+\,
\frac{\Delta_\pi}{96\pi^2F_\pi^2} (14 \beta + \alpha)\,{\sigma(s)}L(s)
\,+\,{\cal O}(\Delta_\pi^2) .\end{aligned}$$
Numerical evaluation {#numerics}
====================
This section is devoted to the numerical evaluation of the isospin breaking corrections $\Delta_4 \delta_\ell (s)$ keeping the full dependence on $\Delta_\pi$. For this, we first need to know how the subtraction parameters that appear in the amplitudes and form factors after the first iteration are related to the corresponding ones in the isospin limit.
Determination of the subtraction parameters {#sub_csts_at_NLO}
-------------------------------------------
From the dispersive representations of the form factors and scattering amplitudes, we have obtained the isospin-breaking corrections in the phases of the pion form factors beyond leading order. These expressions involve the normalizations $F_S^\pi (0)$ and $F_S^{\pi^0} (0)$ and the two subtraction parameters $a_S^\pi$ and $a_S^{\pi^0}$ in the one-loop expressions of the form factors, and only a subset of the 15 subtraction constants that appear in the $\pi\pi$ amplitudes, namely $\alpha_{00}$, $\alpha_x$, $\alpha_{{\mbox{\tiny{$+-$}}}}$, $\beta_{x}$, and $\beta_{{\mbox{\tiny{$+-$}}}}$ on the one hand, $\lambda^{(1)}_{00}$, $\lambda^{(i)}_x$ and $\lambda_{{\mbox{\tiny{$+-$}}}}^{(i)}$, $i = 1,2$, on the other hand. In the isospin limit, the latter are given, as indicated in equation (\[lambdas\_iso\]), in terms of the constants $\lambda_i$ discussed and evaluated in refs. [@KMSF95; @KMSF96]. More accurate determinations have appeared since then [@Colangelo:2001df; @DescotesGenon:2001tn], see below. We thus merely need to evaluate the size of the isospin-breaking deviations like, say, $\lambda^{(i)}_x - \lambda_i$. The subset $\alpha_{00}$...$\beta_{{\mbox{\tiny{$+-$}}}}$ is likewise related to the subthreshold parameters $\alpha$ and $\beta$ in the isospin limit. At lowest order, these relations were given in eq. (\[alphabetaLO\]), but in order to evaluate the phases at next-to-leading order, it is necessary to go beyond this approximation. Again, we only need to know the size of the deviations from the isospin-limit quantities $\alpha$ and $\beta$. As discussed at the end of subsection \[FF\_and\_Amp\_tree\], $\alpha$ and $\beta$ represent the observables that we eventually would like to pin down from a phenomenological analysis of experimental data, so we have to trace down the dependence on these parameters beyond the lowest-order expressions.
Let us now explain how we proceed with these tasks. For this purpose, we briefly come back to the discussion in subsection \[sec:em\_corr\]. The framework that we have presented there can be described by an “effective” lagrangian, whose form is similar to the chiral lagrangian used to treat electromagnetic corrections, but without including photons as dynamical degrees of freedom, as their effect is supposed to be treated by other means or otherwise to be negligible. The leading-order (strong) lagrangian ${\mathcal{L}}_2$ is then supplemented with a contribution of the form [@Colangelo:2008sm]: $${\mathcal{L}}_2 \to {\mathcal{L}}_2 + \widehat{C}\langle QUQU^\dag \rangle \,,\ Q={\rm diag}(2e/3,-e/3)\,,
\label{LO_eff_lag}$$ where $\widehat{C}$ is a low-energy constant that breaks isospin symmetry among the pion masses. The last term, through its transformation properties under chiral symmetry, encodes the information about the electromagnetic origin of the pion mass difference. Although we could have absorbed it into the definition of $\widehat{C}$, we have left the electric charge $e$ apparent, in order to make the comparison with the usual effective theory in presence of electromagnetic interactions more convenient. We call (\[LO\_eff\_lag\]) an “effective" lagrangian since there is no identifiable fundamental theory of which it would constitute the effective theory in the usual sense [^1], the quotation marks serving as a reminder of this limitation. Nevertheless, eq. (\[LO\_eff\_lag\]) constitutes a suitable starting point for a low-energy expansion, with a well-defined and consistent power counting, which reproduces the features of the framework adopted here as far as isospin-violating corrections are concerned. Thus, the “effective" lagrangian at next-to-leading order is supplemented with the terms described in ref. [@Knecht:1997jw], but with the corresponding low-energy constants denoted with a hat, to distinguish them from those obtained in the theory with virtual photons included. Indeed, the absence of virtual photons modifies the structure of the one-loop divergences, and the scale dependence of the renormalized low-energy constants ${\widehat k}^r_i(\mu)$ is given by $$e^2 \mu \frac{d}{d\mu} {\widehat k}^r_i(\mu)\,=\,
- \frac{1}{16\pi^2}\,{\widehat\sigma}_i
,$$ with (for the low-energy constants of interest in our case) $$\begin{aligned}
{\widehat\sigma}_1 &=& {\widehat\sigma}_5 \,=\, -\frac{1}{10}\,\frac{\Delta_\pi}{F^2}
\,,\quad
{\widehat\sigma}_8 \,=\, -\frac{1}{2}\,\frac{\Delta_\pi}{F^2}
\nonumber\\
{\widehat\sigma}_2 &=& {\widehat\sigma}_4 \,=\,{\widehat\sigma}_6\,=\,
\frac{\Delta_\pi^2}{F^2}
\,,\quad {\widehat\sigma}_3 \,=\, {\widehat\sigma}_7 \,=\, 0
.
\label{sigma_hat}\end{aligned}$$ One has also a contribution quadratic in the difference $M_\pi^2 - M_{\pi^0}^2$ from the low-energy constant ${\widehat k}^r_{14}(\mu)$: $$e^4 \mu \frac{d}{d\mu} {\widehat k}^r_{14}(\mu)\,=\,
- \frac{3}{16\pi^2}\,\frac{\Delta_\pi^2}{F^4}
.$$ We emphasize that these scale dependences are different from the ones of the equivalent low-energy counterterms discussed in ref. [@Knecht:1997jw], since we have considered a theory where no virtual photons are included. Furthermore, they follow in a straightforward manner from the expressions given in eqs. (3.9)-(3.11) of [@Knecht:1997jw] upon dropping the terms that do not contain the parameter $Z=C/F^4$, with $M_\pi^2 - M_{\pi^0}^2 = 2 e^2 F^2 Z$ at lowest order, in the notation of that reference \[in the case of ${\widehat\sigma}_{14}$, only the terms in $Z^2$ must be retained, the constant ${\widehat k}^r_{14}$ being multiplied by $e^4$\].
The relevant subtraction constants are then obtained upon matching the one-loop expressions obtained within the framework we have just described with the representations obtained in section \[1stIteration\]. The outcome of this exercise is displayed in appendix \[app:subtraction\]. Let us recall here that there exist explicit one-loop calculations of the various $\pi\pi$ amplitudes [@Knecht:1997jw; @KnechtNehme02; @Meissner:1997fa] and form factors [@Kubis:1999db] considered here, obtained within the full QCD+QED effective theory [@Urech:1994hd; @Neufeld:1994eg; @Neufeld:1995mu; @Knecht:1997jw; @Meissner:1997fa; @Schweizer:2002ft]. As mentioned in subsection \[sec:em\_corr\], these calculations also include isospin-violating contributions arising from the exchanges of virtual low-energy photons, which are however not considered here. In order to make a comparison with these one-loop calculations, we must therefore remove the contributions of virtual photons from the expressions given in these references, and only keep the effects due to the difference of the pion masses [@Colangelo:2008sm]. From a practical point of view, this can be done as described above: contributions proportional to $e^2$, but without the appropriate $Z$ factor, arise from the exchange of virtual photons and are discarded, while at the same time the low-energy constants $k^r_i(\mu)$ are replaced by ${\widehat k}^r_i(\mu)$. Furthermore, since we want to display the dependence on the two independent parameters $\alpha$ and $\beta$, we have, in the computation of the loop contributions, explicitly kept the quantities $F$ and ${\widehat m} B$ defining the leading-order amplitude (\[pipiAmp\_LO\]), for which we have then substituted the lowest-order expressions given in (\[alpha\_beta\_LO\]). This brings in another difference with the one-loop calculations available in the literature.
Numerical input values {#num_input}
----------------------
We wish to investigate the size of the isospin-breaking corrections as functions of $\alpha$ and $\beta$, for fixed values of $\lambda_{1,2}$ and of the ${\widehat k}^r_i$’s. As mentioned above, the former parameters have been evaluated before [@KMSF95; @KMSF96; @Colangelo:2001df; @DescotesGenon:2001tn] using sum rules and medium-energy $\pi\pi$ data. For the numerical evaluations below, we use the values from the “extended fit” of [@DescotesGenon:2001tn]: $$\lambda_1 \,=\, ( -4.18 \pm 0.63 ) \cdot 10^{-3} \ ,\quad \lambda_2 \,=\, ( 8.96 \pm 0.12 ) \cdot 10^{-3} .
\label{lambda_1_2_values}$$ Let us notice that the sum rules that lead to this determinations of $\lambda_1$ and $\lambda_2$ also exhibit a mild dependence with respect to $\alpha$ and $\beta$ [@KMSF95; @KMSF96]. This dependence is, however, covered by the quoted uncertainties, and we will therefore not consider it further.
As far as the constants ${\widehat k}^r_i$ are concerned, we will assume that they take the same numerical values as the low-energy constants ${k}^r_i$. Even though this identification constitutes an approximation whose precision is difficult to assess, it is not obvious to consider simple alternatives to this choice at the time being. Incidentally, this is also the option that was retained in ref. [@Gasser:2007de].
To obtain numerical estimates of the ${k}^r_i$s, we proceed in several step. First, we make use of the relation between these two-flavour constants and their three-flavour counterparts $K_i^r$ [@Urech:1994hd], as worked out at one-loop level in ref. [@Haefeli:2007ey]. For the constants $K_1^r$...$K_6^r$ we then use the evaluation of ref. [@Ananthanarayan:2004qk], as given by the last line of Table 1 in this reference. These determinations rely on a set of sum-rules [@Moussallam:1997xx] involving QCD four-point functions, that are saturated by the lowest-mass resonances in the corresponding channels. This kind of minimal hadronic ansatz, which finds some justification in the limit of a large number of colors $N_C$ [@'tHooft:1973jz; @Witten:1979kh], is usually a good approximation [@Eckeretal89]. We therefore endow the numbers of ref. [@Ananthanarayan:2004qk] with a relative error of 33% (1/$N_C$ for $N_C = 3$) accounting for neglected subleading effects in the $1/N_C$ expansion. We assign the same relative error to the constants $K_{10}^r$ and $K_{11}^r$ that were estimated along similar lines in ref. [@Moussallam:1997xx]. Next, the relation between $k_8^r$ and $K_{11}^r$ also involves [@Haefeli:2007ey] the $SU(3)$ low-energy constants [@Gasser:1984gg] $L_4^r$ and $L_5^r$. For the latter we have taken the ${\cal O}(p^4)$ determination $10^3 \cdot L_5^r(M_\rho) = 1.46 \pm 0.15$ from ref. [@Amoros:2001cp]. $L_4^r$ is not so well determined, and can induce significant differences between the patterns of chiral symmetry breaking for two and three massless flavours [@DescotesGenon:1999uh; @DescotesGenon:2000ct; @DescotesGenon:2000di; @DescotesGenon:2003cg]. For our present purposes, we take $10^3 \cdot L_4^r(M_\rho) = 0 \pm 0.5$, a value which was advocated on the basis of the Zweig rule [@Gasser:1984gg; @Amoros:2001cp], even though later fits and discussions favour larger values [@DescotesGenon:2007ta; @Bernard:2010ex; @Bijnens:2011tb]. In any case, it turns out that $k_8$ plays only a minor role in the numerical evaluation of isospin breaking in the phases. Finally, the relation between the ${k}^r_i$s of interest to us and the $K_i^r$s also involve four constants, $K_{7,8,9,15}^r$ for which there exist no reliable determinations. We have assigned an overall uncertainty of $\pm 1/(16 \pi^2) = \pm 6.3 \cdot 10^{-3}$, based on naive dimensional analysis, to the values of $k_{5,6,7,14}^r$ where these constants occur. The results of this analysis are displayed in table \[tab:LECs\]. Our values reproduce those given in ref. [@Haefeli:2007ey], where however an overall uncertainty of $\pm 6.3 \cdot 10^{-3}$ was assigned uniformly to all the constants $k_i^r(M_\rho)$.
For the sake of illustration, we let the parameters $\alpha$ and $\beta$ vary within the intervals $1.0 \le \alpha \le 1.8$, $1.04 \le \beta \le 1.12$, suggested by the analysis of ref. [@DescotesGenon:2001tn]. These intervals cover a reasonable range of possibilities, but of course, if necessary, other values can be considered.
To summarize, for the numerical analysis that follows we thus use as inputs the values of the constants $\lambda_{1,2}$ in (\[lambda\_1\_2\_values\]), the values of the constants ${\widehat k}^r_i(\mu)$ as given in table \[tab:LECs\], together with [@pdg2010] $$M_\pi \,=\, 139.57\ {\mbox{MeV}}\, , \ M_{\pi^0} \,=\, 134.98\ {\mbox{MeV}}\, , \ F_\pi \,=\, 92.2\ {\mbox{MeV}} .
\label{inputs}$$
i 1 2 3 4 5 6 7 8 14
---------------------------------------------------- --------------- --------------- --------------- --------------- ---------------- --------------- --------------- ---------------- ----------------
$\stackrel{ }{\widehat{k}_i^r}(M_\rho) \cdot 10^3$ $8.4 \pm 2.8$ $3.4 \pm 1.2$ $2.7 \pm 0.9$ $1.4 \pm 0.5$ $-0.8 \pm 6.3$ $3.9 \pm 6.3$ $3.7 \pm 6.3$ $-1.3 \pm 2.5$ $-0.4 \pm 6.3$
: Values of the low-energy constants ${\widehat{k}_i^r}$ used for the estimate of the subtraction constants. The values of the $\widehat{k}_i^r$ correspond to the renormalized constants at the scale $\mu = M_\rho = 770$ MeV. The constants $\stackrel{ }{\widehat{k}_{3}}$ and $\stackrel{ }{\widehat{k}_{7}}$ do not depend on the renormalization scale, cf. eq. (\[sigma\_hat\]). \[tab:LECs\]
Size of isospin corrections to the phases
-----------------------------------------
Let us start with the phase of $F^\pi_S(s)$. We will from now on restrict ourselves to the range of energies above the cusp. On figure \[fig9\] we show the isospin-violating correction $\Delta\delta_0^\pi (s) = \Delta_2 \delta_0^\pi (s) + \Delta_4 \delta_0^\pi (s)$, defined as explained after equation (\[phases\_NLO\]), i.e., for $s > 4M_\pi^2$, $$\begin{aligned}
\Delta\delta_{0}^{\pi}(s) &=&
\sigma (s) \left[
\varphi_0^{\mbox{\tiny{$+-$}}}(s) - \stackrel{{ }_{\mbox{\scriptsize{$o$}}}}{\varphi} \stackrel{{\mbox{\tiny{$+-$}}}}{_{\!\!\!\! 0}} \! \! (s)
\right]
\, -\,
\frac{1}{2}\,\sigma_{0}(s) \left[
\psi_0^{x}(s) - \stackrel{{ }_{\mbox{\scriptsize{$o$}}}}{\psi} \stackrel{x}{_{0}} (s)
\right]
\nonumber\\
&& \!\!\!\!\! + \,
\sigma (s) \left[
\psi_0^{\mbox{\tiny{$+-$}}}(s) - \stackrel{{ }_{\mbox{\scriptsize{$o$}}}}{\psi} \stackrel{{\mbox{\tiny{$+-$}}}}{_{\!\!\!\! 0}} \! \! (s)
\right]
\, -\,
\frac{1}{2}\,\sigma_{0}(s) \left[
\psi_0^{x}(s) - \stackrel{{ }_{\mbox{\scriptsize{$o$}}}}{\psi} \stackrel{x}{_{0}} (s)
\right]
\nonumber\\
&& \!\!\!\!\! - \,
\frac{1}{2}\,
\sigma_{0}(s)\varphi_0^{x}(s)
\left[\left(\frac{F_S^{\pi^0}(0)}{F_S^{\pi}(0)} - 1\right)
-\left(\Gamma_S^{\pi}(s) - \Gamma_S^{\pi^0}(s)\right)
\right]
\nonumber\\
&\equiv&
\Delta_2 \delta_0^\pi (s) \,+\,\Delta_4^U \delta_0^\pi (s) \,+\,\Delta_4^F \delta_0^\pi (s)
.\end{aligned}$$ The three terms of the decomposition in the second equality correspond, in succession, to the three lines of the first one, see the discussion after equation (\[Delta\_U\_F\]). In the cusp region, $\Delta\delta_{0}^{\pi}(s) $ is rather well described by $\Delta_2 \delta_0^\pi (s) \,+\,\Delta_4^U \delta_0^\pi (s)$, the contribution of $\Delta_4^F \delta_0^\pi (s)$ is only marginal. At energies above $\sim 300$ MeV, $\Delta_4^F \delta_0^\pi (s)$ starts to provide a sizeable negative contribution that more and more compensates for $\Delta_4^U \delta_0^\pi (s)$, so that eventually $\Delta \delta_0^\pi (s) \sim \Delta_2 \delta_0^\pi (s)$. This situation is reproduced if we take values of $\alpha$ and $\beta$ different from the ones adopted for figure \[fig9\], but in the range considered here. We stress that it is $\Delta_2 \delta_0^\pi (s)$, and not $\Delta_2^{\mbox{\scriptsize{LO}}} \delta_0^\pi (s)$, that provides a good description of the total effect in the region of higher energies. In this region $\Delta_2 \delta_0^\pi (s)$ and $\Delta_2^{\mbox{\scriptsize{LO}}} \delta_0^\pi (s)$ differ by more than 2 milliradians, see figure \[fig5\].
Turning next to the phase of $F^{\pi^0}_S (s)$, the definition of the different relevant quantities reads $$\begin{aligned}
\Delta \delta_{0}^{\pi^0}(s) &=&
\frac{1}{2}\,\sigma_0 (s) \left[
\varphi_0^{00}(s) - \stackrel{{ }_{\mbox{\scriptsize{$o$}}}}{\varphi} \stackrel{00}{_{\!\! 0}} \! (s)
\right]
\, -\,
\sigma (s) \left[
\varphi_0^{x}(s) - \stackrel{{ }_{\mbox{\scriptsize{$o$}}}}{\varphi} \stackrel{x}{_{0}} (s)
\right]
\nonumber\\
&& \!\!\!\!\! + \,
\frac{1}{2}\,
\sigma_0 (s) \left[
\psi_0^{00}(s) - \stackrel{{ }_{\mbox{\scriptsize{$o$}}}}{\psi} \stackrel{00}{_{\!\! 0}} \! (s)
\right]
\, -\,
\sigma (s) \left[
\psi_0^{x}(s) - \stackrel{{ }_{\mbox{\scriptsize{$o$}}}}{\psi} \stackrel{x}{_{0}} (s)
\right]
\nonumber\\
&& \!\!\!\!\! -\,
\sigma (s)\varphi_0^{x}(s)
\left[\left(\frac{F_S^{\pi}(0)}{F_S^{\pi^0}(0)} - 1\right)
+\left(\Gamma_S^{\pi}(s) - \Gamma_S^{\pi^0}(s)\right)
\right]
\nonumber\\
&\equiv&
\Delta_2 \delta_0^{\pi^0} (s) \,+\,\Delta_4^U \delta_0^{\pi^0} (s) \,+\,\Delta_4^F \delta_0^{\pi^0} (s)
.\end{aligned}$$ We find a rather different situation than in the previous case. As can be seen on figure \[fig10\], the correction $\Delta_4\delta_0^{\pi^0} (s)$ is large, with $\Delta_4^U \delta_0^{\pi^0} (s)$ and $\Delta_4^F \delta_0^{\pi^0} (s)$ both contributing in a substantial way. Here, $\Delta_2 \delta_0^{\pi^0} (s)$, and even less so $\Delta_2^{\mbox{\scriptsize{LO}}} \delta_0^{\pi^0} (s)$, do not provide a decent representation of the full isospin-violating contribution. Again, the situation shown on figure \[fig10\] for specific values of $\alpha$ and $\beta$ is actually generic for all values of these parameters in the ranges considered.
Both $\Delta \delta_{0}^{\pi}(s)$ and $\Delta \delta_{0}^{\pi^0}(s)$ receive contributions from $\psi_0^{x}(s)$, which contains a piece $\Delta_2 \psi_0^{x}(s)$ of order ${\cal O}(\Delta_\pi^2)$. We have checked that its numerical value is indeed tiny, in the range between $-3 \cdot 10^{-3}$ milliradian and $-2 \cdot 10^{-3}$ milliradian for all values of $s$ and of the parameters $\alpha$ and $\beta$ considered here.
In the case of $F_V^\pi (s)$, the form factor effects are absent from the isospin-violating contribution to the phase, which reads simply $$\begin{aligned}
\Delta \delta_{1}^{\pi}(s) &=&
\sigma (s) \left[
\varphi_1^{\mbox{\tiny{$+-$}}}(s) - \stackrel{{ }_{\mbox{\scriptsize{$o$}}}}{\varphi} \stackrel{{\mbox{\tiny{$+-$}}}}{_{\!\!\!\! 1}} \! \! (s)
\right]
\, -\,
\sigma (s)\left[
\psi_1^{\mbox{\tiny{$+-$}}}(s) - \stackrel{{ }_{\mbox{\scriptsize{$o$}}}}{\psi} \stackrel{{\mbox{\tiny{$+-$}}}}{_{\!\!\!\! 1}} \! \! (s)
\right]
\nonumber\\
&\equiv&
\Delta_2 \delta_1^\pi (s) \,+\,\Delta_4 \delta_1^\pi (s)
.\end{aligned}$$ As can be inferred from figure \[fig11\], the overall effect is small in this case, but the unitarity correction gives a substantial decrease of the lowest-order correction.
$\sqrt{s}$ (MeV) $\Delta_2 \delta_0^{\pi}(s)$ $\Delta_4^U \delta_0^{\pi}(s)$ $\Delta_4^F \delta_0^{\pi}(s)$ $\Delta \delta_0^{\pi}(s)$ $\Delta_2 \delta_0^{\pi^0}(s)$ $\Delta_4^U \delta_0^{\pi^0}(s)$ $\Delta_4^F \delta_0^{\pi^0}(s)$ $\Delta \delta_0^{\pi^0}(s)$ $\Delta_2 \delta_1^{\pi}(s)$ $\Delta_4 \delta_1^{\pi}(s)$ $\Delta \delta_1^{\pi}(s)$
------------------ ------------------------------ -------------------------------- -------------------------------- ---------------------------- -------------------------------- ---------------------------------- ---------------------------------- ------------------------------ ------------------------------ ------------------------------ ----------------------------
286.07 11.10 1.42 -0.29 12.23 3.53 2.37 0.38 6.29 0.007 -0.001 0.006
295.97 9.80 1.24 -0.38 10.66 2.70 2.08 0.61 5.40 0.027 -0.005 0.022
304.89 9.54 1.19 -0.45 10.28 2.41 2.04 0.78 5.23 0.050 -0.009 0.041
313.47 9.53 1.17 -0.51 10.19 2.27 2.06 0.91 5.25 0.077 -0.015 0.062
322.02 9.64 1.15 -0.57 10.23 2.21 2.11 1.04 5.36 0.107 -0.022 0.085
330.78 9.82 1.13 -0.62 10.33 2.20 2.18 1.16 5.54 0.141 -0.030 0.111
340.17 10.05 1.11 -0.68 10.48 2.23 2.25 1.28 5.76 0.180 -0.041 0.139
350.92 10.34 1.06 -0.74 10.67 2.29 2.34 1.41 6.04 0.228 -0.055 0.173
364.52 10.75 1.00 -0.82 10.93 2.41 2.45 1.57 6.42 0.295 -0.076 0.218
389.71 11.55 0.80 -0.95 11.40 2.70 2.61 1.85 7.16 0.430 -0.126 0.304
: The isospin-breaking corrections at order ${\cal O}(E^4)$ to the phases (in milliradians) of the scalar and vector form factors for several values of the energy in the range between the $2M_\pi$ threshold and the kaon mass. The break-up into the various contributions discussed in the text is also given. []{data-label="table_num"}
In table \[table\_num\], we have also summarized our result in numerical form, for some values of the energy in the range between the $2 M_\pi$ threshold and the kaon mass. The presence of the form-factor dependent contributions in $\Delta_4 \delta^{\pi}_{0}(s)$ and in $\Delta_4 \delta^{\pi^0}_{0}(s)$ however precludes a direct application of our results to the experimentally more interesting case of the $K^+_{e4}$ decay modes. This necessitates a dedicated study, which will be reported on elsewhere [@wip]. Nevertheless, it is interesting to investigate the kind of conclusions that such a study might lead to from the analysis presented so far for the scalar and vector form factors of the pion. The quantity that comes closest to the observable of interest in the context of the decay mode $K^\pm \rightarrow \pi^+ \pi^- e^\pm
{\stackrel{(-)}{\nu_e}}$ is the difference between the $S$ and $P$ phases, $\delta_0^\pi (s) - \delta_1^\pi (s)$, for which the total isospin-breaking correction reads $$\begin{aligned}
\Delta_{\mbox{\scriptsize{tot}}}^\pi (s) & \equiv & \Delta \delta_0^\pi (s) - \Delta \delta_1^\pi (s)
\nonumber\\
& = & \Delta_2 \delta_0^\pi (s) + \Delta_4^U \delta_0^\pi (s)
+ \Delta_4^F \delta_0(^\pi s) - \Delta_2 \delta_1^{\pi}(s)-\Delta_4 \delta_1^{\pi}(s)
.\end{aligned}$$ This correction in the phase difference is shown on figure \[fig12\], together with the error band induced by the uncertainties on the various input parameters, for fixed values of $\alpha$ and $\beta$. The main contributors to these error bars are the low-energy constants ${\widehat k}_i$, and in particular ${\widehat k}_{1,2,3,4}$, which enter in the correction $\Delta_4^U \delta_0^\pi (s)$, through the isospin-breaking differences such as $\alpha_{00} - \alpha$, $\alpha_x - \alpha$, and so on. Similar error bands have to be associated to the curves for $\Delta_4 \delta_0^\pi (s)$ and $\Delta_4 \delta_0^{\pi^0} (s)$ shown on figs. \[fig9\] and \[fig10\], respectively. Except in the vicinity of the cusp, the correction is thus relatively constant, around 11 milliradians for $(\alpha,\beta)=(1.4,1.08)$, with an uncertainty that is somewhat less than $\pm$1 milliradian. We show the correction, with the associated error band, for three sets of values for $(\alpha, \beta)$. Despite the uncertainties, there remains a sensitivity with respect to these parameters.
We have also compared the exact results obtained in this section with the approximation where only corrections of first order in $\Delta_\pi$ are retained, as discussed in subsection \[IB\_in\_phases\_NLO\]. For the range of parameters considered here, we find that using the approximate expressions for $\Delta_4 \delta_0^{\pi^0} (s)$, $\Delta_4 \delta_0^\pi (s)$, and $\Delta_4 \delta_1^\pi (s)$ does not modify the values of $\Delta \delta_0^{\pi^0} (s)$, $\Delta \delta_0^\pi (s)$, and $\Delta \delta_1^\pi (s)$ by more than a few percents, as soon as the energy is more than $\sim 20$ MeV higher than the $2 M_\pi$ threshold, i.e.$\sqrt{s}\ge 300$ MeV. For practical purposes, one possible option consists therefore in keeping the exact expressions for $\Delta_2 \delta_0^{\pi^0} (s)$, $\Delta_2 \delta_0^\pi (s)$, and $\Delta_2 \delta_1^\pi (s)$, and in using the expressions truncated at first order in $\Delta_\pi$ for the next-to-leading contributions.
Summary and conclusions {#conclusion}
=======================
In this article, we have addressed the issue of isospin breaking due to the difference between the charged and neutral pion masses in the pion form factors and $\pi\pi$ scattering amplitudes in the low-energy domain. We have implemented a dispersive approach to obtain representations of the various $\pi\pi$ scattering amplitudes and pion form factors that are valid at next-to-next-to-leading order in the low-energy expansion. These representations rely on general properties such as relativistic invariance, analyticity, crossing, and unitarity, combined with the chiral counting for the form factors and partial waves, and provide an extension of the general frameworks developped previously in the isospin-symmetric case to the situation where isospin-breaking effects are taken into account. This construction needs as inputs the lowest-order expressions of the pion form factors and $\pi\pi$ $S$ and $P$ partial waves, and proceeds through a two-step iterative process, the partial wave projections obtained from the one-loop representation after the first step serving as inputs for the second step. At the two-loop level, we have obtained partially analytical expressions only, due to the difficulty of performing the dispersive integrals related to contributions of the non-factorizing type. We have nevertheless shown that in the limit where the pion mass difference vanishes, we reproduce known two-loop results for the scattering amplitudes and form factors in the isospin limit. On the other hand, we have obtained explicit analytical expressions for the phases of the two-loop form factors, on which we have focused. We have also provided somewhat more tractable expressions of the isospin-breaking corrections in the phases valid at first order in the difference $M_\pi^2 - M_{\pi^0}^2$. This approximation provides a very good description of isospin-breaking effects at next-to-leading order in the phases, for all energies lying between $\sim 300$ MeV and the kaon mass.
The dispersive representations involve a limited number of subtraction constants, which have to be fixed from experimental data or theoretical sources. We have related the subtraction constants involved in the phases of the two-loop form factors to their counterparts in the isospin limit, and we have provided a numerical evaluation of the isospin-breaking corrections in these subtraction constants. This has allowed us to perform a quantitative study of the size of isospin-violating effects in the phases of the form factors.
We have displayed our results in terms of the subthreshold parameters $\alpha$ and $\beta$ of the pion scattering amplitude in the isospin limit. These parameters represent the unknown quantities that have to be extracted from low-energy data. Equivalently, one may take the two $S$-wave scattering lengths $a_0^0$ and $a_0^2$ as unknowns, and we have provided the formulae necessary in order to operate the translation between the two formulations.
As far as the phases of the form factors are concerned, the main difference with the isospin-symmetric situation is that Watson’s theorem is no longer operative when the various $\pi\pi$ intermediate states that contribute to the unitarity sum become distinguishable, as a consequence of the explicit breaking of isospin symmetry through the pion mass-difference. The phase is still given by a universal contribution, expressed entirely in terms of data related to the $\pi\pi$ scattering amplitudes, but at next-to-leading order there appears a second contribution, that explicitly depends on the form factors under consideration. The numerical size of this form-factor dependent part is relatively small in the case of the $\pi^+\pi^-$ scalar form factor, but definitely more important in the $\pi^0\pi^0$ case.
We have also investigated the sensitivity of the isospin-breaking correction to two subthreshold parameters $\alpha$ and $\beta$, or equivalently to the two $S$-wave scattering lengths $a_0^0$ and $a_0^2$, which represent the quantities that have to be determined from data. Despite the uncertainties induced by the various other parameters, we find that the correction remains sensitive to the values of the subthreshold parameters. For the range of values that we have considered, and depending on the value of the energy, the effect in the total correction can represent up to 5 milliradians. The issue raised in the introduction about the possibility of a bias if isospin-breaking corrections are evaluated for fixed values of these scattering lengths remains therefore, in our opinion, open. Obviously, the situation that prevails after the present study devoted to the scalar form factors of the pions need not be representative of the one encountered in the case of the $K_{\ell 4}$ form factors. To settle the issue, a dedicated study of this experimentally more interesting case is needed. This will be the subject of a forthcoming article [@wip].
During the long course of this work, we have benefited from numerous discussions and/or from correspondence with several colleagues, to whom we wish to express our gratitude: V. Bernard, B. Bloch-Devaux, J. Gasser, K. Kampf, B. Kubis, H. Leutwyler, B. Moussallam, J. Novotný, U.-G. Mei[ß]{}ner, A. Rusetsky, and M. Zdráhal. We are also indebted to V. Bernard for a very careful and critical reading of the manuscript.
Indefinite integrals of the scalar loop function {#app:integJbar}
================================================
In this appendix, we list the integrals involving the scalar loop function ${\bar J}(t)$ that are used for the computation of the one-loop partial wave projections in section \[2ndIteration\]. For unequal masses $m_1\neq m_2$ \[when necessary, we assume that $m_1>m_2$\], the loop function ${\bar J}(t)$ is given by $${\bar J}(t) = \frac{1}{16\pi^2}\,
\bigg\{1 + \frac{m_1^2 - m_2^2}{t}\ln\frac{m_2}{m_1} -
\frac{m_1^2 + m_2^2}{m_1^2 - m_2^2}\ln\frac{m_2}{m_1} +
\frac{\lambda^{1/2}(t)}{2t}
\ln\frac{[t-\lambda^{1/2}(t)]^2 - (m_1^2 - m_2^2)^2}{[t+\lambda^{1/2}(t)]^2 - (m_1^2 - m_2^2)^2}
\bigg\}$$ where $\lambda(t)=t^2-2t(m_1^2 + m_2^2) + (m_1^2 - m_2^2)^2$. The above expression holds for $t<(m_1 - m_2)^2$. For $m_1 = m_2 = M_\pi$, or $m_1 = m_2 = M_{\pi^0}$, this expression corresponds to the functions ${\bar J}(t)$ and ${\bar J}_0(t)$, respectively, defined in eqs. (\[JbarDisp\]) and (\[Jbar1loop\]). Finally, the function ${\bar J}_{\mbox{\tiny{$\! + $}} 0} (s)$ defined in eqs. (\[Jbar\_+0\_Disp\]) and (\[Jbar\_+0\_Loop\]) corresponds to the choice $m_1 = M_\pi$ and $m_2 = M_{\pi^0}$.
The partial wave projections of the one-loop $\pi\pi$ amplitudes discussed require the computation of integrals of the type $$\int dt t^n {\bar J}(t)$$ where $n$ can take negative or positive integer values. In the present context, the range $-2\le n\le 3$ is particularly relevant. It proves convenient to define the variable $\chi(t)$ through $$t\,=\,m_1^2 + m_2^2 - m_1m_2\left(\chi\,+\,\frac{1}{\chi}\right),$$ so that $0\le\chi\le1$ when $-\infty < t <(m_1 - m_2)^2$. The expression of $\chi$ in terms of $t$ is given in eq. (\[chi\_def\]), upon replacing $M_\pi$ by $m_1$, and $M_{\pi^0}$ by $m_2$, $$\chi(t) \,=\,
\frac{\sqrt{(m_1 + m_2)^2 - t} - \sqrt{(m_1 - m_2)^2 - t}}{\sqrt{(m_1 + m_2)^2 - t} + \sqrt{(m_1 - m_2)^2 - t}}\, .$$ For strictly positive values of the integer $n$, the corresponding (indefinite) integrals take a relatively simple form, even when the masses $m_1$ and $m_2$ are not equal (irrelevant integration constants have been discarded), $$\begin{aligned}
16\pi^2 \int dt\, t {\bar J}(t) &=&
\left[\frac{3}{2} - \frac{m_1^2 + m_2^2}{m_1^2 - m_2^2}\ln\frac{m_2}{m_1}\right]\times\frac{t^2}{2}
\,+\, \left[(m_1^2 - m_2^2)\ln\frac{m_2}{m_1} - \frac{m_1^2 + m_2^2}{2}\right]\times t
\nonumber\\
&&
-\, \frac{1}{2}(t-m_1^2 - m_2^2)\lambda^{1/2}(t)\ln\chi(t) + m_1^2 m_2^2 \ln^2\chi(t)
\\
\nonumber\\
16\pi^2 \int dt\, t^2 {\bar J}(t) &=&
\left[\frac{4}{3} - \frac{m_1^2 + m_2^2}{m_1^2 - m_2^2}\ln\frac{m_2}{m_1}\right]\times\frac{t^3}{3}
\,+\, \left[(m_1^2 - m_2^2)\ln\frac{m_2}{m_1} - \frac{m_1^2 + m_2^2}{6}\right]\times\frac{t^2}{2}
\nonumber\\
&&
-\,\frac{t}{6}\,(m_1^4 + m_2^4 + 10 m_1^2 m_2^2)
\nonumber\\
&&
-\, \frac{1}{6}\left[
2t^2 - (m_1^2 + m_2^2)t - (m_1^4 + m_2^4 + 10 m_1^2 m_2^2)
\right] \lambda^{1/2}(t)\ln\chi(t)
\nonumber\\
&&
+\, m_1^2 m_2^2 (m_1^2 + m_2^2) \ln^2\chi(t)
\\
\nonumber\\
16\pi^2 \int dt\, t^3 {\bar J}(t) &=&
\left[\frac{5}{4} - \frac{m_1^2 + m_2^2}{m_1^2 - m_2^2}\ln\frac{m_2}{m_1}\right]\times\frac{t^4}{4}
\,+\, \left[(m_1^2 - m_2^2)\ln\frac{m_2}{m_1} - \frac{m_1^2 + m_2^2}{12}\right]\times\frac{t^3}{3}
\nonumber\\
&&
-\,\frac{t^2}{24}\,(m_1^4 + m_2^4 + 8 m_1^2 m_2^2)\,-\,
\frac{t}{12}\,(m_1^2 + m_2^2)( m_1^4 + m_2^4 + 28 m_1^2 m_2^2)
\nonumber\\
&&
-\, \frac{1}{12}\left[
3t^3 - (m_1^2 + m_2^2)t^2 - (m_1^4 + m_2^4 + 8 m_1^2 m_2^2)t
\right.
\nonumber\\
&&\qquad\qquad\qquad\left.
- (m_1^2 + m_2^2)(m_1^4 + m_2^4 + 28 m_1^2 m_2^2)
\right] \lambda^{1/2}(t)\ln\chi(t)
\nonumber\\
&&
+\, m_1^2 m_2^2 (m_1^4 + m_2^4 + 3 m_1^2 m_2^2) \ln^2\chi(t)
.\end{aligned}$$ For $n\le 0$, the corresponding expressions are more complicated, at least when the masses are different, but can be given a simpler form when expressed in terms of the function $H_{1,0} (x) \,=\, - {\mbox{Li}}_2 (x) - \ln x \ln (1-x)$, which belongs to the family of harmonic polylogarithms [@HPL]: $$\begin{aligned}
16\pi^2 \int dt\, {\bar J}(t) &=&
\left[2 - \frac{m_1^2 + m_2^2}{m_1^2 - m_2^2}\ln\frac{m_2}{m_1}\right]\times t
\,-\, \lambda^{1/2}(t)\ln\chi(t) \,+\, \frac{m_1^2 + m_2^2}{2} \ln^2\chi(t)
\nonumber\\
&&
+\, (m_1^2 - m_2^2)
\left[
H_{1,0} \left(\frac{m_1}{m_2}\chi(t)\right) \,-\, H_{1,0} \left(\frac{m_2}{m_1}\chi(t)\right)
\,+\,\ln\frac{m_1}{m_2} \, \ln\chi(t)
\right]
\\
\nonumber\\
16\pi^2 \int \frac{dt}{t}\, {\bar J}(t) &=&
\left[m_1^2 + m_2^2 - 2m_1m_2\chi(t)\right]\frac{\ln\chi(t)}{t} - \frac{1}{2}\ln^2\chi(t)
- \ln \chi (t)
- (m_1^2 - m_2^2)\ln\frac{m_2}{m_1}\times\frac{1}{t}
\nonumber\\
&&
-\, \frac{m_1^2 + m_2^2}{m_1^2 - m_2^2}
\left[
H_{1,0} \left(\frac{m_1}{m_2}\chi(t)\right) \,-\, H_{1,0} \left(\frac{m_2}{m_1}\chi(t)\right)
\,+\,\ln\frac{m_1}{m_2} \, \ln\chi(t)
\right]
\\
\nonumber\\
16\pi^2 \int \frac{dt}{t^2}\,\, {\bar{\!\!{\bar J}}}(t) &=&
- \left[\frac{1}{2} - \frac{m_1^2 + m_2^2}{m_1^2 - m_2^2}\ln\frac{m_2}{m_1}\right]\times\frac{1}{t}
\,-\, (m_1^2 - m_2^2)\ln\frac{m_2}{m_1}\times\frac{1}{2t^2}
\nonumber\\
&&
-\,
\frac{\lambda^{1/2}(t)}{2 t^2}
\left[ \frac{m_1^2 + m_2^2}{(m_1^2 - m_2^2)^2} t\,-\,1 \right]
\ln\chi(t)
\nonumber\\
&&
-\, 2\frac{m_1^2 m_2^2}{(m_1^2 - m_2^2)^3}
\left[
H_{1,0} \left(\frac{m_1}{m_2}\chi(t)\right) \,-\, H_{1,0} \left(\frac{m_2}{m_1}\chi(t)\right)
\,+\,\ln\frac{m_1}{m_2} \, \ln\chi(t)
\right]\end{aligned}$$ The range of integration, $t_{\mbox{\tiny $-$}} (s)\le t\le t_{\mbox{\tiny $+$}} (s)$, with $$t_{\mbox{\tiny $\pm$}}(s) \,=\, - \frac{1}{2}\,(s - 2 m_1^2 - 2 m_2^2) \,\pm\,
\frac{1}{2}\,\sqrt{(s - 4 m_1^2)(s - 4 m_2^2)} ,$$ will depend on the process under consideration. As the masses become equal, $m_2 \rightarrow m_1$, one has \[$\Delta_{12}\equiv m_1^2 - m_2^2$\] $$\begin{aligned}
t_{\mbox{\tiny $-$}} (s) & = & - (s - 4 m_1^2) - 2 \Delta_{12} + {\cal O}(\Delta_{12}^2)
\nonumber\\
t_{\mbox{\tiny $+$}} (s) & = & - \frac{\Delta_{12}^2}{s - 4 m_1^2} \,+\, {\cal O}(\Delta_{12}^3) .\end{aligned}$$ Then $$\begin{aligned}
\chi_{\mbox{\tiny $-$}} (s) &=& \frac{1 - \sigma (s)}{1 + \sigma (s)}\,+\, {\cal O}(\Delta_{12})
\nonumber\\
\chi_{\mbox{\tiny $+$}} (s) &=& 1 \,-\, \frac{1}{2} \frac{\Delta_{12}}{m_1^2} \frac{1}{\sigma (s)} \,+\, {\cal O}(\Delta_{12}^2)
\nonumber\\
\lambda( t_{\mbox{\tiny $-$}} (s) ) &=& s (s - 4 m_1^2)\,+\, {\cal O}(\Delta_{12})
\nonumber\\
\lambda( t_{\mbox{\tiny $+$}} (s) ) &=& \frac{s}{s - 4 m_1^2}\,\Delta_{12}^2 \,+\, {\cal O}(\Delta_{12}^3) ,
\label{m1_m2_limit}\end{aligned}$$ where $\chi_{\mbox{\tiny $\pm $}} (s) \equiv \chi (t_{\mbox{\tiny $\pm $}} (s) )$.
Polynomials of the next-to-leading-order $\pi\pi$ partial waves {#app:polynomials}
===============================================================
The expressions of the one-loop the partial-wave projections displayed in eqs. (\[psi\_00\]), (\[psi\_+-\_0\]), (\[psi\_+-\_1\]), and (\[psi\_x\_0\]) involve a certain number of polynomials whose expressions are given in this appendix. In the case of $\psi^{00}_0(s)$, these polynomials read $$\begin{aligned}
\xi_{00}^{(0)}(s) &=&
\frac{\lambda_{00}^{(1)}}{2 M_\pi^4}\,
\left({5}s^2\,-\,{16}M_{\pi^0}^2 s\,+\,20M_{\pi^0}^4\right)
\nonumber\\
&+&
\frac{1}{16\pi^2 M_\pi^4}\bigg\{
\frac{16}{9}\beta_{x}^2 s^2 \,+\,
\frac{1}{18}\beta_{x} s\left[\beta_{x}\left(
9 M_{\pi}^2 - 106 M_{\pi^0}^2\right) \,-\,3\alpha_{x}M_{\pi^0}^2\right]
\nonumber\\
&&
+\,\frac{2}{9}M_{\pi^0}^4\left(2 \alpha_{x}^2 + 9 \alpha_{00}^2\right)
\,-\,\frac{2}{9}\beta_{x}\alpha_{x}M_{\pi^0}^2\left(11 M_{\pi}^2 - 7 M_{\pi^0}^2 \right)
\,+\,\frac{2}{9}\beta_{x}^2\left(34 M_{\pi^0}^4 + 5 M_{\pi}^4
-11 M_{\pi^0}^2 M_{\pi}^2 \right)
\bigg\}
\nonumber\\
\xi^{(1;0)}_{00}(s) &=&\frac{\alpha_{00}^2 M_{\pi^0}^4}{64\pi^2 M_\pi^4}
\nonumber\\
\xi^{(1;{\mbox{\tiny$\nabla$}})}_{00}(s) &=&\frac{1}{32\pi^2 M_\pi^4}\bigg\{
\frac{1}{3}\beta_{x}^2 s^2 \,+\,
\frac{1}{3}\beta_{x} s\left[\beta_{x}\left(
3 M_{\pi}^2 - 6 M_{\pi^0}^2\right) \,-\,\alpha_{x}M_{\pi^0}^2\right]
\nonumber\\
&&
+\,\frac{1}{9} \alpha_{x}^2 M_{\pi^0}^4
\,-\,\frac{2}{9}\beta_{x}\alpha_{x}M_{\pi^0}^2\left(5 M_{\pi}^2 - 4 M_{\pi^0}^2 \right)
\,-\,\frac{1}{9}\beta_{x}^2\left( 2 M_{\pi}^4
+ 16 M_{\pi^0}^2 M_{\pi}^2 - 28 M_{\pi^0}^4\right)
\bigg\}
\nonumber\\
\xi^{(2;0)}_{00}(s) &=&\frac{\alpha_{00}^2 M_{\pi^0}^4}{128\pi^2 M_\pi^4}
\nonumber\\
\xi^{(2;{\mbox{\tiny{$\pm$}}})}_{00}(s)
&=&
\frac{1}{64\pi^2 M_\pi^4}\bigg[
\beta_{x} \left( s - \frac{2}{3} M_{\pi}^2 - \frac{2}{3} M_{\pi^0}^2\right)
\,+\,\frac{1}{3}\alpha_{x} M_{\pi^0}^2
\bigg]^2
\nonumber\\
\xi^{(3;0)}_{00}(s) &=& -\frac{\alpha_{00}^2 M_{\pi^0}^4}{96\pi^2 M_\pi^4}
\nonumber\\
\xi^{(3;{\mbox{\tiny$\nabla$}})}_{00}(s) &=& -\frac{1}{432\pi^2 M_\pi^4}\,
\left[
2\beta_{x}^2(5 M_{\pi}^4 - 2 M_{\pi}^2 M_{\pi^0}^2 + 2 M_{\pi^0}^4)
\,+\,2\beta_{x}\alpha_{x} M_{\pi^0}^2 (M_{\pi}^2 - 2 M_{\pi^0}^2)
\,+\, \alpha_{x}^2 M_{\pi^0}^4
\right]
.\end{aligned}$$
The polynomials involved in the expression for $\psi^{\mbox{\tiny{$+-$}}}_0 (s)$ read $$\begin{aligned}
\xi_{{\mbox{\tiny{$+-$}}};S}^{(0)}(s) &=&
\frac{\lambda_{\mbox{\tiny{$+-$}}}^{(1)} + \lambda_{\mbox{\tiny{$+-$}}}^{(2)}}{2 M_{\pi}^4}\,\left(s-2 M_{\pi}^2\right)^2
\,+\,
\frac{\lambda_{\mbox{\tiny{$+-$}}}^{(1)} + 3\lambda_{\mbox{\tiny{$+-$}}}^{(2)} }{6 M_{\pi}^4}\,
\left(s^2 - 2 s M_{\pi}^2 + 4 M_{\pi}^4
\right)
\nonumber\\
&&
+\,\frac{1}{864\pi^2 M_\pi^4}\,
\left[
\frac{s^2}{8}\left(203\beta_{\mbox{\tiny{$+-$}}}^2 + 300 \beta_{x}^2\right)\,+\,
s M_{\pi^0}^2 \left(36 \beta_{\mbox{\tiny{$+-$}}}\alpha_{\mbox{\tiny{$+-$}}} + \frac{27}{4} \beta_{x} \alpha_{x}\right)
\right.
\nonumber\\
&&\left.
-\,\frac{s}{4} \left( 473 \beta_{\mbox{\tiny{$+-$}}}^2 M_{\pi}^2 + 390 \beta_{x}^2 M_{\pi}^2 +
45 \beta_{x}^2 M_{\pi^0}^2 \right)
\,+\,172 \beta_{\mbox{\tiny{$+-$}}}^2 M_{\pi}^4 \,+\, 108 \beta_{x}^2 M_{\pi}^4
\,+\, 21 \beta_{x}^2 M_{\pi^0}^4
\right.
\nonumber\\
&&\left.
-\, 21 \beta_{x}^2 M_{\pi^0}^2 M_{\pi}^2
\,+\,78 \alpha_{\mbox{\tiny{$+-$}}}^2 M_{\pi^0}^4 \,+\, \frac{15}{2} \alpha_{x}^2 M_{\pi^0}^4
\,-\, M_{\pi^0}^2 \left(
48 \beta_{\mbox{\tiny{$+-$}}} \alpha_{\mbox{\tiny{$+-$}}} M_{\pi}^2 + 39 \beta_{x} \alpha_{x} M_{\pi^0}^2
- 15 \beta_{x} \alpha_{x} M_{\pi}^2\right)
\right]
\nonumber\\
\xi^{(1;{\mbox{\tiny{$\pm$}}})}_{{\mbox{\tiny{$+-$}}};S}(s) &=& \frac{1}{576\pi^2 M_\pi^4}\,
\left[\frac{7}{4}\beta_{\mbox{\tiny{$+-$}}}^2 s^2 - 10 \beta_{\mbox{\tiny{$+-$}}}^2 s M_{\pi}^2 +
15 \beta_{\mbox{\tiny{$+-$}}}^2 M_{\pi}^4 + 6 \alpha_{\mbox{\tiny{$+-$}}}^2 M_{\pi^0}^4
\right]
\nonumber\\
\xi^{(1;{\mbox{\tiny$\Delta$}})}_{{\mbox{\tiny{$+-$}}}; S}(s) &=& \frac{1}{1152\pi^2 M_\pi^4}\,
\left[ 3\beta_{x}^2 s^2 \,-\,9 \beta_{x}^2 s \left(2 M_{\pi}^2 - M_{\pi^0}^2\right)
\,+\, 2 \beta_{x}^2 \left( 14 M_{\pi}^4 - 8 M_{\pi^0}^2 M_{\pi}^2
- M_{\pi^0}^4\right)
\right.
\nonumber\\
&&\left.
-\, 3 \beta_{x} \alpha_{x} s M_{\pi^0}^2
+ 2 \beta_{x} \alpha_{x} M_{\pi^0}^2 \left( 4 M_{\pi}^2 - 5 M_{\pi^0}^2\right)
\,+\, \alpha_{x}^2 M_{\pi^0}^4
\right]
\nonumber\\
\xi^{(2;0)}_{{\mbox{\tiny{$+-$}}};S}(s) &=&
\frac{1}{128 \pi^2 M_\pi^4}\,
\bigg[
\beta_{x} \left( s - \frac{2}{3} M_{\pi}^2 - \frac{2}{3} M_{\pi^0}^2\right)
\,+\,\frac{1}{3}\alpha_{x} M_{\pi^0}^2
\bigg]^2
\nonumber\\
\xi^{(2;{\mbox{\tiny{$\pm$}}})}_{{\mbox{\tiny{$+-$}}};S}(s) &=&
\frac{1}{64 \pi^2 M_\pi^4}\,
\left[
\frac{\beta_{\mbox{\tiny{$+-$}}}}{2} \left( s - \frac{4}{3} M_{\pi}^2 \right)
\,+\,\frac{2}{3} \alpha_{\mbox{\tiny{$+-$}}} M_{\pi^0}^2
\right]^2
\nonumber\\
\xi^{(3;{\mbox{\tiny{$\pm$}}})}_{{\mbox{\tiny{$+-$}}};S}(s) &=& - \frac{1}{288\pi^2 M_\pi^4}\,
\left[
5 \beta_{\mbox{\tiny{$+-$}}}^2 M_{\pi}^4 \,+\,
2 \alpha_{\mbox{\tiny{$+-$}}}^2 M_{\pi^0}^4\,-\,\frac{3}{2} \beta_{\mbox{\tiny{$+-$}}}^2 s M_{\pi}^2
\right]
\nonumber\\
\xi^{(3;{\mbox{\tiny$\Delta$}})}_{{\mbox{\tiny{$+-$}}};S}(s) &=& \frac{1}{1728 \pi^2 M_\pi^4}\,
\left[
2{\beta_{x}^2} \left(
-2 M_{\pi}^4 + 2 M_{\pi}^2 M_{\pi^0}^2 - 5 M_{\pi^0}^4 \right)
\,+\,
2 \beta_{x} \alpha_{x} M_{\pi^0}^2 \left(2 M_{\pi}^2 - M_{\pi^0}^2\right)
\,-\,
\alpha_{x}^2\, M_{\pi^0}^4
\right]
,\end{aligned}$$ while for $\psi^{\mbox{\tiny{$+-$}}}_1 (s)$ we obtain $$\begin{aligned}
\xi_{{\mbox{\tiny{$+-$}}};P}^{(0)}(s) &=&
- \frac{\lambda_{\mbox{\tiny{$+-$}}}^{(1)} - \lambda_{\mbox{\tiny{$+-$}}}^{(2)}}{12 M_{\pi}^4}\,s\left(s-4 M_{\pi}^2\right)
\nonumber\\
&&
+\,\frac{1}{3456\pi^2 M_\pi^4}\,
\left[
\frac{s^2}{4}\left(71\beta_{\mbox{\tiny{$+-$}}}^2 - 75 \beta_{x}^2\right)\,+\,
s M_{\pi^0}^2 \left(44 \beta_{\mbox{\tiny{$+-$}}}\alpha_{\mbox{\tiny{$+-$}}} + 11 \beta_{x} \alpha_{x}\right)
\right.
\nonumber\\
&&\left.
-\,s\left( 169 \beta_{\mbox{\tiny{$+-$}}}^2 M_{\pi}^2 - 128 \beta_{x}^2 M_{\pi}^2 +
19 \beta_{x}^2 M_{\pi^0}^2 \right)
\,+\,341 \beta_{\mbox{\tiny{$+-$}}}^2 M_{\pi}^4 \,-\, 200 \beta_{x}^2 M_{\pi}^4 \,-\, 3 \beta_{x}^2 M_{\pi^0}^4
\right.
\nonumber\\
&&\left.
\,+\, 148 \beta_{x}^2 M_{\pi^0}^2 M_{\pi}^2
\,+\,12 \alpha_{\mbox{\tiny{$+-$}}}^2 M_{\pi^0}^4 \,+\, 3 \alpha_{x}^2 M_{\pi^0}^4
\,-\,4 M_{\pi^0}^2 \left(
92 \beta_{\mbox{\tiny{$+-$}}} \alpha_{\mbox{\tiny{$+-$}}} M_{\pi}^2 + 9 \beta_{x} \alpha_{x} M_{\pi^0}^2
+ 14 \beta_{x} \alpha_{x} M_{\pi}^2\right)
\right]
\nonumber\\
\xi^{(1;{\mbox{\tiny{$\pm$}}})}_{{\mbox{\tiny{$+-$}}};P}(s) &=& \frac{1}{1152\pi^2 M_\pi^4}\,
\left[\beta_{\mbox{\tiny{$+-$}}}^2 s^2 - 7 \beta_{\mbox{\tiny{$+-$}}}^2 s M_{\pi}^2 + 21 \beta_{\mbox{\tiny{$+-$}}}^2 M_{\pi}^4
+ 4 \beta_{\mbox{\tiny{$+-$}}} \alpha_{\mbox{\tiny{$+-$}}} s M_{\pi^0}^2
- 24 \beta_{\mbox{\tiny{$+-$}}} \alpha_{\mbox{\tiny{$+-$}}} M_{\pi^0}^2 M_{\pi}^2
\right]
\nonumber\\
\xi^{(1;{\mbox{\tiny$\Delta$}})}_{{\mbox{\tiny{$+-$}}}; P}(s) &=& \frac{1}{576\pi^2 M_\pi^4}\,
\left[ - \frac{3}{4}\beta_{x}^2 s^2 \,+\, \beta_{x}^2 s \left(5 M_{\pi}^2 - M_{\pi^0}^2\right)
\,-\, \beta_{x}^2 \left( 8 M_{\pi}^4 - 6 M_{\pi^0}^2 M_{\pi}^2
- \frac{1}{2} M_{\pi^0}^4\right)
\right.
\nonumber\\
&&\left.
+\, \frac{1}{2} \beta_{x} \alpha_{x} s M_{\pi^0}^2
- \beta_{x} \alpha_{x} M_{\pi^0}^2 \left( 2 M_{\pi}^2 + M_{\pi^0}^2\right)
\right]
\nonumber\\
\xi^{(2;{\mbox{\tiny{$\pm$}}})}_{{\mbox{\tiny{$+-$}}};P}(s) &=&
\frac{1}{2304\pi^2 M_\pi^4}\,\beta_{\mbox{\tiny{$+-$}}}^2 \left( s - 4 M_{\pi}^2 \right)^2
\nonumber\\
\xi^{(3;{\mbox{\tiny{$\pm$}}})}_{{\mbox{\tiny{$+-$}}};P}(s) &=& \frac{1}{864\pi^2 M_\pi^4}\,
\left[
\beta_{\mbox{\tiny{$+-$}}}^2 M_{\pi}^4 \,+\,
4 \beta_{\mbox{\tiny{$+-$}}} \alpha_{\mbox{\tiny{$+-$}}} M_{\pi}^2 M_{\pi^0}^2\,-\,
2 \alpha_{\mbox{\tiny{$+-$}}}^2 M_{\pi^0}^4\,+\,\frac{9}{2} \beta_{\mbox{\tiny{$+-$}}}^2 s M_{\pi}^2
\right]
\nonumber\\
\xi^{(3;{\mbox{\tiny$\Delta$}})}_{{\mbox{\tiny{$+-$}}}; P}(s) &=& \frac{1}{864\pi^2 M_\pi^4}\,
\left[
{\beta_{x}^2} \left(
-2 M_{\pi}^4 + 2 M_{\pi}^2 M_{\pi^0}^2 - 5 M_{\pi^0}^4 \right)
\,+\,
\beta_{x} \alpha_{x} M_{\pi^0}^2 \left(2 M_{\pi}^2 - M_{\pi^0}^2\right)
\,-\,
\frac{\alpha_{x}^2}{2}\, M_{\pi^0}^4
\right]
\nonumber\\
\xi^{(4;{\mbox{\tiny{$\pm$}}})}_{{\mbox{\tiny{$+-$}}};P}(s) &=& \frac{1}{288\pi^2 M_\pi^4}\,
\left[
23 \beta_{\mbox{\tiny{$+-$}}}^2 M_{\pi}^4 \,-\,
16 \beta_{\mbox{\tiny{$+-$}}} \alpha_{\mbox{\tiny{$+-$}}} M_{\pi}^2 M_{\pi^0}^2\,-\,
4 \alpha_{\mbox{\tiny{$+-$}}}^2 M_{\pi^0}^4
\right]
\nonumber\\
\xi^{(4;{\mbox{\tiny$\Delta$}})}_{{\mbox{\tiny{$+-$}}}; P}(s) &=& \frac{1}{288\pi^2 M_\pi^4}\,
\left[
{\beta_{x}^2} \left(
-4 M_{\pi}^4 +16 M_{\pi}^2 M_{\pi^0}^2 - 25 M_{\pi^0}^4 \right)
\,-\,
{4} \beta_{x} \alpha_{x} M_{\pi^0}^2 \left( 2 M_{\pi^0}^2 - M_{\pi}^2\right)
\,-\,
\alpha_{x}^2 M_{\pi^0}^4
\right]
.\end{aligned}$$
Finally, for $\psi^{x}_0(s)$ the polynomials are given by $$\begin{aligned}
\nonumber\\
\xi_{x}^{(0)}(s) &=& -\,\frac{\lambda_{x}^{(1)}}{2 M_{\pi}^4}\,\left(s-2 M_{\pi}^2\right)\left(s-2 M_{\pi^0}^2\right)
\,-\,
\frac{\lambda_{x}^{(2)} }{3 M_{\pi}^4}\,
\left(s^2 - s M_{\pi}^2 - s M_{\pi^0}^2 + 4 M_{\pi}^2 M_{\pi^0}^2
\right)
\nonumber\\
&& - \frac{1}{864\pi^2 M_\pi^4}\,
\left\{
\frac{s^2}{4}\beta_{x}\left(108\beta_{\mbox{\tiny{$+-$}}} + 11 \beta_{x}\right)\,+\,
s \beta_{x} M_{\pi^0}^2 \left(36 \alpha_{\mbox{\tiny{$+-$}}} + \frac{45}{2} \alpha_{x}
+ 27 \alpha_{00}\right)
\right.
\nonumber\\
&&\left.
-\,s \beta_{x}\left[ 18 \beta_{\mbox{\tiny{$+-$}}}\left( 3M_{\pi}^2 + M_{\pi^0}^2\right)
+ \frac{91}{2} \beta_{x} \left( M_{\pi}^2 + M_{\pi^0}^2\right) \right]
\,+\,9 s\beta_{\mbox{\tiny{$+-$}}}\alpha_{x} M_{\pi^0}^2
\right.
\nonumber\\
&&\left.
+\, \beta_{x}\left[ \beta_{x}\left(144 M_{\pi}^4 + 144 M_{\pi^0}^4
- 112 M_{\pi^0}^2 M_{\pi}^2\right)\,+\,
24 \beta_{\mbox{\tiny{$+-$}}} M_{\pi}^2 \left( M_{\pi}^2 + M_{\pi^0}^2\right) \right]
\right.
\nonumber\\
&&\left.
-\, 6 \beta_{x} M_{\pi^0}^2 \left(
4 \alpha_{\mbox{\tiny{$+-$}}}
+ 3 \alpha_{00} \right) \left( M_{\pi}^2 + M_{\pi^0}^2\right)
\,-\, 12\beta_{\mbox{\tiny{$+-$}}}\alpha_{x} M_{\pi^0}^2 M_{\pi}^2
\,+\,3 \alpha_{x} M_{\pi^0}^4 \left(
3 \alpha_{00} + 4 \alpha_{\mbox{\tiny{$+-$}}} + 6 \alpha_{x} \right)
\right\}
\nonumber\\
\xi_{x}^{(1)}(s) &=& - \frac{1}{288\pi^2 M_\pi^4}\,
\left\{
\frac{1}{4}\beta_{x}^2 s^2\,-\,\beta_{x} s \left[
\beta_{x} \left( 3 M_{\pi}^2 + \frac{5}{2} M_{\pi^0}^2\right)
- \frac{3}{2} \alpha_{x} M_{\pi^0}^2 \right]
\right.
\nonumber\\
&&
\left.
\,+\,\alpha_{x}^2 M_{\pi^0}^4
\,+\,\beta_{x}\alpha_{x} M_{\pi^0}^2 \left( 2 M_{\pi^0}^2 - M_{\pi}^2\right)
\,+\,\beta_{x}^2
\left( 7 M_{\pi}^4 + 10 M_{\pi^0}^4 - 7 M_{\pi}^2 M_{\pi^0}^2 \right)
\right\}
\nonumber\\
\nonumber\\
\xi_{x}^{(2;0)}(s) &=& - \frac{1}{128 \pi^2 M_\pi^4}\,
\alpha_{00} M_{\pi^0}^2 \left[\beta_{x} s \,-\,
\frac{2}{3}\beta_{x} \left( M_{\pi}^2 + M_{\pi^0}^2\right)\,+\,
\frac{1}{3}\alpha_{x} M_{\pi^0}^2 \right]
\nonumber\\
\nonumber\\
\xi_{x}^{(2;\pm)}(s) &=& - \frac{1}{128\pi^2 M_\pi^4}
\left[\beta_{x} s \,-\,
\frac{2}{3}\beta_{x} \left( M_{\pi}^2 + M_{\pi^0}^2\right)\,+\,
\frac{1}{3}\alpha_{x} M_{\pi^0}^2 \right]
\left[
\beta_{\mbox{\tiny{$+-$}}} \left( s - \frac{4}{3} M_{\pi}^2 \right)
+ \frac{4}{3} \alpha_{\mbox{\tiny{$+-$}}} M_{\pi^0}^2
\right]
\nonumber\\
\nonumber\\
\xi_{x}^{(3)}(s) &=& \frac{1}{864 \pi^2 M_\pi^4}\,\left[
-3 \beta_{x}^2 \frac{s}{M_{\pi}^2}\,
\left(M_{\pi}^4 + M_{\pi^0}^4 + M_{\pi}^2 M_{\pi^0}^2\right)\,
\right.
\nonumber\\
&&
\left. +\,
2 \beta_{x}\alpha_{x} M_{\pi^0}^2 \left( M_{\pi}^2 - M_{\pi^0}^2
+ \frac{M_{\pi^0}^4}{M_{\pi}^2}\right)
\,+\,\alpha_{x}^2 M_{\pi^0}^4 \left( 1 \,+\,\frac{M_{\pi^0}^2}{M_{\pi}^2}\right)
\right.
\nonumber\\
&&
\left. +\,
10 \beta_{x}^2 \left( 1 \,+\,\frac{M_{\pi^0}^2}{M_{\pi}^2}\right)
\left( M_{\pi}^4 + M_{\pi^0}^4 - M_{\pi}^2 M_{\pi^0}^2 \right)
\right]\end{aligned}$$ In addition, eq. (\[psi\_x\_0\]) involves two other contributions, $$\begin{aligned}
16\pi \Delta_1 \psi^{x}_0(s) &=&
\frac{1}{96\pi^2 F_\pi^4} \left[
\frac{1}{6} \beta_{x}^2 s \,+\,\beta_{x} \alpha_{x} M_{\pi^0}^2
\,+\, \beta_{x}^2 \left( M_{\pi}^2 + M_{\pi^0}^2 \right)
\right]
\nonumber\\
&&
\times
\left[
\left(\sqrt{\frac{s - 4 M_{\pi}^2}{s- 4 M_{\pi^0}^2}}\,-\,1\right)
\lambda^{1/2}(t_{\mbox{\tiny $-$}}(s)) {\cal L}_{\mbox{\tiny $-$}} (s)
\,-\,
\left(\sqrt{\frac{s - 4 M_{\pi}^2}{s- 4 M_{\pi^0}^2}}\,+\,1\right)
\lambda^{1/2}(t_{\mbox{\tiny $+$}}(s)) {\cal L}_{\mbox{\tiny $+$}} (s)
\right]
,\qquad{ }\end{aligned}$$ and $$\begin{aligned}
16\pi \Delta_2 \psi^{x}_0(s) &=& \frac{1}{144\pi^2 F_\pi^4}\,\left[
1 \,-\, \frac{M_{\pi}^2 + M_{\pi^0}^2}{M_{\pi}^2 - M_{\pi^0}^2}\,
\ln\frac{M_{\pi}}{M_{\pi^0}}
\right]\times
\bigg[
\frac{1}{2} \beta_{x}^2 s^2
\,-\, 5\beta_{x}^2 s \left(M_{\pi}^2 + M_{\pi^0}^2\right)
\nonumber\\
&&
\,+\, 2 \beta_{x}^2 \left( 7 M_{\pi}^4 + 7 M_{\pi^0}^4 - 6 M_{\pi}^2 M_{\pi^0}^2\right)
\,+\,3\beta_{x}\alpha_{x} M_{\pi^0}^2 s \,-\,
2\beta_{x}\alpha_{x} M_{\pi^0}^2 \left(M_{\pi}^2 + M_{\pi^0}^2\right) \,
\,+\,2 \alpha_{x}^2 M_{\pi^0}^4
\bigg]
\nonumber\\
&&\, -
\frac{1}{144\pi^2 F_\pi^4}\,
\frac{1}{\sqrt{(s- 4 M_{\pi}^2)(s - 4 M_{\pi^0}^2)}}\,
\bigg\{
\left[4 \beta_{x}^2(5 M_{\pi}^4 - 2 M_{\pi}^2 M_{\pi^0}^2 + 5 M_{\pi^0}^4)
- 6 \beta_{x}^2 s (M_{\pi}^2 + M_{\pi^0}^2)
\right.
\nonumber\\
&&
\left.
\,+\,4 \beta_{x} \alpha_{x} M_{\pi^0}^2 (M_{\pi}^2 + M_{\pi^0}^2)\,+\,
2 \alpha_{x}^2 M_{\pi^0}^4
\right]
\left[ {\cal F}_{\mbox{\tiny $+$}}(s) - {\cal F}_{\mbox{\tiny $-$}}(s) \right]
\nonumber\\
&&
+\, 6 \beta_x\left[
2 \beta_{x} (M_{\pi}^2 + M_{\pi^0}^2) + \alpha_{x} M_{\pi^0}^2
- \frac{s}{2} \beta_{x} \right]
\left[ {\cal G}_{\mbox{\tiny $+$}}(s) - {\cal G}_{\mbox{\tiny $-$}}(s) \right]
\nonumber\\
&&
+\,3 \beta_x^2
\left[ {\cal H}_{\mbox{\tiny $+$}}(s) - {\cal H}_{\mbox{\tiny $-$}}(s) \right]
\bigg\} ,\end{aligned}$$ where ${\cal F}_{\mbox{\tiny $\pm $}}(s) \equiv {\cal F} (t_{\mbox{\tiny $\pm $}}(s))$, and similar definitions for ${\cal G}_{\mbox{\tiny $\pm $}}(s)$ and ${\cal H}_{\mbox{\tiny $\pm $}}(s)$, with $$t_{\mbox{\tiny{$\pm$}}}(s) \,=\, -\frac{1}{2} (s - 2 M_{\pi}^2 - 2 M_{\pi^0}^2)
\,\pm\,
\frac{1}{2}\,\sqrt{(s- 4 M_{\pi}^2)(s - 4 M_{\pi^0}^2)} ,$$ and $$\begin{aligned}
{\cal F}(t) &=& (M_{\pi}^2 - M_{\pi^0}^2)\left[
H_{1,0} \left(\frac{M_{\pi}}{M_{\pi^0}}\chi(t)\right) \,-\, H_{1,0} \left(\frac{M_{\pi^0}}{M_{\pi}}\chi(t)\right)
\,+\,\ln\frac{M_{\pi}}{M_{\pi^0}} \, \ln\chi(t)
\right]
\nonumber\\
{\cal G}(t) &=& (M_{\pi}^2 + M_{\pi^0}^2){\cal F}(t) + (M_{\pi}^2 - M_{\pi^0}^2) t \ln\frac{M_{\pi}}{M_{\pi^0}}
\nonumber\\
&&\!\!\!
- (M_{\pi}^2 - M_{\pi^0}^2)^2 \left[
\left( M_{\pi}^2 + M_{\pi^0}^2 - 2 M_{\pi} M_{\pi^0} \chi(t) \right) \frac{\ln\chi(t)}{t}
- \frac{1}{2} \ln^2\chi(t) - \ln\chi(t) + \frac{M_{\pi}^2 - M_{\pi^0}^2}{t} \ln\frac{M_{\pi}}{M_{\pi^0}}
\right]
\nonumber\\
{\cal H}(t) &=& - 4 M_{\pi}^2 M_{\pi^0}^2 {\cal F}(t)
- (M_{\pi}^2 - M_{\pi^0}^2) t^2 \ln\frac{M_{\pi}}{M_{\pi^0}}
\nonumber\\
&&\!\!\!
- (M_{\pi}^2 - M_{\pi^0}^2)^4 \left[
2 \left(
\frac{1}{2} + \frac{M_{\pi}^2 + M_{\pi^0}^2}{M_{\pi}^2 - M_{\pi^0}^2} \ln\frac{M_{\pi}}{M_{\pi^0}}
\right) \frac{1}{t}
- \frac{M_{\pi}^2 - M_{\pi^0}^2}{t^2} \ln\frac{M_{\pi}}{M_{\pi^0}}
\right.
\nonumber\\
&&
\left.
\qquad\qquad
+ \frac{1}{t^2}
\left(
\frac{M_{\pi}^2 + M_{\pi^0}^2}{(M_{\pi}^2 - M_{\pi^0}^2)^2} t - 1
\right) \lambda^{1/2}(t) \ln \chi(t) \right]
.\end{aligned}$$ Notice that individual terms in $\Delta_2 \psi^{x}_0(s)$ may behave as ${\cal O}[\Delta_\pi^2 \times \ln(\Delta_\pi / M_\pi^2)]$ in the isospin limit, but cancellations occur between these terms, so that overall $\Delta_2 \psi^{x}_0(s) = {\cal O}(\Delta_\pi^2)$.
Two-loop form factors in the isospin limit {#app:IsoLimit}
==========================================
In this appendix, we discuss the scalar and vector form factors at two loops in the isospin limit, where the complications due to the mass difference between neutral and charged pions are absent, and the dispersive integrals can all be expressed in terms of the known function ${\bar J}(s)$. The two form factors $F_S^{\pi^0}$ and $F_S^{\pi^\pm}$ then become identical, since in both cases the two-pion states are projected on their $I=0$, S-wave components, with identical Clebsch-Gordan coefficients. Furthermore, this exercise will provide a check of the calculation in the general case, which has to reduce to the expressions to be found below in the limit $M_{\pi^0} \to M_{\pi}$. Let us recall that ref. [@GasserMeissner91] did not give analytical expressions for the form factors at two loops. Analytical two-loop expressions for the scalar and the vector form factors were given in [@Bijnens98]. The analytical expression for the vector form factor only had also been given earlier in ref. [@Colangelo96].
For the first iteration, the discontinuities of the form factors reduce to \[in this appendix, we omit the superscript $\pi$ most of the time\] $$\begin{aligned}
{\mbox{Im}}F_S(s) &=&
\sigma(s)F_S (0)
\varphi_0(s)
\theta(s-4M_{\pi}^2)\,+\,{\cal O}(E^6)
\nonumber\\
{\mbox{Im}}F_V (s) &=&
\sigma(s)
\varphi_1(s)
\theta(s-4M_{\pi}^2)\,+\,{\cal O}(E^4)\,,\end{aligned}$$ with $$\begin{aligned}
\varphi_0 (s)&=& \varphi_0^{\mbox{\tiny{$+-$}}}(s) \,-\,\frac{1}{2}\,\varphi_0^{x}(s)
\,=\,
\frac{1}{2}\,\varphi_0^{00}(s) \,-\, \varphi_0^{x}(s)
\,=\,
\frac{1}{16\pi F_\pi^2}\,\left[
\beta \left( s - \frac{4}{3}\,M_\pi^2 \right) \,+\, \frac{5}{6}\, \alpha M_\pi^2
\right]
\nonumber\\
\varphi_1 (s) &=& \varphi_1^{\mbox{\tiny{$+-$}}}(s)\,=\,\frac{1}{96\pi F_\pi^2}
\,\beta(s - 4M_\pi^2).\end{aligned}$$ The one-loop expressions of the form factors in the isospin limit read $$\begin{aligned}
F_S (s) &=& F_S (0) \left[
1 \,+\, \frac{1}{6}\langle r^2\rangle_S^{\pi}\,{s}
\,+\, c_S^{\pi}\,{s^2}
\,+\, U_S(s)
\right]
\\
\nonumber\\
F_V &=& 1 \,+\, \frac{1}{6}\langle r^2\rangle_V^{\pi}\, {s}
\,+\, c_V^{\pi}\, {s^2}
\,+\, U_V(s)
,\end{aligned}$$ with $$\begin{aligned}
U_S (s) &=& 16\pi\varphi_0(s){\bar J}(s)\,+\,
\frac{M_\pi^2}{16\pi^2 F_\pi^2}\,\bigg\{
\frac{1}{36}\,\frac{s}{M_\pi^2}\,(8\beta - 5\alpha)
\,-\,
\frac{1}{360}\,\left(\frac{s}{M_\pi^2}\right)^2\,(52\beta + 5\alpha)
\bigg\}
\\
\nonumber\\
U_V (s) &=& \frac{M_\pi^2}{16\pi^2 F_\pi^2}\,\beta\,\left\{
\frac{1}{9}\,\frac{s}{M_\pi^2}\,\left[1\,+\,24\pi^2\sigma^2(s){\bar J}(s)\right]
\,-\,\frac{1}{60}\,\left(\frac{s}{M_\pi^2}\right)^2
\right\} .\end{aligned}$$
In order to implement the second iteration, it is necessary to include the next-to-leading contributions to the discontinuities of the form factors, $$\begin{aligned}
{\mbox{Im}}F_S (s) &=&
\sigma(s)F_S (0) \left\{
\varphi_0(s)\left[
1 \,+\,\Gamma_S (s) \right] \,+\,\psi_0(s)
\right\}
\theta(s-4M_{\pi}^2)\,+\,{\cal O}(E^6)
\nonumber\\
{\mbox{Im}}F_V (s) &=&
\sigma(s) \left\{
\varphi_1(s)\left[
1 \,+\,\Gamma_V (s) \right] \,+\,\psi_1(s)
\right\}
\theta(s-4M_{\pi}^2)\,+\,{\cal O}(E^6)
.
\nonumber\end{aligned}$$ The relevant one-loop corrections $\Gamma_S (s)$ and $\Gamma_V (s)$ to the real parts of the form factors are easy to obtain from their expressions given above, $$\begin{aligned}
\Gamma_S (s) &=& \frac{s}{6} \left[
\langle r^2\rangle_S^{\pi}
\,+\,\frac{1}{96\pi^2 F_\pi^2}\,(8\beta - 5\alpha)\right]
\,+\,\frac{1}{\pi}\,\varphi_0(s)[2 + \sigma(s) L(s)]
\\
\nonumber\\
\Gamma_V (s) &=&\frac{s}{6} \left[
\langle r^2\rangle_V^{\pi}
\,+\,\frac{1}{24\pi^2 F_\pi^2}\,\beta \right]
\,+\, \frac{1}{\pi}\,\varphi_1(s)
[2 + \sigma(s) L(s)]
.
\nonumber\end{aligned}$$
As for the one-loop contributions to the real parts of the $S$ and $P$ partial waves, they are conveniently expressed as $$\begin{aligned}
\psi_0 (s)&=& \psi_0^{\mbox{\tiny{$+-$}}}(s) \,-\,\frac{1}{2}\,\psi_0^{x}(s)
\,=\,
\frac{1}{2}\,\psi_0^{00}(s) \,-\, \psi_0^{x}(s)
\nonumber\\
\psi_1 (s) &=& \psi_1^{\mbox{\tiny{$+-$}}}(s),
\nonumber\end{aligned}$$ where the expressions for $\psi_0^{00}(s)$, $\psi_{0,1}^{\mbox{\tiny{$+-$}}}(s)$, and $\psi_0^{x}(s)$ are given in eqs. (\[lim\_psi\_00\]), (\[lim\_psi\_+-\]), and (\[lim\_psi\_x\]), respectively. They involve the polynomials $\xi_{a}^{(n)} (s)$ of reference [@KMSF95]. We reproduce them in a slightly different notation, in terms of the variable $s$ instead of the relative momentum $q = \sqrt{s/4M_\pi^2 -1}$ in the center-of-mass frame, $$\begin{aligned}
\xi_0^{(0)}(s) &=& \frac{1}{432\pi^2}\,(105\alpha^2 - 120 \alpha\beta + 392 \beta^2)\,+\,
\frac{2}{3}\,(11\lambda_1 + 14\lambda_2)
\nonumber\\
&&
+\,\left\{
\frac{1}{864\pi^2}\,(180\alpha - 617 \beta)\beta \,-\,
\frac{20}{3}\,(\lambda_1 + \lambda_2)
\right\}\,\frac{s}{M_\pi^2}
\nonumber\\
&&
+\,\left\{
\frac{311}{1728\pi^2}\,\beta^2 \,+\, \frac{1}{6}\,(11\lambda_1 + 14\lambda_2)
\right\}\,\left(\frac{s }{M_\pi^2}\right)^2
\nonumber\\
\xi_0^{(1)}(s) &=& \frac{5}{192\pi^2}\,(\alpha^2 + 4 \beta^2)\,-\,
\frac{5}{72\pi^2}\,\beta^2\,\frac{s }{M_\pi^2}\,+\,
\frac{7}{576\pi^2}\,\beta^2\,\left(\frac{s }{M_\pi^2}\right)^2
\nonumber\\
\xi_0^{(2)}(s) &=& \frac{1}{1152\pi^2}\,(25\alpha^2 - 80 \alpha\beta + 64 \beta^2)\,+\,
\frac{1}{96\pi^2}\,\beta (5\alpha - 8 \beta)\,\frac{s}{M_\pi^2}\,+\,
\frac{1}{32\pi^2}\,\beta^2\,\left(\frac{s }{M_\pi^2}\right)^2
\nonumber\\
\xi_0^{(3)}(s) &=& -\frac{5}{288\pi^2}\,( \alpha^2 + 4 \beta^2)\,+\,
\frac{1}{48\pi^2}\,\beta^2\,\frac{s }{M_\pi^2}
\nonumber\\
\nonumber\\
\xi_2^{(0)}(s) &=& \frac{1}{864 \pi^2}\,(93\alpha^2 + 48 \alpha\beta + 112 \beta^2)\,+\,
\frac{4}{3}\,(\lambda_1 + 4\lambda_2)
\nonumber\\
&&
-\,\left\{
\frac{1}{1728 \pi^2}\,(207\alpha + 256 \beta)\beta \,+\,
\frac{2}{3}\,(\lambda_1 + 7 \lambda_2)
\right\}\,\frac{s}{M_\pi^2}
\nonumber\\
&&
+\,\left\{
\frac{265}{3456 \pi^2}\,\beta^2 \,+\, \frac{1}{3}\,(\lambda_1 + 4\lambda_2)
\right\}\,\left(\frac{s }{M_\pi^2}\right)^2
\nonumber\\
\xi_2^{(1)}(s) &=& \frac{1}{192\pi^2}\,(3\alpha^2 - 2 \alpha \beta)\,-\,
\frac{1}{576\pi^2}\,\beta (9 \alpha + 7 \beta )\,\frac{s }{M_\pi^2}\,+\,
\frac{11}{1152\pi^2}\,\beta^2\,\left(\frac{s }{M_\pi^2}\right)^2
\nonumber\\
\xi_2^{(2)}(s) &=& \frac{1}{288\pi^2}\,(\alpha^2 + 4 \alpha\beta + 4 \beta^2)\,-\,
\frac{1}{96\pi^2}\,\beta (\alpha + 2 \beta)\,\frac{s}{M_\pi^2}\,+\,
\frac{1}{128\pi^2}\,\beta^2\,\left(\frac{s }{M_\pi^2}\right)^2
\nonumber\\
\xi_2^{(3)}(s) &=& \frac{1}{288\pi^2}\,( -3\alpha^2 + 2 \alpha \beta)\,-\,
\frac{1}{96\pi^2}\,\beta^2\,\frac{s }{M_\pi^2}
\nonumber\\
\nonumber\\
\xi_1^{(0)}(s) &=& \frac{1}{1728\pi^2}\,(15\alpha^2 - 460 \alpha\beta + 286 \beta^2)
\nonumber\\
&&
+\,\left\{
\frac{5}{1728\pi^2}\,(11\alpha - 12 \beta)\beta \,+\,
\frac{2}{3}\,(\lambda_1 - \lambda_2)
\right\}\,\frac{s}{M_\pi^2}
\nonumber\\
&&
-\,\left\{
\frac{1}{1728\pi^2}\,\beta^2 \,+\, \frac{1}{6}\,(\lambda_1 - \lambda_2)
\right\}\,\left(\frac{s }{M_\pi^2}\right)^2
\nonumber\\
\xi_1^{(1)}(s) &=& -\frac{1}{96\pi^2}\,\beta(5\alpha - 3 \beta)\,+\,
\frac{1}{576\pi^2}\,\beta(5\alpha + \beta)\,\frac{s }{M_\pi^2}\,-\,
\frac{1}{1152\pi^2}\,\beta^2\,\left(\frac{s }{M_\pi^2}\right)^2
\nonumber\\
\xi_1^{(2)}(s) &=& \frac{1}{72\pi^2}\,\beta^2\,\left(\frac{s - 4M_\pi^2}{4M_\pi^2}\right)^2
\nonumber\\
\xi_1^{(3)}(s) &=& \frac{1}{864\pi^2}\,
(-5\alpha^2 + 10 \alpha\beta - 8 \beta^2) \,+\,
\frac{1}{96\pi^2}\,\beta^2\,\frac{s }{M_\pi^2}
\nonumber\\
\xi_1^{(4)}(s) &=& - \frac{5}{144\pi^2}\,(\alpha^2 + 4\alpha\beta - 2 \beta^2)
\,.
\nonumber\end{aligned}$$ Putting all elements together leads to $$\begin{aligned}
\frac{1}{F_S (0)}\,{\mbox{disc}}\,F_S (s) &=&
\sum_{n=0}^3 {\cal S}_n(s) k_n(s) \times \theta(s-4M_\pi^2 )
\nonumber\\
\\
{\mbox{disc}}\,F_V (s) &=&
\sum_{n=0}^4 {\cal V}_n(s) k_n(s) \times \theta(s-4M_\pi^2 )\end{aligned}$$ with $$\begin{aligned}
{\cal S}_n(s) &=& 16\pi\varphi_0(s)\delta_{n,0}
\nonumber\\
&& +\,
\left\{
\frac{8\pi s}{3}\left[
\langle r^2\rangle_S^{\pi}
\,+\,\frac{1}{96\pi^2 F_\pi^2}\,(8\beta - 5\alpha)\right] \,+\,
32 \varphi_0(s)\right\} \varphi_0(s)\delta_{n,0}
\nonumber\\
&&
+\,8\left[\varphi_0(s)\right]^2 \delta_{n,2}
\,+\, \frac{M_\pi^4}{F_\pi^4}\, \xi_0^{(n)}(s)
\nonumber\\
{\cal V}_n(s) &=& 16\pi\varphi_1(s)\delta_{n,0}
\nonumber\\
&& +\,
\left\{
\frac{8\pi s}{3}\left[
\langle r^2\rangle_V^{\pi}
\,+\,\frac{1}{24\pi^2 F_\pi^2}\,\beta \right] \,+\,
32 \varphi_1(s)\right\} \varphi_1(s)\delta_{n,0}
\nonumber\\
&&
+\,8\left[\varphi_1(s)\right]^2 \delta_{n,2}
\,+\, \frac{M_\pi^4}{F_\pi^4}\, \xi_1^{(n)}(s)
.\end{aligned}$$ Performing the dispersive integrals gives $$\begin{aligned}
U_S (s) &=&
\sum_{n=0}^3 {\cal S}_n(s) {\bar K}_n(s) \,+\, P_S (s)
\nonumber\\
\\
U_V (s) &=&
\sum_{n=0}^4 {\cal V}_n(s) {\bar K}_n(s) \,+\, P_V (s).\end{aligned}$$ The two second-order polynomials $P_S (s)$ and $P_V (s)$ are obtained as follows. First write $${\bar K}_n(s) \,=\,\kappa_n^{(1)} \frac{s}{M_\pi^2} \,+\,
\kappa_n^{(2)} \left(\frac{s}{M_\pi^2}\right)^2 \,+\,
\frac{s^3}{\pi}\,\int_{4M_\pi^2}^\infty\,\frac{dx}{x^3}\,\frac{k_n(x)}{x-s-i0},$$ and, next, expand the polynomials ${\cal S}_n(s)$ and ${\cal V}_n(s)$, $$\begin{aligned}
{\cal S}_n(s) &=& {\cal S}_n^{(0)} \,+\, {\cal S}_n^{(1)} \frac{s}{M_\pi^2} \,+\,
{\cal S}_n^{(2)}
\left(\frac{s}{M_\pi^2}\right)^2
\nonumber\\
{\cal V}_n(s) &=& {\cal V}_n^{(0)} \,+\, {\cal V}_n^{(1)} \frac{s}{M_\pi^2} \,+\,
{\cal V}_n^{(2)}
\left(\frac{s}{M_\pi^2}\right)^2\,.
\nonumber\end{aligned}$$ The polynomials $P_S (s)$ and $P_V (s)$ are then given by $$\begin{aligned}
P_S (s) &=& -\,\frac{s^2}{M_\pi^4}\,\sum_{n=0}^3
\left[\kappa_n^{(2)}{\cal S}_n^{(0)}\,+\,\kappa_n^{(1)}{\cal S}_n^{(1)}\right]\,-\,
\frac{s}{M_\pi^2}\,\sum_{n=0}^3
\kappa_n^{(1)}{\cal S}_n^{(0)} \,,
\nonumber\\
P_V (s) &=& -\,\frac{s^2}{M_\pi^4}\,\sum_{n=0}^4
\left[\kappa_n^{(2)}{\cal V}_n^{(0)}\,+\,\kappa_n^{(1)}{\cal V}_n^{(1)}\right]\,-\,
\frac{s}{M_\pi^2}\,\sum_{n=0}^4
\kappa_n^{(1)}{\cal V}_n^{(0)}\,.\end{aligned}$$ Using the coefficients $\kappa_n^{(1)}$ and $\kappa_n^{(2)}$ displayed in the following table:
$n$ $0$ $1$ $2$ $3$ $4$
----------------------- ----------------- -------------------- -------------------- --------------------------------------- -------------------------------------------
$\pi^2\kappa_n^{(1)}$ $\frac{1}{96}$ $- \frac{1}{16}$ $- \frac{1}{24}$ $\frac{\pi^2}{192}\,-\,\frac{1}{32}$ $- \frac{\pi^2}{576}\,+\,\frac{1}{64}$
$\pi^2\kappa_n^{(2)}$ $\frac{1}{960}$ $ - \frac{1}{192}$ $- \frac{7}{2880}$ $\frac{\pi^2}{960}\,-\,\frac{1}{128}$ $-\frac{\pi^2}{1920}\,+\,\frac{19}{3840}$
we obtain $$\begin{aligned}
P_S (s) &=&
\frac{M_\pi^2}{16\pi^2 F_\pi^2}\left[
\frac{8\beta - 5\alpha}{36}\,
\left(\frac{s}{M_\pi^2}\right)\,-\,
\frac{52\beta + 5\alpha}{360}\,
\left(\frac{s}{M_\pi^2}\right)^2
\right]
\nonumber\\
&& + \left(\frac{M_\pi^2}{16\pi^2 F_\pi^2}\right)^2
\left\{
\left(\frac{s}{M_\pi^2}\right)
\left[
- \frac{1}{324}(45\alpha^2 + 232\beta^2)
\,+\,\frac{5\pi^2}{216} (\alpha^2 + 4\beta^2)
\,-\,\frac{16\pi^2}{9} (11\lambda_1 + 14\lambda_2)
\right]
\right.
\nonumber\\
&& +
\left.
\left(\frac{s}{M_\pi^2}\right)^2
\left[
\frac{2\pi^2}{27}(8\beta - 5\alpha)F_\pi^2 \langle r^2\rangle_S^{\pi}
\,-\,\frac{1}{3240}(135\alpha^2 + 362\beta^2)
\right.\right.
\nonumber\\
&&\qquad \qquad \qquad
\left.\left.
\,+\,\frac{\pi^2}{216} (\alpha^2 - 2\beta^2)
\,+\,\frac{8\pi^2}{45} (89\lambda_1 + 86\lambda_2)
\right]
\right\}
,
\nonumber\\
P_V (s) &=&
\frac{M_\pi^2}{16\pi^2 F_\pi^2}\left[
\frac{\beta}{9}\,
\left(\frac{s}{M_\pi^2}\right)\,-\,
\frac{\beta}{60}\,
\left(\frac{s}{M_\pi^2}\right)^2
\right]
\nonumber\\
&& + \left(\frac{M_\pi^2}{16\pi^2 F_\pi^2}\right)^2
\left\{
\left(\frac{s}{M_\pi^2}\right)
\left[
\frac{1}{648}(45\alpha^2 + 340\alpha\beta - 94\beta^2)
\,-\,\frac{\pi^2}{648} (5\alpha^2 + 50\alpha\beta - 28\beta^2)
\right]
\right.
\nonumber\\
&& +
\left.
\left(\frac{s}{M_\pi^2}\right)^2
\left[
\frac{8\pi^2}{27}\,\beta F_\pi^2 \langle r^2\rangle_V^{\pi}
\,+\,\frac{1}{6480}(195\alpha^2 + 1650\alpha\beta + 230\beta^2)
\right.\right.
\nonumber\\
&&\qquad \qquad \qquad
\left.\left.
\,-\,\frac{\pi^2}{3240} (10\alpha^2 + 70\alpha\beta + 7\beta^2)
\,-\,\frac{16\pi^2}{9} (\lambda_1 - \lambda_2)
\right]
\right\}
.\end{aligned}$$
The relations [@KMSF95] $$\begin{aligned}
{\bar K}_2(s) &=& \left(1\,-\,\frac{4M_\pi^2}{s}\right) {\bar K}_1(s)
\,-\,\frac{1}{4\pi^2}
\nonumber\\
{\bar K}_3(s) &=& 3\,\frac{s - 4 M_\pi^2}{M_\pi^2}\,{\bar K}_4(s)
\,-\, 3{\bar J}(s) \,-\, \frac{3}{2}\,{\bar K}_1(s) \,+\,
\left(\frac{3}{32\pi^2}\,-\,\frac{1}{64}\right)\,\frac{s }{M_\pi^2}\,,\end{aligned}$$ allow us to eliminate ${\bar K}_2(s)$ from $F_S (s)$, and both ${\bar K}_2(s)$ and ${\bar K}_3(s)$ from $F_V (s)$, thus leading to the following expressions of the form factors $$\begin{aligned}
\lefteqn{
F_S(s)/F_S(0) \,=\,
1 \,+\, \frac{1}{6}\langle r^2\rangle_S^{\pi}\, {s}
\,+\, c_S^{\pi}\,s^2
}
\nonumber\\
&& + \,\frac{M_\pi^2}{16\pi^2 F_\pi^2}
\left\{
16\pi^2 \left(
\frac{s}{M_\pi^2}\beta - \frac{1}{6} (8\beta - 5\alpha)
\right){\bar J}(s)
+ \frac{1}{36} \frac{s}{M_\pi^2} (8\beta - 5\alpha) -
\frac{1}{360}\left(\frac{s}{M_\pi^2}\right)^2
(52\beta + 5\alpha)
\right\}
\nonumber\\
&& + \,
\left(\frac{M_\pi^2}{16\pi^2 F_\pi^2}\right)^2
\left\{
\frac{8\pi^2}{9}\,{\bar J}(s)
\left[
\left(\frac{s}{M_\pi^2}\right)^2
\left(
48\pi^2 (11{\lambda}_1 + 14 {\lambda}_2) + 48\pi^2 F_\pi^2 \beta \langle r^2\rangle_S^{\pi} +
\frac{1}{6}\beta(551\beta - 15 \alpha)
\right)
\right.\right.
\nonumber\\
&& \left.\left.
\qquad\qquad\qquad\qquad\qquad
\,-\,
\left(\frac{s}{M_\pi^2}\right)
\left(
1920 \pi^2 (\lambda_1 + \lambda_2)
+ 8\pi^2 F_\pi^2 (8\beta - 5\alpha) \langle r^2\rangle_S^{\pi}
\right.\right.\right.
\nonumber\\
&& \left.\left.\left.
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
+ \frac{1}{12} (3684\beta^2 - 1520\alpha\beta + 25\alpha^2)
\right)
\right.\right.
\nonumber\\
&& \left.\left.
\qquad\qquad\qquad\qquad\qquad\qquad
\,+\,
192\pi^2 (11{\lambda}_1 + 14 {\lambda}_2)
+ \frac{1}{3} (976\beta^2 - 480\alpha\beta + 285\alpha^2)
\right]
\right.
\nonumber\\
&&
\left.
\qquad +\,
12\pi^2 {\bar K}_1(s)
\left[
\frac{43}{27}\left(\frac{s}{M_\pi^2}\right)^2 \beta^2
- \frac{20}{27}\left(\frac{s}{M_\pi^2}\right) \beta (14\beta - 3\alpha)
+ \frac{4}{27} (127 \beta^2 - 80 \alpha\beta + 10 \alpha^2)
\right.\right.
\nonumber\\
&& \left.\left.
\qquad\qquad\qquad\qquad\qquad\qquad
- \frac{4}{27}\left(\frac{M_\pi^2}{s}\right) (64\beta^2 - 80\alpha\beta + 25\alpha^2)
\right]
\right.
\nonumber\\
&&
\left.
\qquad +\,
\frac{16\pi^2}{3} {\bar K}_3(s)
\left[
\left(\frac{s}{M_\pi^2}\right) \beta^2 - \frac{5}{6} (4\beta^2 + \alpha^2)
\right]
\right.
\nonumber\\
&&
\left.
\qquad +\,
\left(\frac{s}{M_\pi^2}\right)^2
\left[ \frac{8\pi^2}{45} (89 \lambda_1 + 86 \lambda_2)
+ \frac{2\pi^2}{27}(8\beta - 5\alpha) F_\pi^2\langle r^2\rangle_S^{\pi}
- \frac{1}{3240} (13322\beta^2 + 135 \alpha^2)
\right.\right.
\nonumber\\
&& \left.\left.
\qquad\qquad\qquad\qquad
- \frac{\pi^2}{216} (2\beta^2 - \alpha^2)
\right]
\right.
\nonumber\\
&&
\left.
\qquad +\,
\left(\frac{s}{M_\pi^2}\right)
\left[
- \frac{16\pi^2}{9} (11 \lambda_1 + 14 \lambda_2)
+ \frac{1}{324} (3224\beta^2 - 2160 \alpha\beta - 45\alpha^2)
+ \frac{5\pi^2}{216} (4\beta^2 + \alpha^2)
\right]
\right.
\nonumber\\
&&
\left.
\qquad -\,\frac{1}{9} (8\beta - 5\alpha)^2
\right\}
,\end{aligned}$$
$$\begin{aligned}
\lefteqn{
F_V(s) \,=\,
1 \,+\, \frac{1}{6}\langle r^2\rangle_V^{\pi}\, {s}
\,+\, c_V^{\pi}\, s^2
\,+ \,\frac{M_\pi^2}{16\pi^2 F_\pi^2}
\left\{
\frac{8\pi^2}{3}\,\beta \left(
\frac{s}{M_\pi^2} - 4
\right){\bar J}(s)
+ \frac{1}{9} \beta \frac{s}{M_\pi^2} -
\frac{1}{60}\,\beta \left(\frac{s}{M_\pi^2}\right)^2
\right\}
}
\nonumber\\
&& + \,
\left(\frac{M_\pi^2}{16\pi^2 F_\pi^2}\right)^2
\left\{
\frac{8\pi^2}{9}\,{\bar J}(s)
\left[
\left(\frac{s}{M_\pi^2}\right)^2
\left(
[48\pi^2(\lambda_2 - \lambda_1) - \frac{1}{2}]
+ \frac{1}{2} [16\pi^2 F_\pi^2 \beta\langle r^2\rangle_V^{\pi} + 1] + \frac{7}{6}\, \beta^2
\right)
\right.\right.
\nonumber\\
&& \left.\left.
\qquad\qquad\qquad\qquad\qquad
\,-\,
4 \left(\frac{s}{M_\pi^2}\right)
\left(
[48\pi^2(\lambda_2 - \lambda_1) - \frac{1}{2}]
+ \frac{1}{2} [16\pi^2 F_\pi^2 \beta\langle r^2\rangle_V^{\pi} + 1]
+ \frac{5}{24}\beta(34\beta - 11\alpha)
\right)
\right.\right.
\nonumber\\
&& \left.\left.
\qquad\qquad\qquad\qquad\qquad
\,+\,\frac{5}{6}\,(86 \beta^2 - 104 \alpha\beta + 9 \alpha^2)
\right]
\right.
\nonumber\\
&&
\left.
\qquad +\,
\frac{2\pi^2}{9}\, {\bar K}_1(s)
\left[
\left(\frac{s}{M_\pi^2}\right)^2 \beta^2
- 10 \left(\frac{s}{M_\pi^2}\right) \beta(4\beta - \alpha)
+ 2(74\beta^2 - 40\alpha\beta + 5\alpha^2) - 128\left(\frac{M_\pi^2}{s}\right) \beta^2
\right]
\right.
\nonumber\\
&&
\left.
\qquad +\,
8\pi^2 {\bar K}_4(s)
\left[
\left(\frac{s}{M_\pi^2}\right)^2 \beta^2 -
\frac{1}{9}\left(\frac{s}{M_\pi^2}\right) (44\beta^2 - 10\alpha\beta + 5\alpha^2)
+ \frac{2}{9} (26\beta^2 - 40\alpha\beta + 5\alpha^2)
\right]
\right.
\nonumber\\
&&
\left.
\qquad +\,
\left(\frac{s}{M_\pi^2}\right)^2
\left[ \frac{16\pi^2}{9} (\lambda_2 - \lambda_1)
+ \frac{8\pi^2}{27}\beta F_\pi^2\langle r^2\rangle_V^{\pi}
+ \frac{1}{6480} (1130\beta^2 + 1650 \alpha\beta +195\alpha^2)
\right.\right.
\nonumber\\
&& \left.\left.
\qquad\qquad\qquad\qquad
- \frac{\pi^2}{1620} (71\beta^2 + 35\alpha\beta + 5\alpha^2)
\right]
\right.
\nonumber\\
&&
\left.
\qquad +\,
\left(\frac{s}{M_\pi^2}\right)
\left[
\frac{1}{648} (338\beta^2 + 520 \alpha\beta - 45\alpha^2)
+ \frac{\pi^2}{648} (52\beta^2 - 80\alpha\beta + 10\alpha^2)
\right] \,-\,\frac{16}{9} \beta^2
\right\}
.\end{aligned}$$
In order to recover the expressions of [@Colangelo96] from these formulae, one simply needs to replace the various quantities by their expressions at leading or at next-to-leading order, as they can be found in refs. [@KMSF95; @Gasser:1983yg], $$\begin{aligned}
\alpha &=&
1 \,+\, \frac{1}{32\pi^2}\frac{M_\pi^2}{F_\pi^2}\,(4 {\bar \ell}_4 - 3 {\bar \ell}_3 - 1)
\nonumber\\
\beta &=&
1 \,+\, \frac{1}{8\pi^2}\frac{M_\pi^2}{F_\pi^2}\,( {\bar \ell}_4 - 1)
\nonumber\\
\lambda _1 &=&
\frac{1}{48\pi^2} \,\left( {\bar \ell}_1 - \frac{4}{3} \right)
\nonumber\\
\lambda_2 &=&
\frac{1}{48\pi^2} \,\left( {\bar \ell}_2 - \frac{5}{6} \right)
\nonumber\\
\langle r^2\rangle_S^{\pi} &=&
\frac{3}{8\pi^2 F_\pi^2} \,\left( {\bar \ell}_4 - \frac{13}{12} \right)
\nonumber\\
\langle r^2\rangle_V^{\pi} &=&
\frac{1}{16\pi^2 F_\pi^2} \,( {\bar \ell}_6 - 1)
.
\label{alpha_bet_std}\end{aligned}$$ At the end of this process, we then obtain perfect agreement with ref. [@Colangelo96].
First-order isospin-breaking corrections to the one-loop partial waves {#app:delta_xi}
=======================================================================
The purpose of this appendix is to provide the explicit expressions of the functions that describe the isospin-breaking corrections to the one-loop partial waves, as given in eqs. (\[psi\_00\_exp\]) and (\[psi\_+-\_exp\_and\_psi\_x\_exp\]).
In the case of $\psi^0_{00} (s)$ the corrections that appear in eq. (\[psi\_00\_exp\]) read $$\begin{aligned}
\Delta\xi_{00}^{(0)}(s) &=& \frac{4}{3}\,(\lambda_1 + 2 \lambda_2)
\left(2\frac{s}{M_{\pi}^2} - 5\right)\,+\,
\frac{1}{576\pi^2}\,\left(
160\beta^2 \frac{s}{M_{\pi}^2} + 6 \alpha\beta \frac{s}{M_{\pi}^2}
+ 24 \alpha\beta -504 \beta^2 - 203 \alpha^2 \right)
\nonumber\\
&& -\,
\frac{1}{72\pi^2}\,\frac{M_{\pi}^2}{s - 4M_{\pi}^2}\,
(44 \beta^2 - 28 \alpha\beta + 11 \alpha^2)
\nonumber\\
\Delta\xi_{00}^{(1)}(s) &=&
\frac{1}{2304\pi^2}\,\left(12 \beta^2 \frac{s^2}{M_{\pi}^4} +
60 \beta^2 \frac{s}{M_{\pi}^2} + 12 \alpha\beta \frac{s}{M_{\pi}^2}
- 79 \alpha^2 -368 \beta^2 \right)
\nonumber\\
&& -\,
\frac{1}{12\pi^2}\,\frac{M_{\pi}^2}{s - 4M_{\pi}^2}\,
( \beta - \alpha)\beta
\nonumber\\
\Delta\xi_{00}^{(2)}(s) &=&
\frac{1}{2304\pi^2}\,\left( - 12 \beta^2 \frac{s^2}{M_{\pi}^4} +
84 \beta^2 \frac{s}{M_{\pi}^2} - 12 \alpha\beta \frac{s}{M_{\pi}^2}
- 53 \alpha^2 + 48 \alpha\beta - 64 \beta^2 \right)
\nonumber\\
\Delta\xi_{00}^{(3)}(s) &=&
\frac{1}{864\pi^2}\,( 8 \beta^2 - 12 \alpha\beta + 31 \alpha^2 )
\,+\,\frac{1}{216\pi^2}\,\frac{M_{\pi}^2}{s - 4M_{\pi}^2}\,
(20 \beta^2 - 4 \alpha\beta + 11 \alpha^2)
.\end{aligned}$$
In the case of $\pi^+ \pi^-$ scattering, we proceed as described in subsection \[IB\_in\_phases\_NLO\]: one first obtains the functions \[$X=S,P$\] $$\begin{aligned}
&
{\overline\xi}^{\,(1)}_{{\mbox{\tiny{$+-$}}};X} (s) \,=\,
{\overline\xi}^{(1;{\mbox{\tiny{$\!\pm $}}})}_{{\mbox{\tiny{$+-$}}};X} (s) \,+\,
{\overline\xi}^{\,(1;{\mbox{\tiny$\Delta$}})}_{{\mbox{\tiny{$+-$}}};X} (s)
&
\quad
{\overline\xi}^{\,(2)}_{{\mbox{\tiny{$+-$}}};X} (s) \,=\,
{\overline\xi}^{(2;{\mbox{\tiny{$\!\pm $}}})}_{{\mbox{\tiny{$+-$}}};X} (s) \,+\,
{\overline\xi}^{\,(2;0)}_{{\mbox{\tiny{$+-$}}};X} (s)
\nonumber\\
&
{\overline\xi}^{\,(3)}_{{\mbox{\tiny{$+-$}}};X} (s) \,=\,
{\overline\xi}^{(3;{\mbox{\tiny{$\!\pm $}}})}_{{\mbox{\tiny{$+-$}}};X} (s) \,+\,
{\overline\xi}^{\,(3;{\mbox{\tiny$\Delta$}})}_{{\mbox{\tiny{$+-$}}};X} (s)
&
\quad
{\overline\xi}^{\,(4)}_{{\mbox{\tiny{$+-$}}};P} (s) \,=\,
{\overline\xi}^{(4;{\mbox{\tiny{$\!\pm $}}})}_{{\mbox{\tiny{$+-$}}};P} (s) \,+\,
{\overline\xi}^{\,(4;{\mbox{\tiny$\Delta$}})}_{{\mbox{\tiny{$+-$}}};P} (s)
. \qquad { }\end{aligned}$$ The expansion of the remaining functions gives $$\begin{aligned}
2 \frac{\sigma (s)}{\sigma_{\mbox{\tiny$\Delta$}}(s)}\, L_{\mbox{\tiny$\Delta$}}(s) &=&
16\pi k_1(s) \,-\,8\pi\,\frac{\Delta_\pi}{M_{\pi}^2}
\,\left[4 k_0(s) + k_1(s) - k_2(s) \right] \,+\,{\cal O}(\Delta_\pi^2)
\nonumber\\
3 \,
\frac{M_{\pi^0}^2}{\sqrt{s(s - 4M_{\pi}^2)}}\,L_{\mbox{\tiny$\Delta$}}^2(s) &=& 16\pi k_3(s)
\,-\,8\pi\,\frac{\Delta_\pi}{M_{\pi}^2}
\,\left[ \frac{3}{2} k_1(s) - \frac{3}{2} k_2(s) + 2 k_3(s)\right] \,+\,
{\cal O}(\Delta_\pi^2)
,\end{aligned}$$ and $$\begin{aligned}
\!\!\!\!
\frac{M_{\pi^0}^2}{\sqrt{s(s-4 M_{\pi}^2)}}\,
\left[
1\,+\,\frac{1}{\sigma_{\mbox{\tiny$\Delta$}}(s)}\,L_{\mbox{\tiny$\Delta$}}(s)
\,+\,\frac{M_{\pi^0}^2}{s-4 M_{\pi}^2}\,L_{\mbox{\tiny$\Delta$}}^2(s)
\right] &=& 16\pi k_4(s)
\nonumber\\
&&\!\!\!\!\!\!\!
\,-\,8\pi\,\frac{\Delta_\pi}{M_{\pi}^2}
\,\left[4 k_4(s)\,-\,\frac{M_{\pi}^2}{s - 4 M_{\pi}^2}\,k_2(s) \right]\,+\,
{\cal O}(\Delta_\pi^2)
.\quad\qquad{ }\end{aligned}$$ For the $S$ and $P$ partial-wave projections $\psi^{\mbox{\tiny{$+-$}}}_0(s)$ and $\psi^{\mbox{\tiny{$+-$}}}_1(s)$, this then leads to the expression (\[psi\_+-\_exp\_and\_psi\_x\_exp\]), with $$\begin{aligned}
\Delta\xi^{(0)}_{{\mbox{\tiny{$+-$}}};S} (s) &=&
\frac{1}{576\pi^2}\,\left( -12 \beta^2 \frac{s^2}{M_{\pi}^4} +
\frac{81}{2} \beta^2 \frac{s}{M_{\pi}^2} - \frac{63}{2} \alpha\beta \frac{s}{M_{\pi}^2}
+ 84 \alpha\beta - 40 \beta^2 - 116 \alpha^2 \right)
\nonumber\\
\Delta\xi^{(1)}_{{\mbox{\tiny{$+-$}}};S} (s) &=&
\frac{1}{2304\pi^2}\,\left( 6 \beta^2 \frac{s^2}{M_{\pi}^4} -
33 \beta^2 \frac{s}{M_{\pi}^2} + 15 \alpha\beta \frac{s}{M_{\pi}^2}
- 51 \alpha^2 + 16 \alpha\beta + 56 \beta^2 \right)
\nonumber\\
\Delta\xi^{(2)}_{{\mbox{\tiny{$+-$}}};S} (s) &=&
\frac{1}{2304\pi^2}\,\left( - 6 \beta^2 \frac{s^2}{M_{\pi}^4} +
39 \beta^2 \frac{s}{M_{\pi}^2} - 45 \alpha\beta \frac{s}{M_{\pi}^2}
- 37 \alpha^2 + 64 \alpha\beta - 48 \beta^2 \right)
\nonumber\\
\Delta\xi^{(3)}_{{\mbox{\tiny{$+-$}}};S} (s) &=&
\frac{1}{1728\pi^2}\,( 26 \beta^2 - 2 \alpha\beta + 27 \alpha^2 )
\nonumber\\
\Delta\xi^{(0)}_{{\mbox{\tiny{$+-$}}};P} (s) &=&
\frac{1}{1728\pi^2}\,\left(\frac{9}{2} \beta^2 \frac{s^2}{M_{\pi}^4} -
\frac{29}{2} \beta^2 \frac{s}{M_{\pi}^2} - \frac{61}{2} \alpha\beta \frac{s}{M_{\pi}^2}
+ 266 \alpha\beta - 62 \beta^2 - 15 \alpha^2 \right)
\nonumber\\
\Delta\xi^{(1)}_{{\mbox{\tiny{$+-$}}};P} (s) &=&
\frac{1}{2304\pi^2}\,\left( \frac{3}{2} \beta^2 \frac{s^2}{M_{\pi}^4} -
4 \beta^2 \frac{s}{M_{\pi}^2} - 11 \alpha\beta \frac{s}{M_{\pi}^2}
+ \alpha^2 + 68 \alpha\beta - 15 \beta^2 \right)
\nonumber\\
\Delta\xi^{(2)}_{{\mbox{\tiny{$+-$}}};P} (s) &=&
\frac{1}{2304\pi^2}\,\left( - \frac{3}{2} \beta^2 \frac{s^2}{M_{\pi}^4} +
8 \beta^2 \frac{s}{M_{\pi}^2} + \alpha\beta \frac{s}{M_{\pi}^2}
- \alpha^2 - 4 \alpha\beta - 13 \beta^2 \right)
\nonumber\\
&& -\,\frac{1}{576\pi^2}\,
\frac{M_{\pi}^2}{s - 4 M_{\pi}^2}\,\left[
\alpha^2 + 4 \alpha\beta + 13 \beta^2 \right]
\nonumber\\
\Delta\xi^{(3)}_{{\mbox{\tiny{$+-$}}};P} (s) &=&
\frac{1}{1728\pi^2}\,( 26 \beta^2 - 10 \alpha\beta + 11 \alpha^2 )
\nonumber\\
\Delta\xi^{(4)}_{{\mbox{\tiny{$+-$}}};P} (s) &=&
\frac{1}{24\pi^2}\,( 5 \beta^2 + 3 \alpha\beta + \alpha^2 )
.\end{aligned}$$
Turning eventually to the inelastic $\pi^+ \pi^- \to \pi^0 \pi^0$ channel, one first rewrites the polynomials in eq. (\[psi\_x\_0\]) as $\xi^{(n)}_{x}(s) = {\overline\xi}^{(n)}_{x}(s)\,+\,
(\Delta_\pi/F_\pi^2) \delta\xi^{(n)}_{x}(s) + {\cal O}(\Delta_\pi^2)$, with ${\overline\xi}^{(2)}_{x}(s) \,=\, {\overline\xi}^{\,(2;{\mbox{\tiny{$\pm $}}})}_{x}(s) \,+\,
{\overline\xi}^{\,(2;0)}_{x}(s)$. Next, one proceeds with the expansion of the remaining functions, $$\begin{aligned}
2\,\frac{\lambda^{1/2}(t_{\mbox{\tiny{$-$}}}(s))}{\sqrt{s(s - 4 M_{\pi^0}^2)}}\,
{\cal L}_{\mbox{\tiny{$-$}}} (s) &=& 16\pi k_1(s)\,-\,16\pi\,\frac{\Delta_\pi}{M_{\pi^\pm}^2}\,
\left\{k_0(s)\,+\,\frac{M_{\pi}^2}{s - 4 M_{\pi^\pm}^2}\left[
4 k_0(s)\,+\,k_1(s)\right]\right\}\,+\,{\cal O}(\Delta_\pi^2)
\nonumber\\
3 \,\frac{M_{\pi}^2}{\sqrt{s(s - 4M_{\pi^0}^2)}}\,{\cal L}_{\mbox{\tiny{$-$}}}^2(s) &=&
16\pi k_3(s) \,-\,16\pi\,\frac{\Delta_\pi}{s - 4 M_{\pi}^2}\left[
\frac{3}{2} k_1(s)\,+\,2k_3(s)\right] \,+\,
{\cal O}(\Delta_\pi^2) . \end{aligned}$$ Similar expressions with $t_{\mbox{\tiny{$-$}}}(s)$ replaced by $t_{\mbox{\tiny{$+$}}}(s)$ are of order ${\cal O}(\Delta_\pi^2)$, cf. equation (\[m1\_m2\_limit\]), and thus need not be retained in the present context. Finally, there are the two additional pieces to consider, $$\begin{aligned}
\Delta_1 \psi^{x}_0(s) &=& 2\,\frac{M_{\pi}^4}{F_\pi^4}\,
\sqrt{\frac{s}{s - 4M_{\pi}^2}}\times\frac{\Delta_\pi}{M_{\pi}^2}\,
\frac{(-1)}{192\pi^2}\left[
\frac{1}{6}\beta^2\frac{s}{M_{\pi}^2} + \alpha\beta + 2 \beta^2\right] k_1(s) \,+\,
{\cal O}(\Delta_\pi^2) ,
\nonumber\\
\Delta_2 \psi^{x}_0(s) &=&
{\cal O}(\Delta_\pi^2) . \end{aligned}$$ Putting the various parts together then leads to the expression given in (\[psi\_+-\_exp\_and\_psi\_x\_exp\]), with $$\begin{aligned}
\Delta\xi_{x}^{(0)}(s) &=& - \lambda_1 \left(\frac{s}{M_{\pi}^2} - 2\right)\,-\,
\frac{\lambda_2}{3} \left(\frac{s}{M_{\pi}^2} - 4\right)
\nonumber\\
&& +\,
\frac{1}{1728\pi^2}\,\left(\frac{3}{2}\beta^2 \frac{s^2}{M_{\pi}^4}
- 154 \beta^2 \frac{s}{M_{\pi}^2} + 225 \alpha\beta \frac{s}{M_{\pi}^2}
+ 352 \beta^2 - 270 \alpha\beta + 171 \alpha^2 \right)
\nonumber\\
&& -\,
\frac{1}{72\pi^2}\,\frac{M_{\pi}^2}{s - 4M_{\pi}^2}\,
(8 \beta^2 - 7 \alpha\beta - \alpha^2)
\nonumber\\
\Delta\xi_{x}^{(1)}(s) &=&
\frac{1}{2304\pi^2}\,\left( -20 \beta^2 \frac{s}{M_{\pi}^2} + 3 \alpha\beta \frac{s}{M_{\pi}^2}
+ 80 \beta^2 + 36 \alpha\beta + 13 \alpha^2 \right)
\nonumber\\
&& +\,
\frac{1}{48\pi^2}\,\frac{M_{\pi}^2}{s - 4M_{\pi}^2}\,
\alpha\beta
\nonumber\\
\Delta\xi_{x}^{(2)}(s) &=&
\frac{1}{2304\pi^2}\,\left( - 12 \beta^2 \frac{s}{M_{\pi}^2}
+ 57 \alpha\beta \frac{s}{M_{\pi}^2} + 16 \beta^2 - 104 \alpha\beta
+ 31 \alpha^2 \right)
\nonumber\\
\Delta\xi_{x}^{(3)}(s) &=&
\frac{1}{864\pi^2}\,( 9 \beta^2 \frac{s}{M_{\pi}^2} - 12 \beta^2 - 4 \alpha\beta - 5 \alpha^2 )
\,+\,\frac{1}{216\pi^2}\,\frac{M_{\pi}^2}{s - 4M_{\pi}^2}\,
(8 \beta^2 - \alpha\beta - \alpha^2)
.\end{aligned}$$
Let us close this appendix with a remark concerning the occurrence of contributions proportional to ${M_{\pi}^2}/(s - 4M_{\pi}^2)$ in the expressions of the functions $\Delta \xi^{(n)}(s)$. When summed together into the functions $\psi^{00}_0 (s)$, $\psi^x_0 (s)$, and $\psi_1^{{\mbox{\tiny{$+-$}}}} (s)$ \[they are absent in $\psi_0^{{\mbox{\tiny{$+-$}}}} (s)$\], these singularities combine to give a regular behaviour as $s \to 4 M_\pi^2$. Just like their lowest-order counterparts $\varphi^{00}_0 (s)$, $\varphi^x_0 (s)$, $\varphi_0^{{\mbox{\tiny{$+-$}}}} (s)$, and $\varphi_1^{{\mbox{\tiny{$+-$}}}} (s)$, the real parts of the partial-wave projections at next-to-leading order are regular at $s=4 M_\pi^2$, and the expansion in powers of $\Delta_\pi$ preserves this regularity, see also the remark following eq. (\[psi\_00\_exp\]).
Expressions of the subtraction constants {#app:subtraction}
========================================
The phases of the form factors discussed in section \[IB\_in\_phases\] involve a certain number of subtraction constants, whose values are not fixed by the general properties underlying the dispersive relations that form the starting point of our construction. Two sets of parameters, $\alpha_{00}$, $\alpha_{x}$, and $\alpha_{\mbox{\tiny{$ +-$}}}$ on the one hand, and $\beta_{x}$ and $\beta_{\mbox{\tiny{$ +-$}}}$ on the other hand, are directly related to the parameters $\alpha$ and $\beta$ of the isospin-symmetric $\pi\pi$ amplitude, that are themselves related to the two scattering lengths in the $S$ wave, cf. eq. (\[alpha\_beta\_a00\_a02\]). They represent the quantities to be extracted from experiment. What we need to know, however, is what becomes of the relations (\[alphabetaLO\]) at next-to-leading order. For the remaining set of parameters, the $\lambda$’s, the isospin-breaking corrections to their values in the isospin limit, given in eqs. (\[lambdas\_iso\]), also need to be worked out. In order to obtain this information, we have performed a one-loop calculation of the form factors and scattering amplitudes using the “effective" lagrangian approach described in subsection \[sub\_csts\_at\_NLO\]. The results of this calculation are shown in this appendix.
First, at next-to-leading order, the expressions (\[alphabetaLO\]) become \[the definition of the constants ${\widehat{\cal K}}^{00}_1$ and ${\widehat{\cal K}}^{00}_2$ in terms of low-energy constants introduced in [@Knecht:1997jw] is given in eq. (\[cal\_K\]) below\] $$\begin{aligned}
\frac{F_\pi^2}{F^2} \left( 4 - 3 \frac{2 {\widehat m}B}{M_{\pi^0}^2} \right) - \alpha
&=&
\frac{\Delta_\pi}{M_\pi^2} (\beta - \alpha) \,+\,
\frac{1}{96\pi^2} \frac{M_{\pi^0}^2}{ F_\pi^2} ( 11 \alpha^2 -8 \beta^2 ) \,+\,
\frac{1}{48\pi^2} \frac{\Delta_\pi}{F_\pi^2} \beta (\beta + 5 \alpha)
\nonumber\\
&&
+\,
\frac{1}{32\pi^2} \frac{M_{\pi^0}^2}{F_\pi^2} \left( 4 \alpha^2 - 7 \alpha \beta + 6 \beta^2 \right) L_\pi
\,-\,
3 \beta \frac{e^2}{32\pi^2} \left( {\widehat{\cal K}}^{00}_1 + {\widehat{\cal K}}^{00}_2 \right)
\nonumber\\
\frac{F_\pi^2}{F^2} - \beta
&=&
\frac{1}{48\pi^2} \frac{M_\pi^2}{F_\pi^2} \beta (\beta + 5 \alpha)
,\end{aligned}$$ with $L_\pi\equiv \ln(M_\pi^2/M_{\pi^0}^2)$. Notice the occurrence of the term $\left( \beta - \alpha \right) \frac{\Delta_\pi}{M_\pi^2}$ in the first expression. Since $\beta - \alpha \sim {\cal O}(M_\pi^2 \times \ln M_\pi^2)$, this term reveals a logarithmic singularity (at most) in the chiral limit. Actually, it is finite as $M_{\pi^0} \rightarrow 0$. However, as $M_{\pi} \rightarrow 0$, it develops an infrared singular behaviour, $$\left( \beta - \alpha \right) \frac{\Delta_\pi}{M_\pi^2} \sim \frac{1}{32\pi^2} \frac{M_{\pi^0}^2}{F_\pi^2}
\left( 7 \beta - 4 \alpha \right) \alpha \ln M_\pi^2 .
\label{alpha-beta_IR}$$ We then obtain the following identification, at one-loop precision, with the various parameters involved in the polynomial part of these amplitudes: $$\begin{aligned}
\alpha_{00} &=& \alpha \,+\, \left( \beta - \alpha \right) \frac{\Delta_\pi}{M_\pi^2}
\,+\,
\frac{1}{48 \pi^2} \frac{\Delta_\pi}{F_\pi^2} \left( 5 \alpha + \beta \right) \beta
\,+\,
\frac{1}{96 \pi^2} \frac{M_{\pi^0}^2}{F_\pi^2} \left( 2 \beta^2 - 3 \alpha \beta + 10 \alpha^2 \right) L_\pi
\nonumber\\
&&
+\,
\beta \frac{e^2}{32\pi^2}
\left( {\widehat{\cal K}}^{00}_1 + {\widehat{\cal K}}^{00}_2 \right)
\,-\,
\alpha \frac{e^2}{32\pi^2} {\widehat{\cal K}}^{00}_2
\nonumber\\
\alpha_x &=& \alpha \,+\, 2 \beta \frac{\Delta_\pi}{M_{\pi^0}^2} \,+\, \left( \beta - \alpha \right) \frac{\Delta_\pi}{M_\pi^2}
\,+\, \frac{1}{48 \pi^2} \frac{\Delta_\pi}{F_\pi^2}
\left[\beta \left(11 - 18 \frac{\Delta_\pi}{M_{\pi^0}^2} \right) - 17 \alpha \right] \beta
\nonumber\\
&&
-\,
\frac{1}{96 \pi^2} \frac{M_{\pi^0}^2}{F_\pi^2} \left[ 6 \beta^2 \left( 9 + 2 \frac{\Delta_\pi}{M_{\pi^0}^2} \right)
- \alpha \beta \left( 47 + 6 \frac{\Delta_\pi}{M_{\pi^0}^2} \right) - 4 \alpha^2 \right] L_\pi
\nonumber\\
&&
-\,
\frac{1}{24 \pi^2} \frac{M_{\pi^0}^2}{ F_\pi^2}
\left[
\frac{M_{\pi^0}^2}{\Delta_\pi} L_\pi \,-\, 1 \right] \left( \alpha - \beta \right)
\left[
\beta \left( 4 \frac{\Delta_\pi}{M_{\pi^0}^2} + 1 \right) + \alpha \right]
\nonumber\\
&&
+\,
\frac{\Delta_\pi }{M_{\pi^0}^2} \, \beta \frac{e^2}{32\pi^2} {\widehat{\cal K}}^{x}_1
\,+\,
\beta \frac{e^2}{32\pi^2}
\left( 2 {\widehat{\cal K}}^{x}_1 + {\widehat{\cal K}}^{x}_2
+ 4 {\widehat{\cal K}}^{x}_3 - 3 {\widehat{\cal K}}^{00}_1 - 3 {\widehat{\cal K}}^{00}_2 \right)
\,-\,
\alpha \frac{e^2}{32\pi^2} {\widehat{\cal K}}^{x}_3
\nonumber\\
\alpha_{\mbox{\tiny{$ +-$}}} &=& \alpha \,+\, 4 \beta \frac{\Delta_\pi}{M_{\pi^0}^2}
\,+\, \left( \beta - \alpha \right) \frac{\Delta_\pi}{M_\pi^2}
\,+\, \frac{1}{16 \pi^2} \frac{\Delta_\pi}{F_\pi^2}
\left[ \beta \left(3 - 28 \frac{\Delta_\pi}{M_{\pi^0}^2} \right) - 13 \alpha \right] \beta
\nonumber\\
&&
+\,
\frac{1}{96 \pi^2} \frac{M_{\pi^0}^2}{F_\pi^2}
\left[ 2 \beta^2 \left( 6 \frac{\Delta_\pi^2}{M_{\pi^0}^4} - 16 \frac{\Delta_\pi}{M_{\pi^0}^2} - 45 \right)
+ \alpha \beta \left( 97 + 8 \frac{\Delta_\pi}{M_{\pi^0}^2} \right) + 2 \alpha^2 \right] L_\pi
\nonumber\\
&&
+\,
\beta \frac{e^2}{32\pi^2}
\left( {\widehat{\cal K}}^{\mbox{\tiny{$ +-$}}}_1 + {\widehat{\cal K}}^{\mbox{\tiny{$ +-$}}}_2
+ 4 {\widehat{\cal K}}^{\mbox{\tiny{$ +-$}}}_3 - 3 {\widehat{\cal K}}^{00}_1 - 3 {\widehat{\cal K}}^{00}_2 \right)
\,-\,
\alpha \frac{e^2}{32\pi^2} {\widehat{\cal K}}^{\mbox{\tiny{$ +-$}}}_3
\nonumber\\
&&
+\,
\frac{\Delta_\pi}{M_{\pi^0}^2} \,\beta \frac{e^2}{32\pi^2}
\left( {\widehat{\cal K}}^{\mbox{\tiny{$ +-$}}}_1 + {\widehat{\cal K}}^{\mbox{\tiny{$ +-$}}}_2 \right)
\,+\,
\frac{F_\pi^2}{M_{\pi^0}^2}
\left[
24 e^4 {\widehat k}^r_{14}(\mu) - \frac{9}{4\pi^2}\frac{\Delta_\pi^2}{F^4}
\ln\frac{M_\pi^2}{\mu^2}\right]
\nonumber\\
\beta_x &=& \beta \,+\,
\frac{1}{96 \pi^2} \frac{\Delta_\pi}{ F_\pi^2} \left( 10 \alpha - 19 \beta \right) \beta
\,+\,
\frac{1}{96 \pi^2} \frac{M_{\pi^0}^2}{F_\pi^2} \left( 13 \alpha - 10 \beta \right) \beta L_\pi
\,+\,
\frac{1}{48 \pi^2} \frac{M_{\pi^0}^2}{F_\pi^2}
\left[ \frac{M_{\pi^0}^2}{\Delta_\pi} L_\pi \,-\, 1 \right] \left(4 \beta - \alpha \right) \beta
\nonumber\\
&&
+\, \beta \frac{e^2}{32\pi^2} {\widehat{\cal K}}^{x}_1
\nonumber\\
\beta_{\mbox{\tiny{$ +-$}}} &=& \beta \,+\,
\frac{1}{48 \pi^2} \frac{\Delta_\pi}{F_\pi^2}
\left( 5 \alpha - 20 \beta \right) \beta
\,+\,
\frac{1}{24 \pi^2} \frac{M_{\pi^0}^2}{F_\pi^2}
\left[ \beta \left( 3 \frac{\Delta_\pi}{M_{\pi^0}^2} + 2 \right) - 2 \alpha \right] \beta L_\pi
\,+\, \beta \frac{e^2}{32\pi^2} {\widehat{\cal K}}^{\mbox{\tiny{$ +-$}}}_1
\nonumber\\
\lambda_{00}^{(1)} &=& \frac{1}{3} \left( \lambda_1 + 2 \lambda_2 \right)
\nonumber\\
\lambda_{x}^{(1)} &=& \lambda_1 \,+\, \frac{1}{96\pi^2} \left[
\frac{M_{\pi^0}^2}{\Delta_\pi} L_\pi - 1 \right] \beta^2
\nonumber\\
\lambda_{x}^{(2)} &=& \lambda_2 \,-\,\frac{1}{48\pi^2} \left[
\frac{M_{\pi^0}^2}{\Delta_\pi} L_\pi - 1 \right] \beta^2
\nonumber\\
\lambda_{\mbox{\tiny{$ +-$}}}^{(1)} &=& \lambda_1 \,+\, \frac{1}{32\pi^2}\,L_\pi \beta^2
\nonumber\\
\lambda_{\mbox{\tiny{$ +-$}}}^{(2)} &=& \lambda_2
,\end{aligned}$$ $$\begin{aligned}
\frac{e^2}{32\pi^2}\,{\widehat{\cal K}}^{00}_1
&=&
e^2\left[- \frac{20}{9} {\widehat k}^r_1(\mu) - \frac{20}{9} {\widehat k}^r_2(\mu) +
4 {\widehat k}_3 - 2 {\widehat k}^r_4(\mu) \right]
\,+\,
\frac{\beta}{8 \pi^2} \frac{\Delta_\pi}{F_\pi^2}\,\ln\frac{M_\pi^2}{\mu^2}
\nonumber\\
\frac{e^2}{32\pi^2}\,{\widehat{\cal K}}^{00}_2
&=&
e^2\left[ \frac{40}{9} {\widehat k}^r_1(\mu) + \frac{40}{9} {\widehat k}^r_2(\mu) -
8 {\widehat k}_3 + 4 {\widehat k}^r_4(\mu) -
\frac{20}{9} {\widehat k}^r_5(\mu) - \frac{20}{9} {\widehat k}^r_6(\mu) -
\frac{4}{9} {\widehat k}_7
\right]
\,-\,
\frac{3\beta}{16 \pi^2}\,\frac{\Delta_\pi}{F_\pi^2}\,\ln\frac{M_\pi^2}{\mu^2}
\nonumber\\
\frac{e^2}{32\pi^2} {\widehat{\cal K}}^{x}_1
&=&
e^2\left[- \frac{40}{9} {\widehat k}^r_1(\mu) + \frac{32}{9} {\widehat k}^r_2(\mu) -
8 {\widehat k}_3 + 4 {\widehat k}^r_4(\mu) \right]
\,-\,
\frac{\beta}{4\pi^2} \frac{\Delta_\pi}{F_\pi^2}\,\ln\frac{M_\pi^2}{\mu^2}
\nonumber\\
\frac{e^2}{32\pi^2} {\widehat{\cal K}}^{x}_2
&=&
e^2\left[ -24 {\widehat k}^r_2(\mu)\,+\,
24 {\widehat k}_3 - 12 {\widehat k}^r_4(\mu) \right]
\,+\,
\frac{9\beta}{8 \pi^2} \frac{\Delta_\pi}{F_\pi^2}\,\ln\frac{M_\pi^2}{\mu^2}
\nonumber\\
\frac{e^2}{32\pi^2} {\widehat{\cal K}}^{x}_3
&=&
e^2 \left[ \frac{40}{9} {\widehat k}^r_1(\mu) + \frac{40}{9} {\widehat k}^r_2(\mu) -
4 {\widehat k}_3 + 2 {\widehat k}^r_4(\mu) -
\frac{20}{9} {\widehat k}^r_5(\mu) + \frac{52}{9} {\widehat k}^r_6(\mu) -
\frac{4}{9} {\widehat k}_7 + 8 {\widehat k}^r_8(\mu)
\! \right]
-
\frac{\beta}{4 \pi^2} \frac{\Delta_\pi}{F_\pi^2} \ln\frac{M_\pi^2}{\mu^2}
\nonumber\\
\frac{e^2}{32\pi^2} {\widehat{\cal K}}^{\mbox{\tiny{$ +-$}}}_1
&=&
e^2\left[- \frac{20}{9} {\widehat k}^r_1(\mu) + \frac{52}{9} {\widehat k}^r_2(\mu) +
12 {\widehat k}_3 + 6 {\widehat k}^r_4(\mu) \right]
\,-\,
\frac{3\beta}{8\pi^2} \frac{\Delta_\pi}{F_\pi^2}\,\ln\frac{M_\pi^2}{\mu^2}
\nonumber\\
\frac{e^2}{32\pi^2} {\widehat{\cal K}}^{\mbox{\tiny{$ +-$}}}_2
&=&
e^2\left[- \frac{20}{3} {\widehat k}^r_1(\mu) - \frac{92}{3} {\widehat k}^r_2(\mu) -
12 {\widehat k}_3 - 6 {\widehat k}^r_4(\mu) \right]
\,+\,
\frac{9\beta}{8\pi^2} \frac{\Delta_\pi}{F_\pi^2}\,\ln\frac{M_\pi^2}{\mu^2}
\nonumber\\
\frac{e^2}{32\pi^2} {\widehat{\cal K}}^{\mbox{\tiny{$ +-$}}}_3
&=&
e^2 \left[ \frac{40}{9} {\widehat k}^r_1(\mu) + \frac{40}{9} {\widehat k}^r_2(\mu) -
\frac{20}{9} {\widehat k}^r_5(\mu) + \frac{124}{9} {\widehat k}^r_6(\mu) -
\frac{4}{9} {\widehat k}_7 + 16 {\widehat k}^r_8(\mu)
\right]
-
\frac{5\beta}{16 \pi^2} \frac{\Delta_\pi}{F_\pi^2}\,\ln\frac{M_\pi^2}{\mu^2}
.
\label{cal_K}\end{aligned}$$ From the one-loop expressions of the form factors, we obtain the following information on the subtraction constants in eq. (\[1loopFF\]): at this level of accuracy, $a_S^{\pi^0}$ and $a_V^\pi$ are unchanged as compared to the isospin limit, while $$\begin{aligned}
a_S^{\pi} - a_S^{\pi^0}
&=& \frac{\beta}{32\pi^2 F_\pi^2} \, L_\pi
.
\label{aSpi-aSpi0}\end{aligned}$$ Finally \[we have discarded contributions proportional to $e^2 (m_u - m_d)$ or to $(m_u - m_d)^2$\] $$F_S^{\pi}(0) \,=\, F_S^{\pi^0}(0)\left[
1 \,+\,\frac{e^2}{32\pi^2} \left(
{\widehat{\cal K}}^x_1 + \frac{1}{3} {\widehat{\cal K}}^x_2 + {\widehat{\cal K}}^x_3 - 2 {\widehat{\cal K}}^{00}_1 - {\widehat{\cal K}}^{00}_2
\right)
\,-\,
\frac{1}{8\pi^2} \frac{\Delta_\pi}{F_{\pi}^2} \beta
\,-\, \frac{1}{48\pi^2} \frac{M_{\pi^0}^2}{F_{\pi}^2} \left( 5 \beta + \alpha \right) L_\pi
\right] .$$ As far as comparison is possible, we find agreement with the existing results in the literature quoted at the beginning of this appendix, except in two instances. The expressions for charged pion scattering given in ref. [@KnechtNehme02] only included corrections of first order in isospin breaking, with which we agree. The formulae we give here are not limited to this approximation. Furthermore, we found a slight disagreement with the result of [@Kubis:1999db] for $F_V(s)$: the radius $\langle r \rangle_V^\pi$ exhibits an infrared divergence proportional to $\ln M_{\pi^0}^2$ as $M_{\pi^0}$ goes to zero, whereas we find that $\langle r \rangle_V^\pi$ remains finite in this limit, but diverges as $\sim\ln M_{\pi}^2$ if we send the charged pion mass to zero, keeping $M_{\pi^0}$ fixed. This is also what follows from our analysis in subsection \[Mto0\_1loop\] [^2]. In this context, it is important to stress that in the expressions given above, the scale-independent low-energy constants ${\bar \ell}_i$ are defined as $$\ell_i^r(\mu) \,=\,\frac{\gamma_i}{32\pi^2}\left( {\bar \ell}_i + \ln\frac{M_\pi^2}{\mu^2}\right) ,$$ i.e. the normalization of the logarithm is provided by the charged pion mass. We have also checked that the results given in this appendix display infrared behaviours in agreement with the ones obtained in section \[Mto0\_1loop\], provided one takes $$\lambda_1 \,=\, \frac{1}{48 \pi^2} \left( {\bar \ell}_1 - \frac{4}{3} \right)
\beta^2
, \quad
\lambda_2 \,=\, \frac{1}{48 \pi^2} \left( {\bar \ell}_2 - \frac{5}{6} \right)
\beta^2
.
\label{lambdas_l1_l2}$$ Notice that these expressions differ from the ones given at the end of appendix \[app:IsoLimit\], see eq. (\[alpha\_bet\_std\]), by the factor $\beta^2$. Both are compatible at one-loop order, where one would take $\beta =1$ in the above formula.
Two-loop phases in terms of scattering lengths {#app:scatt_lengths}
===============================================
It is perfectly possible, within the framework adopted in this article, to write down expressions that involve the scattering lengths instead of the subthreshold parameters. This is achieved by choosing a parameterization of the lowest-order amplitudes in terms of the scattering lengths, i.e. the value of the amplitudes at their respective thresholds, rather than in terms of their values at the center of the Mandelstam triangle, as done in the rest of the present article. The expressions (\[AmpTree1\]) and (\[AmpTree2\]) are thus replaced by $$\begin{aligned}
A^{x}(s,t) &=& 16\pi \left[ a_x
\,+\, b_x\, \frac{s - 4 M_{\pi}^2}{F_{\pi}^2}
\right]
\nonumber\\
A^{\mbox{\tiny{$+-$}}}(s,t) &=& 16\pi \left[ a_{\mbox{\tiny{$+-$}}}
\,+\, b_{\mbox{\tiny{$+-$}}} \frac{s-4M_{\pi}^2}{F_{\pi}^2}\,+\, c_{\mbox{\tiny{$+-$}}} \frac{t-u}{F_{\pi}^2}
\right]
\nonumber\\
A^{00}(s,t) &=& 16\pi a_{00}
\nonumber\\
A^{{\mbox{\tiny{$+$}}}0}(s,t) &=& 16\pi \left[ a_{{\mbox{\tiny{$+$}}} 0}
\,+\, b_{{\mbox{\tiny{$+$}}}0}\,\frac{s-(M_{\pi} + M_{\pi^0})^2}{F_{\pi}^2}\,
+\, c_{{\mbox{\tiny{$+$}}} 0}\,\frac{t-u+(M_{\pi} - M_{\pi^0})^2}{F_{\pi}^2}
\right]
\nonumber\\
A^{\mbox{\tiny{$++$}}}(s,t) &=& 16\pi \left[ a_{\mbox{\tiny{$++$}}}
\,+\, b_{\mbox{\tiny{$++$}}} \frac{s - 4 M_{\pi}^2}{F_{\pi}^2}
\right]
.
\label{AmpTree_a}\end{aligned}$$
At this order, the relation between the two sets of parameters is simple, $$\begin{aligned}
a_{00}\,=\,\frac{\alpha_{00} M_{\pi^0}^2}{16\pi F_{\pi}^2}
\, ,
\quad a_{x}\,=\,\frac{\beta_{x}}{24\pi F_{\pi}^2}
( M_{\pi^0}^2 - 5 M_{\pi}^2 )\,-\,
\frac{\alpha_{x} M_{\pi^0}^2}{48\pi F_{\pi}^2}
\, ,
\quad b_x\,=\,- \frac{\beta_{x}}{16\pi}
\nonumber\end{aligned}$$ $$\begin{aligned}
{a}_{{\mbox{\tiny{$+$}}}0}\,=\,-\frac{\beta_{x}}{24\pi F_{\pi}^2}
( M_{\pi^0}^2 + M_{\pi}^2 )\,+\,
\frac{\alpha_{x} M_{\pi^0}^2}{48\pi F_{\pi}^2}
\ ,\quad b_{{\mbox{\tiny{$+$}}}0}\,=\,- c_{{\mbox{\tiny{$+$}}}0}\,=\,- \frac{\beta_{x}}{32\pi}
\nonumber\end{aligned}$$ $$\begin{aligned}
a_{\mbox{\tiny{$+-$}}}\,=\,\frac{\beta_{\mbox{\tiny{$+-$}}}}{12\pi F_{\pi}^2}
\, M_{\pi}^2 \,+\,
\frac{\alpha_{\mbox{\tiny{$+-$}}} M_{\pi^0}^2}{24\pi F_{\pi}^2}
\, ,
\quad b_{\mbox{\tiny{$+-$}}}\,=\,c_{\mbox{\tiny{$+-$}}}\,=\, \frac{\beta_{\mbox{\tiny{$+-$}}}}{32\pi}
\nonumber\end{aligned}$$ $$\begin{aligned}
{a}_{\mbox{\tiny{$++$}}}\,=\,-\frac{\beta_{\mbox{\tiny{$+-$}}}}{6\pi F_{\pi}^2}
\, M_{\pi}^2 \,+\,
\frac{ \alpha_{\mbox{\tiny{$+-$}}} M_{\pi^0}^2}{24\pi F_{\pi}^2}
\, ,
\quad b_{\mbox{\tiny{$++$}}}\,=\, - \frac{\beta_{\mbox{\tiny{$+-$}}}}{16\pi}
.\end{aligned}$$
The quantities $a = a_x, a_{\mbox{\tiny{$+-$}}}, a_{00}$ etc., are scattering lengths to the extent that the tree-level amplitudes (\[AmpTree\_a\]) satisfy $$\label{eq:threshold}
{\mbox{Re}}\,A(s,t,u)\big\vert_{\mbox{\scriptsize{thr}}}\,=\,16\pi a
.$$ The parameters $a$ will keep their meaning up to next-to-next-to-leading order if the above relation still holds for the two-loop amplitudes. This can be achieved upon adjusting the subtraction polynomials accordingly. In practice, this is done through the following choice: $$\begin{aligned}
P^{00}(s,t,u) &=& 16\pi a_{00} \,-\,w_{00}
+\,
\frac{3\lambda_{00}^{(1)}}{F_{\pi}^4}\left[
s(s-4M_{\pi^0}^2) + t(t-4M_{\pi^0}^2) + u(u-4M_{\pi^0}^2)
\right]
\nonumber\\
&& \!\!\!\!\!\!\!
+\,
\frac{3\lambda_{00}^{(2)}}{F_{\pi}^6}\left[
s(s-4M_{\pi^0}^2)(s-2M_{\pi^0}^2) + t(t-4M_{\pi^0}^2)(t-2M_{\pi^0}^2) + u(u-4M_{\pi^0}^2)(u-2M_{\pi^0}^2)
\right]
\nonumber\end{aligned}$$ $$\begin{aligned}
P^{x}(s,t,u) &=& 16\pi a_x \,+\,w_x
\,+\,16\pi b_x\, \frac{s - 4 M_{\pi}^2}{F_{\pi}^2}
-\,
\frac{\lambda_{x}^{(1)}}{F_{\pi}^4}\,s(s-4M_{\pi}^2)
\nonumber\\
&& \!\!\!\!\!\!\!\!\!\!\!\!\!
-\,
\frac{\lambda_{x}^{(2)}}{F_{\pi}^4}\left[
(t + M_{\pi}^2 - M_{\pi^0}^2)(t - 3 M_{\pi}^2 - M_{\pi^0}^2) +
(u + M_{\pi}^2 - M_{\pi^0}^2)(u - 3 M_{\pi}^2 - M_{\pi^0}^2)
\right]
\nonumber\\
&& \!\!\!\!\!\!\!\!\!\!\!\!\!
-\,
\frac{\lambda_{x}^{(3)}}{F_{\pi}^6}\,2s(s-4 M_{\pi}^2)(s - M_{\pi}^2 - M_{\pi^0}^2)
\nonumber\\
&& \!\!\!\!\!\!\!\!\!\!\!\!\!
-\,
\frac{\lambda_{x}^{(4)}}{F_{\pi}^6}\left[
(t + M_{\pi}^2 - M_{\pi^0}^2) \lambda (t)
\,+\,
(u + M_{\pi}^2 - M_{\pi^0}^2) \lambda (u)
\right]
\nonumber\end{aligned}$$
$$\begin{aligned}
P^{{\mbox{\tiny{$+$}}}0}(s,t,u) &=& 16\pi a_{{\mbox{\tiny{$+$}}}0} \,-\, w_{{\mbox{\tiny{$+$}}}0}
\,+\, 16\pi b_{{\mbox{\tiny{$+$}}}0}\,\frac{s-(M_{\pi} + M_{\pi^0})^2}{F_{\pi}^2}\,
+\,16\pi c_{{\mbox{\tiny{$+$}}}0}\,\frac{t-u+(M_{\pi} - M_{\pi^0})^2}{F_{\pi}^2}
\nonumber\\
&& \!\!\!\!\!\!\!\!\!\!\!\!\!
+\,
\frac{\lambda_{x}^{(1)}}{F_{\pi}^4}\,t (t - 4M_{\pi}^2)
\,+\,
\frac{\lambda_{x}^{(2)}}{F_{\pi}^4}\left[
\lambda (s)\,+\,\lambda (u)
\right]
\nonumber\\
&& \!\!\!\!\!\!\!\!\!\!\!\!\!
+\,
\frac{\lambda_{x}^{(3)}}{F_{\pi}^6}\,2t (t - 4M_{\pi}^2)(t - M_{\pi}^2 - M_{\pi^0}^2)
+\,
\frac{\lambda_{x}^{(4)}}{F_{\pi}^6}\left[
(s + M_{\pi}^2 - M_{\pi^0}^2) \lambda (s)\,+\,
(u + M_{\pi}^2 - M_{\pi^0}^2) \lambda (u)
\right]
\nonumber\end{aligned}$$
$$\begin{aligned}
P^{\mbox{\tiny{$+-$}}}(s,t,u) &=& 16\pi a_{\mbox{\tiny{$+-$}}} \,-\, w_{\mbox{\tiny{$+-$}}}
\,+\,16\pi b_{\mbox{\tiny{$+-$}}} \frac{s-4M_{\pi}^2}{F_{\pi}^2}\,+\, 16\pi c_{\mbox{\tiny{$+-$}}} \frac{t-u}{F_{\pi}^2}
\nonumber\\
&& \!\!\!\!\!\!\!\!\!\!\!\!\!
+\,\frac{\lambda_{\mbox{\tiny{$+-$}}}^{(1)} + \lambda_{\mbox{\tiny{$+-$}}}^{(2)}}{F_{\pi}^4}
\left[s ( s - 4 M_{\pi}^2 )\,+\, t( t - 4 M_{\pi}^2 )\right] \,+\,
\frac{2 \lambda_{\mbox{\tiny{$+-$}}}^{(2)}}{F_{\pi}^4}\,u( u - 4 M_{\pi}^2 )
\nonumber\\
&& \!\!\!\!\!\!\!\!\!\!\!\!\!
+\,\frac{\lambda_{\mbox{\tiny{$+-$}}}^{(3)} + \lambda_{\mbox{\tiny{$+-$}}}^{(4)}}{F_{\pi}^6}
\left[s ( s - 4 M_{\pi}^2 ) (s - 2 M_{\pi}^2) \,+\,
t ( t - 4 M_{\pi}^2 ) (t - 2 M_{\pi}^2)
\right]
+\,\frac{2 \lambda_{\mbox{\tiny{$+-$}}}^{(4)}}{F_{\pi}^6}\,u ( u - 4 M_{\pi}^2 ) (u - 2 M_{\pi}^2)
\nonumber\end{aligned}$$
$$\begin{aligned}
P^{\mbox{\tiny{$++$}}}(s,t,u) &=& 16\pi a_{\mbox{\tiny{$++$}}} \,-\, w_{\mbox{\tiny{$++$}}}
\,+\,16\pi b_{\mbox{\tiny{$++$}}} \frac{s - 4 M_{\pi}^2}{F_{\pi}^2}
\nonumber\\
&& \!\!\!\!\!\!\!\!\!\!\!\!\!
+\,\frac{\lambda_{\mbox{\tiny{$+-$}}}^{(1)} + \lambda_{\mbox{\tiny{$+-$}}}^{(2)}}{F_{\pi}^4}
\left[t ( t - 4 M_{\pi}^2 )\,+\, u( u - 4 M_{\pi}^2 )\right] \,+\,
\frac{2 \lambda_{\mbox{\tiny{$+-$}}}^{(2)}}{F_{\pi}^4}\,s( s - 4 M_{\pi}^2 )
\nonumber\\
&& \!\!\!\!\!\!\!\!\!\!\!\!\!
+\,\frac{\lambda_{\mbox{\tiny{$+-$}}}^{(3)} + \lambda_{\mbox{\tiny{$+-$}}}^{(4)}}{F_{\pi}^6}
\left[ t ( t - 4 M_{\pi}^2 ) (t - 2 M_{\pi}^2)\,+\,
u ( u - 4 M_{\pi}^2 ) (u - 2 M_{\pi}^2)
\right]
+\,\frac{2 \lambda_{\mbox{\tiny{$+-$}}}^{(4)}}{F_{\pi}^6}\,s ( s - 4 M_{\pi}^2 ) (s - 2 M_{\pi}^2) \end{aligned}$$
where we have subtracted the values of the one-loop integrals at the appropriate kinematical points to ensure eq. (\[eq:threshold\]) $$\begin{aligned}
w_{00} &=& {\mbox{Re}}\,
\left[W^{00}_0(4M_{\pi^0}^2)\,+\, W^{00}_0(0)\,+\, W^{00}_0(0)\right]
\nonumber\\
w_x &=& {\mbox{Re}}\,\left[
W^x_{0}(4M_{\pi}^2) + 2 W^{{\mbox{\tiny{$+$}}}0}_{0}(M_{\pi^0}^2 - M_{\pi}^2)
+ 6 (5 M_{\pi}^2 - M_{\pi^0}^2) W^{{\mbox{\tiny{$+$}}}0}_{1}(M_{\pi^0}^2 - M_{\pi}^2)
\right]
\nonumber\\
w_{{\mbox{\tiny{$+$}}}0} &=&
{\mbox{Re}}\,\left[ W^{{\mbox{\tiny{$+$}}}0}_{0}((M_{\pi^0} + M_{\pi})^2)
- 3 (M_{\pi^0} - M_{\pi})^2 W^{{\mbox{\tiny{$+$}}}0}_{1}((M_{\pi^0} + M_{\pi})^2)
\right]
\nonumber\\
&&
\!\!\!\!\!\!
+\,{\mbox{Re}}\,\left[ W^{{\mbox{\tiny{$+$}}}0}_{0}((M_{\pi^0} - M_{\pi})^2)
- 3 (M_{\pi^0} + M_{\pi})^2 W^{{\mbox{\tiny{$+$}}}0}_{1}((M_{\pi^0} - M_{\pi})^2)
\right]
\,+\, {\mbox{Re}}\,W^x_0(0)
\nonumber\\
w_{\mbox{\tiny{$+-$}}} &=& {\mbox{Re}}\left[W^{\mbox{\tiny{$++$}}}_0(0)
\,+\,
W^{\mbox{\tiny{$+-$}}}_{0}(4 M_{\pi}^2)
\,+\, W^{\mbox{\tiny{$+-$}}}_{0}(0) \,+\, 12 M_{\pi}^2 W^{\mbox{\tiny{$+-$}}}_{1}(0)\right]
\nonumber\\
w_{\mbox{\tiny{$++$}}} &=& {\mbox{Re}}\,\left[ 2 W^{\mbox{\tiny{$+-$}}}_{0}(0)
\,+\, W^{\mbox{\tiny{$++$}}}_{0}(4 M_{\pi}^2)
\,-\,24 M_{\pi}^2 W^{\mbox{\tiny{$+-$}}}_{1}(0)
\right]
.\end{aligned}$$
These expressions should involve the same number (fifteen) of independent subtraction constants (among them now the scattering lengths) as the ones given in eqs. (\[polynomials\_P\]). This means that there exist six relations between the twenty-one parameters occurring in the above polynomials, which stem from crossing symmetry \[by construction, the unitarity parts satisfy separately the crossing relations\], $P^{x}(t,s,u)+P^{{\mbox{\tiny{$+$}}}0}(s,t,u)=0$ and $P^{\mbox{\tiny{$+-$}}}(u,t,s)-P^{\mbox{\tiny{$++$}}}(s,t,u)=0$. This yields $$\begin{aligned}
b_{{\mbox{\tiny{$+$}}}0} \,+\, c_{{\mbox{\tiny{$+$}}}0}\,=\,0\ ,\quad b_x\,-\,2 b_{{\mbox{\tiny{$+$}}}0}\,=\, 0
\ ,\quad b_{\mbox{\tiny{$+-$}}} \,-\, c_{\mbox{\tiny{$+-$}}}\,=\,0\ ,
\quad b_{\mbox{\tiny{$++$}}}\,+\,2 b_{\mbox{\tiny{$+-$}}}\,=\,0
,\end{aligned}$$ and $$\begin{aligned}
16\pi \left( a_{x} + a_{{\mbox{\tiny{$+$}}}0} - 4\,\frac{M_{\pi}^2}{F_\pi^2}\,b_{x} \right) &=&
w_{{\mbox{\tiny{$+$}}}0}\,-\,w_{x}
\,-\,8\,\frac{M_{\pi}^2 (M_{\pi}^2 - M_{\pi^0}^2)}{F_\pi^4}\,\lambda_{{\mbox{\tiny{$+$}}}0}^{(2)}
\nonumber\\
16\pi \left( a_{\mbox{\tiny{$+-$}}} - a_{\mbox{\tiny{$++$}}} + 4\,\frac{M_{\pi}^2}{F_\pi^2}\,b_{\mbox{\tiny{$++$}}} \right) &=&
w_{\mbox{\tiny{$+-$}}}\,-\,w_{\mbox{\tiny{$++$}}}
.\end{aligned}$$ For the time being, it is convenient not to make use of the two last relations, and to treat all the $S$-wave scattering lengths as independent. Notice also that in the chiral counting the scattering lengths are of order ${\cal O}(E^2)$, whereas $b_{x}$, $b_{\mbox{\tiny{$+-$}}}$ and $b_{\mbox{\tiny{$++$}}}$ are of order ${\cal O}(E^0)$. One may now repeat the computation of the relevant partial-wave projections starting with the expressions of the lowest-order amplitudes in terms of the scattering lengths $a_i$ and effective range parameters $b_i$ $(i= 00 , \pm 0 , x , +- , ++)$, considered as independent quantities, following the procedure outline in sec. \[general\] and fig. \[iterconst\]. The results can still be brought into the representations (\[psi\_00\]), (\[psi\_+-\_0\]), (\[psi\_+-\_1\]), or (\[psi\_x\_0\]), but the expressions of the polynomials involved are different from the ones given in appendix \[app:polynomials\]. For the scattering of neutral pions, the polynomials for $\psi^{00}_0(s)$ now read $$\begin{aligned}
\xi_{00}^{(0)}(s) &=&
\frac{\lambda_{00}^{(1)}}{2 M_\pi^4}\,
({5}s\,+\,{4}M_{\pi^0}^2) (s\,-\,4M_{\pi^0}^2)\,-\,128 \pi^2 \frac{F_\pi^4}{M_\pi^4}
\left( a_x - 4 b_x\frac{\Delta_\pi}{F_\pi^2}\right)^2 {\mbox{Re}}\,{\bar J}(4 M_{\pi^0}^2)
\nonumber\\
&+&
\frac{8 b_x^2}{9 M_\pi^4}\left( 32 s^2 - 112 s M_{\pi^0}^2 + 39 s M_{\pi}^2
+ 224 M_{\pi^0}^4 -732 M_{\pi^0}^2 M_{\pi}^2 + 1260 M_{\pi}^4\right)
\nonumber\\
&&
\!\!\!\!\!
-\,
\frac{8 a_x b_x F_\pi^2}{M_\pi^4}\left( s - 20 M_{\pi^0}^2 + 68 M_{\pi}^2 \right) \,+\,
8\,\frac{F_\pi^4}{M_\pi^4} \left( 3 a_{00}^2 + 8 a_x^2 \right)
\nonumber\\
\xi^{(1;0)}_{00}(s) &=& 4 a_{00}^2 \,\frac{F_\pi^4}{M_\pi^4}
\nonumber\\
\xi^{(1;{\mbox{\tiny$\nabla$}})}_{00}(s) &=&
\frac{8 b_x^2}{3 M_\pi^4}\left( s^2 - 8 s M_{\pi^0}^2 + 13 s M_{\pi}^2
+ 16 M_{\pi^0}^4 - 52 M_{\pi^0}^2 M_{\pi}^2 + 66 M_{\pi}^4\right)
\nonumber\\
&&
\!\!\!\!\!
-\,
\frac{8 a_x b_x F_\pi^2}{M_{\pi}^4}\left( s - 4 M_{\pi^0}^2 + 10 M_{\pi}^2 \right) \,+\, 8 a_x^2 \,\frac{F_\pi^4}{M_\pi^4}
\nonumber\\
\xi^{(2;0)}_{00}(s) &=& 8 a_{00}^2 \,\frac{F_\pi^4}{M_\pi^4}
\nonumber\\
\xi^{(2;{\mbox{\tiny{$\pm$}}})}_{00}(s) &=& 4 \,\frac{F_\pi^4}{M_\pi^4}
\left[
b_x \,\frac{ s - 4 M_{\pi}^2}{F_\pi^2} \,+\, a_x
\right]^2
\nonumber\\
\xi^{(3;0)}_{00}(s) &=& -\,\frac{8}{3}\, a_{00}^2 \,\frac{F_\pi^4}{M_\pi^4}
\nonumber\\
\xi^{(3;{\mbox{\tiny$\nabla$}})}_{00}(s) &=& -\,\frac{160}{3}\, b_x^2
\,+\,32 a_x b_x\,\frac{F_{\pi}^2}{M_\pi^2}\,-\,\frac{16}{3}\, a_x^2 \,\frac{F_\pi^4}{M_\pi^4}
.\end{aligned}$$
For the scattering of charged pions, the polynomials involved in the expression for $\psi^{\mbox{\tiny{$+-$}}}_0 (s)$ ($S$-wave) read $$\begin{aligned}
\xi_{{\mbox{\tiny{$+-$}}};S}^{(0)}(s) &=& \frac{\lambda_{\mbox{\tiny{$+-$}}}^{(1)}}{3 M_{\pi}^4}\,
(s - 4 M_{\pi}^2)(2 s + M_{\pi}^2)
+
\frac{\lambda_{\mbox{\tiny{$+-$}}}^{(2)} }{M_{\pi}^4}\,(s - 4 M_{\pi}^2)(s + M_{\pi}^2)
- 64 \pi^2 \frac{F_\pi^4}{M_\pi^4}\, a_x^2 \, {\mbox{Re}}\,{\bar J}_{0}(4 M_{\pi}^2)
\nonumber\\
&&
+\,\frac{4s^2}{9 M_\pi^4} \left(25 b_x^2 + 7 b_{\mbox{\tiny{$++$}}}^2 + \frac{119}{3}b_{\mbox{\tiny{$+-$}}}^2\right)
\,+\,\frac{2 s M_{\pi^0}^2}{3 M_\pi^4}\,b_x^2 +
\frac{2 s }{9 M_\pi^2} \left(71 b_{\mbox{\tiny{$++$}}}^2 - 220 b_x^2 - \frac{1736}{3} b_{\mbox{\tiny{$+-$}}}^2 \right)
\nonumber\\
&&
+\,\frac{2 s F_\pi^2}{M_\pi^4} \left(3 a_x b_x - 5 b_{\mbox{\tiny{$++$}}} a_{\mbox{\tiny{$++$}}} +
6 b_{\mbox{\tiny{$+-$}}} a_{\mbox{\tiny{$+-$}}} \right)
+\frac{32}{9} \left(29 b_{\mbox{\tiny{$++$}}}^2 + 59 b_x^2 + \frac{448}{3} b_{\mbox{\tiny{$+-$}}}^2 \right)
-\,\frac{8 M_{\pi^0}^4}{ M_\pi^4}\,b_x^2 + \,\frac{88 M_{\pi^0}^2}{3 M_\pi^2}\,b_x^2
\nonumber\\
&&
-\,\frac{8 F_{\pi}^2}{M_\pi^2} \left(8 a_{\mbox{\tiny{$++$}}} b_{\mbox{\tiny{$++$}}} + 15 a_x b_x +
32 a_{\mbox{\tiny{$+-$}}} b_{\mbox{\tiny{$+-$}}} \right)
-\,\frac{8 M_{\pi^0}^2 F_\pi^2}{M_\pi^4}\,a_x b_x \,+\, 4 \left( 3 a_{\mbox{\tiny{$++$}}}^2 + 5 a_x^2 +
6 a_{\mbox{\tiny{$+-$}}}^2 \right)\frac{F_\pi^4}{M_\pi^4}
\nonumber\\
\xi^{(1;{\mbox{\tiny{$\pm$}}})}_{{\mbox{\tiny{$+-$}}};S}(s) &=& \frac{2s^2}{9 M_\pi^4}
\left(3 b_{\mbox{\tiny{$++$}}}^2 + 2 b_{\mbox{\tiny{$+-$}}}^2\right) +
\,\frac{2 s }{9 M_\pi^2} \left(15 b_{\mbox{\tiny{$++$}}}^2 + 4 b_{\mbox{\tiny{$+-$}}}^2 \right)
-\,\frac{2 s F_\pi^2}{M_\pi^4} \left( b_{\mbox{\tiny{$++$}}} a_{\mbox{\tiny{$++$}}}
+ 2 b_{\mbox{\tiny{$+-$}}} a_{\mbox{\tiny{$+-$}}} \right)
\nonumber\\
&&
+\,\frac{20 }{3 }
\left(3 b_{\mbox{\tiny{$++$}}}^2 + 8 b_{\mbox{\tiny{$+-$}}}^2 \right)
-\, \frac{12 F_{\pi}^2}{M_\pi^2} \left(a_{\mbox{\tiny{$++$}}} b_{\mbox{\tiny{$++$}}}
+ 2 a_{\mbox{\tiny{$+-$}}} b_{\mbox{\tiny{$+-$}}} \right)
\,+\, 2 \left( a_{\mbox{\tiny{$++$}}}^2 + 2 a_{\mbox{\tiny{$+-$}}}^2 \right) \frac{F_\pi^4}{M_\pi^4}
\nonumber\\
\xi^{(1;{\mbox{\tiny$\Delta$}})}_{{\mbox{\tiny{$+-$}}}; S}(s) &=& \frac{4s^2}{3 M_\pi^4} \, b_x^2 +\,
\frac{2 s (M_{\pi^0}^2 + 4 M_{\pi}^2)}{3 M_\pi^4}\,b_x^2 -\,\frac{2 s F_\pi^2}{M_\pi^4}\,a_x b_x
+ \,\frac{4 (8 M_{\pi}^4 + 10 M_{\pi}^2 M_{\pi^0}^2 - 3 M_{\pi^0}^4)}{3 M_\pi^4} \, b_x^2
\nonumber\\
&&
-\, \frac{4 F_\pi^2 (2 M_{\pi}^2 + M_{\pi^0}^2)}{M_\pi^4}\,a_x b_x \,+\, 2 a_x^2\,\frac{F_\pi^4}{M_\pi^4}
\nonumber\\
\xi^{(2;0)}_{{\mbox{\tiny{$+-$}}};S}(s) &=&
2\,\frac{F_\pi^4}{M_\pi^4} \left[b_{x}\,\frac{s - 4 M_{\pi}^2}{F_\pi^2}
\,+\ a_{x}
\right]^2
\nonumber\\
\xi^{(2;{\mbox{\tiny{$\pm$}}})}_{{\mbox{\tiny{$+-$}}};S}(s) &=&
4\,\frac{F_\pi^4}{M_\pi^4} \left[b_{\mbox{\tiny{$+-$}}}\,\frac{s - 4 M_{\pi}^2}{F_\pi^2}
\,+\ a_{\mbox{\tiny{$+-$}}}
\right]^2
\nonumber\\
\xi^{(3;{\mbox{\tiny{$\pm$}}})}_{{\mbox{\tiny{$+-$}}};S}(s) &=& \frac{16 s}{3 M_\pi^2} \, b_{\mbox{\tiny{$+-$}}}^2
\,-\, \frac{40 }{9 }
\left(3 b_{\mbox{\tiny{$++$}}}^2 + 8 b_{\mbox{\tiny{$+-$}}}^2 \right)
\,+\,\frac{8 F_{\pi}^2}{M_\pi^2} \left(a_{\mbox{\tiny{$++$}}} b_{\mbox{\tiny{$++$}}} +
2 a_{\mbox{\tiny{$+-$}}} b_{\mbox{\tiny{$+-$}}} \right)
-\, \frac{4}{3} \left( a_{\mbox{\tiny{$++$}}}^2 + 2 a_{\mbox{\tiny{$+-$}}}^2 \right) \frac{F_\pi^4}{M_\pi^4}
\nonumber\\
\xi^{(3;{\mbox{\tiny$\Delta$}})}_{{\mbox{\tiny{$+-$}}};S}(s) &=&
-\,\frac{4}{3} \left[
2 \,\frac{M_{\pi^0}^4 - 4 M_{\pi^0}^2 M_{\pi}^2 + 8 M_{\pi}^4}{ M_\pi^4}\,b_x^2
\,+\, 2 \,\frac{(M_{\pi^0}^2 - 4 M_{\pi}^2)F_\pi^2 }{ M_\pi^4}\, a_x b_x \,+\, a_x^2\,\frac{F_\pi^4}{M_\pi^4}
\right]
,\end{aligned}$$ while for the $P$-wave contribution, $\psi^{\mbox{\tiny{$+-$}}}_1 (s)$, we obtain $$\begin{aligned}
\xi_{{\mbox{\tiny{$+-$}}};P}^{(0)}(s) &=&
- \frac{\lambda_{\mbox{\tiny{$+-$}}}^{(1)} - \lambda_{\mbox{\tiny{$+-$}}}^{(2)}}{12 M_{\pi}^4}\,s\left(s-4 M_{\pi}^2\right)
+\,\frac{s^2}{18 M_\pi^4} \left(25 b_{\mbox{\tiny{$++$}}}^2 - 25 b_x^2 - \frac{16}{3}b_{\mbox{\tiny{$+-$}}}^2\right)
-\,
\frac{4 s }{3 M_\pi^2} \left(\frac{7}{6} b_{\mbox{\tiny{$++$}}}^2 - b_x^2 + \frac{208}{9} b_{\mbox{\tiny{$+-$}}}^2 \right)
\nonumber\\
&&
+\,\frac{2 s M_{\pi^0}^2}{9 M_\pi^4}\,b_x^2
+\,\frac{22 s F_\pi^2}{9 M_\pi^4} \left( a_x b_x - b_{\mbox{\tiny{$++$}}} a_{\mbox{\tiny{$++$}}}
+ 2 b_{\mbox{\tiny{$+-$}}} a_{\mbox{\tiny{$+-$}}} \right)
+\,\frac{8}{9}
\left( 55 b_x^2 + 250 b_{\mbox{\tiny{$+-$}}}^2 - \frac{291}{4} b_{\mbox{\tiny{$++$}}}^2 \right)
\nonumber\\
&&-\,\frac{14 M_{\pi^0}^4}{3 M_\pi^4}\,b_x^2
+\, \frac{184 M_{\pi^0}^2 }{9 M_\pi^2}\,b_x^2
+\,\frac{8 F_{\pi}^2}{9 M_\pi^2} \left(35 a_{\mbox{\tiny{$++$}}} b_{\mbox{\tiny{$++$}}}
- 29 a_x b_x - 70 a_{\mbox{\tiny{$+-$}}} b_{\mbox{\tiny{$+-$}}} \right)
-\,\frac{16 M_{\pi^0}^2 F_\pi^2}{3 M_\pi^4}\,a_x b_x
\nonumber\\
&&
-\, 2 \left( a_{\mbox{\tiny{$++$}}}^2 - a_x^2 - 2 a_{\mbox{\tiny{$+-$}}}^2 \right) \frac{F_\pi^4}{M_\pi^4}
\nonumber\\
\xi^{(1;{\mbox{\tiny{$\pm$}}})}_{{\mbox{\tiny{$+-$}}};P}(s) &=& \frac{ s^2}{3 M_\pi^4}
\left( b_{\mbox{\tiny{$++$}}}^2
-\,\frac{4}{3}\,b_{\mbox{\tiny{$+-$}}}^2\right)
-\,\frac{8 s }{3 M_\pi^2}\,b_{\mbox{\tiny{$+-$}}}^2
+\,\frac{2 s F_\pi^2}{3 M_\pi^4}\,(2a_{\mbox{\tiny{$+-$}}} b_{\mbox{\tiny{$+-$}}}
- a_{\mbox{\tiny{$++$}}} b_{\mbox{\tiny{$++$}}})
- 10 b_{\mbox{\tiny{$++$}}}^2
+\,
\frac{112}{3}\,b_{\mbox{\tiny{$+-$}}}^2
\nonumber\\
&&
-\,\frac{4 F_{\pi}^2 }{M_\pi^2}\,(2a_{\mbox{\tiny{$+-$}}} b_{\mbox{\tiny{$+-$}}}
- a_{\mbox{\tiny{$++$}}} b_{\mbox{\tiny{$++$}}})
\nonumber\\
\xi^{(1;{\mbox{\tiny$\Delta$}})}_{{\mbox{\tiny{$+-$}}}; P}(s) &=& -\frac{ s^2}{3 M_\pi^4}\,b_{x}^2
+\,\frac{2 s F_\pi^2}{3 M_\pi^4}\,a_x b_x +\,
2 b_x^2\,\frac{8 M_{\pi}^4 + 8 M_{\pi}^2 M_{\pi^0}^2 - M_{\pi^0}^4}{3 M_\pi^4}
\,-\, 4 a_x b_x\,\frac{(2 M_{\pi}^2 + M_{\pi^0}^2) F_\pi^2}{3 M_\pi^4}
\nonumber\\
\xi^{(2;{\mbox{\tiny{$\pm$}}})}_{{\mbox{\tiny{$+-$}}};P}(s) &=& \frac{4}{9}\,
\left(\frac{s}{M_\pi^2} \,-\, 4 \right)^2 b_{\mbox{\tiny{$+-$}}}^2
\nonumber\\
\xi^{(3;{\mbox{\tiny{$\pm$}}})}_{{\mbox{\tiny{$+-$}}};P}(s) &=& \frac{16 s}{3 M_\pi^2}\,b_{\mbox{\tiny{$+-$}}}^2
+\,\frac{40}{3}\,b_{\mbox{\tiny{$++$}}}^2 \,-\,\frac{256 }{9}\,b_{\mbox{\tiny{$+-$}}}^2
+ \,8\,\frac{F_{\pi}^2}{ M_\pi^2}\,(2 a_{\mbox{\tiny{$+-$}}} b_{\mbox{\tiny{$+-$}}}
- a_{\mbox{\tiny{$++$}}} b_{\mbox{\tiny{$++$}}})
+\,\frac{4}{3}\left( a_{\mbox{\tiny{$++$}}}^2 - 2 a_{\mbox{\tiny{$+-$}}}^2 \right) \frac{F_\pi^4}{M_\pi^4}
\nonumber\\
\xi^{(3;{\mbox{\tiny$\Delta$}})}_{{\mbox{\tiny{$+-$}}}; P}(s) &=& - 8 \,
\frac{8 M_{\pi}^4 - 4 M_{\pi}^2 M_{\pi^0}^2 + M_{\pi^0}^4}{3 M_\pi^4}\, b_x^2
+\, 8 \,\frac{(4 M_{\pi}^2 - M_{\pi^0}^2) F_\pi^2}{3 M_\pi^4} a_x b_x
-\, \frac{4}{3}\, a_x^2 \frac{F_\pi^4}{M_\pi^4}
\nonumber\\
\xi^{(4;{\mbox{\tiny{$\pm$}}})}_{{\mbox{\tiny{$+-$}}};P}(s) &=& 40 b_{\mbox{\tiny{$++$}}}^2
\,-\,\frac{64 }{3 }\,b_{\mbox{\tiny{$+-$}}}^2
\,+\,32\,\frac{F_{\pi}^2}{ M_\pi^2}\,(2 a_{\mbox{\tiny{$+-$}}} b_{\mbox{\tiny{$+-$}}}
- a_{\mbox{\tiny{$++$}}} b_{\mbox{\tiny{$++$}}})
\,+\,8 \left( a_{\mbox{\tiny{$++$}}}^2 - 2 a_{\mbox{\tiny{$+-$}}}^2 \right) \frac{F_\pi^4}{M_\pi^4}
\nonumber\\
\xi^{(4;{\mbox{\tiny$\Delta$}})}_{{\mbox{\tiny{$+-$}}}; P}(s) &=& -8 \,
\frac{16 M_{\pi}^4 - 16 M_{\pi}^2 M_{\pi^0}^2 + 5 M_{\pi^0}^4}{M_\pi^4} \,b_x^2
+\, 32 \,\frac{(2 M_{\pi}^2 - M_{\pi^0}^2) F_\pi^2}{M_\pi^4} \,a_x b_x
-\, 8 a_x^2 \,\frac{F_\pi^4}{M_\pi^4}
.\end{aligned}$$
Finally, in the case of a scattering involving two neutral pions and two charged pions, we obtain the expression for the polynomials describing $\psi^x_0$ $$\begin{aligned}
\xi_{x}^{(0)}(s) &=&
-\,\frac{\lambda_{x}^{(1)}}{2 M_{\pi}^4}\,s(s - 4 M_{\pi}^2)
\,-\,
\frac{\lambda_{x}^{(2)} }{3 M_{\pi}^4}\,(s - 4 M_{\pi}^2)
(s + 3 M_{\pi^\pm}^2 - M_{\pi^0}^2)
-\,64\pi^2 \frac{F_\pi^4}{M_\pi^4}\,a_x a_{00}\, {\mbox{Re}}\,{\bar J}_{0}(4 M_{\pi}^2)
\nonumber\\
&&
+\,\,(16\pi)^2 \left[
4\,\frac{7 M_{\pi}^4 + 2 M_{\pi}^2 M_{\pi^0}^2 - M_{\pi^0}^4}{3 M_\pi^4}\,b_{{\mbox{\tiny{$+$}}} 0}^2
-\,8 a_{{\mbox{\tiny{$+$}}} 0} b_{{\mbox{\tiny{$+$}}} 0}\,\frac{F_{\pi}^2}{M_\pi^2}
\,+\, a_{{\mbox{\tiny{$+$}}} 0}^2\,\frac{F_\pi^4}{M_\pi^4}
\right]
{\mbox{Re}}\,
{\bar J}_{{\mbox{\tiny{$+$}}} 0}(- \Delta_\pi)
\nonumber\\
&&
+\, \frac{ \Delta_\pi^2}{3 M_{\pi}^4}\,(32\pi)^2 \frac{F_\pi^4}{M_\pi^4}\, b_{{\mbox{\tiny{$+$}}} 0}^2
\, {\mbox{Re}}\ \, {\bar{\!\!{\bar J}}}_{{\mbox{\tiny{$+$}}} 0}(- \Delta_\pi)
+\,
\frac{8 s^2}{27 M_\pi^4} \left(54 b_x b_{{\mbox{\tiny{$+-$}}} } - 11 b_{{\mbox{\tiny{$+$}}} 0}^2 \right)
\nonumber\\
&&
+\,\frac{16 s M_{\pi^0}^2 }{27 M_\pi^4}\,b_{{\mbox{\tiny{$+$}}} 0}^2 \,+\,
\frac{16 s}{27 M_\pi^2} \left( b_{{\mbox{\tiny{$+$}}} 0}^2 - 216 b_x b_{{\mbox{\tiny{$+-$}}}} \right)
+\,\frac{8 s F_\pi^2}{M_\pi^4} \left( b_x a_{00} + 2 b_x a_{{\mbox{\tiny{$+-$}}}} + 2 a_{x} b_{{\mbox{\tiny{$+-$}}}} +
5 a_{{\mbox{\tiny{$+$}}} 0} b_{{\mbox{\tiny{$+$}}} 0} \right)
\nonumber\\
&&
+\,32 \left(8 b_{{\mbox{\tiny{$+-$}}}} b_x - 7 b_{{\mbox{\tiny{$+$}}} 0}^2 \right)
-\,\frac{224 M_{\pi^0}^4}{ M_\pi^4}\,b_{{\mbox{\tiny{$+$}}} 0}^2
-\,\frac{2752}{27}\,\frac{M_{\pi^0}^2 }{M_\pi^2}\,b_{{\mbox{\tiny{$+$}}} 0}^2
+\,128\,\frac{M_{\pi^0}^2 F_\pi^2}{M_\pi^4}\,a_{{\mbox{\tiny{$+$}}} 0} b_{{\mbox{\tiny{$+$}}} 0}
\nonumber\\
&&
+\,\frac{32 F_{\pi}^2}{M_\pi^2} \left(4 a_{{\mbox{\tiny{$+$}}} 0} b_{{\mbox{\tiny{$+$}}} 0}
- 2 a_x b_{{\mbox{\tiny{$+-$}}}}
- 2 b_{x} a_{{\mbox{\tiny{$+-$}}}} - b_x a_{00}\right)
+\, 8 \left( a_x a_{00} - 6 a_{\pm 0}^2 \right) \frac{F_\pi^4}{M_\pi^4}
\nonumber\\
\xi_{x}^{(1)}(s) &=& -\, \frac{8 s (s + 2 M_{\pi^0}^2 )}{9 M_\pi^4}\,b_{{\mbox{\tiny{$+$}}} 0}^2
+\,\frac{8 s F_\pi^2}{M_\pi^4} \, a_{\pm 0} b_{\pm 0}
-\,\frac{16}{3}\, \frac{ 5 M_{\pi}^4 + 4 M_{\pi}^2 M_{\pi^0}^2 + 11 M_{\pi^0}^4}{M_\pi^4}\,b_{{\mbox{\tiny{$+$}}} 0}^2
\nonumber\\
&&
+\,16\,\frac{(M_{\pi}^2 + 2 M_{\pi^0}^2) F_\pi^2}{M_\pi^4}\, a_{{\mbox{\tiny{$+$}}} 0} b_{{\mbox{\tiny{$+$}}} 0}
-\, 8 \, a_{{\mbox{\tiny{$+$}}} 0}^2 \,\frac{F_\pi^4}{M_\pi^4}
\nonumber\\
\xi_{x}^{(2;0)}(s) &=& 2 a_{00}\,\frac{F_\pi^4}{M_\pi^4}
\left[ \frac{s - 4 M_\pi^2}{F_\pi^2}\,b_x \,+\, a_x \right]
\nonumber\\
\xi_{x}^{(2;\pm)}(s) &=& 4\,\frac{F_\pi^4}{M_\pi^4}
\left[ \frac{s - 4 M_\pi^2}{F_\pi^2}\,b_x \,+\, a_x \right]
\left[ \frac{s - 4 M_\pi^2}{F_\pi^2}\,b_{\mbox{\tiny{$+-$}}} \,+\, a_{\mbox{\tiny{$+-$}}} \right]
\nonumber\\
\xi_{x}^{(3)}(s) &=&
-\,\frac{32 s}{9 M_\pi^2} \, b_{{\mbox{\tiny{$+$}}} 0}^2
\left(1 + \frac{M_{\pi^0}^2}{M_{\pi}^2} + \frac{M_{\pi^0}^4}{M_{\pi}^4} \right)
+\,\frac{176 }{9 }\,b_{{\mbox{\tiny{$+$}}} 0}^2 \left(1 + \frac{M_{\pi^0}^6}{M_{\pi}^6}\right)
\,+\,\frac{16 M_{\pi^0}^2(M_{\pi}^2 + M_{\pi^0}^2)}{ M_\pi^4}\,b_{{\mbox{\tiny{$+$}}} 0}^2
\nonumber\\
&&
-\,32 a_{{\mbox{\tiny{$+$}}} 0} b_{{\mbox{\tiny{$+$}}} 0}\,\frac{F_{\pi}^2}{3 M_\pi^2}
\left(1 + \frac{M_{\pi^0}^2}{M_{\pi}^2} + \frac{M_{\pi^0}^4}{M_{\pi}^4} \right)
\,+\, 8 a_{{\mbox{\tiny{$+$}}} 0}^2 \left(1 + \frac{M_{\pi^0}^2}{M_{\pi}^2} \right)\frac{F_\pi^4}{M_\pi^4} \end{aligned}$$ In addition, eq. (\[psi\_x\_0\]) involves two other contributions, one of order $O(\Delta_\pi)$ which reads $$\begin{aligned}
16\pi \Delta_1 \psi^{x}_0(s) &=&
\frac{16}{F_\pi^2}
\left\{
\frac{ s}{9 F_\pi^2}\, b_x^2 \,-\, b_x a_x
\,+\, 2 b_x^2 \, \frac{M_{\pi}^2 + M_{\pi^0}^2}{F_\pi^2}\right\}
\nonumber\\
&&
\times
\left[
\left(\sqrt{\frac{s - 4 M_{\pi}^2}{s- 4 M_{\pi^0}^2}}\,-\,1\right)
\lambda^{1/2}(t_{\mbox{\tiny $-$}}(s)) {\cal L}_{\mbox{\tiny $-$}} (s)
\,-\,
\left(\sqrt{\frac{s - 4 M_{\pi}^2}{s- 4 M_{\pi^0}^2}}\,+\,1\right)
\lambda^{1/2}(t_{\mbox{\tiny $+$}}(s)) {\cal L}_{\mbox{\tiny $+$}} (s)
\right]
.\qquad{ }\end{aligned}$$ We do not give the explicit expression of $\Delta_2 \psi^{x}_0$, since it represents a tiny contribution of order $O(\Delta_\pi^2)$ which can be neglected for practical purposes, as indicated in section \[numerics\] [@wip].
At this stage, one can follow the discussion of section \[IB\_in\_phases\] and determine the isospin-breaking differences $\Delta\delta_0^{\pi}$, $\Delta\delta_0^{\pi^0}$, $\Delta\delta_1^{\pi}$ in terms of the different scattering lengths and effective range parameters. Even though one might hope to determine all these parameters from high-precision data on the different channels involved, it seems more realistic to express them in terms of the subthreshold parameters $\alpha_i, \beta_i,\lambda^{(n)}_i$ with $i= 00 , \pm 0 , x , +- , ++$ $$\begin{aligned}
a_{00} &=& \frac{\alpha_{00} M_{\pi^0}^2}{16\pi F_{\pi}^2} \,+\,
\frac{9}{4\pi}\,\lambda_{00}^{(1)}\,\frac{M_{\pi^0}^4}{F_\pi^4}
\,+\,\frac{1}{32 \pi}\left(\frac{\alpha_{00} M_{\pi^0}^2}{F_{\pi}^2}\right)^2 {\bar J}_0(4M_{\pi^0}^2)
\nonumber\\
&&\!\!\!\!\!
+\,\frac{1}{144 \pi F_\pi^4}
\left[2 \beta_{x}(5 M_{\pi^0}^2 - M_{\pi}^2)\,+\,\alpha_{x} M_{\pi^0}^2\right]^2
{\bar J} (4 M_{\pi^0}^2)
\nonumber\\
a_x &=& \frac{\beta_{x}}{24\pi F_{\pi}^2}
( M_{\pi^0}^2 - 5 M_{\pi}^2 )\,-\,
\frac{\alpha_{x} M_{\pi^0}^2}{48\pi F_{\pi}^2}
\,
-\,\frac{\lambda_{x}^{(1)}}{4 \pi}\,\frac{M_{\pi}^2 (2 M_{\pi}^2 - M_{\pi^0}^2)}{F_\pi^4}
\,-\,\frac{\lambda_{x}^{(2)}}{2 \pi}\,\frac{M_{\pi}^4 }{F_\pi^4}
\nonumber\\
&&\!\!\!\!\!
-\,\frac{1}{72 \pi F_\pi^4} ( 2 \beta_{\mbox{\tiny{$+-$}}} M_{\pi}^2 + \alpha_{\mbox{\tiny{$+-$}}} M_{\pi^0}^2 )
\left[2 \beta_{x} ( 5M_{\pi}^2 - M_{\pi^0}^2) + \alpha_{x} M_{\pi^0}^2 \right]
{\bar J}(4 M_{\pi}^2)
\nonumber\\
&&\!\!\!\!\!
-\,\frac{1}{96 \pi F_\pi^4} \,\alpha_{00} M_{\pi^0}^2
\left[2 \beta_{x} ( 5 M_{\pi}^2 - M_{\pi^0}^2) + \alpha_{x} M_{\pi^0}^2 \right]
{\mbox{Re}} \,{\bar J}_{0}(4 M_{\pi}^2)
\nonumber\\
&&\!\!\!\!\!
-\,\frac{1}{72 \pi F_\pi^4} \left[\beta_{x}^2\,(M_{\pi}^4 + M_{\pi^0}^4 - 10 M_{\pi}^2 M_{\pi^0}^2)
\,+\,4 \beta_{x} \alpha_{x} M_{\pi^0}^2 ( 2 M_{\pi}^2 - M_{\pi^0}^2) \,+\,
\alpha_{x}^2 M_{\pi^0}^4 \right] {\bar J}_{{\mbox{\tiny{$+$}}} 0} (M_{\pi^0}^2 - M_{\pi}^2)
\nonumber\\
&&\!\!\!\!\!
-\,\frac{1}{24 \pi F_\pi^4} \,\beta_{x}^2\,(M_{\pi}^2 - M_{\pi^0}^2)^2
\ {\bar{\!\!{\bar J}}}_{{\mbox{\tiny{$+$}}} 0} (M_{\pi^0}^2 - M_{\pi}^2)
\nonumber\\
{a}_{{\mbox{\tiny{$+$}}}0} &=& -\frac{\beta_{x}}{24\pi F_{\pi}^2}
( M_{\pi^0}^2 + M_{\pi}^2 )\,+\,
\frac{\alpha_{x} M_{\pi^0}^2}{48\pi F_{\pi}^2}\,+\,
\frac{1}{4 \pi F_\pi^4}\,( \lambda_{x}^{(1)} + 2 \lambda_{x}^{(2)}) M_{\pi}^2 M_{\pi^0}^2
\nonumber\\
&&\!\!\!\!\!
+\,\frac{1}{144 \pi F_\pi^4} \left[\beta_{x}^2\,(M_{\pi}^4 + M_{\pi^0}^4 - 10 M_{\pi}^2 M_{\pi^0}^2
+ 12 M_{\pi}^3 M_{\pi^0} + 12 M_{\pi} M_{\pi^0}^3)
\right.
\nonumber\\
&&\qquad
\left.
\,-\,4 \beta_{x} \alpha_{x} M_{\pi^0}^2 ( M_{\pi}^2 + M_{\pi^0}^2) \,+\,
\alpha_{x}^2 M_{\pi^0}^4 \right] {\bar J}_{{\mbox{\tiny{$+$}}}0} \left( (M_{\pi} + M_{\pi^0})^2\right)
\nonumber\\
&&\!\!\!\!\!
+\,\frac{1}{144 \pi F_\pi^4} \left[\beta_{x}^2\,(M_{\pi}^4 + M_{\pi^0}^4 - 10 M_{\pi}^2 M_{\pi^0}^2
- 12 M_{\pi}^3 M_{\pi^0} - 12 M_{\pi} M_{\pi^0}^3)
\right.
\nonumber\\
&&\qquad
\left.
\,-\,4 \beta_{x} \alpha_{x} M_{\pi^0}^2 ( M_{\pi}^2 + M_{\pi^0}^2) \,+\,
\alpha_{x}^2 M_{\pi^0}^4 \right] {\bar J}_{{\mbox{\tiny{$+$}}}0} \left( (M_{\pi} - M_{\pi^0})^2\right)
\nonumber\\
&&\!\!\!\!\!
+\,\frac{\beta_{x}^2}{48\pi F_{\pi}^4}\,( M_{\pi} - M_{\pi^0})^4
\, {\bar{\!\!{\bar J}}}_{{\mbox{\tiny{$+$}}}0} \!\left( (M_{\pi} + M_{\pi^0})^2\right) \,+\,
\frac{\beta_{x}^2}{48\pi F_{\pi}^4}\,( M_{\pi} + M_{\pi^0})^4
\, {\bar{\!\!{\bar J}}}_{{\mbox{\tiny{$+$}}}0} \!\left( (M_{\pi} - M_{\pi^0})^2\right)
\nonumber\\
{a}_{\mbox{\tiny{$+-$}}} &=& \frac{\beta_{\mbox{\tiny{$+-$}}}}{12\pi F_{\pi}^2}
\, M_{\pi}^2 \,+\,
\frac{\alpha_{\mbox{\tiny{$+-$}}} M_{\pi^0}^2}{24\pi F_{\pi}^2}\,+\,
\frac{1}{2 \pi F_\pi^4}\,( \lambda_{\mbox{\tiny{$+-$}}}^{(1)} + 2 \lambda_{\mbox{\tiny{$+-$}}}^{(2)}) M_{\pi}^4
\nonumber\\
&&\!\!\!\!\!
+\,\frac{1}{36\pi F_\pi^4}\,(2\beta_{\mbox{\tiny{$+-$}}} M_{\pi}^2 +
\alpha_{\mbox{\tiny{$+-$}}} M_{\pi^0}^2 )^2 {\bar J} (4 M_{\pi}^2)
\,+\,\frac{1}{288\pi F_\pi^4}\,(8\beta_{x} M_{\pi}^2 + \alpha_{x} M_{\pi^0}^2 )^2
{\mbox{Re}} \,{\bar J}_{0}(4 M_{\pi}^2)
\nonumber\\
{a}_{\mbox{\tiny{$++$}}} &=& -\frac{\beta_{\mbox{\tiny{$+-$}}}}{6\pi F_{\pi}^2}
\, M_{\pi}^2 \,+\,
\frac{ \alpha_{\mbox{\tiny{$+-$}}} M_{\pi^0}^2}{24\pi F_{\pi}^2}\,+\,
\frac{1}{2 \pi F_\pi^4}\,( \lambda_{\mbox{\tiny{$+-$}}}^{(1)} + 2 \lambda_{\mbox{\tiny{$+-$}}}^{(2)}) M_{\pi}^4
\,+\,\frac{1}{72\pi F_\pi^4}\,(4\beta_{\mbox{\tiny{$+-$}}} M_{\pi}^2 - \alpha_{\mbox{\tiny{$+-$}}} M_{\pi^0}^2 )^2 {\bar J}(4 M_{\pi}^2)
\,.\qquad{ }\end{aligned}$$ These expressions can be exploited, by relying on appendix \[app:subtraction\] and expressing the subthreshold parameters $\alpha_i, \beta_i,\lambda^{(n)}_i$ with $i= 00 , \pm 0 , x , +- , ++$ in terms of the isospin-limit parameters $\alpha,\beta,\lambda^{(n)}$. The latter could be taken as the fundamental parameters of the analysis, but they can also be traded for the two $\pi\pi$ scattering lengths $a_0^0$ and $a_0^2$ (up to higher-order corrections that can be estimated using Chiral Perturbation Theory). This series of matching will be indeed the point of view adopted for the analysis of $K_{\ell 4}$ decays, allowing us to reexpress the isospin-breaking correction to be applied to the phase-shift difference in terms of the two scattering lengths $a_0^0$ and $a_0^2$ [@wip].
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[^1]: One of us (M. K.) wishes to thank J. Gasser for numerous interesting discussions on this and on related issues.
[^2]: The authors of ref. [@Kubis:1999db] have informed us that they agree with us on this point. We thank B. Kubis for correspondence regarding this issue.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Teleoperation is necessary when the robot is applied to real missions, for example surveillance, search and rescue. We proposed teleoperation system using past image records (SPIR). SPIR virtually generates the bird’s-eye view image by overlaying the CG model of the robot at the corresponding current position on the background image which is captured from the camera mounted on the robot at a past time. The problem for SPIR is that the communication bandwidth is often narrow in some teleoperation tasks. In this case, the candidates of background image of SPIR are few and the position of the robot is often delayed. In this study, we propose zoom function for insufficiency of candidates of the background image and additional interpolation lines for the delay of the position data of the robot. To evaluate proposed system, an outdoor experiments are carried out. The outdoor experiment is conducted on a training course of a driving school.'
author:
- 'Noritaka Sato, Masataka Ito and Fumitoshi Matsuno[^1][^2] [^3]'
title: |
**Teleoperation System Using Past Image Records\
Considering Narrow Communication Band**
---
INTRODUCTION
============
Background
----------
Recently, robots are used in order to complete tasks instead of humans in the place where a human cannot enter. Some of them are full autonomous, but even in the case of an autonomous robot, teleoperation is often necessary when the robot is applied to real missions, for example surveillance, search and rescue.
Generally, in the human-robot interface for the teleoperation, images from a camera mounted on the robot are displayed. However, if the camera is mounted on the front of the robot and its axis is fixed to the forward direction, it is difficult for an operator to recognize the width of the robot from the image and understand the relation between the robot and the environment around the robot. Therefore, the operator has heavy workloads and a beginner cannot remote control the robot well. In addition, in the teleoperation interface, when the communication is unstable, bad-quality images may be displayed and the images may have the delays, because the information volume of an image is large. When a robot runs on rough terrain, the images from the mounted camera on it are not stable because of robot attitude changes. It is very hard for the operator to watch these nonsteady vibrately images for remote control. Therefore we should develop an interface to reduce the operator’s workloads.
Related work
------------
To solve these drawbacks, some interfaces for teleoperation of a robot are proposed. Yanco et al. analyze the real teleoperation tasks and find situation awareness is important to reduce workload of an operator [@Yanco]. To improve situation awareness in the teleoperation, Yanco et al. recommends providing the more spatial information to the operator. To display the robot and its surroundings, Shiroma et al. attach a long pole on the robot whose top end has a camera [@EI]. Murphy et al. take vision supports from other robots’ cameras where the operated robot is on the image from a camera mounted on other robots [@multi]. However these interfaces have disadvantages that the robot’s size is increased or the mission often became slow because the operator should control two or more robots. Nguyen et al. and Saitoh et al. proposed the interface in which a virtual 3D environment including a CG model of the robot is displayed [@VR; @MVE]. And Nielsen et al. proposed the interface in which a CG model of the robot and an image from the camera mounted on the robot are displayed on the environment map [@Map]. However, these methods need 3D modeling or map building of the environment and it often takes much time to generate them. Therefore, these systems are difficult to implement to robots in the disaster area where the robot has error of self localization and the communication is unstable.
To develop the interface which is robust against bad communication conditions and does not need heavy computational power and high cost sensors, Matsuno et al. proposed an original idea of the teleoperation system using past image records (SPIR), which is effective to an unknown environment [@patent]. This system uses an image captured at a past time as a background image (Fig. \[IM\]) and overlays a CG model of the robot on it at the corresponding current position and orientation. In this way, the system virtually displays the current robot situation and its surroundings from the past point of view of the camera mounted on the robot (Fig. \[overview\]). The system can generate a bird’s-eye view image which includes the teleoperated robot and its surroundings, and real time vibrations of the image is eliminated by using a fixed background image captured in the past time.
![Using a past image as a background image.[]{data-label="IM"}](eps/IM_new.eps){width="0.8\hsize"}
![Overview of SPIR.[]{data-label="overview"}](eps/principle.eps){width="0.8\hsize"}
Principle of SPIR
-----------------
The algorithm of SPIR is following:\
\
(Step 1) Get robot’s position and orientation\
The system gets current robot’s position and orientation based on odometry, GPS, SLAM algorithm and so on. In SPIR, the error of the position is canceled when the background image is switched, because the position of the virtual robot (CG model) is depends on relative positions between current robot position and past robot position that the background image was captured.\
\
(Step 2) Save the image and its position\
The system stores images from the camera mounted on the robot on the buffer (temporary memory) as candidates of the background image. At this time, the system also stores the position of the robot as the camera position where the image is captured. The set of the stored image and position is called “past image record.”\
\
(Step 3) Select the background image\
The system selects an optimal image as a background. In this study, the system switches the background image if the distance between the current robot position and the past robot position where the background image was taken is larger than a threshold value.\
\
(Step 4)Generate bird’s-eye view scene\
The system generates a bird’s-eye view scene by overlaying a CG model of the robot at the corresponding current position and orientation on the background image selected in Step(3).\
\
By iterating above algorithm, the system displays the bird’s-eye view image to the operator (Fig. \[overview\]). Detail descriptions of the implementation of SPIR are reported in existing papers [@kagotani2; @sice].
Research purpose
----------------
In this study, we focus on operability in the case of the narrow communication bandwidth. Because SPIR uses discrete images, the load of transmission is reduced compared with traditional system which the operator controls the robot by using the images sending in the real time from the robot. However, if the communication bandwidth is narrow, stored images in the database of the candidate of the background image may be too few. Therefore, the system may provide a feeling of strangeness for the operator in the case of the narrow communication band, because the size of the CG model is changed significantly when the background image is switched. The longer the distance from the position where the background image is captured is, the smaller the size of CG model is. If the candidates of the background image are few, the distance may be long. In this study, we proposed zoom function to overcome this problem.
Moreover, in the case of the narrow communication band, transmission delay will occur. In this case, since the current position data of the robot also have delay, the position data of the CG model on the background image is not correct. Therefore it is difficult for the operator to teleoperate the robot, because the operator misses the current position of the robot. In this study, we propose additional interpolation lines that the operator can predict the robot position easily. And the operator can use them as indicators for generating driving control commands of the robot.
In the section 2, we proposed zoom function and additional interpolation lines in SPIR for improving the robustness against the communication delay. In the section 3, we show evaluation experiments in the outdoor environments. The section 4 is conclusion.
SPIR considering narrow communication band
==========================================
Zoom function
-------------
When the background image is changed, the relative relation between the current position of the robot and the past viewpoint where the selected background image was captured is discretely changed. If the distance of the past viewpoints before and after the background image is updated is small, the system makes a frequent small change of the size of the CG model of the robot on the monitor when the background image is switched, as shown in the upper images of Fig. \[fig:zoom\]. In the case of the narrow communication band, because many images cannot be sent, the relative distance of two positions may be large. The big change of the size of the CG model of the robot in the background image as shown in the middle images of Fig. \[fig:zoom\] is a source of the sense of incongruity. To overcome this problem, a zoom function is installed in SPIR in order to keep the size of CG model on the background image for every sampling time, as shown in the bottom images of Fig. \[fig:zoom\].
![Snapshot images of SPIR with and without zoom fanction.[]{data-label="fig:zoom"}](eps/zoom_image.eps){width="0.9\hsize"}
We define a vertical angle $\theta$ of the field of view (FOV) as a parameter of a zooming ratio. We explain about the calculation method of the vertical angle $\theta$ to keep the ratio of CG models on the displayed background images.
As shown in Fig. \[fig:calcAngle\], $d(t)$ is the distance between the viewpoint of the background image and the current position of the robot, $\theta(t)$ is the vertical angle of FOV of the background image, $h$ is the constant height of the robot, and $H(t)$ is the vertical length of FOV positioned $d(t)$ away from the viewpoint. The system should keep $h/H(t)$ to be a constant value $k (0<k<1)$ for every sampling, because if this ratio is kept, the size of the CG model on the background image is also not changed regardless of the motion of the robot. From the geometric relationship, $$d(t)\tan(\theta(t)/2)=H(t)/2, \ \ k=h/H(t).$$ We obtain $$\label{thetaZoom}
\theta(t)=2\tan^{-1}(h/2d(t)k).$$
In the case that the background image does not change ($0 \le t < t_1$ in Fig. \[fig:zoom\]), the distance $d(t)$ changes continuously, then $\theta(t)$ changes continuously based on Eq.(\[thetaZoom\])to keep $h/H(t)$. In the case that the background image is updated ($t=t_1$ in Fig. \[fig:zoom\]), the distance $d(t)$ changes discretely, then $\theta(t)$ changes discretely to keep $h/H(t)=k$. By changing the angle $\theta(t)$ of the field of view to keep $k$ constant according to the Eq. (\[thetaZoom\]), even if the background image is switched, the size of the CG model on the background image is not changed.
In proposed system, we used the OpenGL and OpenCL functions for image mapping and image extraction.
![Calculation of angle of field.[]{data-label="fig:calcAngle"}](eps/zoom_new.eps){width="0.99\hsize"}
Additional line
---------------
If the transmission delay occurs, the current position data of the robot also has delay and the position of the CG model on the background image in SPIR is not correct. Therefore it is difficult for the operator to understand the current position of the robot and control the robot.
To overcome this problem, we add lines on the displayed image generated by SPIR. By displaying additional lines, the operator can easily predict the coming position of the robot and control it.
In this research, we introduce two types of additional lines;
\(a) Extended line of front wheel axis (Solid lines (a) in Fig. \[fig:additional\_line\])\
This line is an extension of the front wheel axis of the robot. This line is overlaid on the background image according to the steering angle of the robot. Because the operator can easily understand the center of the rotation of the robot, the robot can smoothly turn by fixing this line on the center of the corner.
\(b) Predictive trajectory (Dotted lines (b) in Fig. \[fig:additional\_line\])\
This line shows the predictive trajectory of the wheel of the robot. By using the predictive trajectories, an operator can easily estimate the motion of the robot. By collimating this line with the edge of the course or the center line of the road, the robot can run without swerving from the road.
![Additional lines.[]{data-label="fig:additional_line"}](eps/addLine.eps){width="0.8\hsize"}
Evaluation experiment
=====================
Experimental method
-------------------
we valid the effectiveness of the zoom function and the additional lines in the case of narrow communication band as explained in the section 3. The number of subjects is 8, and we compared three methods: (1)normal front camera whose angle of FOV is 60 degrees(Front Camera), (2)existing SPIR without use zoom function and not add lines (Existing SPIR) and (3) extended SPIR with zoom function and add lines(Proposed SPIR2).
The outdoor experiment environment is shown in Fig. \[fig:course\]. This is a training course of a driving school whose size is about 120\[m\] $\times$ 80\[m\] and the length of the course is about 250\[m\]. One subject remote controls the robot three times for each system and each system is chosen randomly. in order to cancel the influence of the order of trials. The system configuration of this experiment is almost same as the previous experiment in the section 4.1. We use a UGV (Unmanned Ground Vehicle) as a mobile robot developed by YAMAHA Motor Co., Ltd. as shown in Fig. \[fig:UGV\]. Maximum transrational velocity of the UGV is 1.0 \[m/s\] in this experiment. To control the UGV easily, we use a handle and a pedal instead of a joystick in the previous experiment.
![Course of examination.[]{data-label="fig:course"}](eps/course2.eps){width="0.85\hsize"}
![UGV.[]{data-label="fig:UGV"}](eps/UGV.eps){width="0.55\linewidth"}
As we focus on the effectiveness of the zoom function and the additional lines, the experiments has been carried out without moving objects. In the experiment, we set the limitation of the communication band with the assumption of using mobile phone communication. Table \[tbl:setting\] shows the parameters of image and data which are sent from the robot to the operator station in each system. These parameters in each system are set as the best values to remote control the robot with the limitation of the bandwidth according to results of preliminary experiments.
Front Camera Existing SPIR and Proposed SPIR2
-------------------------------------------- ------------------ ----------------------------------
Jpeg quality 15 50
Image size 640x480\[pixel\] 640x480\[pixel\]
Transmission interval of image 0.7\[sec\] 1.4\[sec\]
Transmission delay of image 1.2\[sec\] 1.9\[sec\]
Transmission interval of data except image 0.02\[sec\] 0.02\[sec\]
Transmission delay of data except image 0.5\[sec\] 0.5\[sec\]
We record average speeds and position errors as the indicates of the operator’s workload, because if the operator feels complexity of the teleoperation the speed of the robot will reduce and the position error will increase. We define the position error as the sum of the nearest distance between each sampling position of the robot and the center line of the road as the target trajectory.
Results and consideration
-------------------------
Fig. \[fig:average\_speed\] and Fig. \[fig:running\_position\_error\] show the result of average speeds and average position errors for three methods, respectively. We evaluate the obtained results with LSD (Least Significant Difference) test [@LSD]. In the relation of average speeds, Proposed SPIR2 has significant difference to Existing SPIR, and Existing SPIR has significant difference to Front Camera. In the position errors, Front Camera has significant difference to Existing SPIR and Proposed SPIR2, but there is no significant difference between Existing SPIR and Proposed SPIR2.
Because Existing SPIR and Proposed SPIR2 mark high speeds and less errors than Front Camera, SPIR can apply effectively under the situation of the narrow communication band. Proposed SPIR2 is better than Existing SPIR at the average speed. The difference between them is only zoom function and additional lines. Therefore these proposed improvements are effective for the operator to teleoperate the robot under the communication limitation.
![Average speeds.[]{data-label="fig:average_speed"}](eps/SPIR2_speed.eps){width="0.7\hsize"}
![Position errors.[]{data-label="fig:running_position_error"}](eps/SPIR2_error.eps){width="0.7\hsize"}
In the experiment, because each subject teleoperated the robot with three comparing systems, we use with-in subjects ANOVA. In order to counterbalance order, each comparing method was randomly taken. Table \[tbl:average\_speeds\] and Table \[tbl:position\_errors\] shows the ANOVA table. In our experimental, because the F score 34.47 and 7.33 for average speed and position error are also higher than 6.51, the results are significant at the 0.01 significance level.
Now we apply LSD method for multiple comparison to three comparing systems. In our experiment, the difference of average is higher than 0.088 or 0.149, the result is significant at the 0.05 or 0.01 significance level. We can find the differences over 0.088 among the average speeds of three methods. In the position errors, the difference between System B and System C is not enough to be significance level, but System A has enough difference to System B and C.
From the results of statistical analysis, the speed of the robot is faster in the following order: System C (Proposed SPIR2), System B (Existing SPIR) and System A (Front Camera) and the all differences are enough to be 0.05 significance level. On the other hand, the position errors also decrease in same order. However System A (Proposed SPIR2) has enough difference to be significance level, but there is not enough difference between System B (Existing SPIR) and System C (Front Camera).
From these results, we make consideration about each system. With Front Camera, because the field of view is narrow and the frame rate is decreased, the teleoperation of the robot is very hard. Therefore the trajectory of the robot is meandering shape and the position error is increased. Moreover the operator may teleoperate the robot slowly in order to keep the course.
In the Existing SPIR, the position error is not larger than proposed SPIR2, but slower. If the background image is not updated, the CG model of the robot on the image is smaller and smaller. However when the image is updated, the size of the CG model is suddenly changed. Because this large change makes the feeling of strangeness to the operator, he cannot input faster command to the robot.
Compared with these systems, Proposed SPIR2 keeps the course and the average speed is very high. The average speed in whole course is 80
SV SS df MS F
------- -------- ---- -------- -------------
A 0.4453 2 0.2226 32.47 \*\*
Sub 0.1402 7 0.0200
SxA 0.0959 14 0.0068
Total 0.6815 23 \*\*p`<`.01
: ANOVA table (average speeds)[]{data-label="tbl:average_speeds"}
SV SS df MS F
------- -------- ---- -------- -------------
A 0.2834 2 0.1417 7.33\*\*
Sub 0.3901 7 0.0557
SxA 0.2704 14 0.0193
Total 0.9440 23 \*\*p`<`.01
: ANOVA table (position errors)[]{data-label="tbl:position_errors"}
Conclusion
==========
In this study, we proposed a solution for problem of existing teleoperation system using past image records (SPIR). To solve the problem of existing SPIR that is occurred under narrow communication bandwidth, zoom function and additional lines are installed in SPIR. By the outdoor evaluation experiment, we can find that the proposed system is useful under narrow communication band. From the experimental results, we find that the proposed SPIR reduces the operator workloads of teleoperation comparing to existing SPIR.
We would like to extend this system to the multiple-robots system in the future.
Acknowledgment {#acknowledgment .unnumbered}
==============
A part of the results in this research was collaboratively conducted with Yamaha Motor, Co., Ltd.
[99]{}
M. Baker, R. Casey, B. Keyes, and H. A. Yanco: Improved interfaces for human-robot interaction in urban search and rescue, Proc. 2004 IEEE International Conference on Systems, Man and Cybernetics, pp.2960-2965 (2004)
Naoji Shiroma, and Noritaka Sato, and Yu-huan Chiu, and Fumitoshi Matsuno: “Study on Effective Camera Images for Mobile Robot Teleoperation”, Proceedings of 13th IEEE International Workshop on Robot and Human Interactive Communication, 2004.
Robin R. Murphy , and Erika Rogers , and School Mathematical , and Computer Sciences: “Cooperative Assistance for Remote Robot Supervision”, Presence, Vol. 5, No. 2, pp. 224-240, 1996.
Laurent A. Nguyen, and Maria Bualat, and Laurence J. Edwards, and Lorenzo Flueckiger, and Charles Neveu, and Kurt Schwehr, and Michael D. Wagner, and Eric Zbinden: “Virtual Reality Interfaces for Visualization and Control of Remote Vehicles”, Autonomous Robots, Vol. 11, No. 1, pp. 59-68, 2001.
Saitoh, Kensaku and Machida, Takashi and Kiyokawa, Kiyoshi and Takemura, Haruo: “A 2D-3D integrated interface for mobile robot control using omnidirectional images and 3D geometric models”, Proceedings of the 5th IEEE and ACM International Symposium on Mixed and Augmented Reality, 2006.
Curtis W. Nielsen, and Michael A.Goodrich: “Comparing hte Usefulness of Video and Map Information in Navigation Tasks”, Proceedings of the 1st ACM SIGCHI/SIGART Conference on Human-robot interaction, 2006.
Fumitoshi Matsuno, and Masahiko Inami, and Naoji Shiroma: “Method for generating image”,Patent Entry No. 4348468, 2004.
Maki Sugimoto, and Georges Kagotani, and Hideaki Nii, and Naoji Shiroma, and Masahiko Inami, and Fumitoshi Matsuno: “Time Follower’s Vision: A Teleoperation Interface with Past Images”, IEEE Computer Graphics and Applications, Vol. 25, No. 1, pp.54-63, 2005.
Naoji Shiroma, and Hirokazu Nagai, and Maki Sugimoto, and Masahiko Inami and Fumitoshi Matsuno: “Synthesized Scene Recollection for Robot Teleoperation”, Field and Service Robotics, Vol. 25, pp.403-414, 2006.
Rupert G. Miller: “Simultaneous Statistical Inference”, Springer-Verlag, 1981.
[^1]: N. Sato is with department of Computer Science and Engineer, Nagoya Institute of Technology, Gokiso-cho, Showa-ku, Nagoya, Aichi 466-8555, Japan. [[email protected]]{}
[^2]: M. Ito was with department of Mechanocal Engineering and Intelligent Systems, The University of Electro-Communications, Chofugaoka, Cofu, Tokyo, Japan.
[^3]: F. Matsuno is with department of Mechanical Engineering and Science , Kyoto University, Kyotodaigaku-Katsura, Nishikyo-ku, Kyoto, 615-8540, Japan.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We combine bare perturbation theory with the imaginary time evolution technique to study one-loop radiative corrections to various components of angular momentum of the electron. Our investigations are based on the canonical decomposition of angular momentum, where spin and orbital components, associated with fermionic and electromagnetic degrees of freedom, are individually approached. We use for this purpose quantum electrodynamics in the general covariant gauge and develop a formalism, based on the repeated use of the Sochocki-Plemelj formula, for proper enforcement of the imaginary time limit. It is then shown that careful implementation of imaginary time evolutions is crucial for getting a correct result for total angular momentum of the electron in the bare perturbative expansion. We also analyze applicability of the Pauli-Villars regularization to our problem, developing a variant of this technique based on modifications of studied observables by subtraction of their ghost operator counterparts. It is then shown that such an approach leads to the consistent regularization of all angular momenta that we compute.'
author:
- Bogdan Damski
title: 'Angular momentum of the electron: One-loop studies'
---
Introduction {#Intro_sec}
============
The electron, undoubtedly one of the most fundamental constituents of matter, is characterized by a set of physical properties such as the mass, charge, magnetic moment, and spin.
Experimental studies of its mass and charge, $m$ and $e$ below, started in the late nineteenth century in a series of experiments conducted by Thomson [@Thomson1897b]. They have been successfully continued ever since. By contrast, progress in theoretical characterization of these parameters is rather uninspiring, if we notice that dimensionless quantities involving them–such as the fine structure constant or ratios of the electron mass to other lepton masses–have never been convincingly estimated.
The electron’s intrinsic magnetic moment was introduced by Uhlenbeck and Goudsmit [@Uhlenbeck1926] about a century ago in an attempt to explain the anomalous Zeeman effect, which was discovered by Preston at the same time Thomson conducted his electron experiments [@Preston1898]. Its understanding rapidly progressed soon after thanks to Dirac [@Dirac1928], whose theory predicted \[Dirac\] for the electron’s magnetic moment. Two decades later [@SchwingerPR1948], Schwinger found a more accurate approximation through a perturbative quantum electrodynamics (QED) calculation replacing (\[Dirac\]) with , \[Schw\] where = \[alfa\] is the fine structure constant written here in the Heaviside-Lorentz system of units combined with $\hbar=c=1$ (we use such units throughout this work). This prediction immediately explained spectroscopic “anomalies” found in measurements of Nafe and Nelson [@NafeNelson1948] and Foley and Kusch [@FoleyKusch1948] that were done concurrently with Schwinger’s calculations. Ever since perturbative calculations of the electron’s magnetic moment have gone hand in hand with various experimental measurements reaching astonishing accuracy [@ComminsAnnRev2012]. These efforts allowed for some of the most stringent tests of QED.
The electron’s spin was introduced together with its intrinsic magnetic moment in [@Uhlenbeck1926]. It was then put on a firm theoretical basis by Dirac [@Dirac1928], whose relativistic quantum mechanics leads to the following expression for the angular momentum operator [@Greiner] - \^[imn]{}z\^m, \[JDirac\] where $\psi$ is the Dirac field operator, $\N{ \ }$ denotes normal ordering, \^i=\^[imn]{}\^m\^n/2, and $\gamma$ are Dirac matrices. The first (second) operator in (\[JDirac\]) is the fermionic spin (orbital) angular momentum operator. Consider now the electron at rest, whose spin is polarized in the $\pm z$ direction. The expectation value of operator (\[JDirac\]), in the corresponding quantum state $|\Psi\ra$, is $\spinz$, where = reflects the fact that the electron’s spin equals one-half. The orbital component of the angular momentum operator does not contribute to such an expectation value \_=0, \[Diraco\] and so one finds \_=. \[Diracs\]
The situation is considerably more complex in QED, where the total angular momentum operator is built of not only fermionic but also electromagnetic operators. The question of how one can attribute angular momentum to different degrees of freedom is non-trivial and it lead to the so-called angular momentum controversy involving various issues such as the lack of gauge invariant definition of spin and orbital angular momentum of photons and the question of experimental relevance of gauge non-invariant quantities [@LeaderPhysRep2014]. More importantly, in the context of this work, all components of total angular momentum of the electron receive radiative corrections [@BurkardtPRD2009; @LiuPRD2015; @JiPRD2016; @BDfield].
The interest in angular momentum decompositions of the electron in particular and other subatomic particles in general comes from the fact that they provide fundamental insights into properties of these particles. This statement is perhaps best illustrated by experimental and theoretical studies of angular momentum decompositions of nucleons performed over last four decades and comprehensively summarized in [@Deur2019].
It is the purpose of this work to compute radiative corrections to right-hand sides of (\[Diraco\]) and (\[Diracs\]) as well as to remaining components of total angular momentum of the electron. Similar studies were performed not long ago [@LiuPRD2015; @JiPRD2016]. These calculations were done in the light-front formalism, employed the light-cone gauge, and used renormalized perturbation theory. They are, on the technical level, very different from our studies as we use imaginary time evolution formalism, work in the general covariant gauge, and employ bare perturbation theory. Therefore, we see our work as complementary to previous efforts. Among other things, this paper discusses non-trivial results on implementation of imaginary time evolutions, it presents gauge non-invariant angular momenta from the covariant-gauge perspective, and it conclusively describes intricacies of proper application of the Pauli-Villars regularization to the studied problem. Its outline is the following.
We explain in Sec. \[Basics\_sec\] the approach that we use to carry out computations. Next, we describe in Sec. \[Fer\_spin\_sec\] different contributions to fermionic spin angular momentum of the electron. Remaining angular momenta–fermionic orbital, electromagnetic spin and orbital, and gauge-fixing ones–are discussed in Sec. \[Other\_sec\]. Then, a proper way of imposing the Pauli-Villars regularization onto all these expressions is presented in Sec. \[PVV\_sec\]. One-loop radiative corrections are computed in Sec. \[Regularized\_sec\]. The discussion of obtained results is presented in Sec. \[Discussion\_sec\]. Several appendices are added to this paper to make its main body better readable and to facilitate verification of our calculations. We explain our notation in Appendix \[Conventions\_sec\] and collect all bispinor matrix elements in Appendix \[Matrix\_sec\]. Intricacies associated with implementation of imaginary time evolutions are discussed in Appendix \[Implementation\_sec\], while adaptation of the Pauli-Villars regularization technique to our problem is presented in Appendix \[Pauli\_sec\]. Finally, some integrals from Sec. \[Regularized\_fer\_spin\] are evaluated in Appendix \[Integrals\_app\].
Basics {#Basics_sec}
======
The starting point for our considerations is the QED Lagrangian density [@Greiner]
[L]{} =& - F\_F\^+ A\_A\^-\^2 + - \^A\_,
\[LL\] where the second term is employed to regulate the infrared (IR) sector of the theory, while the third one, the so-called gauge-fixing term, facilitates quantization of the electromagnetic field in the general covariant gauge (the term general refers to the arbitrary greater than zero value of $\xi$). The bare mass and charge of the electron are denoted by $\mo$ and $\eo$, the photon mass is written as $\lambda$, and remaining symbols follow all standard conventions (Appendix \[Conventions\_sec\]).
We compute total angular momentum through the formula from Sec. 2.4 of [@Greiner] J\^i=\^[imn]{} M\^[0mn]{}, \[Js\] where the canonical angular momentum tensor density is given by the following sum of the orbital term, expressed through the canonical energy-momentum tensor density $\vartheta^{\mu\nu}$, and the spin term
M\^ = \^z\^-\^z\^+M\^,
\[emtensor\] \^=&\^A\^ +\^- \^[L]{}\
=&-F\^ \^A\_-\_A\^\^A\^ +\^\^- \^[L]{},
M\^=& A\_ + \[\^,\^\]\
=&F\^A\^-F\^A\^+\_A\^(\^A\^-\^A\^) + \^,
where $[\,,]$ stands for the commutator. These expressions lead to
J\^i=& \^\^i -\^[imn]{}z\^m \^\_n + \^[imn]{} F\_[m0]{}A\_n\
+& \^[imn]{} z\^m F\_[j0]{}\_n A\_j + \^[imn]{} z\^m\_A\^\_n A\_0.
\[Jclass\]
First two terms in (\[Jclass\]), fermionic spin and orbital angular momenta, have already been introduced in Sec. \[Intro\_sec\]. The third and fourth term are known as electromagnetic spin and orbital angular momenta. Finally, we will refer to the last term of (\[Jclass\]) as gauge-fixing angular momentum because it originates from the gauge-fixing term in (\[LL\]). Such a term is a unique feature of the covariant gauge approach, and so it is quite interesting to see how it contributes to total angular momentum of the electron.
The sum of first four expressions in (\[Jclass\]) is known as the Jaffe-Manohar decomposition of total angular momentum [@JaffeNPB1990; @LeaderPhysRep2014]. As we have shown above, it follows directly from the canonical formalism, which makes it quite distinctive. Such a decomposition, however, is not unique as one can try to modify the density of angular momentum through either Euler-Lagrange equations or through addition of $3$-divergence terms. Since advantages and disadvantages of different angular momentum decompositions are comprehensively discussed in [@LeaderPhysRep2014], we will not dwell on them.
Angular momentum operators are now obtained by replacing classical fields in (\[Jclass\]) with Heisenberg-picture operators and by imposing normal ordering. In the form suitable for perturbative calculations, we write them as $$\begin{aligned}
&J^i_\sp = \int \d{z}\N{\overline{\psi}\,\Gamma^i\psi}, \ \Gamma^i=\frac{\ii}{4}\varepsilon^{imn}\gamma^0\gamma^m\gamma^n,
\label{Jspin}\\
&J^i_\orb= \int \d{z}\N{\overline{\psi}\,\nabla^i_\z\psi}, \
\nabla^i_\z=-\ii\gamma^0\varepsilon^{imn}z^m\frac{\partial}{\partial z^n},
\label{Jorb}\\
&J^i_\spel=\int \d{z} \varepsilon^{imn} \N{F_{m0}A_n},
\label{Jspinel}\\
&J^i_\orbel=\int \d{z} \varepsilon^{imn} z^m
\N{F_{j0}\partial_n A_j},
\label{Jorbel}\\
&J^i_\xi=\xi \int \d{z} \varepsilon^{imn} z^m \N{\partial_\sigma A^\sigma\partial_n A_0},
\label{Jxi}\end{aligned}$$ where we have used the bullet $\bullet$ and the wavy line $\thicksim$ to distinguish fermionic operators from electromagnetic ones. The total angular momentum operator is then J\^i=J\^i\_+J\^i\_+J\^i\_+J\^i\_+J\^i\_. \[totalJ\]
We will compute expectation values of operators (\[Jspin\])–(\[Jxi\]) in the QED ground state with one net electron,[^1] which we denote as $\Oprim$. As such quantities are time-independent, we set z=(0,) \[z0\] to simplify the discussion in intermediate steps (as a self-consistency check, we have verified that $z^0$ eventually drops out from all expectation values if it is not set to zero). Calculations will be performed in the framework of bare perturbation theory combined with the imaginary time evolution technique.
Imaginary time evolutions start from the one-electron ground state of the free Hamiltonian | s=a\^\_[ s]{}|0, \[0s\] where $|0\ra$ is the vacuum state of the free theory and the operator $a_{\0 s}$ is introduced in Appendix \[Conventions\_sec\]. Such a state describes the electron at rest whose spin is polarized such that $\exval{J^i}{\0s}=\spinz$ (the same polarization has been employed in Sec. \[Intro\_sec\]). Its $4$-momentum f=(,) \[f0\] frequently appears in the following discussion. State (\[0s\]) is then evolved in time (its non-trivial dynamics is induced by the interaction Hamiltonian $\int\d{x}{\cal H}_\IN$). Enforcement of the imaginary time limit leads to [@PS]
$$\begin{aligned}
\label{fghj_a}
&\exval{\J_\chi}{\Opr}=\limT\expval{\J_\chi}{\Opr}{\LT}, \\
&\expval{\J_\chi}{\Opr}{\LT}=
\frac{\la\0s|\T\J^I_\chi\exp(-\ii\intT\dd{x}{\cal H}^I_\IN)|\0s\ra}{
\la\0s|\T\exp(-\ii\intT\dd{x}{\cal H}^I_\IN)|\0s\ra}, \label{fghj_b}\\
&\intT\dd{x}=\intTT dx^0\int\d{x},\\
&\chi=\sp, \orb, \spel, \orbel, \xi,
\label{chi_list}\end{aligned}$$
\[fghj\]
where interaction-picture operators are labeled with the index $I$, $\J^I_\chi$ operators are obtained by replacing Heisenberg-picture fields with their interaction-picture counterparts,[^2] \^[I]{}\_= A\^[I]{}\_, \[Hint\] and $\T$ is the time-ordering operator.
To proceed with (\[fghj\]), we will need fermionic S(x-y)= \_[I]{}(x)\_[I]{}(y)= 0|\_I(x)\_I(y)|0= e\^[-p(x-y)]{} \[prop\_fer\] and electromagnetic
$$\begin{aligned}
&D_{\mu\nu}(x-y)=
\contraction{}{A}{^{I}_\mu(x)}{A}A^{I}_\mu(x)A^{I}_\nu(y)=
\la0|\T A^I_\mu(x)A^I_\nu(y)|0\ra=
-\ii\int\frac{\dd{p}}{(2\pi)^4}\frac{d_{\mu\nu}(p)}{p^2-\lambda^2+\izero}e^{-\ii p\cdot (x-y)},\\
&d_{\mu\nu}(p)= \eta_{\mu\nu}+\xxii\frac{p_\mu p_\nu}{p^2-\lambda^2/\xi+\izero}
\label{prop_mod}%\end{aligned}$$
\[prop\_el\]
propagators. The former expression is given by the standard formula, while the latter one can be either derived using the trick from Sec. 7.6 of [@Greiner] or taken from Sec. 33.4 of [@ColemanBook].
Evaluation of (\[fghj\_b\]) will be performed with $T>0$ and then the limit \[limitT\] will be taken. Proper evaluation of this limit is no trivial matter in some of our computations. To illustrate the subtle point here, we note that integration over time in (\[fghj\_b\]) leads to expressions of the form e\^[x\^0 P\^0]{}=, \[dx0sin\] where $P^0$ is some combination of timelike components of $4$-momenta. Limit (\[limitT\]) cannot be taken on (\[dx0sin\]). The standard textbook solution of this complication is to transfer the $-\izero$ from the limit to the imaginary part of propagators’ denominators (see e.g. Sec. 4.4 of [@PS]). After that, the limit $T\to\infty$ is taken. This leads to the Dirac delta function due to the following well-known identity (P\^0)=\_[T]{}. \[deltac\] Such a procedure presumably greatly simplifies calculations. However, it leads to the incorrect result for fermionic spin angular momentum of the electron and it actually complicates a bit the discussion of its fermionic orbital angular momentum. Therefore, a more rigorous approach is needed and we develop it in Appendix \[Implementation\_sec\]. Among other things, such an approach can be used for showing that the above-mentioned heuristic procedure provides correct results for other angular momenta that we discuss.
Next, we note that due to the commutation of the total angular momentum operator with the Hamiltonian, angular momentum in states $|\0 s\ra$ and $\Oprim$ is the same, consequently =. \[toot\] Furthermore, the expectation value of each individual angular momentum operator, say $J^i_\chi$ with $\chi$ given by (\[chi\_list\]), must be either directly proportional to $\spinz$ or vanish. It is so because after averaging over spin projections of the electron, there is no preferred direction in the three-dimensional space, where $\exval{J^i_\chi}{\Opr}$ is discussed. Hence, $\exval{J^i_\chi}{\Opr}$ cannot have the $\sz$-independent component.[^3] We will use this observation over and over again to simplify calculations.
Moreover, since we will be doing the perturbative expansion around the one-electron state, we will be encountering the normalizing constant =s|s=. \[Vfact\] While such a constant is formally infinite, it gets unambiguously cancelled during computations. This happens because all expressions that contribute to the final result describe processes that happen homogeneously in space. As a result, the outermost spatial integral in every such expression is done over a function that is constant in space, and so it exactly cancels down normalizing constant (\[Vfact\]) appearing in the denominator of such an expression. Needless to say, factors like (\[Vfact\]) are frequently encountered in studies involving delocalized states (see e.g. above-cited [@LeaderPhysRep2014]).
We also mention that we will draw position-space Feynman diagrams in Figs. \[0th\_order\]–\[2nd\_order\_orbital\] to illustrate contributions to fermionic spin and orbital angular momenta of the electron. The diagram from Fig. X will be referred to as Diag. X. Rules for drawing these diagrams can be deduced without much effort by comparing them to analytical expressions that we list for them. There is no need to linger over these rules because all diagrams will be drawn only after analytical expressions will be worked out.
Finally, for the sake of brevity, we will drop the term O(\^4) from all expressions for expectation values of angular momentum operators.
Perturbative expansion for fermionic spin angular momentum {#Fer_spin_sec}
==========================================================
We will derive here the IR-regularized expression for fermionic spin angular momentum of the electron. To proceed, we expand (\[fghj\_b\]) in the series in $\eo$
$$\begin{aligned}
\expval{\J_\sp}{\Opr}{\LT}
&= \frac{\la\0s|\J^I_\sp|\0s\ra}{\Vol} \label{Jpert1} \\
&-\frac{1}{2}
\frac{\la\0s|\T\J^I_\sp
\intT\dd{x}{\cal H}^{I}_\IN\intT\dd{y}{\cal H}^{I}_\IN|\0s\ra}{\Vol} \label{Jpert2}\\
&+\frac{1}{2}
\frac{\la\0s|\J^I_\sp|\0s\ra}{\Vol}
\frac{\la\0s|\T\intT\dd{x}{\cal H}^{I}_\IN\intT\dd{y}{\cal H}^{I}_\IN|\0s\ra}{\Vol}. \label{Jpert3} \end{aligned}$$
\[pertLT\]
Zeroth-order contribution (\[Jpert1\]) is illustrated in Fig. \[0th\_order\]. We obtain after using (\[ext\_contractions\]) and (\[u\]) = =. \[Jpert1\_prim\]
![Diagrammatic illustration of $\ous\Gamma^i\us/(2\pi)^3$ from (\[Jpert1\_prim\]). The grey box stands for the operator $\Gamma^i$ from (\[Jspin\]). External lines are for zero-momentum electrons (the same notation is used in all our figures). []{data-label="0th_order"}](test2.eps){width="0.35\columnwidth"}
To compute (\[Jpert2\]), we need the following matrix element that can be obtained through Wick’s theorem combined with (\[ext\_contractions\])
$$\begin{aligned}
\nonumber
\la\0s|\T\N{\overline{\psi}_I(z)&\Gamma^i\psi_I(z)}
\N{\overline{\psi}_I(x)\gamma^\mu\psi_I(x)}
\N{\overline{\psi}_I(y)\gamma^\nu\psi_I(y)} |\0s\ra =\\
& \frac{e^{\ii f\cdot (x-y)}}{(2\pi)^3} \ous\gamma^\mu S(x-z)\Gamma^i S(z-y)\gamma^\nu\us
\label{q1} \\
+&\frac{e^{\ii f\cdot (z-y)}}{(2\pi)^3}\ous\Gamma^i S(z-x)\gamma^\mu S(x-y)\gamma^\nu \us
\label{q2} \\
+&\frac{e^{\ii f\cdot (x-z)}}{(2\pi)^3}\ous\gamma^\mu S(x-y)\gamma^\nu S(y-z)\Gamma^i \us
\label{q3} \\
-&\frac{1}{2(2\pi)^3}
\trr{S(y-x)\gamma^\mu S(x-y)\gamma^\nu}\ous\Gamma^i \us
\label{q4}\\
-&\frac{1}{(2\pi)^3}
\trr{S(y-z)\Gamma^i S(z-y)\gamma^\nu}\ous\gamma^\mu \us
\label{q5}\\
-&\Vol \trr{S(x-z)\Gamma^iS(z-y)\gamma^\nu S(y-x)\gamma^\mu}
\label{q6}\\
+&(\barexymunu \ \text{on all terms}).
\label{q7}\end{aligned}$$
\[q\]
Matrix element (\[q\]) can be additionally simplified with (\[z0\]) and (\[f0\]) leading to $e^{\ii f\cdot z}=1$. Its contractions with the photon propagator are diagrammatically depicted in Fig. \[2nd\_order\_licznik\].
![The (a)–(f) panels illustrate photon-propagator contractions with expressions (\[q1\])–(\[q6\]), respectively. []{data-label="2nd_order_licznik"}](test.eps){width="\pref\columnwidth"}
To compute (\[Jpert3\]), we proceed similarly as in (\[q\]) getting
$$\begin{aligned}
&\frac{\la\0s|\N{\overline{\psi}_I(z)\Gamma^i\psi_I(z)}|\0s\ra}{\Vol}
\la\0s|\T\N{\overline{\psi}_I(x)\gamma^\mu\psi_I(x)}
\N{\overline{\psi}_I(y)\gamma^\nu\psi_I(y)}
|\0s\ra=\nonumber \\
&\phantom{+}\frac{e^{\ii f\cdot (x-y)}}{(2\pi)^6}\frac{\ous\Gamma^i \us}{\Vol}
\ous\,\gamma^\mu S(x-y)\gamma^\nu \us
\label{qq1}\\
&-\frac{1}{2(2\pi)^3}
\trr{S(y-x)\gamma^\mu S(x-y)\gamma^\nu}\ous\Gamma^i \us \label{qq2}\\
&+(\barexymunu \ \text{on all terms}),
\label{qq3}\end{aligned}$$
\[qq\]
whose contractions with the photon propagator are diagrammatically shown in Fig. \[2nd\_order\_mianownik\]. Replacements (\[q7\]) and (\[qq3\]) produce a factor of $2$ during evaluation of diagrams, which cancels down a prefactor of $1/2$ from (\[Jpert2\]) and (\[Jpert3\]).
To correctly evaluate contributions of different diagrams to fermionic spin angular momentum of the electron, one must properly enforce limit (\[limitT\]). This has to be carefully done because the standard procedure outlined between (\[dx0sin\]) and (\[deltac\]) leads to incorrect results when Diags. \[2nd\_order\_licznik\]b, \[2nd\_order\_licznik\]c, and \[2nd\_order\_mianownik\]a are considered. The comprehensive discussion of the appropriate way of handling the imaginary time limit can be found in Appendix \[Implementation\_sec\]. We will frequently refer the reader to it quoting below only its final outcomes for individual diagrams.
Finally, to make equations a bit more compact, we introduce the following notation $$\begin{aligned}
\label{tildeq_a}
&\text{Diag. X}=\limT \TT{\text{Diag. X}},\\
\label{tildeq}
&\tilde q=(q^0,\p), \ \bar{k}=(k^0,-\p).\end{aligned}$$ We are now ready to discuss diagrams.
![The (a) and (b) panels illustrate photon-propagator contractions with expressions (\[qq1\]) and (\[qq2\]), respectively. []{data-label="2nd_order_mianownik"}](test1.eps){width="0.55\columnwidth"}
[**Diagram \[2nd\_order\_mianownik\]a**]{}. We start with = \^i D\_(x-y) \^S(x-y)\^\
= d\_(k)\
=2\^2dk\^0 F(k\^0,p\^0) , \[3A\] where identities (\[ubaru2\]) and (\[xiubaru4\]) have been employed to get F(k\^0,p\^0)=&\
+& . \[Fcov\] Note that we only list those arguments of the function $F$ that are most relevant for enforcement of the imaginary time limit. Using (\[chi\]), we get
$$\begin{aligned}
\text{Diag. \ref{2nd_order_mianownik}a}=&
\label{diagram3a_1}
2\pi \eo^2\spinz\int \dddd{p} F(\mo-p^0,p^0) \limT T \\
\label{diagram3a_2}
+&\frac{\eo^2\spinz}{2} \int \dddd{p}dk^0
\BB{\frac{F(k^0,p^0)}{(k^0+p^0-\mo+\izero)^2}+ \frac{F(k^0,p^0)}{(k^0+p^0-\mo-\izero)^2}}.\end{aligned}$$
\[diagram3a\]
It is now worth to stress that the procedure described between (\[dx0sin\]) and (\[deltac\]) produces the following ill-defined factor under the integral sign \^2, which gives a warning sign that such a simplification is meaningless in this case. By ignoring this fact, one ends with term (\[diagram3a\_1\]) after a formal identification of $\delta(0)$ with \_[-T]{}\^T . Leaving aside the discussion of this dubious substitution, such a procedure [*misses*]{} crucially-important term (\[diagram3a\_2\]), whose derivation requires a more sophisticated analytical approach (Appendix \[Implementation\_sec\]). We also mention that (\[diagram3a\_1\]) cancels out with similar terms from Diags. \[2nd\_order\_licznik\]b and \[2nd\_order\_licznik\]c.[^4]
[**Diagrams \[2nd\_order\_licznik\]b and \[2nd\_order\_licznik\]c**]{}. Now, we compute =- e\^[f(z-y)]{} D\_(x-y) \^i S(z-x)\^S(x-y)\^\
=-\
d\_(k). \[diagram2b\_1\]
Employing (\[ubaru4\]) and (\[xiubaru2\]), (\[diagram2b\_1\]) can be written as $$\begin{gathered}
\TT{\text{Diag. \ref{2nd_order_licznik}b}}=
-2\ii\eo^2\spinz \int \dddd{p} \frac{dq^0dk^0}{2\pi}
\frac{F(k^0,p^0)}{q^0-\mo+\izero}\\
\cdot\frac{\sin[T(k^0+p^0-q^0)]}{k^0+p^0-q^0} \frac{\sin[T(k^0+p^0-\mo)]}{k^0+p^0-\mo}.
\label{diagram2b_2}\end{gathered}$$
We now note that the procedure outlined between (\[dx0sin\]) and (\[deltac\]) leads to $\delta(q^0-\mo)$ producing a meaningless factor of $1/\izero$ in the expression for $\text{Diag. \ref{2nd_order_licznik}b}$. This leaves no doubts that careful implementation of the imaginary time limit is necessary.
So, using (\[2chii\]), we find =&-\^2 F(-p\^0,p\^0) T\
&- dk\^0 . \[diagram2b\_3\]
Computation of =- e\^[f(x-z)]{} D\_(x-y) \^S(x-y)\^S(y-z)\^i \[diagram2c\_1\] follows now straightforwardly as through formal manipulations one can show that = \[diagram2c\_2\] if (\[z0\]) holds.
[**Diagram \[2nd\_order\_licznik\]a**]{}. We compute here =- e\^[f(x-y)]{} D\_(x-y) \^S(x-z)\^i S(z-y)\^\
=-\
d\_(k)\
=-4\^2 d\_(|[k]{})\
. \[diagram2a\_1\] With the help of (\[ubaru6\]), (\[xiubaru6\]), and (\[chiii\]) we arrive at $$\begin{gathered}
\text{Diag. \ref{2nd_order_licznik}a}=-\ii\eo^2\spinz\int\dddd{p}
\left[
\frac{2(p^2+\mo^2)+4(p_3)^2}{(p^2-\mo^2+\izero)^2[(p-f)^2-\lambda^2+\izero]}\right.\\
\left.
+\xxii\frac{1}{[(p-f)^2-\lambda^2+\izero][(p-f)^2-\lambda^2/\xi+\izero]}\right].
\label{diagram2a_2}\end{gathered}$$ We mention in passing that the procedure discussed between (\[dx0sin\]) and (\[deltac\]) gives a correct result here (no singularities are encountered during its implementation).
There are no other one-loop contributions to fermionic spin angular momentum of the electron in covariantly quantized QED. Indeed, disconnected vacuum bubble Diags. \[2nd\_order\_licznik\]d and \[2nd\_order\_mianownik\]b immediately cancel out due to the difference in overall signs of (\[Jpert2\]) and (\[Jpert3\]). Therefore, there is no need to write down expressions for them. Moreover, = D\_(x-y) \^\[diagram2e\] and =\^2 D\_(x-y) \[diagram2f\] also do not contribute because they are both $\sz$-independent–see identity (\[ubaru0\]) and the discussion below (\[toot\]).
The final IR-regularized result for fermionic spin angular momentum of the electron comes from Diags. \[0th\_order\], \[2nd\_order\_licznik\]a–\[2nd\_order\_licznik\]c, and \[2nd\_order\_mianownik\]a = + + + +, \[unJspin\] where the superscript $\lambda$ indicates the fact that the IR regularization is present in (\[unJspin\]). This expression can be obtained by adding (\[Jpert1\_prim\]) and (\[diagram2a\_2\]) to $$\begin{gathered}
\text{Diag. \ref{2nd_order_licznik}b}+
\text{Diag. \ref{2nd_order_licznik}c}+
\text{Diag. \ref{2nd_order_mianownik}a}=\\
2\ii\eo^2\spinz\int\frac{\dd{p}}{(2\pi)^4}
\BB{
\frac{\om{p}^2(p^2-\mo^2)+\lambda^2[3(p^0-\mo)^2-\om{p}^2]}{\lambda^2(p^2-\mo^2+\izero)[(p-f)^2-\lambda^2+\izero]^2}
-\frac{\om{p}^2+\lambda^2/\xi}{\lambda^2[(p-f)^2-\lambda^2/\xi+\izero]^2}
}.
\label{suma3}\end{gathered}$$ Note that there is no singularity in the integrand of (\[suma3\]) at $\lambda=0$ despite a factor of $\lambda^2$ in denominators, which can be shown by rearranging terms.
Perturbative expansion for other angular momenta {#Other_sec}
================================================
We will derive here IR-regularized expressions for fermionic orbital angular momentum, electromagnetic spin and orbital angular momenta, and gauge-fixing angular momentum.
Such an expression for fermionic orbital angular momentum can be obtained through straightforward modifications of calculations reported in Sec. \[Fer\_spin\_sec\]. We will discuss its derivation in Sec. \[Fer\_orb\_sec\].
Results for electromagnetic spin, electromagnetic orbital, and gauge-fixing angular momenta have to be derived from scratch, which is simplified by the following observation. Namely, it can be easily shown with (\[chiii\]), that IR-regularized expressions for these angular momenta can be obtained from (\[fghj\]) through the replacement \_T . \[qwwww\] The hint that such a simplification is going to work comes from the fact that (\[qwwww\]), which amounts to the procedure described between (\[dx0sin\]) and (\[deltac\]), does not lead to singular expressions here. By combining (\[qwwww\]) with the following observation s|\^I\_|s=, =, , , we find from (\[fghj\]) that \[Jpert\_chia\] =- s|\^I\_\^[I]{}\_(x) [H]{}\^[I]{}\_(y)|s, which will be used in Secs. \[El\_spin\_sec\]–\[G\_fix\_sec\].
Fermionic orbital angular momentum {#Fer_orb_sec}
----------------------------------
We begin by noting that s|\^I\_|s=, which simplifies a bit the following discussion based on (\[fghj\]). Another matrix element that we need to know is
$$\begin{aligned}
\nonumber
\la\0s|\T\N{\overline{\psi}_I(z)&\nabla^i_\z\psi_I(z)}
\N{\overline{\psi}_I(x)\gamma^\mu\psi_I(x)}
\N{\overline{\psi}_I(y)\gamma^\nu\psi_I(y)} |\0s\ra =\\
&\phantom{+}\frac{e^{\ii f\cdot (x-y)}}{(2\pi)^3}\ous\gamma^\mu S(x-z)\nabla_\z^i S(z-y)\gamma^\nu \us
\label{o1} \\
&+\frac{e^{\ii f\cdot (z-y)}}{(2\pi)^3}\ous\nabla_\z^i S(z-x)\gamma^\mu S(x-y)\gamma^\nu \us
\label{o2} \\
&-\frac{1}{(2\pi)^3}
\trr{S(y-z)\nabla_\z^i S(z-y)\gamma^\nu} \ous\gamma^\mu \us
\label{o3}\\
&-\Vol \trr{S(x-z)\nabla_\z^iS(z-y)\gamma^\nu S(y-x)\gamma^\mu}
\label{o4}\\
&+(\barexymunu \ \text{on all terms}),
\label{o5}\end{aligned}$$
\[o\]
whose contractions with the photon propagator are diagrammatically depicted in Fig. \[2nd\_order\_orbital\]. Such an expression can be obtained by replacing $\Gamma^i$ in (\[q\]) by $\nabla^i_\z$ and by noting that the latter operator gives zero when acting on bispinors $\us$ (\[u\]). Replacements (\[o5\]) produce a factor of $2$ during evaluation of diagrams, which cancels down a prefactor of $1/2$ coming from the second order expansion of the exponential function in the numerator of (\[fghj\_b\]).
Armed with (\[o\]), we can proceed similarly as in Sec. \[Fer\_spin\_sec\] discussing each diagram separately. We start from the only diagram, which yields a non-zero contribution to fermionic orbital angular momentum of the electron.
![The (a)–(d) panels illustrate photon-propagator contractions with expressions (\[o1\])–(\[o4\]), respectively. The grey triangle stands for the operator $\nabla^i_\z$, which is defined in (\[Jorb\]). It acts on the fermionic propagator attached to its vertex. []{data-label="2nd_order_orbital"}](test3.eps){width="0.55\columnwidth"}
[**Diagram \[2nd\_order\_orbital\]a**]{}. We employ notation (\[tildeq\_a\]) and compute =- e\^[f(x-y)]{}D\_(x-y)\^S(x-z)\_\^i S(z-y)\^\
=-\
()\^i d\_(k). Next, we use e\^[(p-q)z]{} ()\^i =\^[imn]{}q\^n (-), \[pq\_derz\] and integrate by parts to move derivatives acting on $\delta(\p-\q)$ to the rest of the integrand. Boundary terms from integration by parts disappear. For example, because the integrand of the resulting surface integral is proportional to \^[imn]{}q\^mq\^n=0. \[ijn\_delta\] Derivatives of propagators’ denominators lead to the same factors and so they also do not contribute. A similar thing can be said about derivatives of the exponential term because (-) e\^[(qy-px)]{} \~ (+)\^m e\^[(-)]{}=0. \[exp\_vanish\] In the end, after spacetime integrations and employment of (\[chiii\]), we arrive at = d\_(f-p), \[4aa\] where $\{\,,\}$ stands for the anticommutator.
Finally, we use (\[ubaru12\]) and (\[ubaru12next\]) to get $$\begin{gathered}
\text{Diag. \ref{2nd_order_orbital}a}
=-4\ii\eo^2\spinz \int\dddd{p}
\frac{(p_1)^2+(p_2)^2}{(p^2-\mo^2+\izero)^2[(p-f)^2-\lambda^2+\izero]}\\
\cdot\B{1+\xxiidwa\frac{p^2-\mo^2}{(p-f)^2-\lambda^2/\xi+\izero}}.
\label{4a4a}\end{gathered}$$ It is perhaps worth to mention that the procedure discussed between (\[dx0sin\]) and (\[deltac\]) leads to the same result for this diagram.
[**Diagram \[2nd\_order\_orbital\]b**]{}. We study now =- e\^[f(z-y)]{}D\_(x-y)\_\^i S(z-x)\^S(x-y)\^\
=-\
()\^i d\_(k). \[xdse\] Next, we note that e\^[(f-q)z]{} ()\^i=-\^[imn]{}\[q\^n()\], which after integration by parts, where boundary terms trivially vanish, immediately shows that $\TT{\text{Diag. \ref{2nd_order_orbital}b}}=0$. This implies =0. \[llllp\] We mention in passing that such a derivation of this result avoids singular expressions that may be encountered after employment of (\[qwwww\]).
[**Diagrams \[2nd\_order\_orbital\]c and \[2nd\_order\_orbital\]d**]{}. These diagrams, $$\begin{aligned}
&\text{Diag. \ref{2nd_order_orbital}c}=\limT \frac{\eo^2}{\Vol}
\intT\dd{x}\dd{y}\int\ddd{z}D_{\mu\nu}(x-y) \trr{S(y-z)\nabla_\z^i S(z-y)\gamma^\nu}
\ous\gamma^\mu \us,
\label{4c}\\
&\text{Diag. \ref{2nd_order_orbital}d}=\limT\eo^2
\intT\dd{x}\dd{y}\int\d{z}
D_{\mu\nu}(x-y)\trr{S(x-z)\nabla_\z^iS(z-y)\gamma^\nu S(y-x)\gamma^\mu},
\label{4d}\end{aligned}$$ do not contribute to fermionic orbital angular momentum because they are $\sz$-independent–see identity (\[ubaru0\]) and the discussion below (\[toot\]).
The final IR-regularized result for fermionic orbital angular momentum is = . \[z1234\]
Electromagnetic spin angular momentum {#El_spin_sec}
-------------------------------------
We set $\chi=\spel$ in (\[Jpert\_chia\]) and note that the matrix element, which we need to compute, factorizes into the product of electromagnetic and fermionic matrix elements
$$\begin{aligned}
&\la\0s|\T(\J^I_\spel)^i {\cal H}^{I}_\IN(x){\cal H}^{I}_\IN(y)|\0s\ra=
\eo^2{\cal A}^i_{\mu\nu}(x,y){\cal F}^{\mu\nu}(x,y),\\
\label{Aimunu}
&{\cal A}^i_{\mu\nu}(x,y)=\varepsilon^{imn}\int\d{z}\langle0|\T\N{F^I_{m0}(z) A^I_n(z)} A^I_\mu(x)A^I_\nu(y)|0\rangle,\\
&{\cal F}^{\mu\nu}(x,y)=\la\0s|\T\N{\overline{\psi}_I(x)\gamma^\mu\psi_I(x)}\N{\overline{\psi}_I(y)\gamma^\nu\psi_I(y)}|\0s\ra.
\label{Fmunu}\end{aligned}$$
Evaluation of its fermionic part was done in [@BDfield], and we quote the final result for completeness here
\^(x,y)=[F]{}\_\^(x,y)+[F]{}\_\^(x,y), \[Fsymasym\] \_\^(x,y)=& e\^[(f-p)(x-y)]{}\
+&2V e\^[(p-q)(x-y)]{}\
+& (xy ), \[Fsym\] \^\_(x,y)= e\^[(f-p)(x-y)]{} - (xy). \[ploik\]
The above splitting is based on symmetry (anti-symmetry) of ${\cal F}^{\mu\nu}_\sym$ (${\cal F}^{\mu\nu}_\asym$) with respect to the transformation $\mu\leftrightarrow\nu$. Another important difference between ${\cal F}^{\mu\nu}_\sym$ and ${\cal F}^{\mu\nu}_\asym$ is that the former is $\sz$-independent, and so it cannot contribute to the final result due to reasons explained below (\[toot\]). We will thus replace ${\cal F}^{\mu\nu}$ below by ${\cal
F}^{\mu\nu}_\asym$.
Electromagnetic matrix element (\[Aimunu\]) is easily obtained through Wick’s theorem combined with the following identity 0|\_A\^I\_(x)A\^I\_(y)|0=D\_(x-y), \[bgt\] which can be shown with canonical commutation relations. It reads
[A]{}\^i\_(x,y)&= \^[imn]{} F\^I\_[m0]{}(z) A\^I\_(x) A\^I\_n(z)A\^I\_(y)+()\
&= ,
\[mat\_spel\] a\^i\_(p,q\_0)=&\^[imn]{}p\_m\
+&\^[imn]{} \_[m]{}\_[n]{}(p\_0+q\_0), \[uyuyuy\] \[aimunu\]
where $\tilde{q}$ is defined in (\[tildeq\]).
The IR-regularized expression for electromagnetic spin angular momentum of the electron can be then written as =-\^i\_(x,y)[F]{}\^\_(x,y). After simple algebra, we end up with a rather surprisingly compact formula =-4\^2. \[bareJel1\]
Electromagnetic orbital angular momentum {#El_orb_sec}
----------------------------------------
We set $\chi=\orbel$ in (\[Jpert\_chia\]) and again notice that the resulting matrix element, which has to be computed, factorizes into the product of electromagnetic and fermionic matrix elements s|(\^I\_)\^i [H]{}\^[I]{}\_(x)[H]{}\^[I]{}\_(y)|s= \^2\^(x,y), \[Bimunu\] where ${\cal B}^i_{\mu\nu}$ and ${\cal C}^i_{\mu\nu}$ will be defined below.
To compute the electromagnetic matrix element, equal to the expression in square brackets in (\[Bimunu\]), we need to evaluate
&z\^m0| A\^I\_(x)A\^I\_(y)|0\
&= z\^m \_A\^I\_(z) A\^I\_(x) \_A\^I\_(z) A\^I\_(y)+()\
&=- z\^m e\^[-px +qy +(p-q)z]{} + ()\
&= (+)\^m e\^[-px+y]{}\
&+ e\^[-px+y]{} ,
\[ddAAAA\] where contractions have been computed as in (\[bgt\]), $\d{z}$ integration has been done with z\^m e\^[(p-q)z]{} = (-), \[dkdq\] integration by parts has been employed, and $$\begin{aligned}
\label{box12a}
&{_{\alpha\beta\gamma\delta}\Box_{\mu\nu}}(p,q)=
-\frac{1}{2}
\frac{p_\alpha d_{\beta\mu}(p)}{p^2-\lambda^2+\izero} \frac{q_\gamma d_{\delta\nu}(q)}{q^2-\lambda^2+\izero}, \\
&{_{\alpha\beta\gamma\delta m}\cancel{\Box}_{\mu\nu}}(p,q)=
\frac{\ii}{2}
\B{\frac{\partial}{\partial p^m}-\frac{\partial}{\partial q^m}}
\B{\frac{p_\alpha d_{\beta\mu}(p)}{p^2-\lambda^2+\izero}
\frac{q_\gamma d_{\delta\nu}(q)}{q^2-\lambda^2+\izero}}
\label{box12b}\end{aligned}$$ have been introduced. We mention in passing that there are no boundary terms from such integration by parts.
We obtain by combining (\[Jorbel\]), (\[Bimunu\]), and (\[ddAAAA\])
$$\begin{aligned}
&{\cal B}^i_{\mu\nu}(x,y)= \int \dddd{p} \frac{dq^0}{2\pi}
(\x+\y)^m\, b^i_{m\mu\nu}(p,\tilde{q})
e^{-\ii p\cdot x+\ii\tilde{q}\cdot y},\\
&b^i_{m\mu\nu}(p,q)=\varepsilon^{imn}\BB{
{_{j0nj}}\Box_{\mu\nu}(p,q) - {_{0jnj}}\Box_{\mu\nu}(p,q)} + (\mu\leftrightarrow\nu,p\leftrightarrow q),\end{aligned}$$
\[BBmunu\]
$$\begin{aligned}
&{\cal C}^i_{\mu\nu}(x,y)= \int \dddd{p} \frac{dq^0}{2\pi}
c^i_{\mu\nu}(p,\tilde{q})
e^{-\ii p\cdot x+\ii\tilde{q}\cdot y},\\
&c^i_{\mu\nu}(p,q)=\varepsilon^{imn}\BB{
{_{j0njm}}\cancel{\Box}_{\mu\nu}(p,q)
-{_{0jnjm}}\cancel{\Box}_{\mu\nu}(p,q)} - (\mu\leftrightarrow\nu,p\leftrightarrow q).\end{aligned}$$
\[CCmunu\]
Proceeding similarly as in Sec. \[El\_spin\_sec\], we write the IR-regularized expression for electromagnetic orbital angular momentum of the electron as =- \_\^(x,y), \[p1234p\] where the contribution of ${\cal B}^i_{\mu\nu}$ to (\[p1234p\]) vanishes because it is proportional to the term that has the same structure as the right-hand side of (\[exp\_vanish\]). We get after simple algebra $$\begin{gathered}
\expval{J^i_\orbel}{\Opr}{\lambda}=-\frac{2\eo^2\sz}{\Vol}\int\dd{x}\dd{y}\int\dddd{k}\dddd{p}\frac{dq^0}{(2\pi)^4}
\frac{\varepsilon^{0\mu\nu3}\mo-\varepsilon^{\sigma\mu\nu3}k_\sigma}{k^2-\mo^2+\izero}
c^i_{\mu\nu}(p,\tilde{q})\\
\cdot
e^{\ii(f-k-p)\cdot x+\ii(k+\tilde{q}-f)\cdot y}.
\label{qqeenn}\end{gathered}$$
Finally, with the help of c\^i\_(p,p)= , \[fpp\]we obtain $$\begin{gathered}
\expval{J^i_\orbel}{\Opr}{\lambda}
=-4\ii\eo^2\spinz\int\dddd{p}
\frac{(p_1)^2+(p_2)^2}{(p^2-\mo^2+\izero)[(p-f)^2-\lambda^2+\izero]^2}\\
\cdot
\BB{1-\frac{1}{2\xi} \frac{(p-f)^2-\lambda^2}{(p-f)^2-\lambda^2/\xi+\izero} }.
\label{tytyty}\end{gathered}$$
Gauge-fixing angular momentum {#G_fix_sec}
-----------------------------
We set $\chi=\xi$ in (\[Jpert\_chia\]) and note that the resulting expression can be obtained by straightforward modifications of calculations from Sec. \[El\_orb\_sec\]. Namely, $\exval{J^i_\xi}{\Opr}$ is given by the right-hand side of (\[qqeenn\]) with $c^i_{\mu\nu}$ being replaced by $\tilde c^{\,i}_{\mu\nu}$, whose diagonal components are given by c\^[i]{}\_(p,p)=. This leads to the following IR-regularized expression for gauge-fixing angular momentum of the electron =-2\^2 . \[JxiEx\]
Pauli-Villars regularization {#PVV_sec}
============================
We will discuss here implementation of the Pauli-Villars regularization in our calculations (see [@RayskiPR1949; @PauliRMP1949] for early works on this technique as well as [@BogolubovBook; @ColemanBook] for its variations). In its simplest version, it is based on the following modifications of either fermionic propagator (\[prop\_fer\]) - , \[repl\_fer\] where $M=\mo,\Lambda$, or electromagnetic propagator (\[prop\_el\]) - , \[repl\_el\] where the replacement $\lambda\to\Lambda$ is also applied to $d_{\mu\nu}(p)$, which depends on $\lambda$ too. The parameter $\Lambda$ is supposed to be taken to infinity upon removal of the regularization. We have implemented these three ad hoc replacements, finding that none of them leads to total angular momentum of the electron that is independent of $\xi$. Calculations leading to such a conclusion can be performed by technically straightforward extensions of studies presented in this paper and so we will not linger over them.
Failure of these popular yet somewhat arbitrary regularization attempts means that we need a systematic approach, imposing the Pauli-Villars regularization consistently all across calculations. One may thus consider modifications of the Lagrangian density (see [@Schwartz; @Gupta] for textbook introduction to this technique). Such a bottom-up approach introduces ghost fields, say $\tilde A^\mu$ and $\tilde\psi$, through the replacement
[L]{}=&-F\_F\^ -\^2 +A\_A\^ +\
&+F\_F\^ +\^2 -A\_A\^ +\
&-(\^+ \^)(A\_+A\_). \[PVL\]
This leads to the interaction-picture density of the interaction Hamiltonian \^I\_= ( +) (A\^I\_+A\^I\_), \[Htilde\] which has to be used in imaginary time evolutions. Such evolutions in our studies start from the state |=| s|0, \[bullet\] where $|\tilde 0\ra$ contains no ghost particles.
As we discuss in Appendix \[Pauli\_sec\], replacements \^I\_\^I\_, | s|\[repl\] performed on (\[fghj\]) regularize only expectation values of $J^i_\sp$ and $J^i_\orb$. They are equivalent to modification (\[repl\_el\]) of the electromagnetic propagator in calculations from Secs. \[Fer\_spin\_sec\] and \[Fer\_orb\_sec\]. The problem now is that replacements (\[repl\]), when imposed on (\[fghj\]), [*do not*]{} regularize expectation values of $J^i_\spel$, $J^i_\orbel$, and $J^i_\xi$.
To overcome this difficulty, we first introduce ghost angular momentum operators $\tilde J_\chi^i$, which are obtained from $J_\chi^i$ by replacing all fields with their ghost counterparts. Next, we consider J\^i-J\^i=\_(J\^i\_- J\^i\_), \[JmJ\] where $\chi$ is given by (\[chi\_list\]). The expectation value of the left-hand side of (\[JmJ\]), upon removal of the regularization, should yield total angular momentum of the electron. It should be so because ghost angular momentum should not contribute in such a limit (there are no ghost particles in the unperturbed state of the system and the $\Lambda\to\infty$ limit suppresses addition of such particles to the perturbed state).
The idea now is to compute the expectation value of J\^i\_-J\^i\_\[jjj\] in the system described by modified Lagrangian density (\[PVL\]), and to treat the resulting expression, say $\expval{J^i_\chi}{\Opr}{\lambda\Lambda}$, as both the IR- and UV-regularized expectation value of the operator $J^i_\chi$. According to remarks presented below (\[JmJ\]), such a regularization procedure should not affect the value of total angular momentum of the electron, and so it may be considered as a prospective solution to regularization challenges that we face.
To put such a scheme to the test, we marry up (\[fghj\]) with (\[repl\]), and replace $J^i_\chi$ in the resulting formula by (\[jjj\]) getting =- \[lL\] for all angular momenta that we study (see Appendix \[Pauli\_sec\] for derivation of this formula). For $\chi=\sp,\orb$ this is exactly what one obtains through replacements (\[repl\]) imposed on (\[fghj\]) because those angular momenta are linear in electromagnetic propagators–see the comment below (\[repl\]). For $\chi=\spel,\orbel,\xi$, (\[lL\]) does not correspond to any of above-mentioned modifications of propagators. For example, (\[lL\]) is not equivalent to (\[repl\_el\]) because expressions for those angular momenta are quadratic in electromagnetic propagators. It is thus evident that such a ghost subtraction technique extends the standard Pauli-Villars approach based solely on modifications of Lagrangian density (\[PVL\]). We find it quite reassuring that these two methods agree for fermionic spin and orbital angular momenta, where the standard approach works.
All in all, (\[lL\]) delivers the consistent Pauli-Villars regularization of all angular momenta that we study. Such a procedure, when individual regularized angular momenta are added up, leads to the $\xi$-independent value of total angular momentum of the electron (Sec. \[Regularized\_sec\]). It is perhaps worth to stress that the fact that we work with arbitrary $\xi>0$ allows us for a rather stringent test of reliability of the regularization procedure that we use. Indeed, the requirement of gauge invariance, within the family of all covariant gauges, eliminates a great deal of presumably sensible Pauli-Villars-like regularizations.
One-loop radiative corrections {#Regularized_sec}
==============================
To compute one-loop radiative corrections, we will use subtraction procedure (\[lL\]) to impose ultraviolet (UV) regularization onto expressions (\[diagram2a\_2\]), (\[suma3\]), (\[4a4a\]), (\[bareJel1\]), (\[tytyty\]), and (\[JxiEx\]). This step is necessary because without it those expressions do not have definite values. To simplify such obtained formulae, products of propagators’ denominators will be joined with the following identities
$$\begin{aligned}
&\frac{1}{AB}=\int_0^1 ds\frac{1}{[sA + (1-s)B]^2},\\
&\frac{1}{AB^2}=\int_0^1 ds \frac{2(1-s)}{[s A + (1-s) B]^3},\\
&\frac{1}{A^2B^2}=\int_0^1 ds\,\frac{6(1-s)s}{[s A + (1-s)B]^4},\\
&\frac{1}{ABC}=\int_0^1ds\int_0^{1-s}du\frac{2}{[sA+uB+(1-s-u)C]^3},\end{aligned}$$
\[p3p3\]
the timelike component of the $4$-vector $p$ will be shifted to make resulting denominators $p^2$ dependent, Lorentz averaging of numerators will be implemented through replacements $p_\mu
p_\nu\to\eta_{\mu\nu}p^2/4$, and finally Wick rotation will be performed followed by straightforward evaluation of resulting Euclidean integrals. Such obtained expressions will be compactly written after introduction of the following functions $$\begin{aligned}
&\Delta_\chi=(1-s)^2+s(\chi/\mo)^2,\\
&\tilde\Delta_\chi=(1-s-u)^2 + (s+u/\xi)(\chi/\mo)^2.\end{aligned}$$ Above-mentioned calculations will be done under tacit assumptions that these functions are greater than zero for $\chi=\lambda,\Lambda$.
Fermionic spin angular momentum {#Regularized_fer_spin}
-------------------------------
We will apply here procedure (\[lL\]) to individual diagrams introducing = -\[iiooiioo\] as the Pauli-Villars-regularized version of IR-regularized only $\text{Diag. X}$ from Sec. \[Fer\_spin\_sec\]. Note that limit (\[limitT\]) is already taken in (\[iiooiioo\]).
Following steps outlined around (\[p3p3\]), we get $$\begin{gathered}
\LMPH{\text{Diag. \ref{2nd_order_licznik}a}}=
\frac{\eo^2\spinz}{8\pi^2}\int_0^1 ds (1-s)
\BB{
\ln\frac{\Delta_\Lambda}{\Delta_\lambda} + (1+s^2) \B{\frac{1}{\Delta_\Lambda} -
\frac{1}{\Delta_\lambda}}}+
\frac{\eo^2\spinz}{8\pi^2}\xxii\ln\frac{\Lambda}{\lambda}
\label{d1_s}\end{gathered}$$ and $$\begin{gathered}
\LMPH{\text{Diag. \ref{2nd_order_licznik}b}}+
\LMPH{\text{Diag. \ref{2nd_order_licznik}c}}+
\LMPH{\text{Diag. \ref{2nd_order_mianownik}a}} = \\
\frac{\eo^2\spinz}{8\pi^2}\int_0^1 ds\left[
s\ln\frac{\Delta_\lambda}{\Delta_\Lambda}
+2(2-s)(1-s)s\B{\frac{1}{\Delta_\lambda} -\frac{1}{\Delta_\Lambda}}\right]
-\frac{\eo^2\spinz}{8\pi^2}\xxii\ln\frac{\Lambda}{\lambda}.
\label{d2new}\end{gathered}$$
Integrals in these equations can be analytically evaluated, but resulting expressions are not compact. We list them in Appendix \[Integrals\_app\]. Among other things, they can be used for showing that unless $\xi$ is fine-tuned, (\[d1\_s\]) and (\[d2new\]) are logarithmically divergent in both IR and UV upon removal of the regularization. For $\xi=\infty$, the Landau gauge, these expressions are still IR divergent but UV finite. For $\xi=1/3$, the Fried-Yennie gauge, (\[d1\_s\]) and (\[d2new\]) are IR finite but UV divergent. Both features are typical of covariant gauge calculations.
Next, we take limits of $\mph\to0$ and $\Lambda\to\infty$ on the sum of (\[Jpert1\_prim\]), (\[d1\_s\]), and (\[d2new\]) getting \[earlyJspin\] =. Using $\eo=e+O(e^3)$, this can be written as \[alphaJspin\] =+O(\^2). This one-loop result for fermionic spin angular momentum of the electron agrees with earlier studies [@LiuPRD2015; @JiPRD2016]. Several remarks are in order now.
To begin, our calculations show that (\[earlyJspin\]) is $\xi$-independent, i.e., one and the same in the family of all covariant gauges. This becomes apparent even before removal of the regularization due to trivial cancellation of last terms in (\[d1\_s\]) and (\[d2new\]) when the sum of all diagrams is considered. We find it interesting that $\xi$-dependence in these equations takes such a simple form despite the fact that $\xi$ shows up in the denominator of electromagnetic propagator (\[prop\_el\]). Indeed, one would naturally expect that after joining propagators’ denominators through (\[p3p3\]), $\xi$-dependence will be transferred to the $\Delta_\chi$-like function appearing under the integral over the auxiliary parameter $s$. This is actually what happens in intermediate stages of calculations, but then unforeseen simplifications occur allowing for trivial evaluation of $\xi$-dependent parts of (\[d1\_s\]) and (\[d2new\]).
Next, we remark that (\[d2new\]) is equal to $\spinz(Z_2-1)$, where $Z_2$ is the renormalization constant of the Dirac field. One can easily verify this statement in the Feynman gauge by looking at Sec. 7.1 of [@PS], where $Z_2(\xi=1)$ is computed. In the general covariant gauge, one can repeat calculations from [@PS] with propagator (\[prop\_el\]). Such obtained expression for $Z_2(\xi)$ is complicated, but it can be easily numerically checked that it also supports the above remark. Appearance of $Z_2$ in (\[d2new\]) is expected. For example, a quick look at Figs. \[2nd\_order\_licznik\]b, \[2nd\_order\_licznik\]c, and \[2nd\_order\_mianownik\]a reveals that diagrams depicted there are similar in structure to the ones encountered during evaluation of $Z_2$ from the study of the electron propagator in the QED vacuum state [@PS]. Finally, we mention that the $\xi\neq1$ correction to $Z_2(\xi)$, which can be extracted from the last term in (\[d2new\]), appears also in [@JohnsonPRL1959], where calculations are Pauli-Villars-regularized in a slightly different way.[^5]
Other angular momenta {#Regularized_other_sec}
---------------------
We will apply here regularization procedure (\[lL\]) to angular momenta studied in Sec. \[Other\_sec\]. This results in the following set of equations $$\begin{gathered}
\expval{J^i_\orb}{\Opr}{\lambda\Lambda}=-4\ii\eo^2\spinz \int\dddd{p}
\frac{(p_1)^2+(p_2)^2}{(p^2-\mo^2+\izero)^2}\\
\cdot\BB{\frac{1}{(p-f)^2-\lambda^2+\izero}
\B{1+\xxiidwa\frac{p^2-\mo^2}{(p-f)^2-\lambda^2/\xi+\izero}}-\ltoL},
\label{qqqw}\end{gathered}$$ $$\begin{gathered}
\expval{J^i_\spel}{\Opr}{\lambda\Lambda}
=-4\ii\eo^2\spinz\int\dddd{p}\frac{2(p^0-\mo)^2-(p_1)^2-(p_2)^2}{p^2-\mo^2+\izero}
\\ \cdot\BB{\frac{1}{[(p-f)^2-\lambda^2+\izero]^2}-\ltoL},
\label{qaz}\end{gathered}$$ $$\begin{gathered}
\expval{J^i_\orbel}{\Opr}{\lambda\Lambda}
=-4\ii\eo^2\spinz\int\dddd{p}
\frac{(p_1)^2+(p_2)^2}{p^2-\mo^2+\izero}\\
\cdot\BB{
\frac{1}{[(p-f)^2-\lambda^2+\izero]^2}
\B{1-\frac{1}{2\xi} \frac{(p-f)^2-\lambda^2}{(p-f)^2-\lambda^2/\xi+\izero}}-\ltoL},
\label{vvvvv}\end{gathered}$$ $$\begin{gathered}
\expval{J^i_\xi}{\Opr}{\lambda\Lambda}
=-2\ii\eo^2\spinz\int\dddd{p}
\frac{(p_1)^2+(p_2)^2}{p^2-\mo^2+\izero}
\\ \cdot\BB{
\frac{1}{[(p-f)^2-\lambda^2+\izero][(p-f)^2-\lambda^2/\xi+\izero]}-\ltoL
}.
\label{JxiExPV}\end{gathered}$$
Even without evaluating these expressions, one can notice that their sum is $\xi$-independent, which is something that we have anticipated in Sec. \[PVV\_sec\]. A bit surprising now is that $\expval{J^i_\spel}{\Opr}{\lambda\Lambda}$ and $\expval{J^i_\orb+J^i_\orbel+J^i_\xi}{\Opr}{\lambda\Lambda}$ are separately $\xi$-independent. Such an observation, however, is formal because we will shortly see that both quantities are actually infinite upon removal of the regularization.
Following the procedure outlined at the beginning of Sec. \[Regularized\_sec\], we get $$\begin{aligned}
&\expval{J^i_\orb}{\Opr}{\lambda\Lambda}
=-\spinz\frac{\eo^2}{4\pi^2}\int_0^1ds (1-s)\ln\frac{\Delta_\Lambda}{\Delta_\lambda}
-\spinz\frac{\eo^2}{8\pi^2}\xxii\int_0^1ds\int_0^{1-s}du
\ln\frac{\tilde\Delta_\Lambda}{\tilde\Delta_\lambda},
\\
&\expval{J^i_\spel}{\Opr}{\lambda\Lambda}
=\spinz\frac{\eo^2}{2\pi^2}
\int_0^1ds\, s \BB{ \ln\frac{\Delta_\Lambda}{\Delta_\lambda}
-(1-s)^2\B{\frac{1}{\Delta_\lambda} -\frac{1}{\Delta_\Lambda} } },
\\
&\expval{J^i_\orbel}{\Opr}{\lambda\Lambda}
=-\spinz\frac{\eo^2}{4\pi^2}\int_0^1ds\, s \ln\frac{\Delta_\Lambda}{\Delta_\lambda}
+\spinz\frac{\eo^2}{8\pi^2}\frac{1}{\xi}\int_0^1ds\int_0^{1-s}du
\ln\frac{\tilde\Delta_\Lambda}{\tilde\Delta_\lambda},
\\
&\expval{J^i_\xi}{\Opr}{\lambda\Lambda}
=-\spinz\frac{\eo^2}{8\pi^2}\int_0^1ds\int_0^{1-s}du
\ln\frac{\tilde\Delta_\Lambda}{\tilde\Delta_\lambda}.\end{aligned}$$
This can be further simplified if we remove the IR regularization. With some extra effort, we get the following results exhibiting rather non-trivial $\xi$-dependence $$\begin{aligned}
&\limlzero\expval{J^i_\orb}{\Opr}{\lambda\Lambda}
\simeq\spinz\frac{\eo^2}{8\pi^2}\B{-\frac{1+\xi}{\xi}\ln\frac{\Lambda}{\mo}+\frac{5}{4}
-\frac{3}{4\xi}+\frac{\ln\xi}{2\xi}},\\
&\limlzero\expval{J^i_\spel}{\Opr}{\lambda\Lambda}\simeq\spinz\frac{\eo^2}{2\pi^2}
\B{\ln\frac{\Lambda}{\mo}+\frac{3}{4}},\\
\label{JorbelAss}
&\limlzero\expval{J^i_\orbel}{\Opr}{\lambda\Lambda}
\simeq\spinz\frac{\eo^2}{8\pi^2}\B{\frac{1-2\xi}{\xi}\ln\frac{\Lambda}{\mo}-\frac{5}{2}
+\frac{3}{4\xi}-\frac{\ln\xi}{2\xi(1-\xi)}},\\
\label{JxiAss}
&\limlzero\expval{J^i_\xi}{\Opr}{\lambda\Lambda}\simeq\spinz\frac{\eo^2}{8\pi^2}\B{-\ln\frac{\Lambda}{\mo}-\frac{3}{4}
+\frac{\ln\xi}{2(1-\xi)}},\end{aligned}$$ where $\simeq$ means that we omit terms that vanish in the limit of $\Lambda\to\infty$. Note that all these expressions are well-defined for any $\xi>0$. Among other things, they allow us to conclude that upon removal of the regularization =. \[qqqmmm\]
Combining (\[qqqmmm\]) with (\[earlyJspin\]), we see that in our one-loop calculations the expectation value of total angular momentum operator (\[totalJ\]) is given by (\[toot\]), which can be seen as a self-consistency check of our studies.
Discussion {#Discussion_sec}
==========
We have teamed the bare perturbative expansion with the imaginary time evolution technique to study radiative corrections to different components of angular momentum of the electron. Our calculations have been done in the general covariant gauge. The results that we have obtained can be summarized as follows.
First, we have carefully discussed implementation of imaginary time evolutions developing a rigorous analytical procedure taking care of singularities that may appear in the course of calculations. Such evolutions are routinely used for generation of ground states, which are then used for computation of expectation values of products of field operators in interacting quantum field theories. Results that we present on this matter are missed in standard textbooks on quantum field theory, where enforcement of the imaginary time limit is trivialized to steps outlined between (\[dx0sin\]) and (\[deltac\]). On the one hand, our calculations show how disastrous such an oversimplification is when bare perturbation theory is employed for evaluation of self-energy-type diagrams. On the other hand, they provide a general framework that can be readily deployed in computations of other expectation values in quantum field theories. This can be useful for either resolving possible issues with “simplified” handling of the imaginary time limit or for rigorous checking whether such a procedure is justified. These remarks are comprehensively illustrated by our studies in Sec. \[Fer\_spin\_sec\], where computations of some diagrams have been only possible after sophisticated enforcement of the imaginary time limit.
Second, we have computed fermionic spin and orbital, electromagnetic spin and orbital, and gauge-fixing angular momenta of the electron. Out of these five quantities, only fermionic spin angular momentum is gauge invariant, and so it can be conclusively compared to earlier studies, which were done in the light-cone gauge [@LiuPRD2015; @JiPRD2016]. It agrees with these works showing equivalence of the light-cone and general covariant gauge calculations. While such an agreement is expected on general grounds, it is perhaps worth to mention that the issue of gauge independence is still quite non-trivial (Sec. 2.5.2 of [@LeaderPhysRep2014]). More importantly, technical comparision between calculations in these completely different gauges should be interesting and our detailed discussion should facilitate it.
Third, the remaining four angular momenta are gauge non-invariant. Out of them, gauge-fixing angular momentum is specific to covariant gauge studies and it is instructive to take a closer look at it. It is so because its presence turns out to be of key importance to assigning spin one-half to the electron in covariantly quantized electrodynamics. Indeed, (\[toot\]) would not hold without it even in the $\xi\to\infty$ limit, where the Lorentz gauge is most transparently enforced (Sec. 15.5 of [@WeinbergII]). This is interesting because $J^i_\xi$ can be seen as a physically meaningless artifact of the quantization procedure and so the question arises why it non-trivially contributes to the physically meaningful quantity such as electron’s spin. We expect that resolution of this puzzle is the following. The gauge-fixing term in Lagrangian density (\[LL\]) not only generates gauge-fixing angular momentum, but it also affects the electromagnetic propagator. The latter impacts computations of expectation values of gauge non-invariant angular momentum operators. As a result, those expectation values get implicitly modified by the presence of the gauge-fixing term and this modification is explicitely cancelled in (\[toot\]) by gauge-fixing angular momentum, so that it has no effect on electron’s spin.
Fourth, we have developed a variant of the Pauli-Villars regularization by requiring that total angular momentum of the electron should be one and the same in the family of all covariant gauges. This obvious condition is violated by the simplest versions of the Pauli-Villars regularization. In our scheme, one subtracts from the observable of interest its ghost operator counterpart, and then calculates the expectation value of such obtained operator through imaginary time evolution. The latter is consistently implemented by the standard addition of ghost fields to the Lagrangian density. The net effect of this procedure is very simple for observables that we study (\[lL\]). We believe that it would be interesting to put this approach to the test in other problems as well.
Finally, to place result (\[alphaJspin\]) in a wider context, we mention that only one more finite gauge invariant individual component of total angular momentum of the electron was identified so far. Namely, electromagnetic angular momentum [@BDfield] \_ = -+O(\^2), \[Jfield\] where $\E$ and $\Bold$ are electric and magnetic field operators.[^6] Gauge invariance and finiteness of (\[alphaJspin\]) and (\[Jfield\]) should make them especially interesting from the experimental point of view. Given the fact that various angular momenta, contributing to nucleons’ spin, have been extensively experimentally studied [@Deur2019], we are hopeful that such quantities can be also measured. The remaining open question is how this can be achieved.
[**Acknowledgements**]{}\
I would like to thank Aneta for being a wonderful sounding board during all these studies. Diagrams in this work have been done in JaxoDraw [@JaxoDraw2]. This work has been supported by the Polish National Science Centre (NCN) grant DEC-2016/23/B/ST3/01152.
Conventions and all that {#Conventions_sec}
========================
We use the Minkowski metric $\eta=\text{diag}(+---)$ and choose $\varepsilon^{0123}=+1=\varepsilon^{123}$. Greek and Latin indices take values $0,1,2,3$ and $1,2,3$, respectively, when they refer to components of $4$- and $3$-vectors. We use the Einstein summation convention. 3-vectors are written in bold, e.g. $x=(x^\mu)=(x^0,\x)$. Electron’s bare and physical charges are both negative.
We introduce =, =||, =, \[expectation\] and write the interaction-picture Dirac field operator as
$$\begin{aligned}
\label{DiracI}
&\psi_I(x)=\int\frac{\d{p}}{(2\pi)^{3/2}} \sqrt{\frac{\mo}{\vareps{p}}}
\sum_s\BB{a_{\p s}u(\p,s)e^{-\ii p\cdot x} + b^\dag_{\p s} v(\p,s)e^{\ii p\cdot x} }, \
(p^\mu)=(\vareps{p},\p),\\
&\{a_{\p s},a^\dag_{\q r}\}=\{b_{\p s},b^\dag_{\q r}\}=\delta_{sr}\delta(\p-\q), \end{aligned}$$
where $a_{\p s}$ annihilates the electron and $b_{\p s}$ annihilates the positron (both of momentum $\p$ and the spin state $s$). All other anticommutators involving those operators are equal to zero. We choose bispinors $u(\p,s)$ and $v(\p,s)$, in the standard representation of $\gamma$ matrices that we use, so that
$$\begin{aligned}
\label{ups}
&u(\p,s)=
\frac{1}{\sqrt{2\mo(\vareps{p}+\mo)}}
\binom{(\vareps{p}+\mo)\phi^s}{\p\cdot\sig\phi^s}, \ v(\p,s)= \frac{1}{\sqrt{2\mo(\vareps{p}-\mo)}}
\binom{(\vareps{p}-\mo)\phi^s}{\p\cdot\sig\phi^s},\\
&\phi^s=\binom{1}{0},\binom{0}{1}.\end{aligned}$$
We define contractions of $\psi_I$ on zero-momentum external lines as \_I(x)| s=e\^[-fx]{}, s|\_I(x)= e\^[fx]{}, =u(,s), \[ext\_contractions\] where $|\0 s\rangle$ and $f$ are given by (\[0s\]) and (\[f0\]), respectively. The $\us$ bispinors are eigenstates of the $z$-component of the one-particle fermionic spin angular momentum operator \^3 = , =(
[l]{} 1\
0\
0\
0
) =+1/2, =(
[l]{} 0\
1\
0\
0
) =-1/2. \[u\]
Finally, we mention that there is no summation over $s$ in matrix elements $\ous\cdots\us$.
Bispinor matrix elements {#Matrix_sec}
========================
Results presented below are obtained in the standard (Dirac) representation of $\gamma$ matrices. It is then a simple exercise to show that the same results are obtained in all representations unitarily similar to the standard one (Weil, Majorana, etc.). This statement is equivalent to saying that they are invariant under $\gamma^\mu\to U\gamma^\mu U^\dag$ and $\us\to U\us$ transformations, where $U$ is an arbitrary unitary matrix of dimension four (see [@PalArxiv2007; @ArminjonBraz2008] for the discussion of representation-independence of various results associated with the Dirac equation).
The following expressions are used in our computations $$\begin{gathered}
\ous\gamma^\mu(\gamma\cdot p+\mo)\gamma_\mu \us=4\mo-2p^0,
\label{ubaru2}\\
\ous\gamma\cdot k(\gamma\cdot p+\mo)\gamma\cdot k\us=2k^0k\cdot p+k^2(\mo-p^0),
\label{xiubaru4}\\
\ous\Gamma^i(\gamma^0q^0+\mo)\gamma^\mu(\gamma\cdot p+\mo)\gamma_\mu \us=
\spinz(\mo+q^0)\ous\gamma^\mu(\gamma\cdot p+\mo)\gamma_\mu \us,
\label{ubaru4}\\
\ous\Gamma^i(\gamma^0 q^0+\mo)\gamma\cdot k (\gamma\cdot
p+\mo)\gamma\cdot k\us=\spinz(\mo+q^0)\ous\gamma\cdot k(\gamma\cdot p+\mo)\gamma\cdot k\us,
\label{xiubaru2}\\
\ous\gamma^\mu(\gamma\cdot p+\mo)\Gamma^i(\gamma\cdot p+\mo)\gamma_\mu
\us=2\sz\BB{\delta^{i3}(p^2+\mo^2)+2p_i p_3},
\label{ubaru6}\\
\ous\gamma\cdot(f- p)(\gamma\cdot p+\mo)\Gamma^i (\gamma\cdot p+\mo)\gamma\cdot(f-p)\us=
\spinz (p^2-\mo^2)^2,
\label{xiubaru6}\\
\label{ubaru0}
\ous\gamma^\mu \us=\eta^{\mu0},\\
\label{ubaru12}
\varepsilon^{imn}p^n\ous\gamma^\mu\{\gamma^m\gamma^0,\gamma\cdot p+\mo\}\gamma_\mu\us=-8\ii\sz(\delta^{i3}\om{p}^2-p_ip_3),\\
\label{ubaru12next}
\varepsilon^{imn}p^n\ous\gamma\cdot(f-p)\{\gamma^m\gamma^0,\gamma\cdot p+\mo\}\gamma\cdot(f-p)\us=-4\ii\sz(\delta^{i3}\om{p}^2-p_ip_3)(p^2-\mo^2).\end{gathered}$$ We mention in passing that we simplify matrix elements (\[ubaru6\]), (\[ubaru12\]), and (\[ubaru12next\]) under integral signs by the replacement $p_ip_3\to\delta^{i3}(p_3)^2$.
It is interesting to note that $\sz$-dependence, in all expectation values that we study, comes from expressions that critically depend on the four-dimensional Levi-Civita symbol, whose extension to a $d\neq4$ dimensional space-time, used in the dimensional regularization, is problematic (see e.g. Appendix B.2 of [@DreinerPhysRep2010] and references therein). This can be proved by combining (\[ubaru0\]) and the following easy-to-verify identities $$\begin{gathered}
\ous\gamma^\mu\gamma^\nu\us=\eta^{\mu\nu}-2\ii\sz\varepsilon^{0\mu\nu3},
\label{2gamma}\\
\ous\gamma^\mu\gamma^\sigma\gamma^\nu\us=
\eta^{\mu\sigma}\eta^{\nu0}+\eta^{\sigma\nu}\eta^{\mu0}-\eta^{\mu\nu}\eta^{\sigma0}-2\ii\sz\varepsilon^{\mu\sigma\nu3},
\label{3gamma}\\
\ous\gamma^0\gamma^1\gamma^2\gamma^3\us=0
\label{gamma5}\end{gathered}$$ with the observation that any product of $\gamma$ matrices can be always reduced to the single term containing at most four $\gamma$ matrices, whose indices are distinct.
Implementation of imaginary time evolutions {#Implementation_sec}
===========================================
In the following, we work out integrals that are necessary for implementation of imaginary time evolutions. While doing so, we will frequently use the Sochocki-Plemelj formula dx = dxf(x), \[SP\] where $\dashint$ stands for the Cauchy principal value. Several things have to be kept in mind in the following discussion.
First, as we have mentioned in Sec. \[Basics\_sec\], $T$ will be greater than zero during evaluation of integrals and then the limit $\barelimT$ will be taken.
Second, we will use below the function G(k\^0,p\^0,…), \[Gkp\] which will be assumed to have poles at k\^0=, p\^0=, \[poles\] etc. Masses $M$, $M'$, etc. will be greater than zero. In other words, poles of (\[Gkp\]) will come from propagators’ denominators: $(k^0)^2-\om{p}^2-M^2+\izero$, $(p^0)^2-\om{p}^2-{M'}^2+\izero$, etc.
Third, as (\[Gkp\]) will vanish for large arguments in our studies, there will be no problems with convergence of contour integrals that we will discuss.
[**Type $\Rzymskie{1}$ integrals**]{}. The integrals of interest here are given by the formula \_ = dp\^0 dk\^0 G(k\^0,p\^0) , \[chistart\] where poles of the function $G$ are characterized by $M>0$ and $M'=\mo$. Such integrals appear in studies of Diag. \[2nd\_order\_mianownik\]a, where $M$ is greater than zero due to the IR regularization provided by either the photon mass term or the ghost photon mass term in Pauli-Villars-regularized calculations.
We rewrite (\[chistart\]) as
\_ =& dp\^0 dk\^0 G(k\^0,p\^0)\
+& dp\^0 dk\^0 G(k\^0,p\^0) .
Using now (\[SP\]), we arrive at
$$\begin{aligned}
\chi_\Rzymskie{1}
&=\pi \int dp^0 G(\mo-p^0,p^0) \limT T
\label{chii_a} \\
&+\frac{1}{4} \limT\int dp^0 dk^0 G(k^0,p^0)\frac{1-e^{2\ii T(k^0+p^0-\mo)}}{k^0+p^0-\mo}\frac{1}{k^0+p^0-\mo+\izero} \label{chii_b}\\
&+\frac{1}{4}\limT\int dp^0 dk^0 G(k^0,p^0)\frac{1-e^{-2\ii T(k^0+p^0-\mo)}}{k^0+p^0-\mo} \frac{1}{k^0+p^0-\mo-\izero}.\label{chii_c}\end{aligned}$$
\[chii\]
Suppose now that we evaluate integrals (\[chii\_b\]) and (\[chii\_c\]) on semicircular contours in upper and lower half-planes of complex $k^0$ and $p^0$, respectively. This turns exponential terms in (\[chii\_b\]) and (\[chii\_c\]) into
$$\begin{aligned}
&e^{\pm2\ii T(k^0+p^0-\mo)} \xrightarrow[\text{integrations}]{\text{contour}} e^{-2\ii T \gamma_\pm},
\\
&\gamma_\pm=\sqrt{\om{p}^2+M^2}+\sqrt{\om{p}^2+{M'}^2}\pm\mo.\end{aligned}$$
\[exp0\]
Next, we note that $\gamma_\pm>0$ for $M$ and $M'$ specified below (\[chistart\]). Therefore, when we take the limit $\barelimT$, exponential terms can be dropped from (\[chii\_b\]) and (\[chii\_c\]) if we properly shift poles of $1/(k^0+p^0-\mo)$, which amounts to \_ =&dp\^0 G(-p\^0,p\^0) T\
+& dp\^0 dk\^0 . \[chi\]
[**Type $\Rzymskie{2}$ integrals**]{}. Next, we introduce G(k\^0,p\^0,q\^0)=, where $G(k^0,p^0)$ is the same as in $\chi_\Rzymskie{1}$, and consider \_ = dp\^0 dk\^0 dq\^0 G(k\^0,p\^0,q\^0) . \[uuu1234\] Integrals of such a form appear in studies of Diags. \[2nd\_order\_licznik\]b and \[2nd\_order\_licznik\]c.
We rewrite (\[uuu1234\]) as \_ = & dp\^0 dk\^0 dq\^0 G(k\^0,p\^0,q\^0) e\^[-T(q\^0-)]{}\
+ & dp\^0 dk\^0 dq\^0 G(k\^0,p\^0,q\^0) e\^[T(q\^0-)]{} .
Employing (\[SP\]), we obtain \_ = & dp\^0 dk\^0 dq\^0 G(k\^0,p\^0,q\^0) e\^[-T(q\^0-)]{}\
+ & dp\^0 dk\^0 dq\^0 G(k\^0,p\^0,q\^0) e\^[T(q\^0-)]{} . \[2chii1\]
Doing the first (second) integral over $q^0$ on the lower (upper) semicircular contour of the complex $q^0$ half-plane, joining integrals, rearranging terms, and then splitting them again we arrive at \_ =&- dp\^0 dk\^0 G(k\^0,p\^0)\
&- dp\^0 dk\^0 G(k\^0,p\^0) . \[2chii2\]
Using again (\[SP\]), we obtain
\_ &=-\^2dp\^0 G(-p\^0,p\^0) T\
&- dp\^0 dk\^0 G(k\^0,p\^0)\
&- dp\^0 dk\^0 G(k\^0,p\^0) .
\[2chii3\]
Repeating now steps around (\[exp0\]), we note that exponential terms in (\[2chii3\]) vanish upon taking the limit, which after proper shifting of the pole of $1/(k^0+p^0-\mo)$ leaves us with \_ =-\^2dp\^0 G(-p\^0,p\^0) T - dp\^0 dk\^0 . \[2chii\]
[**Type $\Rzymskie{3}$ integrals**]{}. Now, we consider \_ =dp\^0 dk\^0 dq\^0 G(k\^0,p\^0,q\^0) , \[chiii1\] where poles of $G(k^0,p^0,q^0)$ are parameterized by $M>0$ and $M'=M''=\mo$ in expressions for Diags. \[2nd\_order\_licznik\]a and \[2nd\_order\_orbital\]a. During evaluation of electromagnetic spin, electromagnetic orbital, and gauge-fixing angular momenta, they are given by $M=\mo$, and $M',M''>0$.
We rewrite (\[chiii1\]) as \_ = &dp\^0 dk\^0 dq\^0 G(k\^0,p\^0,q\^0)\
- &dp\^0 dk\^0 dq\^0 G(k\^0,p\^0,q\^0) , which after using (\[SP\]) leads to \_ = &dp\^0 dq\^0 G(-q\^0,p\^0,q\^0)\
+&dp\^0 dk\^0 dq\^0 G(k\^0,p\^0,q\^0)\
&.
After splitting integrals over sinuses into Cauchy principal value integrals and then one more employment of (\[SP\]), we obtain
$$\begin{aligned}
\chi_\Rzymskie{3}
&=
\pi^2\int dp^0 G(\mo-p^0,p^0,p^0)\\
\label{bb}
+&\frac{\pi}{2\ii}\limT\int dp^0 dq^0 \BB{G(\mo-q^0,p^0,q^0)+G(\mo-p^0,p^0,q^0)}
\frac{e^{\ii T(p^0-q^0)}}{p^0-q^0+\izero}\\
+&\frac{\pi}{2\ii}\limT\int dp^0 dq^0 \BB{G(\mo-q^0,p^0,q^0)+G(\mo-p^0,p^0,q^0)}
\frac{e^{\ii T(q^0-p^0)}}{q^0-p^0+\izero}\\
-&\frac{1}{4}\limT\int dp^0 dk^0 dq^0 G(k^0,p^0,q^0)
\frac{e^{\ii T(2k^0+p^0+q^0-2\mo)}}{(k^0+p^0-\mo+\izero)(k^0+q^0-\mo+\izero)}\\
\label{ee}
-&\frac{1}{4}\limT\int dp^0 dk^0 dq^0 G(k^0,p^0,q^0)
\frac{e^{-\ii
T(2k^0+p^0+q^0-2\mo)}}{(k^0+p^0-\mo-\izero)(k^0+q^0-\mo-\izero)}\\
\label{ff}
+&\frac{1}{4}\limT\int dp^0 dk^0 dq^0 G(k^0,p^0,q^0)
\frac{e^{\ii T(p^0-q^0)}}{(k^0+p^0-\mo+\izero)(k^0+q^0-\mo-\izero)}\\
\label{gg}
+&\frac{1}{4}\limT\int dp^0 dk^0 dq^0 G(k^0,p^0,q^0)
\frac{e^{\ii T(q^0-p^0)}}{(k^0+p^0-\mo-\izero)(k^0+q^0-\mo+\izero)}.\end{aligned}$$
\[chiii4\]
Integrands in terms (\[bb\])–(\[gg\]) involve factors , \[argumenty\] where $h^0$ variables are timelike components of $4$-momenta appearing in expressions for propagators. If we now integrate each term on semicircular contours in upper $(+)$ and lower $(-)$ half-planes of complex $h^0$, we will see that poles of (\[argumenty\]) do not contribute to such contour integrals. Thus, only poles of the $G$ function contribute, but they turn exponential terms into the form similar to (\[exp0\]). For $M$, $M'$, and $M''$ listed below (\[chiii1\]), one can then easily argue that (\[bb\])–(\[gg\]) are removed by the limit $\barelimT$.
All in all, we get \_ = \^2dp\^0 G(-p\^0,p\^0,p\^0). \[chiii\]
Pauli-Villars regularization {#Pauli_sec}
============================
We will discuss here technicalities related to implementation of the Pauli-Villars regularization through introduction of ghost fields, whose interaction-picture propagators are [@Gupta] $$\begin{aligned}
&
\tilde S(x-y)=
\la\tilde0|\T\tilde\psi_I(x)\overline{\tilde\psi}_I(y)|\tilde0\ra=
\ii\int\frac{\dd{p}}{(2\pi)^4}\frac{\gamma\cdot p+\Lambda}{p^2-\Lambda^2+\izero}e^{-\ii p\cdot(x-y)},
\label{vbnml2}\\
&\tilde D_{\mu\nu}(x-y)=\la\tilde0|\T\tilde A^I_\mu(x)\tilde A^I_\nu(y)|\tilde0\ra
=\ii \int\dddd{p}\frac{e^{-\ii p\cdot (x-y)}}{p^2-\Lambda^2+\izero}
\B{\eta_{\mu\nu}+\xxii\frac{p_\mu p_\nu}{p^2-\Lambda^2/\xi+\izero}}.
\label{yhnnhy}\end{aligned}$$ A quick look at (\[prop\_fer\]) and (\[prop\_el\]) reveals that while $S(x-y)$ and $\tilde S(x-y)$ differ only in masses, $D_{\mu\nu}(x-y)$ and $\tilde D_{\mu\nu}(x-y)$ differ in both masses and overall signs.
Modification of (\[fghj\]) by (\[repl\]) asks for evaluation of | O\_I \^I\_(x)\^I\_(y)|= \^2[M]{}\_E[M]{}\_D, \[MM\] where matrix elements involving either real or ghost electromagnetic (Dirac field) operators are denoted as ${\cal M}_E$ (${\cal M}_D$). Their indices are suppressed for the sake of brevity. Expressions for ${\cal M}_E$ and ${\cal M}_D$ can be easily derived with the help of Wick’s theorem. During their evaluation, one must keep in mind that ghost fields follow bosonic statistics. Moreover, it is worth to remember that operators $O_I$ are normal ordered (the same comment applies to their ghost counterparts $\tilde O_I$ and to $\tilde{\cal
H}^I_\IN$). Normal ordering of all these operators substantially simplifies resulting expressions.
[**Dirac field operators**]{}. Taking $O=J^i_\sp,J^i_\orb$, we obtain \[elMfer\] [M]{}\_E= D\_(x-y)+D\_(x-y), \_D=& s|O\_I\_I(x)\^\_I(x)\_I(y)\^\_I(y)|s\
+&s|O\_I|s. These two formulae also hold when the unit operator is substituted for $O$. This observation is useful during studies of fermionic spin angular momentum of the electron, where the denominator of (\[fghj\_b\]) non-trivially contributes.
[**Electromagnetic operators**]{}. For $O=J^i_\spel, J^i_\orbel,J^i_\xi$, we get $$\begin{aligned}
\label{elMel}
&{\cal M}_E=\la0|\T O_I A^I_\mu(x) A^I_\nu(y)|0\ra,\\
&{\cal M}_D= {\cal F}^{\mu\nu}(x,y)+\Vol\trr{\tilde S(y-x)\gamma^\mu\tilde S(x-y)\gamma^\nu},
\label{ferMel}\end{aligned}$$ where ${\cal F}^{\mu\nu}$ is given by (\[ploik\]). Note that the last term of (\[ferMel\]) is $\sz$-independent, and so it has no influence on angular momentum of the electron due to reasons explained below (\[toot\]).
Having these results, one can easily show that replacements (\[repl\]), when performed on (\[fghj\]), lead to $$\begin{aligned}
\label{ferfer}
&\expval{J^i_\chi}{\Opr}{\lambda} \to \expval{J^i_\chi}{\Opr}{\lambda}-\expval{J^i_\chi}{\Opr}{\Lambda} \ \for \ \chi=\sp,\orb,\\
&\expval{J^i_\chi}{\Opr}{\lambda} \to
\expval{J^i_\chi}{\Opr}{\lambda} \ \for \ \chi=\spel,\orbel,\xi,
\label{elel}\end{aligned}$$ where the superscript $\lambda$ reminds us that before introduction of ghost fields our calculations have already been IR-regularized. Thus, while angular momenta listed in (\[ferfer\]) are regularized by modification (\[PVL\]) of the Lagrangian density, the ones from (\[elel\]) are not. We mention in passing that (\[ferfer\]) follows from the fact that (\[elMfer\]) can be written as $D_{\mu\nu}(x-y)-\ltoL$.
To fix the problem caused by (\[elel\]), we consider expectation values of differences of angular momentum operators and their ghost counterparts. This asks for evaluation of the analog of (\[MM\]), | O\_I \^I\_(x)\^I\_(y)|= \^2\_E\_D, \[tMM\] leading to the following set of expressions.
[**Ghost Dirac field operators**]{}. Taking $\tilde O=\tilde J^i_\sp,\tilde J^i_\orb$, we obtain \_E=D\_(x-y)+D\_(x-y), \_D= & 0|O\_I\_I(y)\^\_I(y)|0\
+& 0|O\_I\_I(x)\^\_I(x)|0\
+&0|O\_I\_I(x)\^\_I(x)\_I(y)\^\_I(y)|0, \[ghMfer\] where we have used (\[ext\_contractions\]) and (\[ubaru0\]) to arrive at (\[ghMfer\]).
[**Ghost electromagnetic operators**]{}. For $\tilde O=\tilde J^i_\spel, \tilde J^i_\orbel,\tilde J^i_\xi$, we get $$\begin{aligned}
\label{ghelMel}
&\tilde{\cal M}_E=\la\tilde0|\T \tilde O_I \tilde A^I_\mu(x) \tilde
A^I_\nu(y)|\tilde0\ra,\\
&\tilde{\cal M}_D= {\cal F}^{\mu\nu}(x,y)+\Vol\trr{\tilde S(y-x)\gamma^\mu\tilde S(x-y)\gamma^\nu}.
\label{ghferMfer}\end{aligned}$$
Using (\[MM\])–(\[ferMel\]) and (\[tMM\])–(\[ghferMfer\]), one can show that if we impose on (\[fghj\]) replacements J\^i\_J\^i\_-J\^i\_\[JJtJ\] and (\[repl\]), then such modifications will result in - =,,,,. \[dreg\] Two comments are in order now.
First, ghost operator subtraction (\[JJtJ\]) does not affect expectation values of angular momentum operators built of Dirac fields, which are regularized by mere addition of ghost fields to the Lagrangian density, see (\[ferfer\]). The easiest way to see this is to combine the observation that whole (\[ghMfer\]) is $\sz$-independent with arguments presented below (\[toot\]).
Second, ghost operator subtraction (\[JJtJ\]) leads to regularization of angular momentum operators composed of electromagnetic operators, for which (\[dreg\]) can be understood by noting that (\[ghelMel\]) is obtained by performing the transformation $\lambda\to\Lambda$ on (\[elMel\]).
Evaluation of integrals {#Integrals_app}
=======================
We evaluate here definite integrals from (\[d1\_s\]) and (\[d2new\]). To this aim, we need the following indefinite integrals $$\begin{gathered}
4\int ds (1-s)
\B{
\ln\Delta_\chi + \frac{1+s^2}{\Delta_\chi} }=2s(\tchi^2-4)
-\BB{(\tchi^2-2)^2 + 2(1-s)^2}\ln\BB{(1-s)^2+s\tchi^2}\\+
\frac{2\tchi(\tchi^4-6\tchi^2+12)}{\sqrt{4-\tchi^2}}\arctan\frac{\tchi^2-2(1-s)}{\tchi\sqrt{4-\tchi^2}}+\text{const}
\label{I1}\end{gathered}$$ and $$\begin{gathered}
4\int ds\B{
s\ln\Delta_\chi+\frac{2(2-s)(1-s)s}{\Delta_\chi}}=2s(s-3\tchi^2-6)+
(3\tchi^4+2s^2-6)\ln\BB{(1-s)^2+s\tchi^2}\\-
\frac{6\tchi(\tchi^4-2\tchi^2-4)}{\sqrt{4-\tchi^2}}\arctan\frac{\tchi^2-2(1-s)}{\tchi\sqrt{4-\tchi^2}}+\text{const},
\label{I2}\end{gathered}$$ where $\tchi=\chi/\mo$.
These expressions can be used for any $0<\tchi^2<4$. For $\tchi^2>4$, the following replacements $$\begin{aligned}
&\sqrt{4-\tchi^2}\to\ii\sqrt{\tchi^2-4}, \\
&\arctan\frac{\tchi^2-2(1-s)}{\tchi\sqrt{4-\tchi^2}}\to
-\ii\arctanh\frac{\tchi^2-2(1-s)}{\tchi\sqrt{\tchi^2-4}}\end{aligned}$$ should be employed. They make right-hand sides of (\[I1\]) and (\[I2\]) real. Analogical replacements are also meant to be applied below.
Using (\[I1\]) and (\[I2\]), we find
$$\begin{aligned}
& \int_0^1 ds (1-s)
\BB{
\ln\frac{\Delta_\Lambda}{\Delta_\lambda} + (1+s^2) \B{\frac{1}{\Delta_\Lambda} -
\frac{1}{\Delta_\lambda}}}
=I_1(\tilde\lambda)-I_1(\tilde\Lambda)-2\ln\frac{\Lambda}{\lambda},\\
&I_1(x)=\frac{x^2-4}{2}(x^2\ln x-1)-
\frac{x^4-6x^2+12}{2}\frac{x}{\sqrt{4-x^2}}\arctan\frac{\sqrt{4-x^2}}{x}\end{aligned}$$
\[II1\]
and
$$\begin{aligned}
&\int_0^1 ds\left[
s\ln\frac{\Delta_\lambda}{\Delta_\Lambda}
+2(2-s)(1-s)s\B{\frac{1}{\Delta_\lambda}
-\frac{1}{\Delta_\Lambda}}\right]=I_2(\tilde\lambda)-I_2(\tilde\Lambda)+2\ln\frac{\Lambda}{\lambda},\\
&I_2(x)=\frac{3x^2}{2}(x^2\ln x-1)-\frac{3(x^4-2x^2-4)}{2}\frac{x}{\sqrt{4-x^2}}\arctan\frac{\sqrt{4-x^2}}{x},\end{aligned}$$
\[II2\]
respectively.
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[^1]: The term net refers to the fact that besides electrons in vacuum electron-positron pairs, there is one electron in such a state.
[^2]: This may be less obvious for operators involving time derivatives of the $4$-potential $A_\mu$–$J^i_\spel$, $J^i_\orbel$, and $J^i_\xi$–but it can be proven there as well (see e.g. [@BDfield]).
[^3]: As a self-consistency check, we have directly verified for $\xi=1$ that this is indeed the case in all our calculations. The same explicit verification has been also performed for expectation values of ghost operators $\tilde J^i_\chi$ discussed in Appendix \[Pauli\_sec\].
[^4]: The sum of Diags. \[2nd\_order\_licznik\]b, \[2nd\_order\_licznik\]c, and \[2nd\_order\_mianownik\]a is entirely determined by careful enforcement of limit (\[limitT\]). It cannot be obtained by the simplified procedure mentioned between (\[dx0sin\]) and (\[deltac\]).
[^5]: The difference comes from the fact that our regularization is consistently implemented throughout calculations, whereas the one in [@JohnsonPRL1959] is done “by hand”.
[^6]: Such a result was obtained with ad hoc regularization attempts (\[repl\_fer\]) and (\[repl\_el\]) explored in [@BDfield]. It can be also obtained with the ghost subtraction technique discussed in Sec. \[PVV\_sec\] and Appendix \[Pauli\_sec\] of this paper. Its indifference to details of the Pauli-Villars regularization scheme presumably comes from favorable convergence properties of the expression that is regularized during evaluation of (\[Jfield\]).
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'This article is concerned with the question of whether Marcin- kiewicz multipliers on $\mathbb R^{2n}$ give rise to bilinear multipliers on $\mathbb R^n\times \mathbb R^n$. We show that this is not always the case. Moreover we find necessary and sufficient conditions for such bilinear multipliers to be bounded. These conditions in particular imply that a slight logarithmic modification of the Marcinkiewicz condition gives multipliers for which the corresponding bilinear operators are bounded on products of Lebesgue and Hardy spaces.'
address:
- |
Department of Mathematics\
University of Missouri-Columbia\
Columbia, MO 65211
- |
Department of Mathematics\
University of Missouri-Columbia\
Columbia, MO 65211
author:
- Loukas Grafakos
- 'Nigel J. Kalton'
title: The Marcinkiewicz multiplier condition for bilinear operators
---
[^1]
Introduction {#s-introduction}
============
0
In this article we study bilinear multipliers of Marcinkiewicz type. Recall that a function $\sigma(\xi, \eta)= \sigma(\xi_1, \dots , \xi_n, \eta_1,
\dots , \eta_n)$ defined away from the coordinate axes on $\mathbb R^{2n}$, which satisfies the conditions $$\label{ma}
|\partial^\alpha_\xi \partial^\beta_\eta \sigma(\xi, \eta ) | \le
C_{\alpha, \beta} |\xi_1|^{- \alpha_1}\dots |\xi_n|^{- \alpha_n}
|\eta_1|^{- \beta_1}\dots |\eta_n|^{- \beta_n}$$ for sufficiently large multi-indices $\alpha=(\alpha_1, \dots , \alpha_n)$ and $\beta=(\beta_1, \dots , \beta_n)$, is called a Marcinkiewicz multiplier. It is a classical result, see for instance [@stein-old], that Marcinkiewicz multipliers give rise to bounded linear operators $M_\sigma$ from $L_p(\mathbb R^{2n})$ into itself for $1<p<\infty$. Here $M_\sigma$ is the multiplier operator with symbol $\sigma$, that is $$M_\sigma (F)(x) = \int_{\mathbb R^{2n}} \widehat{F}(\xi) \sigma (\xi) e^{ 2\pi i
\langle x, \xi \rangle } d\xi,$$ where $F$ is a Schwartz function on $\mathbb R^{2n}$ and $\widehat{F}(\xi)$ is the Fourier transform of $F$, defined by $
\widehat{F}(\xi)= \int_{\mathbb R^{2n}} F(x) e^{ -2\pi i
\langle x, \xi \rangle } dx.
$ (We will use the notation $\langle x, y \rangle=\sum_{k=1}^m x_ky_k$ for $x=(x_1, \dots , x_m)$ and $y=(y_1, \dots , y_m)$ elements of $\mathbb R^{m}$.) The Marcinkiewicz condition (\[ma\]) is less restrictive than the Hörmander-Mihlin condition $$\label{ho}
|\partial^\alpha_\xi \partial^\beta_\eta \sigma(\xi, \eta ) | \le
C_{\alpha, \beta} (|\xi|+|\eta|)^{-|\alpha| -|\beta|},$$ which is also known to imply boundedness for the linear operator $W_\sigma$ from $L_p(\mathbb R^{2n})$ into itself when $1<p<\infty$. The advantage of condition (\[ho\]) is that it is supposed to hold for multi-indices up to order $|\alpha|+|\beta|\le n+1$ versus up to order $|\alpha|+|\beta|\le 2n$ for condition (\[ma\]).
In this paper we study bilinear multiplier operators whose symbols satisfy similar conditions. More precisely, we are interested in boundedness properties of bilinear operators $$W_\sigma (f,g) (x) = \int_{\mathbb R^{2n}} \widehat{f}(\xi)
\widehat{g}(\eta) \sigma (\xi, \eta ) e^{ 2\pi i
\langle x, \xi \rangle }e^{ 2\pi i
\langle x, \eta\rangle } \, d\xi\, d\eta ,$$ originally defined for $f,g$ Schwartz functions on $\mathbb R^n$ and $\sigma$ a function on $\mathbb R^{2n}$. A well-known theorem of Coifman and Meyer [@CM] says that if the function $\sigma$ on $\mathbb R^{2n}$ satisfies (\[ho\]) for sufficiently large multi-indices $\alpha$ and $\beta$, then the bilinear map $W_\sigma (f,g)$ extends to a bounded operator from $L_{p_1}(\mathbb R^n)\times L_{p_2}(\mathbb R^n)$ into $L_{p_0,\infty }( \mathbb R^n)$ when $1<p_1, p_2<\infty$, $1/p_1+1/p_2=1/p_0$ and $p_0\ge 1$. ($L_{p_0,\infty}$ here denotes the space weak $L_{p_0}$.) This result was later extended to the range $1> p_0 \ge 1/2$ by Grafakos and Torres [@GT] and Kenig and Stein [@KS]. The extension into $L_{p_0}$ for $p_0< 1$ was stimulated by the recent work of Lacey and Thiele [@lacey-thiele2] who showed that the discontinuous symbol $\sigma(\xi, \eta)=-i \text{sgn }(\xi-\eta) $ on $\mathbb R^2$ gives rise to a bounded bilinear operator $W_\sigma$ from $L_{p_1}(\mathbb R)\times L_{p_2}(\mathbb R)$ into $ L_{p_0}(\mathbb R)$ for $2/3<p_0<\infty$ when $1<p_1,p_2<\infty$ and $1/p_1+1/p_2=1/p_0.$
In this article we address the question of whether the Marcinkiewicz condition (\[ma\]) on $\mathbb R^{2n}$ gives rise to a bounded bilinear operator $W_\sigma$ on $\mathbb R^n \times \mathbb R^n$. We answer this question negatively. More precisely, we show that there exist examples of bounded functions $\sigma (\xi, \eta)$ on $\mathbb R^n \times \mathbb R^n$ which satisfy the stronger condition $$\label{hypothesis}
|\partial^\alpha_\xi \partial^\beta_\eta \sigma(\xi, \eta ) | \le
C_{\alpha, \beta} |\xi|^{-|\alpha|}|\eta|^{-|\beta|}$$ for all multi-indices $\alpha$ and $\beta$, for which the corresponding bilinear operators $W_\sigma$ do not map $L_{p_1} \times L_{p_2} $ into $L_{p_0,\infty}$ for any triple of exponents satisfying $1/p_1+1/p_2=1/p_0$ and $1<p_1, p_2<\infty$.
We reduce this problem to the study of bilinear operators of the type $$\label{ooo}
(f,g)\to \sum_{j\in \mathbb Z}\sum_{k\in \mathbb Z} a_{jk}\,
\widetilde{\Delta}_jf \, \widetilde{\Delta }_kg ,$$ where $a_{jk}$ is a bounded sequence of scalars depending on $\sigma$ and $\widetilde{\Delta}_j$ are the Littlewood-Paley operators given by multiplication on the Fourier transform side by a smooth bump supported near the frequency $|\xi|\sim 2^j$. In section \[s-bilinear+infmatrices\], in particular Theorem \[matrixnorms\], we find a necessary and sufficient condition on the infinite matrix $A=(a_{jk})_{j,k}$ so that the bilinear operator in (\[ooo\]) maps $L_{p_1} \times L_{p_2} $ into $L_{p_0,\infty}$. This condition is expressed in terms of an Orlicz space norm of the sequence $(a_{jk})_{j,k}$. It turns out that this condition is independent of the exponents $p_1,p_2,p_0$ and depends only on quantities intrinsic to the matrix $A$, (although the actual norm of the operator in (\[ooo\]) from $L_{p_1} \times L_{p_2} $ into $L_{p_0,\infty}$ does depend on the indices $p_1,p_2,p_0$).
The results of section \[s-bilinear+infmatrices\] are transferred to multiplier theorems for bilinear operators in section \[s-applications\]. This transference is achieved using a Fourier expansion of the symbol $\sigma$ on products of dyadic cubes. Theorem \[main1\] is the main result of this section and Theorem \[bestposs\] shows that this theorem is best possible. Theorem \[main1\] allows us to derive that the estimates $$\label{above}
|\partial_{\xi}^{\alpha}\partial_{\eta}^{\beta} \sigma(\xi,\eta)| \le
C_{\alpha, \beta }|\xi|^{-|\alpha|} |\eta|^{-|\beta|}
\big(\log(1+|\log\tfrac{|\xi|}{|\eta|}|)\big)^{-\theta}$$ do give rise to a bounded bilinear operator $W_\sigma$ on products of $L_p$ spaces when $\theta >1$, while we show that this is not the case when $0<\theta<\frac12.$ We obtain similar results when the expression $\big(\log(1+|\log\tfrac{|\xi|}{|\eta|}|)\big)^{-\theta}$ in (\[above\]) is replaced by the expression $\big(\log(1+|\log\tfrac{|\xi|}{|\eta|}|)\big)^{-1}
\!\big(\log(1+\log(1+|\log\tfrac{|\xi|}{|\eta|}|))\big)^{-\theta}$ for $\theta>1$.
We find more convenient to work with the martingale difference operators $\Delta_k $ associated with the $\sigma-$algebra of all dyadic cubes of size $2^k$ in $\mathbb R^n$ and later transfer our results to the Littlewood-Paley operators $\widetilde\Delta_k $. This point of view is introduced in the next section.
We end this article with a short discussion on paraproducts, see section \[s-paraproducts\]. These are operators of the type (\[ooo\]) for specific sequences $(a_{jk})_{j,k}$ of zeros and ones.
A maximal operator {#s-maximal}
==================
0
Let $(\Omega,\Sigma,\mathbb P)$ be any probability space and let $(\Sigma_k)_{k\ge 0}$ be a [*filtration*]{} i.e. an increasing sequence of sub-$\sigma-$algebras of $\Sigma$. We say that $(\Sigma_k)$ is a [*dyadic filtration*]{} if each $\Sigma_k$ is atomic and has precisely $2^k$ atoms each with probability $2^{-k}.$ We say $(\Sigma_k)$ is a [*$2^n-$adic filtration*]{} if each $\Sigma_k$ is atomic with precisely $2^{nk}$ atoms each with probability $2^{-nk}.$
Associated to $\Sigma_k$ we define the conditional expectation operators $\mathcal E_kf=\mathbb E(f|\Sigma_k)$ and the martingale difference operators $\Delta_kf =\mathcal E_kf-\mathcal E_{k-1}f$ for $k\ge 1,$ and $f\in L_1(\Omega).$
Let $A=(a_{jk})$ be a complex $M\times N$ matrix, and let $(\Omega,\Sigma,\mathbb P)$ be a probability space with a dyadic filtration $(\Sigma_k)_{k\ge 0}$. For $1\le p<\infty$ we define $h_{p}(A)$ to be the least constant so that for all $f\in L_p(\Omega)$ we have $$\label{defh}
\big\|\max_{1\le j\le M}|\sum_{k=1}^Na_{jk}\Delta_kf| \big\|_{L_p}\le
h_{p}(A)\|f\|_{L_p}.$$ We also define the corresponding weak constants, i.e. the least constants so that for all $f\in L_p(\Omega)$ we have $$\label{defhweak}
\big\|\max_{1\le j\le M}|\sum_{k=1}^Na_{jk}\Delta_kf|\big\|_{L_{p,\infty}}\le
h_p^w(A)\|f\|_{L_p}.$$ Finally for $0<q<p<\infty$ we define the mixed constants $h_{p,q}(A)$ as the least constants such that for all $f\in L_p(\Omega)$ we have $$\label{defhmixed}
\big\|\max_{1\le j\le M}|\sum_{k=1}^Na_{jk}\Delta_kf|\big\|_{L_q}\le
h_{p,q}(A)\|f\|_{L_p}.$$
Note that these definitions are independent of the choice of the probability space and of the dyadic filtration. Indeed if $A$ is fixed, it suffices to take $f\in L_p(\Sigma_N)$ and hence we can consider a finite probability space with $2^N$ points and a finite dyadic filtration $(\Sigma_k)_{k=0}^N.$ We also note that $h_p(A)$ is the operator norm of the map $T_A:L_p(\Omega)\to L_p(\Omega;\ell_{\infty}^M)$ defined by $$T_Af= \big(\sum_{k=1}^Na_{jk}\Delta_k f \, \big)_{j=1}^M.$$ Similarly $h_p^w(A)$ is the norm of the operator $T_A:L_p\to
L_{p,\infty}(\Omega;\ell_{\infty}^M).$
Our first result is that all these constants are mutually equivalent, when $1<p<\infty$:
\[equiv\] If $ 1<p,q<\infty$ then there is a constant $0<C=C(p,q)<\infty$ such that for all complex $M\times N$ matrices $A$ we have $$\frac1C h_p(A)\le h_q^w(A)\le h_q(A)\le Ch_p(A).$$
It suffices to prove an estimate of the type $h_p(A)\le Ch_q^w(A)$ for any choice of $1<p,q<\infty.$ We first prove a weak type $(1,1)$ estimate for $T_A$, i.e. that $h_1^w(A)\le Ch_q^w(A).$ Suppose $f\in L_1$ with $\|f\|_{L_1}=1.$ Then if $\lambda, \gamma >0,$ with $\lambda\gamma>1,$ we can use an appropriate Calderón-Zygmund decomposition to find finite sets $D_1,\ldots ,D_m$ so that each $D_l$ is an atom of some $\Sigma_l , $ $$\gamma \lambda \le
\mathbb P (D_l) ^{-1}
\int_{D_l} |f| \, d \mathbb P =\Ave_{D_l}f \le
2 \gamma \lambda,$$ and $|f(\omega)| \le \gamma\lambda$ if $\omega\notin \cup_{l=1}^mD_l.$ Let $$g=\sum_{l=1}^m (\Ave_{D_l}f)\chi_{D_l}$$ and $E=\cup_{l=1}^mD_l.$ Then $T_A(f\chi_E-g)$ is supported in $E$ and thus $$\label{lou0}
\mathbb
P(\|T_A(f\chi_E-g)\|_{\ell_\infty^M}>\lambda/2) \le \mathbb P(E)\le
(\gamma\lambda)^{-1}.$$ On the other hand $\|f-f\chi_E+g\|_{L_{\infty}}\le 3\gamma\lambda$ and $\|f-f\chi_E+g\|_{L_1} \le 1$. Hence $\|f-f\chi_E+g\|_{L_q} \le 3^{1/q'}(\gamma\lambda)^{1/q'}$ and so $$\label{lou1}
\|T_A(f-f\chi_E+g)\|_{L_{q,\infty}(\ell_\infty^M)}\le h_q^w(A)3^{1/q'}
(\gamma\lambda) ^{1/q'},$$ which implies that $$\label{lou2}
\mathbb P(\|T_A(f-f\chi_E+g)\|_{\ell_\infty^M}> \lambda /2) \le
\frac{2^q}{ \lambda^q} ( h_q^w(A) )^q 3^{q-1} (\gamma \lambda)^{q-1} .$$ Selecting $\gamma=1/h^w_q(A)$ and combining with (\[lou0\]) we obtain (for $\lambda>h_q^w(A))$ $$\label{lou3}
\lambda \, \mathbb P(\|T_Af\|_{\ell_\infty^M}>\lambda)\le Ch_{q}^w(A)$$ where $C=C(p,q)$. This gives the weak-type (1,1) estimate for $T_A.$ Now by the Marcinkiewicz interpolation theorem (applied to the sublinear map $f\mapsto \|T_Af(\omega)\|_{\ell_\infty^M} $) we obtain that $h_p(A)\le
C(p,q)h_q^w(A)$ as long as $1<p<q.$
We now prove that $h_p(A)\le C(p,q)h_q^w(A)$ when $1<q<p<\infty$. We consider the dual map $T_A^*:L_1(\Omega;\ell_1^M)\to L_1$ defined by $$T_A^*\mathbf f= \sum_{j=1}^M \sum_{k=1}^N a_{jk}\Delta_kf_j$$ where $\mathbf
f(\omega)=(f_j(\omega))_{j=1}^M.$ We have that $T_A^*:L_r(\Omega;\ell_1^M)\to L_r$ has norm bounded by $C(q,r)h_q^w(A)$ as long as $1<r'<q$ i.e. $q'<r<\infty.$ Using this $r$ as a starting point, we repeat the argument above to show that $T_A^*:L_1(\Omega;\ell_1^M)\to
L_{1,\infty}$ has norm bounded by $Ch_q^w(A).$ The Marcinkiewicz interpolation theorem can again be used to show that $T_A^*:L_{p'}(\Omega,\ell_1^M)\to L_{p'}$ has norm bounded by $ Ch_q^w(A)$ for all $1<p'<r$, and thus in particular when $1<p'<q'$. Therefore we obtain that $h_p(A)\le Ch_q^w(A)$ when $1<q<p<\infty$.
[**Remark.**]{} From now we will write $h(A)=h_2(A)$ so that each $h_p(A)$ for $1<p<\infty$ is equivalent to $h(A).$
It is of some interest to observe that even the corresponding mixed constants are also equivalent to $h(A).$
\[mixed\] Suppose $0<q<p$ and $1<p<\infty$. Then there is a constant $C=C(p,q)$ so that $$\frac1C h(A)\le h_{p,q}(A)\le Ch(A).$$
This will depend on the following Lemma:
\[Lem1.3\] Suppose $1\le p<\infty$ and $0<q<p.$ Then there is a constant $C=C(p,q)$ so that if $r=\min(p,2)$ we have $$\label{wte}
\|T_A\|_{L_p\to L_{r,\infty}(\ell_{\infty}^M)}\le C h_{p,q}(A).$$
(Lemma \[Lem1.3\]) We may assume $q<r.$ This is a fairly standard application of Nikishin’s theorem, see [@N]. Here we use a version given in [@Pisier]. It is simplest to consider the case when $\Omega$ is finite with $|\Omega|=2^N.$ Consider the map $T_A:L_p\to L_q(\Omega;\ell_{\infty}^M)$. For each $f\in L_p$ with $\|f\|_{L_p}\le 1$, let $F_f(x)=\|T_Af(x)\|_{\ell_{\infty}^M}.$ For $\|f_j\|_{L_p}\le 1$ with $1\le j\le J$, $\sum_{j=1}^J|b_j|^r=1$, and $(\epsilon_j)_{j=1}^{J}$ a sequence of independent Bernoulli random variables on some probability space, we have $$\big\|\max_{1\le j\le J} |b_j|F_{f_j}\big\|_{L_q} \le \mathbb
E \bigg(\big\|\sum_{j=1}^J\epsilon_jb_jT_Af_j
\big\|_{L_q(\ell_{\infty}^M)} \bigg)
\le Ch_{p,q}(A),$$ since $L_p$ has type $r$. It follows from [@Pisier] that there is a function $w\in L_1$, with $\int w\,d\mathbb P=1,$ and $w\ge 0$ a.e such that for any set $E\subset\Omega$ $$\Big(\int_E F_f^qd\mathbb P\Big)^{\frac1q} \le Ch_{p,q}(A) \Big(\int_E
w\, d\mathbb P\Big)^{\frac1q-\frac1r}.$$ Now consider the set $S$ of all permutations of $\Omega$ which induce permutations of the atoms of each $\Sigma_k$ for $1\le k\le
N$; there are $2^{2^N-1}$ such permutations $\varphi.$ For $\varphi\in
S$ we have $$\Big(\int_E F_{f\circ\varphi}^q \, d\mathbb P\Big)^{\frac1q} \le
Ch_{p,q}(A)\Big(\int_E w\, d\mathbb P\Big)^{\frac1q-\frac1r}$$ or equivalently $$\Big(\int_E F_{f}^q \, d\mathbb P\Big)^{\frac1q} \le Ch_{p,q}(A)\Big(\int_E
w\circ\varphi^{-1}\, d\mathbb P\Big)^{\frac1q-\frac1r}.$$ Raising to the power $(\frac1q-\frac1r)^{-1}$, averaging over $S$, and then raising to the power $\frac1q-\frac1r$ gives $$\Big(\int_E F_{f}^q d\mathbb P\Big)^{\frac1q} \le
Ch_{p,q}(A)\bigg(\frac{1}{|S|}\sum_{\varphi\in S}\Big(\int_E
w\circ\varphi^{-1} \, d\mathbb P \Big) \bigg)^{\frac1q-\frac1r}.$$ But this implies $$\Big(\int_E F_{f}^q d\mathbb P\Big)^{\frac1q} \le
Ch_{p,q}(A)\, \mathbb P(E)^{\frac1q-\frac1r}$$ which gives the required weak type estimate (\[wte\]).
We now return to the proof of Theorem \[mixed\]. We first observe that we always have $h_{p,q}(A)\le Ch_p^w(A)$ since $q<p$. If $1<p\le 2$, Lemma \[Lem1.3\] gives that $h_p^w(A)\le Ch_{p,q}(A)$ and the required conclusion follows from Theorem \[equiv\]. Assume therefore that $p>2$ and that $T_A$ maps $L_p\to L_q(\ell_{\infty}^M) $ with norm $h_{p,q}(A)$. Fix $f$ with $\|f\|_{L_1}=1$ and use the Calderón-Zygmund decomposition of Theorem \[equiv\], to obtain (\[lou0\]) as before, but instead of (\[lou1\]) the estimate $$\label{lou1-1}
\|T_A(f-f\chi_E+g)\|_{L_q }\le h_{p,q} (A) 3^{1/p'}
(\gamma\lambda) ^{1/p'},$$ which implies $$\label{lou2-2}
\mathbb P(\|T_A(f-f\chi_E+g)\|_{\ell_\infty^M}> \lambda /2) \le
\frac{2^q}{ \lambda^q} ( h_{p,q} (A) )^q 3^{q/p'} (\gamma \lambda)^{q/p'}.$$ Selecting $\gamma= h_{p,q}(A)^{-s}\lambda^{s-1}$ with $\frac1s=\frac1{p'}+\frac1q$ and combining with (\[lou0\]) we obtain $$\label{lou3-3}
\lambda \, \mathbb P(\|T_Af\|_{\ell_\infty^M}>
\lambda)^{\frac1s}\le C h_{p,q}(A).$$ This says that $T_A$ maps $L_1$ into $ L_{s,\infty}(\ell_{\infty}^M)$ with norm at most $Ch_{p,q}(A)$, in particular that $T_A$ maps $L_1$ into $L_t(\ell_{\infty}^M)$ as long as $0<t<s$. Lemma \[Lem1.3\] gives that $T_A$ maps $L_p $ into $L_{2,\infty}(\ell_{\infty}^M)$ and also $L_1 $ into $L_{2,\infty}(\ell_{\infty}^M)$ with norms at most a multiple of $h_{p,q}(A)$. By interpolation it follows that $T_A$ maps $L_r $ into $L_{2,\infty}(\ell_{\infty}^M) \subset
L_{r,\infty}(\ell_{\infty}^M)$ for $1\le r\le 2.$ We conclude that $h_r^w(A)\le Ch_{p,q}(A)$ for $1< r< 2$ but since $h_r^w(A)$ is comparable to $h_p^w(A)$, we finally obtain $h_p^w(A)\le Ch_{p,q}(A)$. Since the converse inequality is always valid when $q<p$, we apply Theorem \[equiv\] to conclude the proof.
We next prove the elementary observation for $1<p<\infty,$ that $h(A)$ remains unchanged when interpolating extra columns or extra rows of zeros.
\[zeros\] Let $A$ be a complex $M\times N$ matrix and $(m_r)_{r=1}^M,
(n_s)_{s=1}^N$ be two increasing finite sequences of natural numbers. Suppose $M_1\ge m_M$ and $N_1\ge n_N.$ Let $B=(b_{jk})$ be the $M_1\times N_1$-matrix defined by $b_{jk}=a_{rs}$ when $j=m_r$ and $k=n_s$, and $b_{jk}=0$ otherwise. Then $h(A)= h(B).$
Interpolating extra rows of zeros is trivial, so we can assume $m_r=m$ for all $r.$ For the case of columns, we only need to show that $h(B)\le h(A).$ We may suppose that $\Omega$ is a finite set with $2^{N_1}$ points and that $(\Sigma_k)_{k=0}^{N_1}$ is a finite dyadic filtration of $\Omega$. It is then possible to write $\Omega=\Omega_1\times \Omega_2$ where $|\Omega_1|=2^{N_1-N }$ and $|\Omega_2|=2^{N }$, and find a dyadic filtration $(\Sigma_k^{(1)})_{k=0}^{N_1-N}$ of $\Omega_1$ and a dyadic filtration $(\Sigma_k^{(2)})_{k=0}^{N}$ of $\Omega_2$ so that $\Sigma_k^{(1)}\times\Sigma_k^{(2)}=\Sigma_{n_k},$ for $0\le k\le N$ and $\Sigma_{k+1}^{(1)}\times \Sigma_k^{(2)}=\Sigma_{n_{k+1}-1}$ for $0\le
k\le N-1.$ Then for $f\in L_2(\Omega_1\times\Omega_2)$ let $g=\sum_{k=1}^N \Delta_kf$ and note that $$\Delta_{n_k}f(\omega_1,\omega_2)=\Delta_k^{(2)}g_{\omega_1}(\omega_2),$$ where $g_{\omega_1}(\omega_2)=g(\omega_1,\omega_2).$ Hence $$\int_{\Omega_2}
\sup_j \Big|\sum_{k=1}^Na_{j,n_k}\Delta_{n_k}f(\omega_1,\omega_2)
\Big|^2d\omega_2
\le h_p(A)\int_{\Omega_2}|g(\omega_1,\omega_2)|^2d\omega_2.$$ Integrating over $\Omega_1$ gives $$\big\|\sup_j|\sum_{k=1}^Na_{j,n_k}\Delta_{n_k}f|
\big\|_{L_2} \le h_p(A)\|g\|_{L_2}\le
h_p(A)\|f\|_{L_2}.$$ This completes the proof.
We can now extend our definitions, replacing dyadic filtrations by $2^n$-adic filtrations:
\[filtration\] Suppose $n\in\mathbb N$ and $1<p<\infty$. Then there is a constant $C(p,n)$ with the following property. Let $(\Omega,\Sigma,\mathbb P)$ be a probability space and suppose $(\Sigma_k)_{k=0}^{\infty}$ is a $2^n$-adic filtration. Let $A$ be any $M\times N$ matrix and let $h_p(A;n)$ be the least constant so that $$\big\|\sup_j|\sum_{k=1}^Na_{jk}\Delta_kf|\big\|_{L_p}\le h_p(A;n)\|f\|_{L_p} ,$$ and $h_p^w(A;n)$ be the least constant so that $$\big\|\sup_j|\sum_{k=1}^Na_{jk}\Delta_kf|\big\|_{L_{p,\infty}}\le
h^w_p(A;n)\|f\|_{L_p}.$$ Then $h_p^w(A)\!\le\! h_p^w(A;n),\ h_p(A)\!\le\! h_p(A;n)$, and $h_p^w(A;n)\!\le\! h_p(A;n)\!\le\! Ch(A).$
This is essentially trivial; we need only to prove that $h_p(A;n)\le Ch(A).$ To do this note that $h_p(A;n)=h_p(B)$ where $B$ is obtained from $A$ by repeating each column $n$ times. The proposition follows then by the triangle law from Lemma \[zeros\].
Estimates for $h(A)$ {#estimates}
====================
We next turn to the problem of estimating $h(A).$ We shall assume that $(\Omega,\mathbb P)$ is a fixed probability space with a dyadic filtration $(\Sigma_k)_{k=0}^{\infty}.$ Our first estimate is trivial.
\[bv\] There is a constant $C$ so that for any $M\times N$ matrix $A=(a_{jk})$ we have $$h(A)\le C \sup_{1\le j\le M}\sum_{k=0}^N
|a_{jk}-a_{j,k+1}|,$$ where we set $a_{j0}=a_{j,N+1}=0$ for all $1\le j\le M.$
Suppose $f\in L_2.$ Summation by parts gives $$\sum_{k=1}^Na_{jk}\Delta_kf = \sum_{k=0}^N
(a_{jk}-a_{j,k+1}){\mathcal E_k}f,$$ thus $$|\sum_{k=1}^Na_{jk}\Delta_kf|\le (\sup_{1\le j\le M} \sum_{k=0}^N
|a_{jk}-a_{j,k+1}|)\sup_k|{\mathcal E}_kf|,$$ and the result follows because of the maximal estimate $$\|\sup_k|{\mathcal E}_kf|\|_{L_2}\le C\|f\|_{L_2},$$ proved in [@garsia].
We next turn to the problem of getting a more delicate estimate. To do this we need the theory of a certain Lorentz space. Let $w=(w_k)_{k=1}^{\infty}$ be a decreasing sequence of positive real numbers. We will consider the following two conditions on $w:$ $$\label{cdn1}
\exists C>0,\,\,\,\exists\theta>0,\qquad w_k\le
C\Big(\frac{\log(j+1)}{\log(k+1)}\Big)^{\theta}w_j \quad\text{when }\,\, 1\le
j\le k,$$ (where throughout this paper $\log$ denotes the logarithm with base $2$) and $$\label{cdn2}
\sum_{k=1}^{\infty}\frac{w_k}{k}<\infty.$$ Roughly speaking (\[cdn1\]) means that $w_k$ decays logarithmically while (\[cdn2\]) implies that it decays reasonably fast. Note that $w_k=(\log(k+1))^{-\theta}$ satisfies (\[cdn1\]) if $\theta>0$ and (\[cdn2\]) if $\theta>1.$ The sequence $w_k=(\log(k+1))^{-1}(\log\log(k+2))^{-\theta}$ satisfies both (\[cdn1\]) and (\[cdn2\]) when $\theta>1.$
Now let $d=d(w,1)$ be the Lorentz sequence space of all complex sequences $\mathbf u=(u_k)_{k\in\mathbb Z}$ such that $$\|\mathbf u\|_{d}=\sup_{\pi}\sum_{k\in\mathbb Z}w_{\pi(k)}|u_k|
<\infty$$ where the supremum is taken over all one-one maps $\pi:\mathbb
Z \to\mathbb N.$ The dual of $d(w,1)$ can be naturally identified as the space $d^*=d^*(w,1)$ consisting of all sequences $(v_k)_{k\in\mathbb Z}$ so that $$\sup_{k\in\mathbb N}
\frac{ v_1^*+\cdots+v_k^* }{w_1+\cdots+w_k}=\|\mathbf
v\|_{d^*}<\infty$$ where $(v_k^*)_{k=1}^{\infty}$ is the decreasing rearrangement of $(|v_k|)_{k\in\mathbb Z}.$ We refer to [@LT1] p. 175 for properties of Lorentz spaces. Note that under condition (\[cdn1\]), $d(w,1)$ is also an Orlicz sequence space (see [@LT1] p. 176).
The following Lemma is surely well-known to specialists, but we include a proof.
\[cotype\] Under condition (\[cdn1\]), the Lorentz space $d(w,1)$ has cotype two.
By combining Proposition 1.f.3 (p.82) and Theorem 1.f.7 (p.84) of [@LT2] one sees that it is only necessary to show that $d(w,1)$ has a lower $q$-estimate for some $q<2.$ To do this observe that if $\mathbf v_1,\cdots,\mathbf v_N$ are disjointly supported sequences, then $$\|\sum_{j=1}^N\mathbf v_j\|_{d}\ge
\inf_{k\ge 1}\frac{w_k}{w_{kN}}\sum_{j=1}^N\|\mathbf v_j\|_{d}.$$ Hence $$\sum_{j=1}^N\|\mathbf v_j\|_{d(w,1)}\le C(\log
(N+1))^{\theta}\|\sum_{j=1}^N\mathbf v_j\|_{d}.$$ Now suppose $1<q<2$ and $\|\sum\mathbf v_j\|_d=1.$ Then for each $s\in \mathbb N$, let $m_s$ be the number of $j$ so that $2^{-s}<\|\mathbf v_k\|_d \le 2^{-s+1} .$ Then $$m_s2^{-s} \le C(\log (m_s+1))^{\theta}.$$ This in turn implies that $$m_s^{1-\rho} \le C2^s$$ where $\rho>0$ is chosen so that $(1-\rho)^{-1}<q.$ Then we obtain an estimate $$\sum_{j=1}^N\|\mathbf v_j\|_d^q\le C\sum_{s=1}^{\infty}m_s2^{-sq}\le
C'.$$ This establishes a lower $q$-estimate.
The norms $\|\cdot\|_d$ and $\|\cdot\|_{d^*}$ are of course defined for finite sequences with $M$ elements and thus can be thought as norms on $\Bbb C^M.$ We denote these spaces $d(w,1)^{(M)}$ and $d^*(w,1)^{(M)}.$
\[changesign\] If $(w_n)$ satisfies both (\[cdn1\]) and (\[cdn2\]) then given $2<p<\infty$ there is a constant $C$ so that for any sequence $\epsilon_k=\pm1$ and any $M,N\in\mathbb N$ we have the estimate $$\Big(\mathbb E\big( \big\|\sum_{k=1}^N\epsilon_k\Delta_k\mathbf
f\big\|_{\ell_{\infty}}^2\big) \Big)^{\frac12}\le C
\big(\mathbb E \big(\|\mathbf f\|_{d^*} ^p\big)
\big)^{\frac1p},$$ for any $\mathbf f\in L_p(\Omega;d^*(w,1)^{(M)}).$
We start by using an argument due to Muckenhoupt [@muckenhoupt], see also [@Torchinsky]. For any fixed $\epsilon_1,\ldots,\epsilon_N$ let $S=\sum_{k=1}^N\epsilon_k\Delta_k.$ Now fix $f\in L_{\infty}.$ Then by a result of Burkholder [@burkholder], $\|S\|_{L_p\to L_p}=p-1$ if $2\le p<\infty.$ Then for any $\alpha>0$ we have $$\label{ghg}
\mathbb E(\cosh (\alpha |S f|)) \le
1+\sum_{m=1}^{\infty}\frac{\alpha^{2m}}{(2m)!}(2m-1)^{2m}\|f\|_{L_{2m}}^{2m}.$$ Since $\|f\|_{L_{2m}}^{2m}\le \|f\|_{L_2}^2
\|f\|_{L_{\infty}}^{2m-2}$ and since for $m\ge 1$ we have $$\frac{(2m-1)^{2m}}{(2m)!} \le \frac{(2m )^{2m}}{(2m)!} \le e^{2m} ,$$ it follows from (\[ghg\]) that $$\mathbb E(\cosh (\alpha |S f|)-1) \le
(\alpha e)^2\|f\|_{L_2}^2\sum_{k=0}^{\infty}(\alpha e)^{2k}
\|f\|_{L_\infty}^{2k}.$$ In particular if $\alpha e\|f\|_{\infty}\le \frac12$ we have $$\label{firstestimate}
\mathbb E(\cosh (\alpha |S f|)-1) \le 2e^2
\alpha^2\|f\|_{L_2}^2.$$
At this point we return to the Lorentz space $d(w,1)$. Let us define $\gamma_0=0$, $\gamma_1=1$, and $\gamma_k=2^{2^{k-2}}$ for $k\ge 2.$ Let $W_k=w_{\gamma_k}.$ It will be convenient to normalize condition (\[cdn2\]) so that we have $$\label{cdn20}
\sum_{k=1}^{\infty}\gamma_{k}W_k=1.$$ We also note that (\[cdn1\]) implies the existence of a constant $C$ so that we have $$\label{cdn10}|w_1+\cdots+w_k|\le Ckw_k$$ for $k\ge 1.$
Now suppose $\mathbf f=(f_j)_{j=1}^M\in L_{\infty}(\Omega;\mathbb C^M).$ Suppose that $\mathbf f$ is supported on a measurable set $E$ and satisfies $\|\mathbf f(\omega)\|_{d^*}\le 1$ everywhere. Then we can define a measurable map $\pi$ from $\Omega$ into the set of permutations of $\{1,2,\ldots,M\}$ so that $|f_{\pi(\omega)(1)}(\omega)|\ge
|f_{\pi(\omega)(2)}(\omega)|\ge\cdots \ge |f_{\pi(\omega)(M)}(\omega)|$ for all $\omega \in \Omega .$ Thus $$|f_{\pi(\omega)(j)}(\omega)|\le Cw_j$$ for all $1\le j\le M.$ Let $E_{jk}=\{\omega\in E:\ \pi(\omega)(k)=j\} $ when $j,k\in \{1,\dots , M\}$ and $E_{jk}=\emptyset$ otherwise. Now for $1\le j\le M$ and $l=1,2,3,\dots$, let $$f_j^{(l)}=\sum_{k=\gamma_{l-1}}^{\gamma_l-1}f_j\chi_{E_{jk}}$$ so that $f_j=\sum_{l=1}^\infty f_j^{(l)}$. If $0<\alpha e\le \frac1{2C }$ we can estimate $$\begin{align*}
\mathbb E(\cosh (\alpha |S f_j|)-1)
&=\mathbb E
\big(\cosh \big(\big|\sum_{l=1}^{\infty}\alpha Sf_j^{(l)}\big|\big)-1\big)\\
&\le \mathbb E\Big(\max_{l\ge 1}\big(\cosh (\alpha
\gamma_l^{-1}W_l^{-1}|Sf_j^{(l)}| )-1\big)\Big)\\ &\le e^2\alpha ^2
\sum_{l=1}^{\infty}\gamma_l^{-2}W_l^{-2}\|f_j^{(l)}\|_{L_2}^2,
\end{align*}$$ in view of (\[firstestimate\]) since $\|f_j^{(l)}\|_{L_\infty}\le CW_l$ and $\alpha
\gamma_l^{-1}W_l^{-1}\|f_j^{(l)}\|_{L_\infty}\le \frac12. $ Thus $$\mathbb E(\cosh (\alpha |S f_j|)-1) \le
e^2 C^2\alpha^2\sum_{l=1}^{\infty}\gamma_l^{-2}
\sum_{k=\gamma_{l-1}}^{\gamma_l-1} \mathbb P(E_{jk}).$$ It follows that $$\mathbb E(\cosh (\alpha \|S\mathbf f\|_{\ell_{\infty}})-1) \le
e^2C^2\alpha^2\sum_{j=1}^M\sum_{l=1}^{\infty}\gamma_l^{-2}
\sum_{k=\gamma_{l-1}}^{\gamma_l-1} \mathbb P(E_{jk}).$$ Note that for each $k\in\mathbb N,$ $\sum_{j=1}^M\mathbb
P(E_{jk})\le
\mathbb P(E).$ Hence we obtain that if $\mathbf f$ is supported on $E$ with $\|\mathbf f(\omega)\|_{d^*}\le 1$ everywhere and $\alpha e<\frac1{2C},$ then $$\label{firststep}
\mathbb E(\cosh (\alpha \|S\mathbf
f\|_{\ell_{\infty}})-1) \le e^2 C^2\alpha^2
\sum_{l=1}^{\infty}\gamma_l^{-1}\mathbb P(E)=C_1\alpha^2\mathbb
P(E)$$ for a suitable constant $C_1.$ Let us next refine (\[firststep\]). For $n\ge 0,$ let $$G_n=\{\omega\in E: \ 4^{-n-1}< \|\mathbf(\omega)\|_{d^*}\le 4^{-n}\}.$$ Then by (\[firststep\]) we have if $\alpha<(4Ce)^{-1}$ $$\mathbb E(\cosh (2^{n+1}\alpha \|S (\mathbf
f\chi_{G_n})\|_{\ell_{\infty}})-1) \le C_1\alpha^2 4^{-n} \mathbb P(G_n)$$ and as $$\mathbb E(\cosh (\alpha \|S\mathbf f\|_{\ell_{\infty}})-1)\le
\mathbb E\Big(\sup_{n\ge 0}\big(\cosh (2^{n+1}\alpha\| S(\mathbf
f\chi_{G_n})\|_{\ell_{\infty}})-1\big)\Big),$$ we obtain, under the assumptions $\|\mathbf f(\omega)\|_{d^*}\!\le\! 1$ everywhere and $\alpha\!<\! (4C)^{-1}$, $$\label{2step}
\mathbb E(\cosh (\alpha \|S\mathbf f\|_{\ell_{\infty}})-1)\le
C_1\alpha^2\sum_{n=0}^{\infty}4^{-n}\mathbb P(G_n)\le C_2\mathbb
E(\|\mathbf f\|_{d^*}).$$
If we use a fixed value of $\alpha$ and the estimate $x^2\le 2(\cosh
x-1)$ we find that $$\mathbb E(\|S\mathbf f\|_{\ell_{\infty}}^2)\le C_3\mathbb E (\|\mathbf
f\|_{d^*})$$ if $\|\|\mathbf f\|_{d^*}\|_{\infty}\le 1$. This in turn gives us for every $\mathbf f\in L_{\infty}(\Omega; d^*(w,1)^{(M)})$ $$\label{3step}
\mathbb E(\|S\mathbf f\|_{\ell_{\infty}}^2)\le C_3 \|\|\mathbf
f\|_{d^*}\|_{\infty}\, \mathbb E(\|\mathbf f\|_{d^*}).$$
Now let $2<p<\infty$ and fix $\mathbf f$ with $\mathbb E(\|\mathbf f\|_{d^*}^p)= 1$. We set $E_0=\{\|\mathbf f\|_{d^*}\le 1\}$ and $E_n=\{2^{n-1}<\|\mathbf
f\|_{d^*}\le 2^n\}$ for $n\ge 1.$ Applying (\[3step\]) we obtain $$\begin{aligned}
(\mathbb E(\|S\mathbf f\|_{\ell_{\infty}}^2))^{\frac12} \le &
\big( C_3 \sum_{n=0}^{\infty} 2^n
\mathbb P(E_n) \mathbb E ( \|\mathbf f\|_{d^*} )
\big)^{\frac12}\le
C_3^{\frac12}\sum_{n=0}^{\infty}2^{\frac{n}{2}}\mathbb P(E_n)^{\frac12} \\
\le & C_3^{\frac12}\big(\sum_{n=0}^{\infty}
2^{(2-p)n} \big)^{\frac12}
\big(\sum_{n=0}^{\infty} 2^{np}\mathbb
P(E_n) \big)^{\frac12} \le C_4 ,
\end{aligned}$$ which completes the proof under the assumption $\mathbb E(\|\mathbf f\|_{d^*}^p)= 1$. The general case follows by scaling.
We now establish our main estimate for $h(A).$
\[lorentz\] Let $w=(w_n)_{n=1}^{\infty}$ be a sequence satisfying (\[cdn1\]) and (\[cdn2\]). Then there is a constant $C$ so that for any $M\times N$ matrix $A=(a_{kj})_{j,k}$ we have $$h(A) \le C \max_{1\le k\le N}\|\mathbf a_k\|_{d^*}$$ where $\mathbf a_k=(a_{kj})_{j=1}^M.$ In particular we have $$h(A) \le C\max_{j,k}\frac{|a_{jk}|}{w_{|j-k|+1}}.$$
We suppose $p>2$ and that $A$ is a matrix satisfying $\max_{1\le k\le N}\|\mathbf a_k\|_{d^*}\le 1.$ Consider the operator $T_A:L_p(\Omega)\to L_2(\Omega;\ell_{\infty}^M).$ The adjoint operator is $T_A^*:\,\, L_2(\Omega;\ell_1^M)\to L_{p'}(\Omega)$ given by $$T_A^*(\mathbf f)=\sum_{k=1}^N\langle \Delta_k\mathbf f,\mathbf
a_k\rangle.$$ The dual statement of the result in Proposition \[changesign\] gives that for any sequence of $\pm 1$’s, $\epsilon_1,\ldots,\epsilon_N$ we have the estimate $$\label{signs}
\big(\mathbb E \big(\|\sum_{k=1}^N\epsilon_k\Delta_k\mathbf
f\|_{d}^{p'}\big)\big)^{\frac1{p'}}\le
C(\mathbb E(\|\mathbf f\|_{\ell_1}^2))^{\frac12}$$ where $C$ depends only on $(w_n).$ Now let $\epsilon_1,\ldots\epsilon_N$ be a sequence of independent Bernoulli random variables on some probability space $(\Omega',\mathbb P').$ We use $\mathbb E'$ to denote expectations on $\Omega'.$ Using Lemma \[cotype\] we obtain $$\begin{aligned}
(\mathbb E (\|T_A^*\mathbf f\|_d^{p'}))^{\frac1{p'}}
&\le C_0\big(\mathbb
E \big(\sum_{k=1}^N |\langle \Delta_k\mathbf f,\mathbf
a_k\rangle|^2 \big)^{\frac {p'}2}\big)^{\frac1{p'}}\\
&\le C_0 \big(\mathbb E \big(\sum_{k=1}^N\|\Delta_k\mathbf
f\|_d^2\big)^{\frac {p'}2}\big)^{\frac1{p'}} \\
&\le C_1 \big(\mathbb E\mathbb E'\big( \|\sum_{k=1}^N\epsilon_k\Delta_k\mathbf
f\|_d^{p'} \big)\big)^{\frac1{p'}}\\
&\le C_2 (\mathbb E\|\mathbf
f\|_{\ell_1}^{2})^{\frac1{2}}.\end{aligned}$$ This gives $h_{p,2}(A)\le C_2$ which completes the proof by using Theorem \[mixed\].
[**Remark.**]{} Theorem \[lorentz\] implies that given any $\theta>1$ there is a constant $C_{\theta}$ so that $$\label{44444}
h(A)\le C_{\theta}$$ whenever $A =(a_{kj})_{j,k}$ is a matrix satisfying $$\label{logcondition}
|a_{jk}|\le 2 (\log (2+|j-k|))^{-\theta}.$$ We show that this is not the case when $0<\theta<\frac12$. Let $N$ be any natural number and define $A=(a_{jk})$ to be a $2^N\times N$ matrix given by $a_{jk}=b_{jk}N^{-\theta}$, where $b_{jk}= \pm1 $ and the set $(b_{jk})_{j=1}^{2^N}$ runs through all $2^N$ choices of signs. Choose $f$ real so that $|\Delta_k
f|=1$ for $1\le k\le N.$ Then $\|f\|_{L_2}=\sqrt N.$ On the other hand $$\max_{1\le j\le 2^N}|\sum_{k=1}^{N}a_{jk}\Delta_kf|=N^{1-\theta}\,
\chi_{\Omega},$$ which implies that $h(A)\ge N^{\frac12-\theta}$. However $$|a_{jk}|\le N^{-\theta} \le 2(N+1)^{-\theta} \le 2 (\log (2+|j-k| ))^{-\theta}$$ but $h(A)\ge N^{\frac12-\theta}\to \infty$ as $N\to \infty$. Thus (\[44444\]) fails when $0<\theta<\frac12$.
The harmonic version of the maximal operator {#s-harmonic}
============================================
0
We shall now fix $n\in\mathbb N$ and work with $\mathbb R^n.$ Let $\mathcal D_0$ be the collection of all unit cubes of the form $\prod_{j=1}^n[m_j,m_j+1]$ where $m_j\in\mathbb Z$ and let $\mathcal D_k$ be the set of all cubes of the form $\prod_{j=1}^n[2^{-k}m_j,2^{-k}(m_j+1)]$ where $m_j\in\mathbb Z.$ For $k\in \mathbb Z$, let $\Sigma_k$ denote the $\sigma-$algebra generated by the dyadic cubes $\mathcal D_k$. We define the corresponding conditional expectation operators $$\mathcal E_kf=\sum_{Q\in \mathcal D_k}(\Ave_{Q}f)\chi_Q$$ for $f\in
L_1^{loc}(\mathbb R^n)$ and the martingale difference operators $\Delta_kf=\mathcal E_kf-\mathcal E_{k-1}f $ for $k\in \mathbb Z$.
Now let $A=(a_{jk})_{j,k\in\mathbb Z}$ be any infinite complex matrix. We shall call $A$ a $c_{00}-$matrix if it has only finitely many non-zero entries. For a $c_{00}-$matrix define $h_p[A;n]$ to be the least constant such that for all $f\in L_p(\mathbb R^n)$ we have $$\label{defh2}
\|\max_{j\in\mathbb Z}|\sum_{k\in\mathbb Z}a_{jk}\Delta_kf|\|_{L_p}\le
h_{p}[A;n]\|f\|_{L_p}.$$ Also let $h_p^{w}[A;n]$ be the corresponding weak-type constant, i.e. the least constant such that for all $f\in L_p(\mathbb R^n)$ we have $$\label{defhweak2}
\|\max_{j\in\mathbb Z}
|\sum_{k=1}^Na_{jk}\Delta_kf|\|_{L_{p,\infty}}\le
h_p^w[A;n]\|f\|_{L_p}.$$ The following Lemma is easily verified and we omit its proof.
\[lem3.1\] Let $h_p^w(A;n)$ and $h_p(A;n)$ be as in Proposition \[filtration\]. For any $1<p<\infty$ and any infinite $c_{00}$-matrix $A$ we have $h_p[A;n]=h_p(B;n)$ and $h_p^w[A;n]=h_p^w(B;n)$, where $B$ is any $M\times
N$ matrix of the form $b_{jk}=a_{j+r,k+s}$ for some $r,s\in\mathbb Z$ such that $a_{j+r,k+s}=0$ unless $1\le j\le M$ and $1\le k\le N$.
Now for any infinite matrix $A$ we define $$h(A)=\sup_Nh \big((a_{j-N,k-N})_{1\le j\le 2N}^{1\le k\le 2N}\big).$$ The following is an immediate consequence of Lemma \[lem3.1\] and Proposition \[filtration\].
For any $1<p<\infty$ and any $n\in\mathbb N$ there is a constant $C=C(p,N)$ so that for any infinite $c_{00}$-matrix we have $$C^{-1}h(A)\le h_p^w[A;n]\le h_p[A;n]\le Ch(A).$$
We now turn to the harmonic model of the maximal operator studied in section \[s-maximal\]. Let $\mathcal S(\mathbb R^n)$ denote the set of all Schwartz functions on $\mathbb R^n$ and for $f\in \mathcal S(\mathbb R^n)$ let $$\widehat{f}(\xi)= \int_{\mathbb R^n} f(x)e^{- 2\pi i\langle \xi , x\rangle} dx$$ denote the Fourier transform of $f$. We will denote by $f\spcheck (\xi) =\widehat{f}(-\xi)$ the inverse Fourier transform of $f$. We shall fix a radial function $\psi\in\mathcal S(\mathbb R^n)$ whose Fourier transform is real-valued and satisfies $\widehat\psi(\xi)=1$ for $|\xi|\le 1$ and $\widehat\psi(\xi) =0$ for $|\xi|\ge 2.$ We define a Schwartz function $\phi$ by setting $\widehat{\phi}(\xi)=
\widehat{\psi}(\xi)-\widehat{\psi}(2\xi)$. Then $\widehat{\phi}$ is supported in the annulus $2^{-1}\le |\xi |\le 2$. We then define $\psi_j(x)=2^{nj}\psi(2^{j}x)$ and $\phi_j(x)=2^{nj}\phi(2^{j}x) $ for $j\in\mathbb Z.$ Note that $\widehat{\phi_j}(\xi)=\widehat{\phi }(2^{-j}\xi)$ is supported in the annulus $2^{j-1}\le |\xi |\le 2^{j+1}$. We also define operators $$\widetilde S_jf=\psi_j *f \quad\text{and }\,\,\,
\widetilde\Delta _j f=\phi_j*f$$ for $f\in L_1+L_{\infty}.$ The $\widetilde\Delta _j$’s are called the Littlewood-Paley operators. Now if $A=(a_{jk})_{(j,k)\in\mathbb Z^2}$ is an infinite $c_{00}$-matrix and $1<p<\infty,$ we let $\widetilde
h_p(A)$ be the least constant so that for all $f\in L_p$ we have $$\label{tildeh}
\big\|\sup_{j\in\mathbb Z}\big|\sum_{k\in\mathbb
Z}a_{jk}\widetilde\Delta_k f\big| \big\|_{L_p}\le \widetilde h_p(A)\|f\|_{L_p}.$$ We also define $\widetilde h_p^w(A)$ to be the least constant such that for all $f\in L_p$ we have $$\label{tildehw}
\big\|\sup_{j\in\mathbb Z} \big|\sum_{k\in\mathbb
Z}a_{jk}\widetilde\Delta_k f\big| \big\|_{L_{p,\infty}}\le
\widetilde h^w_p(A)\|f\|_{L_p}.$$
We now have the following.
\[translation\] Suppose $r\in\mathbb Z$. Then if $1<p<\infty$ and $A=(a_{jk})$ is any infinite $c_{00}$-matrix, then $\widetilde
h_p(A)=\widetilde h_p(B)$ and $\widetilde h_p^w(A)=\widetilde h_p^w(B)$, where $B=(b_{jk})$ and $b_{jk}=a_{j,k+r}.$
Consider the dilation operator $D_rf(x)=f(2^{-r}x).$ Then $D_r^{-1}\widetilde\Delta_kD_rf= \widetilde\Delta_{k-r}f $ and we have $$\begin{aligned}
&\big\|\sup_j\big|\sum_k a_{j,k+r}\widetilde\Delta_k f\big| \big\|_{L_p}
=\big\|\sup_j\big|\sum_ka_{jk}\widetilde\Delta_{k-r}f\big| \big\|_{L_p}\\
=&2^{-rn/p} \big\|\sup_j\big|\sum_k a_{jk}\widetilde\Delta_kD_rf\big|
\big\|_{L_p}
\le 2^{-rn/p}h_p(A)\|D_rf\|_{L_p}=h_p(A)\|f\|_{L_p},\end{aligned}$$ which implies $\widetilde h_p(B)\le \widetilde h_p(A).$ Likewise we obtain $\widetilde h_p(A)\le \widetilde h_p(B).$ The corresponding result for the weak type constants follows similarly.
Next we prove that the Littlewood-Paley operators $\widetilde \Delta_j$ and the martingale difference operators $\Delta_k$ are essentially orthogonal on $L_2$ when $k\neq j$.
\[LP-martingale\] There exists a constant $C$ so that for every $k,j$ in $\mathbb Z$ we have the following estimate on the operator norm of $\Delta_j\widetilde \Delta_k:\,
L_2(\mathbb R^n)\to L_2(\mathbb R^n)$ $$\label{4444}
\|\Delta_k\widetilde \Delta_j\|_{L_2\to L_2}\le C2^{-|j-k|}.$$
By a simple dilation argument it suffices to prove (\[4444\]) when $k=0$. In this case we have the estimate $$\begin{aligned}
&\|\Delta_0\widetilde \Delta_j\|_{L_2\to L_2} =
\|\mathcal E_0\widetilde \Delta_j-\mathcal E_{-1}
\widetilde \Delta_j \|_{L_2\to L_2} \\
\le &\|\mathcal E_0\widetilde \Delta_j- \widetilde \Delta_j \|_{L_2\to
L_2} + \|\mathcal E_{-1}\widetilde \Delta_j- \widetilde \Delta_j \|_{L_2\to
L_2}\end{aligned}$$ and also by the self-adjointness of the $\Delta_k$’s and $\widetilde\Delta_j$’s we have $$\begin{aligned}
&\|\Delta_0\widetilde \Delta_j\|_{L_2\to L_2} =\|\widetilde
\Delta_j\Delta_0 \|_{L_2\to L_2} =
\|\widetilde \Delta_j\mathcal E_0-
\widetilde \Delta_j \mathcal E_{-1}\|_{L_2\to L_2} \\
\le &\|\widetilde \Delta_j\mathcal E_0- \mathcal E_0 \|_{L_2\to
L_2} + \|\widetilde \Delta_j\mathcal E_{-1}- \mathcal E_{0} \|_{L_2\to
L_2}.\end{aligned}$$ The required estimate (\[4444\]) (when $k=0$) will be a consequence of the pair of inequalities $$\begin{aligned}
& \|\mathcal E_0\widetilde \Delta_j- \widetilde \Delta_j \|_{L_2\to
L_2} + \|\mathcal E_{-1}\widetilde \Delta_j- \widetilde \Delta_j \|_{L_2\to
L_2} \le C 2^{j} &\text{when $j\le 0$,}\label{c1} \\
&\|\widetilde \Delta_j\mathcal E_0- \mathcal E_0 \|_{L_2\to
L_2} + \|\widetilde \Delta_j\mathcal E_{-1}- \mathcal E_{0} \|_{L_2\to
L_2}\le C 2^{-j} &\text{when $j\ge 0$.} \label{c2}\end{aligned}$$ We start by proving (\[c1\]). We only consider the term $\mathcal E_0\widetilde \Delta_j- \widetilde \Delta_j$ since the term $\mathcal E_{-1}\widetilde \Delta_j- \widetilde \Delta_j$ is similar. Let $f\in L_2(\mathbb R^n)$. Then $$\begin{aligned}
&\|\mathcal E_0\widetilde \Delta_j f- \widetilde \Delta_jf \|_{L_2}^2
= \sum_{Q\in \mathcal D_0} \|f*\phi_j -\Ave_Q (f*\phi_j) \|_{L_2(Q)}^2\\
\le & \sum_{Q\in \mathcal D_0} \int_Q\int_Q
|(f*\phi_j)(x)-(f*\phi_j)(t)|^2\, dt\, dx\\
\le & \sum_{Q\in \mathcal D_0} \int_Q\int_Q \Big(\int_{3Q} |f(y)|
|\phi_j (x-y)| \,dy \Big)^2 \, dt\, dx \\
&\quad\quad\,\, + \sum_{Q\in \mathcal D_0}
\int_Q\int_Q \Big(\int_{3Q} |f(y)| |\phi_j (t-y)| \,dy \Big)^2 \, dt\, dx \\
&\quad\quad\,\, +\sum_{Q\in \mathcal D_0}
\int_Q\int_Q \Big(\int_{(3Q)^c} |f(y)| 2^{jn+j} |\nabla \phi(2^j(\xi_{x,t}-y))|
\,dy \Big)^2 \, dt\, dx ,\end{aligned}$$ where $\xi_{x,t}$ lies on the line segment between $x$ and $t$. It is now easy to see that the sum of the last three expressions above is bounded by $$C 2^{2jn} \sum_{Q\in \mathcal D_0} \int_{3Q} |f(y)|^2\, dy + C_M 2^{2j}
\sum_{Q\in \mathcal D_0} \int_Q \Big(\int_{\mathbb R^n} \frac{2^{jn}|f(y)|\,
dy}{(1+ 2^j|x-y|)^M} \Big)^2dx$$ which is clearly controlled by $C 2^{2j} \|f\|_{L_2}^2$. This estimate is useful when $j\le 0$.
We now turn to the proof of (\[c2\]). Since $\widetilde \Delta_j$ is the difference of two $\widetilde S_j$’s, it will suffice to prove (\[c2\]) where $\widetilde \Delta_j$ is replaced by $\widetilde S_j$. We only work with the term $\widetilde S_j\mathcal E_0- \mathcal E_0 $ since the other term can be treated similarly. We have $$\begin{aligned}
&\|\widetilde S_j\mathcal E_0 f- \mathcal E_0f\|_{L_2}^2 =
\big\|\sum_{Q\in \mathcal D_0} (\Ave_Q f) \, (\psi_j *\chi_Q-\chi_Q)
\big\|_{L_2}^2
\\
\le &
2\big\|\sum_{Q\in \mathcal D_0} (\Ave_Q f) \, (\psi_j *\chi_Q-\chi_Q)
\chi_{3Q}\big\|_{L_2}^2 +2\big\|\sum_{Q\in \mathcal D_0} (\Ave_Q f) \,
(\psi_j *\chi_Q )
\chi_{(3Q)^c}\big\|_{L_2}^2.\end{aligned}$$ Since the functions appearing inside the sum in the first term above have supports with bounded overlap we obtain $$\big\|\sum_{Q\in \mathcal D_0} (\Ave_Q f) \, (\psi_j *\chi_Q-\chi_Q)
\chi_{3Q}\big\|_{L_2}^2 \le C\sum_{Q\in \mathcal D_0} (\Ave_Q |f|)^2
\|\psi_j *\chi_Q-\chi_Q\|_{L_2}^2,$$ and the crucial observation is that $$\|\psi_j *\chi_Q-\chi_Q\|_{L_2} \le C 2^{-j} ,$$ which can be easily checked using the Fourier transform. Therefore we obtain $$\big\|\sum_{Q\in \mathcal D_0} (\Ave_Q f) \, (\psi_j *\chi_Q-\chi_Q)
\chi_{3Q}\big \|_{L_2}^2 \le C2^{-2j}\|f\|_{L_2}^2,$$ and the required conclusion will be proved if we can show that $$\label{55}
\big \|\sum_{Q\in \mathcal D_0} (\Ave_Q f) \,
(\psi_j *\chi_Q )
\chi_{(3Q)^c}\big \|_{L_2}^2 \le C2^{-2j} \|f\|_{L_2}^2.$$ We prove (\[55\]) by using a purely size estimate. Let $c_Q$ be the center of the dyadic cube $Q$. For $x\notin 3Q$ we have the easy estimate $$|(\psi_j*\chi_Q)(x)|\le \frac{C_M 2^{jn}}{ (1+2^j|x-c_Q|)^{M}} \le
\frac{C_M 2^{jn}}{ (1+2^j )^{M/2}}\frac{1}{ (1+ |x-c_Q|)^{M/2}}$$ since both $2^j\ge 1, |x-c_Q|\ge 1$. We now control the left hand side of (\[55\]) by $$\begin{aligned}
&2^{j(2n-M)}
\sum_{Q\in \mathcal D_0} \sum_{Q'\in \mathcal D_0} (\Ave_Q |f|)(\Ave_{Q'} |f|)
\int_{\mathbb R^n} \!\frac{C_M\,\,dx}{ (1\!+\! |x\! -\! c_Q|)^{\frac{M}2}
(1\!+\! |x\!-\! c_{Q'}|)^{\frac{M}2}} \\
\le &2^{j(2n-M)}
\sum_{Q\in \mathcal D_0} \sum_{Q'\in \mathcal D_0} \frac{
(\Ave\limits_Q |f|)(\Ave\limits_{Q'} |f|)}{ (1+|c_Q-c_{Q'}|)^{\frac{M}4}}
\int_{\mathbb R^n} \!\frac{C_M\,\,dx}{ (1\!+\! |x\! -\! c_Q|)^{\frac{M}4}
(1\!+\! |x\!-\! c_{Q'}|)^{\frac{M}4}}\\
\le &2^{j(2n-M)}
\sum_{Q\in \mathcal D_0} \sum_{Q'\in \mathcal D_0} \frac{
C_M'}{ (1+|c_Q-c_{Q'}|)^{\frac{M}4}} \bigg( \int_Q |f(y)|^2\, dy+
\int_{Q'} |f(y)|^2\, dy\bigg) \\
\le &C_M'' 2^{j(2n-M)} \sum_{Q\in \mathcal D_0}\int_Q |f(y)|^2\, dy
= C_M'' 2^{j(2n-M)}\|f\|_{L_2}^2.\end{aligned}$$ By taking $M$ large enough we obtain (\[55\]) and thus (\[c2\]).
We have the following result relating $h(A)$ and $\widetilde h_p(A)$.
\[equivalence\] For every $1<p<\infty$, there is a constant $C$ depending only on $\psi$ and $p$ so that for any $c_{00}-$matrix $A$ we have $$\frac1C h(A)\le\widetilde h^w_p(A)\le\widetilde h_p(A)\le Ch (A).$$
Consider the operators $V_r,\ r\in\mathbb Z$ defined by $$V_r=\sum_{j\in\mathbb Z}\Delta_j\widetilde\Delta _{j+r}.$$ Then $$V_rV_r^* =\sum_{j,k} \Delta_j\widetilde\Delta_{j+r}\widetilde\Delta
_{k+r}\Delta_k = \sum_{|j-k|\le 1} \Delta_j
\widetilde\Delta_{j+r}\widetilde\Delta_{k+r}\Delta_k.$$ Hence by splitting into 3 pieces and using Proposition \[LP-martingale\] we obtain the estimate $$\|V_r\|_{L_2\to L_2} \le C2^{-|r|}.$$
Next pick $q$ so that $1<q<\infty$ and $\frac1p=\frac{\theta}q+\frac{1-\theta}2$ where $0<\theta<1.$ Let $(\epsilon_j)_{j\in\mathbb Z}$ be a sequence of independent Bernoulli random variables on some probability space $(\Omega,\mathbb P).$ Then for $f\in L_q(\Omega) $ we have $$V_rf =\int_{\Omega} \sum_{j\in\mathbb Z}\sum_{k\in\mathbb Z}
\epsilon_j(\omega)\epsilon_{k-r}(\omega)\Delta_j\widetilde\Delta_kf\,
d\mathbb P.$$ Averaging now gives $$\|V_rf\|_{L_q}\le (\max_{\omega}\|\sum_{j\in\mathbb
Z}\epsilon_j(\omega)\Delta_j\|_{L_q\to L_q})(\max_{\omega}\|\sum_{k\in\mathbb Z}
\epsilon_{k-r}(\omega)\widetilde \Delta_k\|_{L_q\to L_q})\|f\|_{L_q}.$$ Hence $\|V_r\|_{L_q\to L_q}\le C$ where $C$ depends only on $q $ and $\psi$. Similarly $\|V_r^*\|_{L_q\to L_q}\le C.$ By interpolation we obtain $\|V_r\|_{L_p\to L_p},
\|V_r^*\|_{L_p\to L_p}\le C2^{-|r|(1-\theta)}.$
Finally let us write $$\begin{aligned}
\sup_{j\in \mathbb Z}\big|\sum_{k\in \mathbb Z}a_{jk}\widetilde\Delta_kf\big|
&= \sup_{j\in
\mathbb Z} \big|\sum_{k\in \mathbb Z}a_{jk} \sum_{r\in \mathbb Z}
\Delta_{k-r}\widetilde\Delta_kf\big|\\ &\le
\sum_{r\in \mathbb Z}\sup_{j\in
\mathbb Z}\big|\sum_{k\in \mathbb Z} a_{j,k+r} \Delta_{k}
\widetilde\Delta_{k+r}f\big| .\end{aligned}$$ Thus by Proposition \[filtration\], $$\|\sup_{j\in \mathbb Z}|\sum_{k\in \mathbb Z}a_{jk}
\widetilde\Delta_kf|\|_{L_p} \le
Ch(A)\sum_{r\in
\mathbb Z} \|V_rf\|_{L_p} \le Ch_p(A) \|f\|_{L_p}.$$ This shows that $\widetilde h_p(A)\le Ch(A).$
For the converse direction we use $V_r^*$ and Lemma \[translation\]. We have $$\sup_{j\in \mathbb Z} \big|\sum_{k\in \mathbb Z}a_{jk}\Delta_kf\big|
\le \sum_{r\in \mathbb Z}\sup_{j\in \mathbb Z}\big|\sum_{k\in \mathbb Z} a_{j,k+r}
\widetilde\Delta_{k}\Delta_{k+r}f\big|$$ and so $$\big\|\sup_{j\in\mathbb Z}\big|\sum_{k\in\mathbb Z}a_{jk}\Delta_kf
\big|\big\|_{L_{p,\infty}}\le C\widetilde h_p^w(A)\sum_{r\in\mathbb Z}
\|V_r^*f\|_{L_p}$$ which leads to the estimate $h(A)\le C\widetilde h_p^w(A).
$
We next extend the definition of $\widetilde h_p(A)$ to the case when $0<p\le
1$. For such $p$’s we define $\widetilde h_p(A)$ to be the least constant so that for $f\in\mathcal S $ we have $$\label{pless1}
\|\sup_{j\in\mathbb Z}|\sum_{k\in\mathbb
Z} a_{jk}\widetilde \Delta f|\|_{L_p}\le C\|f\|_{H_p}.$$ The space $H_p$ that appears on the right of (\[pless1\]) when $0<p\le 1$ is the classical real Hardy space of Fefferman and Stein [@FS] and the expression $\|\,\,\|_{H_p}$ is its quasi-norm.
\[hardy\] If $0<p<1$ then there is constant $C=C(p,\psi)$ so that $C^{-1}h(A)\le \widetilde h_p(A)\le Ch(A).$
First we show the estimate $\widetilde h_p(A)\le C h(A).$ Using the atomic characterization of $H_p$, [@coifman], we note that it suffices to get an estimate for a function $f\in \mathcal S$ supported in a cube $Q$ so that $|f(x)|\le |Q|^{-\frac1p}$ for $x\in Q$ and $\int x^{\alpha}f(x)=0$ if $|\alpha|\le N=[n(\frac1p-1)].$ It is then easy to see that if $x\notin 2Q$ $$|\sum_{k\in\mathbb Z}a_{jk}\widetilde\Delta_kf(x)|\le
C h(A)|x-c_Q|^{-n-N-1}$$ since $|a_{jk}|\le Ch(A)$ for each $j,k$. (Here $2Q$ is the cube with twice the length and the same center $c_Q$ as usually.) This gives the estimate $$\int_{\mathbb R^n\setminus 2Q}\sup_j|\sum_{k}a_{jk}\widetilde\Delta_kf(x)|^pdx\le
C^ph(A)^p.$$ On the other hand, $$\int_{2Q}
\sup_j \big|\sum_{k}a_{jk}\widetilde\Delta_kf(x)\big|^pdx\le
C|Q|^{1-\frac{p}{2}}h(A)^p\bigg(\int_Q|f(x)|^2dx\bigg)^{\frac{p}{2}}$$ and combining with the previous estimate we obtain $\widetilde h_p(A)\le Ch(A).$
Complex interpolation gives that $\widetilde
h_q(A)\le \widetilde h_2(A)^{\theta}\widetilde h_p(A)^{1-\theta}$ when $1<q<2$ and $\frac1q=\frac{1-\theta}{p}+\frac{\theta}{2}.$ Since $\widetilde h_q(A)\ge
C^{-1}h(A)$ we deduce the estimate $\widetilde h_p(A)\ge C^{-1}h(A).$
Bilinear operators {#bilinear}
==================
0
Let $\sigma$ be a bounded measurable function on $\mathbb
R^n\times\mathbb R^n$. For $f,g\in\mathcal S(\mathbb R^n)$ we define a bilinear operator $W_{\sigma}(f,g)$ with multiplier $\sigma$ by setting $$\label{bilineardef}
W_{\sigma}(f,g)(x) =\int_{\mathbb R^n}\int_{\mathbb R^n}
\sigma (\xi ,\eta)\widehat f(\xi )\widehat
g(\eta)e^{2\pi i \langle x , \xi+\eta \rangle }\, d\xi d\eta.$$ If (\[bilineardef\]) is satisfied we say that $\sigma$ is the bilinear symbol (or multiplier) of $W_\sigma$. Now suppose $1<p_1,p_2<\infty$ and let $p_0$ be defined by $\frac1{p_0}=\frac1{p_1}+\frac1{p_2}.$ We say that $W_{\sigma}$ is strongly $(p_1,p_2)-$bounded if $W_{\sigma}$ extends to a bounded bilinear operator from $L_{p_1}\times L_{p_2}\to L_{p_0}$. In this case we denote its norm by $\|W_{\sigma}\|_{L_{p_1}\times L_{p_2}\to L_{p_0}}$ (we define this be expression to be $\infty$ if $W_{\sigma}$ is not bounded). Similarly we say $W_{\sigma}$ is weakly $(p_1,p_2)-$bounded if it extends to a bounded bilinear operator from $L_{p_1}\times L_{p_2}\to L_{p_0,\infty}$ and its norm is then denoted $\|W_{\sigma}\|_{L_{p_1}\times L_{p_2}\to L_{p_0,\infty}}.$
We extend these definitions to the case $0< p_1,p_2<\infty$ by replacing the $L_p $ spaces by the corresponding Hardy spaces when $0<p_j\le 1$. In the definition below we set $H_p=L_p$ for $1<p<\infty$. Given $0<p_1, <p_2<\infty$ and $p_0$ defined by $\frac1{p_0}=\frac1{p_1}+\frac1{p_2},$ we say that $W_{\sigma}$ is strongly $(p_1,p_2)-$bounded if it extends to a bounded bilinear operator from $H_{p_1}\times H_{p_2}\to L_{p_0}$, and we denote its norm by $\|W_{\sigma}\|_{H_{p_1}
\times H_{p_2}\to L_{p_0}}$. We say that $W_{\sigma}$ is weakly $(p_1,p_2)-$bounded if it extends to a bounded bilinear operator from $H_{p_1}\times H_{p_2}\to L_{p_0,\infty}$, and in this case we denote its norm by $\|W_{\sigma}\|_{H_{p_1}\times H_{p_2}\to
L_{p_0,\infty}}$. Now for a bounded function $\sigma$ on $\mathbb
R^n\times \mathbb R^n$ and $ 0<p_1,p_2<\infty$ we define its corresponding strong and weak $(p_1,p_2)$-multiplier norm by $$\|\sigma\|_{\mathcal M_{p_1,p_2}}=\|W_{\sigma}\|_{H_{p_1}\times H_{p_2}\to
L_{p_0}}\quad\text{and}
\quad \|\sigma\|_{\mathcal M^w_{p_1,p_2}}=\|W_{\sigma}\|_{H_{p_1}\times
H_{p_2}\to L_{p_0,\infty}},$$ where $1/p_0=1/p_1+1/p_2$.
This definition of multiplier norm is analogous to that in the linear case. If $\upsilon\in L_{\infty}(\mathbb R^n)$, $\|\upsilon\|_{\mathcal M_{p}}$ denotes the norm of $\upsilon$ as a multiplier from $H_p$ into $L_{p }$ that is $$\|\upsilon\|_{\mathcal M_{p }}= \|
M_\upsilon \|_{H_p\to L_p }, \quad\quad\text{where}\quad
M_\upsilon f= (\upsilon
\widehat{f}\,)
\spcheck,$$ when $0<p<\infty$. Next we mention a few properties of multipliers.
\[inv\] Suppose $\sigma\in L_{\infty}(\mathbb R^n\times
\mathbb R^n)$ and $0<p_1,p_2<\infty.$ Then:(i) If $\sigma'(\xi,\eta)\!=\!\sigma(\xi\!-\!\xi_0,\eta
\!-\!\eta_0)$ for some fixed $\xi_0,\eta_0$ then $\|\sigma'\|_{\mathcal
M_{p_1,p_2}}=\|\sigma\|_{\mathcal M_{p_1,p_2}}.$(ii) If $L:\mathbb R^n\to\mathbb R^n$ is an invertible linear operator and $\sigma_L(\xi,\eta)=\sigma(L\xi,L\eta)$ then $\|\sigma_L\|_{\mathcal
M_{p_1,p_2}}=\|\sigma\|_{\mathcal M_{p_1,p_2}}.$(iii) If $\mu,\upsilon\in L_{\infty}(\mathbb R^n)$ and $\sigma'(\xi,\eta)=\mu(\xi)\sigma(\xi,\eta)\upsilon(\eta),$ then $$\|\sigma'\|_{\mathcal M_{p_1,p_2}}\le
\|\mu\|_{\mathcal
M_{p_1}}\|\sigma\|_{\mathcal
M_{p_1,p_2}}\|\upsilon\|_{\mathcal M_{p_2}}.$$
For (i) note that $W_{\sigma'}(f,g)=e^{2\pi i\langle
x,\xi_0+\eta_0\rangle}W(e^{-2\pi i\langle x,\xi_0\rangle}f,e^{-2\pi i\langle
x,\eta_0\rangle}g).$ For (ii) note that $W_{\sigma_L}(f,g)\!\circ\!
(L^t)^{-1}= W(f\!\circ\!
(L^t)^{-1},g\!\circ \!(L^t)^{-1}).$ (iii) is trivial.
\[bounded2\] Let $\sigma\in L_{\infty}(\mathbb R^n\times\mathbb R^n)$. Suppose that either $p_0\ge 1,$ or that $\sigma$ is locally Riemann-integrable (i.e. continuous except on a set of measure zero). Then $\|\sigma\|_{L_\infty}\le
\|\sigma\|_{\mathcal M_{p_1,p_2}}$ whenever $p_0= p_1p_2/(p_1+p_2)$ and $0<p_1,p_2<\infty.$
Suppose that $\sigma$ is locally Riemann-integrable and let $(\xi_0,\eta_0)$ be a point of continuity of $\sigma.$ Then if we put $\sigma'_{\lambda}(\xi,\eta)= \sigma
(\xi_0+\lambda\xi,\eta_0+\lambda\eta),$ Proposition \[inv\] gives that $\|W_{\sigma'_{\lambda}}\|_{H_{p_1}\times H_{p_2} \to
L_{p_0}} =\|W_{\sigma}\|_{H_{p_1}\times H_{p_2} \to
L_{p_0}}.$ Now if $f,g\in\mathcal S$ it is easy to see that as $\lambda\to 0$ we have convergence in $L_2$ (and even pointwise) of $W_{\sigma'_{\lambda}}(f,g)$ to $ \sigma(\xi_0,\eta_0)f(x)g(x).$
If $p_0\ge 1$ let $Q_k$ be a cube of side $2^{-k}$ centered at $(0,0)$ in $\mathbb R^n\times \mathbb R^n.$ Let $$\sigma_k(\xi,\eta)=\frac{1}{|Q_k|}\int_{Q_k}
\sigma(\xi+\xi_0,\eta+\eta_0)d\xi_0\,d\eta_0.$$ Proposition \[inv\] and the fact that $p_0\ge 1$ easily imply that $\|W_{\sigma_k}\|_{L_{p_1}\times L_{p_2} \to
L_{p_0}}\le \|W_{\sigma}\|_{L_{p_1}\times L_{p_2} \to
L_{p_0}}.$ Since $\sigma_k$ is continuous we have $\|\sigma_k\|_{L_\infty}\le \|W_{\sigma}\|_{L_{p_1}\times L_{p_2} \to
L_{p_0}}$. Taking limits as $k\to \infty$ yields the conclusion.
Next we require a lemma on series in $L_p.$
\[easy\] Let $0<p<\infty.$ Suppose that for some $(f_{jk})_{(j,k)\in\mathbb Z^2 }$ sequence of $L_p$ functions and for all pairs of sequences $(\delta_j)_{j\in\mathbb Z}, (\delta'_k)_{k\in\mathbb Z}$ with $\sup_{j\in\mathbb Z}|\delta_j|\le 1$ and $\sup_{j\in\mathbb
Z}|\delta_j'|\le 1,$ we have $$\sup_{N\in \mathbb N} \big\|\sum_{|j|\le N}\sum_{|k|\le
N}\delta_j\delta'_kf_{jk}\big\|_{L_p} \le M.$$ Then there is a constant $C=C(p)$ such that (i) $\sup_{|\delta_j|\le 1}\|\sum_{j\in\mathbb
Z}\delta_jf_{jj}\|_{L_p}\le CM$ (and the series converges unconditionally),(ii) $\|(\sum_{j\in\mathbb Z}\sum_{k\in\mathbb
Z}|f_{jk}|^2)^{\frac12}\|_{L_p} \le CM.$
In fact (ii) follows immediately from Khintchine’s inequality by taking $\epsilon_j,\epsilon'_k$ two mutually independent sequences of Bernoulli random variables. To obtain (i), take $\epsilon_j$ be a sequence of Bernoulli random variables and for any finite subset $\mathcal F\subset\mathbb Z $ write $$\label{kkkk-ll}
\sum_{j\in \mathcal F}\delta_jf_{jj} = \sum_{j\in\mathcal
F}\sum_{k\in\mathcal
F}\delta_j\epsilon_j\epsilon_kf_{jk}-\sum_{\substack{j,k\in\mathcal F\\
j\neq k}}\delta_j\epsilon_j\epsilon_kf_{jk}.$$ Now for all $|\delta_j|\le 1$, (see also [@K1], proof of Theorem 4.6), $$\begin{aligned}
&\mathbb E(\|\sum_{\substack{j,k\in\mathcal F\\ j\neq
k}}\delta_j\epsilon_j\epsilon_kf_{jk}\|_{L_p}^p)^{1/p}\le
C\big\|(\sum_{\substack{j,k\in\mathcal F\\
j<k}} |\delta_jf_{jk}+\delta_kf_{kj}|^2)^{\frac12}\big\|_{L_p} \\ \le
&C\big\|(\sum_{j\in\mathbb Z}\sum_{k\in\mathbb
Z}|f_{jk}|^2)^{\frac12}\big\|_{L_p} \le CM\end{aligned}$$ by a generalization of Khintchine’s inequality due to Bonami [@bonami] and part (ii). The same estimate is also valid for $\sum\limits_{j\in\mathcal
F}\sum\limits_{k\in\mathcal
F}\delta_j\epsilon_j\epsilon_kf_{jk}$ by our assumptions. These estimates together with (\[kkkk-ll\]) give (i).
We now introduce some notation that will be useful in the sequel. For $(j,k)\in\mathbb Z$ let $D_{jk}=
\{(\xi,\eta):2^{j-1}\le |\xi|\le
2^{j+1},\ 2^{k-1}\le |\eta|\le 2^{k+1}\}.$ Also for $\theta>0$ let $D_{jk}(\theta)=
\{(\xi,\eta):2^{j-\theta}\le |\xi|\le
2^{j+\theta},\ 2^{k-\theta}\le |\eta|\le 2^{k+\theta}\}.$
\[subdiagonal\] For any $1<p_1,p_2<\infty$ there is a constant $C=C(p_1,p_2)$ so that whenever $(\sigma_{jk})_{j,k\in\mathbb Z}$ is a family of bilinear symbols with $\supp
\sigma_{jk}\subset D_{jk}$ which satisfy $$\sup_{ |\delta_j|\le 1}
\sup_{ |\delta_k'|\le 1}\|\sum_{j}\sum_k\delta_j\delta'_k
\sigma_{jk}\|_{\mathcal M_{p_1,p_2}}\le M,$$ then the following statements are valid:(i) For any scalar sequence $(\delta_j)$ with $\sup_j|\delta_j|\le 1$ and any $r\in\mathbb Z$ we have $$\|\sum_{j\in\mathbb Z}\delta_j \sigma_{j,j+r}\|_{\mathcal M_{p_1,p_2}}\le CM.$$ (ii) For all $r\ge 3$ we have, $$\|\sum_{j\in\mathbb Z}\sum_{k\le
j-r}\sigma_{jk}\|_{\mathcal M_{p_1,p_2}}+ \|\sum_{k\in\mathbb
Z}\sum_{j\le k-r}\sigma_{jk}\|_{\mathcal M_{p_1,p_2}}\le
C(1+r^{\max(\frac1{p_0},1)})M.$$ (iii) For every $r\ge 3$, $p_0\le 1$ and for all $f,g\in\mathcal S,$ we have $$\begin{aligned}
&\big\|\sum_{j\in\mathbb Z}\sum_{k\le j-r}W_{\sigma_{jk}} \big\|_{
L_{p_1}\times L_{p_2} \to H_{p_0}}
\le C(1+r^{\max(\frac1{p_0},1)})M \\
&\big\|\sum_{k\in\mathbb
Z}\sum_{j\le k-r} W_{\sigma_{jk}} \big\|_{
L_{p_1}\times L_{p_2} \to H_{p_0}}
\le C(1+r^{\max(\frac1{p_0},1)})M .\end{aligned}$$
For simplicity we write $W_{jk}=W_{\sigma_{jk}} $ below. (i) follows directly from Lemma \[easy\]. To prove (ii) and (iii) it is enough to consider the case $r=3$, since the other cases follow trivially by applying (i) and the known case $r=3.$ We therefore suppose $r\ge 3$ and establish both (ii) and (iii). An easy calculation gives that for $f,g$ Schwartz, the function $W_{jk}(f,g)$ has Fourier transform supported in the annulus $ 2^{j-2}\le |\zeta |\le 2^{j+2}$ when $k\le j-3$. It follows that $$\begin{aligned}
\begin{split}\label{kl}
\|\sum_{j\in\mathbb Z}\sum_{k\le j-3}W_{jk}(f,g)\|_{L_{p_0}} &\le
\|\sum_{j\in\mathbb Z}\sum_{k\le j-3}W_{jk}(f,g)\|_{H_{p_0}} \\
&\le C\|(\sum_{j\in\mathbb Z}|\sum_{k\le j-3}W_{jk}(f,g)|^2
)^{\frac12}\|_{L_{p_0}} \\
&\le C\mathbb E (\|\sum_{j\in\mathbb Z}\epsilon_j
\sum_{k\le j-3}W_{jk}(f,g)\|_{L_{p_0}}^{p_0})^{1/p_0}
\end{split}\end{aligned}$$ where as usual $(\epsilon_j)_{j\in\mathbb Z}$ is a sequence of independent Bernoulli random variables. (If $p_0>1$ then $H_{p_0}=L_{p_0}$.) We need to control the last term in (\[kl\]).
Our hypothesis gives the estimate $$\label{aaaa}
\mathbb E(\|\sum_{j\in\mathbb Z}\sum_{k\in\mathbb
Z}\epsilon_jW_{jk}(f,g)\|_{L_{p_0}}^{p_0})^{1/p_0} \le
CM\|f\|_{L_{p_1}}\|g\|_{L_{p_2}},$$ while we can apply (i) to obtain $$\label{bbbb}
\mathbb E(\|\sum_{j\in\mathbb Z}\sum_{|k-j|\le
2}\epsilon_jW_{jk}(f,g)\|_{L_{p_0}}^{p_0})^{1/p_0}\le
CM\|f\|_{L_{p_1}}\|g\|_{L_{p_2}}.$$ It remains to estimate $$\begin{aligned}
\mathbb E \big(\|\sum_{j\in\mathbb Z}\sum_{k\ge
j+3}\epsilon_jW_{jk}(f,g)\|_{L_{p_0}}^{p_0}\big)^{1/p_0}&\le
\mathbb E\big(\|\sum_{j\in\mathbb Z}\sum_{k\ge
j+3}\epsilon_jW_{jk}(f,g)\|_{H_{p_0}}^{p_0}\big)^{1/p_0}\\ &\le
C \mathbb E\big(\|\sum_{k\in\mathbb Z}(\sum_{j\le
k-3}\epsilon_jW_{jk}(f,g)\|_{L_{p_0}}^{p_0}\big)^{1/p_0}\\
&\le C\mathbb E\big(\|\sum_{k\in\mathbb Z}\epsilon'_k\sum_{j\le
k-3}\epsilon_j W_{jk}(f,g)\|_{L_{p_0}}^{p_0}\big)^{1/p_0}\end{aligned}$$ where $\epsilon_k'$ is a second (independent) sequence of independent Bernoulli random variables. Hence using again Khintchine’s inequality we have $$\begin{aligned}
\begin{split}\label{kkkk}
\mathbb E \big(\|\sum_{j\in\mathbb Z}\sum_{k\ge
j+3}\epsilon_jW_{jk}(f,g)\|_{L_{p_0}}^{p_0}\big)^{1/p_0}&\le
C\|(\sum_{k\in\mathbb
Z}\sum_{j\le k-3}|W_{jk}(f,g)|^2)^{\frac12}\|_{L_{p_0}}\\
&\le C\|(\sum_{k\in\mathbb Z}\sum_{j\in \mathbb Z}
|W_{jk}(f,g)|^2)^{\frac12}\|_{L_{p_0}}\\
&\le CM\|f\|_{L_{p_1}}\|g\|_{L_{p_2}}
\end{split}\end{aligned}$$ in view of Lemma \[easy\]. Using (\[aaaa\]), (\[bbbb\]), and (\[kkkk\]) we obtain $$\mathbb E \big(\|\sum_{j\in\mathbb Z}\epsilon_j\sum_{k\le
j-3}W_{jk}(f,g)\|^{p_0}_{L_{p_0}} \big)^{1/p_0}
\le CM\|f\|_{L_{p_1}}\|g\|_{L_{p_2}}$$ which combined with (\[kl\]) gives the first of the assertions (ii) and (iii) for $r=3$. The second assertions are derived similarly by symmetry.
We will need one further preliminary lemma.
\[factors\] For any $1<p_1,p_2<\infty$ there is a constant $C=C(p_1,p_2)$ such that for any family of symbols $(\sigma_{jk})_{j,k\in\mathbb Z }$ with $\supp \sigma_{jk}\subset D_{jk}$ and for any $\mu,\upsilon$ $C^{\infty} $ functions on the annulus $\frac14\le |\xi|\le 4 $ we have $$\sup_{|\delta_j|}\sup_{|\delta_k'|\le 1} \big\|
\sum_{j\in\mathbb Z}\sum_{k\in\mathbb Z}
\delta_j\delta_k' \tau_{jk} \big\|_{\mathcal M_{p_1,p_2}}
\le CK_{\mu}K_{\upsilon} \sup_{|\delta_j|}\sup_{|\delta_k'|\le 1}
\big\|\sum_{j}\sum_k\delta_j\delta'_k
\sigma_{jk}\big\|_{\mathcal M_{p_1,p_2}},$$ where $\tau_{jk}(\xi,\eta)=\mu(2^{-j}\xi)\sigma_{jk}(\xi,\eta)\upsilon(2^{-k}\eta)$, $$K_{\mu}=\sup_{\substack{|\alpha|\le m\\
\frac14\le |\xi|\le 4}}\left|\frac{\partial^{\alpha}\mu}{\partial
\xi^{\alpha}}\right|,\qquad\text{ }\qquad
K_{\upsilon}=\sup_{\substack{|\alpha|\le m\\
\frac14\le |\xi|\le 4}}\left|\frac{\partial^{\alpha}\upsilon}{\partial
\xi^{\alpha}}\right|,$$ and $m=[(n+1)/2].$
Recalling the definition of $\phi$ from section \[s-harmonic\] we note that the function $$\big(\sum_{l=j-2}^{j+2}\widehat\phi_l(\xi) \big)
\big(\sum_{l=j-2}^{k+2}\widehat\phi_l(\eta) \big)$$ is compactly supported and is equal to $1$ on the support of $\sigma_{jk}(\xi , \eta)$. For any sequence $\delta_j$ with $\sup|\delta_j|\le 1$ we observe that $$\begin{aligned}
\label{ftre}
& \big\|\big(\sum_{j\in\mathbb
Z}\delta_j\mu(2^{-j} \xi )\big)\big(\sum_{l=j-2}^{j+2}\widehat\phi_l(\xi)\big)
\big\|_{\mathcal M_{p_1}}\le CK_{\mu}\\
& \big\|\big(\sum_{k\in\mathbb
Z}\delta_k'\mu(2^{-jk} \eta )\big)\big(\sum_{l=k-2}^{k+2}\widehat\phi_l(\eta)\big)
\big\|_{\mathcal M_{p_2}}\le CK_{\upsilon}\label{ftre2}\end{aligned}$$ by the Hörmander multiplier theorem. Let $ U_{j_1,j_2,k_1,k_2}$ be the bilinear operator with symbol $$\Big(\delta_{j_1} \mu(2^{-j_1} \xi)\sum_{l=j_1-2}^{j_1+2}
\widehat\phi_l(\xi)\Big)
\sigma_{j_2,k_2}(\xi,\eta) \Big(\delta'_{k_1}
\upsilon(2^{-k_1}\eta)\sum_{l=k_1-2}^{k_1+2}\widehat\phi_l(\eta)\Big),$$ for some fixed $|\delta_j|, |\delta_k'|\le 1$. Let $$M= \sup_{|\delta_j|}\sup_{|\delta_k'|\le 1}
\big\|\sum_{j}\sum_k\delta_j\delta'_k
\sigma_{jk}\big\|_{\mathcal M_{p_1,p_2}}$$ and let $(\epsilon_j),(\epsilon_k')$ be two sequences of mutually independent Bernoulli random variables. Then for $f,g\in\mathcal S$ we have $$\begin{aligned}
&\mathbb E \big(\|\sum_{j_1\in\mathbb Z}\sum_{j_2\in\mathbb
Z}\sum_{k_1\in\mathbb Z}\sum_{k_2\in\mathbb
Z}\epsilon_{j_1}\epsilon_{j_2}\epsilon_{k_1}'\epsilon_{k_2}'
U_{j_1,j_2,k_1,k_2} (f,g)\|_{L_{p_0}}^{p_0} \big)^{\frac{1}{ p_0}} \\
&\quad\quad\quad\quad\quad\quad\quad
\quad\quad\quad\quad\quad\quad\quad
\le CMK_{\mu}K_{\upsilon}\|f\|_{L_{p_1}}\|f\|_{L_{p_2}}\end{aligned}$$ by our hypothesis, (\[ftre\]), and (\[ftre2\]). We now use Lemma \[easy\] twice to deduce that $$\|\sum_{j\in\mathbb Z}\sum_{k\in\mathbb Z}U_{j,j,k,k}(f,g)\|_{L_{p_0}}\le
CK_{\mu}K_{\upsilon}M \|f\|_{L_{p_1}}\|g\|_{L_{p_2}}.$$ This proves the required assertion.
Bilinear operators and infinite matrices {#s-bilinear+infmatrices}
========================================
0
Recall from section \[s-harmonic\] that $\phi_j(x)=2^{nj}\phi(2^jx)$ are smooth bumps whose Fourier transforms are supported in the annuli $2^{j-1}\le |\xi |\le 2^{j+1}$. In this section we will consider symbols $\sigma$ of the form $$\label{elemsymbols}
\sigma_A(\xi,\eta)=\sum_{j\in\mathbb Z}\sum_{k\in\mathbb
Z}a_{jk}\widehat{\phi_j}(\xi)\widehat{\phi_k}(\eta)$$ where $A=(a_{jk})_{(j,k)\in\mathbb Z^2}$ is a bounded infinite matrix. We let $W_A=W_{\sigma_A}$ and $\|A\|_{\infty}=\sup_{j,k}|a_{jk}|.$
If $A$ is such an infinite matrix we define $A_L$ to be its lower-triangle and $A_U$ to be its upper-triangle i.e. $A_L=(a_{jk}\theta_{jk})_{j,k}$ and $A_U=(a_{jk}\theta_{kj})_{j,k}$ where $\theta_{jk}=1$ if $k<j$ and $0$ otherwise. We let $A_D$ be the diagonal $A-A_U-A_L.$ Now define $$\label{defH}
H(A)=h(A_L)+h(A_U^t)+\|A\|_{\infty}$$ Notice that $H(A)\ge \|A\|_{\infty}$ and that $H$ is a norm on the space of $\{A:H(A)<\infty\}$ which makes it a Banach space.
Our objective will be to show that for any choice of $0<p_1,p_2<\infty$ we have $\|W_A\|_{H_{p_1}\times H_{p_2}\to L_{p_0}}\approx H(A).$ This will provide us with an equivalent expression for the norm of the multiplier $\sigma_A$ defined in (\[elemsymbols\]).
We start by proving the simple upper estimate below.
\[upper\] If $0<p_1,p_2<\infty$ there is a constant $C=C(p_1,p_2)$ so that for any matrix $A$ we have $\|\sigma_A\|_{\mathcal M_{p_1,p_2}}\le CH(A).$
We give the proof in the case $p_1,p_2>1$; the only real alteration for the other cases would be to replace the appropriate $L_{p_j}-$norm with the $H_{p_j}-$ norm and use Theorem \[hardy\]. Suppose $f,g\in\mathcal S$ and consider $$\begin{aligned}
\begin{split}\label{no0}
W_{A}(f,g)=&\sum_{j\in\mathbb
Z}\sum_{k\le j-3}a_{jk}\widetilde\Delta_jf\widetilde\Delta_kg
+\sum_{k\in\mathbb Z}\sum_{j\le
k-3}a_{jk}\widetilde\Delta_jf\widetilde\Delta_kg \\ &\quad\quad\quad
+\sum_{j\in\mathbb Z}\sum_{k=j-2}^{j+2}
a_{jk}\widetilde\Delta_jf\widetilde\Delta_kg.
\end{split}\end{aligned}$$ We estimate the first term by noticing that for fixed $j$ the Fourier transform of $\widetilde\Delta_jf \sum_{k\le j-3}a_{jk}\widetilde\Delta_kg$ is contained in the set $\{\zeta:\,\, 2^{j-2}\le |\zeta |\le 2^{j+2}\}.$ Hence if $p_0>1$ we have $$\|\sum_{j\in\mathbb Z}\widetilde\Delta_jf\sum_{k\le
j-3}a_{jk}\widetilde\Delta_kg\|_{L_{p_0}} \le C\|(\sum_{j\in\mathbb
Z}|\widetilde\Delta_jf|^2)^{\frac12} (|\sum_{k\le
j-3}a_{jk}\widetilde\Delta_k|^2)^{\frac12}\|_{L_{p_0}}.$$ If $0<p_0\le 1$ we obtain the same estimate by noticing that $$\|\sum_{j\in\mathbb Z}\widetilde\Delta_jf\sum_{k\le
j-3}a_{jk}\widetilde\Delta_kg\|_{L_{p_0}} \le
\|\sum_{j\in\mathbb Z}\widetilde\Delta_jf\sum_{k\le
j-3}a_{jk}\widetilde\Delta_kg\|_{H_{p_0}}$$ and using the corresponding square-function estimates in $H_{p_0}$. Now we have $$\begin{aligned}
\begin{split}\label{no1}
& \big\|(\sum_{j\in\mathbb Z}|\widetilde\Delta_jf|^2| \sum_{k\le
j-3}a_{jk}\widetilde\Delta_kg|^2)^{\frac12}\big\|_{L_{p_0}} \\
&\quad\quad\quad\quad\quad\quad\quad\quad \le \big\|(\sum_{j\in
\mathbb Z}|\widetilde\Delta_jf|^2)^{\frac12}\sup_{j\in\mathbb Z}|\!\!
\sum_{k\le j-3}a_{jk}\widetilde\Delta_kg|\big\|_{L_{p_0}}.
\end{split}\end{aligned}$$ If we let $A_{LL}$ be the matrix with entries $a_{jk}$ if $k\le j-3$ and $0$ otherwise, then $h(A_{LL})\le h(A_L)+h(B)$ where $B$ is the matrix with entries $a_{jk}$ if $j-2\le k\le j-1$ and $0$ otherwise. It is trivial to see that one has the estimate $h(B)\le 2\|A\|_{\infty}
$ so that $h(A_{LL})\le Ch(A_L).$ Hence (\[no1\]) and Theorem \[equivalence\] give $$\begin{aligned}
\| \sum_{j\in\mathbb Z}\sum_{k\le
j-3}a_{jk}\widetilde\Delta_jf\widetilde\Delta_kg\|_{L_{p_0}}
&\le C\|(\sum_{j\in\mathbb
Z}\widetilde\Delta_jf)^{\frac12}\|_{L_{p_1}}\|\sup_{j\in\mathbb Z}|
\sum_{k\in\mathbb
Z}a_{jk}\widetilde\Delta_kg|\|_{L_{p_2}}\\
&\le Ch(A_L)\|f\|_{L_{p_1}}\|g\|_{L_{p_2}}.\end{aligned}$$
The same argument shows that the third term in (\[no0\]) is controlled by $Ch(A_U^t)\|f\|_{L_{p_1}}\|g\|_{L_{p_2}}.$ The middle term in (\[no0\]) is easy. For $-2\le r\le 2 $ we have $$\begin{aligned}
&\big\|\sum_{j\in\mathbb
Z}a_{j,j+r}\widetilde\Delta_jf\widetilde\Delta_{j+r}g\big\|_{L_{p_0}} \\
\le & \big\|(\sum_{j\in\mathbb Z}|a_{j,j+r}||\widetilde\Delta_j
f|^2)^{\frac12}\big\|_{L_{p_1}}
\big\|(\sum_{k\in\mathbb Z}|a_{j,j+r}||\widetilde\Delta_{j+r}
g|^2)^{\frac12}\big\|_{L_{p_2}}
\\ \le &
C\max_{j}|a_{j,j+r}|\|f\|_{L_{p_1}}\|g\|_{L_{p_2}}.\end{aligned}$$ Combining we obtain the required upper estimate: $\|\sigma_A\|_{\mathcal M_{p_1,p_2}}\le CH(A).$
To obtain the converse is somewhat more complicated. First we prove a general result which we will use in other situations as well.
\[averaging\] For any $1<p_1,p_2<\infty$ with $p_0=(1/p_1+1/p_2)^{-1}\!\ge\! 1$, there is a constant $C=C(p_1,p_2)$ with the following property. Whenever $(\sigma_{jk})_{(j,k)\in\mathbb Z^2}$ is a family of symbols with $\supp \sigma_{jk}\subset D_{jk}$ which satisfy $$\sup_{|\delta_j|\le 1} \sup_{|\delta'_k|\le 1}
\|\sum_{j}\sum_k\delta_j\delta'_k W_{\sigma_{jk}}\|_{L_{p_1}
\times L_{p_2}\to L_{p_0}}\le M,$$ then $$\|\sigma_A\|_{\mathcal M_{p_1,p_2}} \le CM,$$ where $A=(a_{jk})_{j,k}$ and $$a_{jk} =\int_{\mathbb R^n}\int_{\mathbb
R^n}\sigma_{jk}(2^j\xi,
2^k\eta)d\xi\,d\eta .$$
As before we write $W_{jk}=W_{\sigma_{jk}}.$ Let us consider first the case when $\sigma_{jk}=0$ unless $k\le j-5.$ Let $\upsilon$ be a $C^{\infty}-$function on $\mathbb R^n$ supported on $2^{-4}\le |\xi|\le 2^4$ and such that $\upsilon(\xi)=1$ on $2^{-3}\le |\xi|\le 2^3.$ Fix $\xi_0\in\mathbb R^n$ and consider the symbol $$\tau_{jk}(\xi_0;\xi,\eta)=\upsilon
(2^{-j}\xi)\sigma_{jk}(\xi+2^j\xi_0,\eta).$$ Note that $\tau_{jk}$ is supported in $D_{jk}(4).$ Let $T_{jk}$ be bilinear operator with symbol $\tau_{jk}$. For any sequences $(\delta_j)_{j\in\mathbb Z},(\delta'_k)_{k\in\mathbb
Z}$ with $\sup|\delta_j|,\sup|\delta'_k|\le 1$ and $f,g\in\mathcal S $ we have $$\|\sum_{j\in\mathbb Z}\sum_{k\in\mathbb Z}\delta_j\delta_k'
T_{jk}(f,g)\|_{L_{p_0}} \le C\|(\sum_{j\in\mathbb Z}|\sum_{k\in\mathbb
Z}\delta_k'T_{jk}(f,g)|^2)^{\frac12}\|_{L_{p_0}}$$ by considering the supports of the Fourier transforms. But then for fixed $j,$ $$\sum_{k\in\mathbb Z}\delta_k'T_{jk}(f,g)(x)=e^{- 2\pi i \langle
x,2^j\xi_0\rangle}\sum_{k\in\mathbb Z}\delta_kW_{jk}(f,g)(x),$$ hence $$\begin{aligned}
\|\sum_{j\in\mathbb Z}\sum_{k\in\mathbb Z}\delta_j\delta_k'
T_{jk}(f,g)\|_{L_{p_0}} &\le C\|(\sum_{j\in\mathbb Z}|\sum_{k\in\mathbb
Z}\delta_k'W_{jk}(f,g)|^2)^{\frac12}\|_{L_{p_0}}\\
& \le C \|\sum_{j\in\mathbb Z}\sum_{k\in\mathbb
Z}\delta_k'W_{jk}(f,g)\|_{H_{p_0}}\\
&\le CM\|f\|_{L_{p_1}}\|g\|_{L_{p_2}}\end{aligned}$$ using Proposition \[subdiagonal\].
Now note that if $|\xi_0|> 18$ then all $T_{jk}$ vanish. Since $p_0\ge
1$, we integrate over $|\xi_0|\le 18$ to obtain symbols $$\tau'_{jk}(\xi,\eta)=\int_{|\xi_0|\le 18}
\tau_{jk}(\xi,\eta)\,d\xi_0= \upsilon(2^{-j}\xi)\int_{\mathbb
R^n}\sigma_{jk}(\xi+2^j\xi_0,\eta)d\xi_0$$ with corresponding bilinear operators $T_{jk}'$ satisfying $$\|\sum_{j\in\mathbb Z}\sum_{k\in\mathbb
Z}\delta_j\delta'_kT'_{jk}\|_{L_{p_1}
\times L_{p_2}\to L_{p_0}}\le CM$$ whenever $|\delta_j|,|\delta_k'|\le 1.$
Note that $\tau'_{jk}$ is supported on $D_{jk}(3).$ Also if $2^{j-3}\le
|\xi|\le 2^{j+3}$ we have that $\tau_{jk}'(\xi,\eta)$ is constant in $\xi.$
Next let $O_n$ be the orthogonal group of $\mathbb R^n$ and let $dL$ denote the Haar measure on this group. Define $$\tau^{\#}_{jk}(\xi,\eta)=
\int_{\frac14}^{4}\lambda^{n-1}\int_{O_n}\tau'_{jk}(\lambda L\xi,\lambda
L\eta)dL\, d\lambda ,$$ and let $T^{\#}_{jk}$ be the corresponding bilinear operator. If $(\xi,\eta)\in
D_{jk}$ we can compute that $$\tau^{\#}_{jk}(\xi,\eta)= c2^{nk}|\eta|^{-n}a_{jk}$$ where $c$ is a constant depending only on dimension. On the other hand, since $p_0\ge 1$, Proposition \[inv\] (ii) gives that $$\|\sum_{j\in\mathbb Z}\sum_{k\in\mathbb
Z}\delta_j\delta'_kT^{\#}_{jk}\|_{L_{p_1}
\times L_{p_2}\to L_{p_0}}\le CM$$ whenever $|\delta_j|,|\delta_k'|\le 1.$
Note that $\supp\tau^{\#}_{jk}\subset D_{jk}(6).$ Let us take $\mathbb M_1$ and $\mathbb M_2$ to be residue classes modulo 10. Then if we replace $\delta_j$ by $\delta_j\chi_{\mathbb M_1}(j)$ and $\delta'_k$ by $\delta'_k\chi_{\mathbb M_2}(k)$ we obtain a bilinear operator whose symbol coincides with $a_{jk}2^{nk}|\eta|^{-n}\delta_j\delta_k'$ on $D_{jk}$ for $(j,k)\in\mathbb M_1\times\mathbb M_2.$ Using Proposition \[inv\] (iii) and the multipliers $\sum_{j\in\mathbb M_1}\widehat{\phi_j}$ and $\sum_{k\in\mathbb M_2}\widehat{\phi_k}$ we obtain that the bilinear operator $V$ with symbol $$\sum_{j\in\mathbb M_1}\sum_{k\in\mathbb
M_2}\delta_j\delta_k'2^{nk}|\eta|^{-n}a_{jk}\widehat{\phi_j}(\xi)\widehat
\phi_k(\eta),$$ satisfies $\|V\|_{L_{p_1}
\times L_{p_2}\to L_{p_0}}\le CM.$ Summing over 100 different pairs of residue classes gives a similar estimate for the symbol $$\sum_{j\in\mathbb Z}\sum_{k\in\mathbb
Z}\delta_j\delta_k'2^{nk}|\eta|^{-n}a_{jk}\widehat{\phi_j}(\xi)\widehat{
\phi_k}(\eta).$$ The last step is to remove the factor $2^{nk}|\eta|^{-n}.$ But this can be done by using Lemma \[factors\] since $|\eta|^{-n}$ is $C^{\infty}$ on $\frac14\le |\eta|\le 4.$
We will use this result to make an important estimate on the effect of translation in the computation of $\|W_A\|_{L_{p_1}
\times L_{p_2}\to L_{p_0}}.$ Let us define $A^{[r,s]}$ to be the matrix $(a_{j+r,k+s})_{j,k}.$
\[translation2\](i) There is a constant $C$ so that for all matrices $A$ we have $$\|\sigma_{A^{[r,s]}}\|_{\mathcal M_{2,2}}\le
C^{|r-s|}\|\sigma_A\|_{\mathcal M_{2,2}}$$ (ii) For all $1<p_1,p_2<\infty$ with $p_0=p_1p_2/(p_1+p_2)\ge 1$, there is a constant $C=C(p_1,p_2)$ so that if $|\delta_j|,|\delta'_k|\le 1$ then $$\|\sum_{j\in\mathbb Z}\sum_{k\in\mathbb
Z}\delta_j\delta_k'a_{jk}\widehat\phi(2^{-j}\xi)
\widehat\phi(2^{-k}\eta)\|_{\mathcal
M_{p_1,p_2}}\le C \|\sigma_A\|_{\mathcal M_{p_1,p_2}},$$ i.e. $\|\sigma_D\|_{M_{p_1,p_2}}\le C \|\sigma_A\|_{\mathcal M_{p_1,p_2}},$ where $D=(d_{jk})_{j,k}=(\delta_j\delta_k'a_{jk})_{j,k}.$
It is clear from Proposition \[inv\] that for any $r\in\mathbb Z$ we have $$\|W_{A^{[r,r]}}\|_{L_2\times L_2\to L_1}=\|W_A\|_{L_2\times L_2\to L_1 }.$$ Thus it suffices to consider the case $r=0$ and $s=\pm 1$ and establish a bound in this case. To do this we consider the symbols $$\sigma_{jk}(\xi,\eta)=\sigma_A(\xi,\eta)\mu(2^{-j}\xi)\upsilon(2^{-k}\eta)
\widehat{\phi_j}(\xi)\widehat{\phi_k}(\eta),$$ where $\mu,\upsilon$ are $C^{\infty}-$functions satisfying $|\mu(\xi)|,|\upsilon(\eta)|\le 1$ for all $\xi,\eta.$ Since $\|\sum_{j\in\mathbb
Z}\delta_j\mu(2^{-j}\xi)\widehat{\phi_j}(\xi)\|_{\mathcal M_{2}}$ is bounded by $3$ whenever $\sup_j|\delta_j|\le 1,$ and there is a similar bound for $\sum_{k\in\mathbb Z}\delta'_k\upsilon(2^{-k}\eta)\widehat{\phi_k}(\eta)$ we have an immediate estimate; $$\|\sum_{j\in\mathbb Z}\sum_{k\in\mathbb
Z}\delta_j\delta'_kW_{\sigma_{jk}}\|_{L_2\times L_2\to L_1} \le
9\|W_A\|_{L_2\times L_2\to L_1}.$$ Now let $$b_{jk}= \int_{\mathbb R^n}\int_{\mathbb
R^n}\sigma_{jk}(2^j\xi,2^k\eta)d\xi\,d\eta.$$ Then we can compute $$b_{jk} = \sum_{r=-1}^1\sum_{s=-1}^1 c_{rs}a_{j+r,k+s}$$ where $$c_{rs}= \int_{\mathbb R^n}\int_{\mathbb R^n}\mu(\xi)\upsilon(\eta)
\widehat{\phi_{-r}}(\xi)\widehat{\phi_{-s}}(\eta)\widehat{\phi_0}(\xi)
\widehat{\phi_0}(\eta)
d\xi\,d\eta.$$
Since the functions $\widehat{\phi_{r}}$ for $-1\le r\le 1$ are linearly independent on the support of $\widehat{\phi_0}$ we can use the above estimate for a linear combination of a finite number of choices of $\upsilon$ and $\xi$ so that $c_{rs}=0$ except when $r=0$ and $s=1$, so that $B=cA^{[0,1]}$ for some fixed constant $c\neq 0.$ By Proposition \[averaging\] we have $\|W_B\|_{L_2\times L_2\to L_1}\le C\|W_A\|_{L_2\times
L_2\to L_1}.$ This and the similar argument for the case $s=-1$ gives the result (i).
For (ii) we observe that the above argument actually also yields a bound on $\|W_D\|_{L_2\times L_2\to L_1}$ when $D=(d_{jk})=(\delta_j\delta_k'b_{jk})$ (since $\delta_j\delta'_k\sigma_{jk}$ also verifies the hypotheses of Proposition \[averaging\]. By choosing a similar linear combination we can then ensure that $b_{jk}=ca_{jk}$ and obtain the desired result.
The next step is to consider a discrete model of the bilinear operator $W_{\sigma_A}.$ We restrict ourselves to $p_1=p_2=2$ for this, although our calculations can be done in more generality. If $A$ is a $c_{00}-$matrix we define $V_{A}:\,\, L_2\times L_2\to L_1$ by $$V_{A}(f,g)=\sum_{j\in\mathbb Z}\sum_{k\in\mathbb
Z}a_{jk}\Delta_jf\Delta_kg,$$ where $\Delta_j$ are the martingale difference operators as defined in section 4. We then have
\[discrete\] There is a constant $C$ so that if $A$ is a (strictly) lower-triangular matrix we have $ h(A)\le C\|V_{A}\|_{L_2\times L_2\to L_1}.$
This is a stopping time argument. Suppose $f\in L_2$ with $\|f\|_{L_2}=1.$ Note that for each $j$ the function $f_j=\sum_{k\in\mathbb Z}a_{jk}\Delta_kf$ is $\Sigma_{j-1}$-measurable where $\Sigma_{j-1}$ is the $\sigma-$algebra generated by the dyadic cubes in $\mathcal D_{j-1}.$ Fix $\lambda>0$. For each $j$ let $\mathcal Q_j$ be the collection of cubes $Q\in\mathcal D_{j-1}$ so that $|f_j|>\lambda$ on $Q$ and for each $j_1<j$ we have $|f_{j_1}|\le
\lambda$ on $Q.$ It is not difficult to see that $$\{x:\ \max_{j\in\mathbb Z}|f_j(x)|>\lambda\}= \bigcup\limits_{j\in\mathbb
Z}\bigcup\limits_{Q\in\mathcal Q_j}Q$$ and this is a disjoint union. Also note the left-hand side has finite measure.
For each $j$ be $u_j$ be a $\Sigma_j-$measurable function such that $|u_j|=1$ everywhere and $\mathcal E_{j-1}u_j=0.$ Let $$g= \sum_{j\in\mathbb Z}u_j\sum_{Q\in\mathcal Q_j}\chi_Q.$$ Then $$\|g\|_{L_2}^2= |\{x:\max_{j\in\mathbb Z}|f_j(x)|\}|$$ and $$V_A(f,g) = \sum_{j\in\mathbb Z}f_j\Delta_jg
= \sum_{j\in\mathbb Z}f_ju_j\sum_{Q\in\mathcal Q_j}\chi_Q.$$ Hence $$|V_A(f,g)|\ge \lambda \chi_{(\max_j|f_j|>\lambda)}$$ so that we have $$\lambda|\{\max_j|f_j|>\lambda\}\le \|V_A\|_{L_2\times L_2\to L_1}.$$ This implies that $h_2^w(A)\le \|V_A\|_{L_2\times L_2\to L_1}$ and the result follows from Theorem \[equiv\].
We are now ready for the main result:
\[matrixnorms\] Suppose $0<p_1,p_2<\infty$. Then there is a constant $C=C(p_1,p_2)$ so that for any infinite matrix $A$ we have $$\frac1CH(A)\le \|\sigma_A\|_{\mathcal M_{p_1,p_2}^w}\le
\|\sigma_A\|_{\mathcal M_{p_1,p_2}}\le CH(A).$$
The upper bound is proved in Lemma \[upper\] so we only need to prove the lower bound. It suffices to prove the results for the case when $A$ is a $c_{00}-$matrix. We start by considering the case $p_1=p_2=2,$ when $A$ is strictly lower-triangular.
In this case let us estimate the norm of the discrete model $V_A.$ In fact $$\begin{aligned}
V_A(f,g) &= \sum_{j\in\mathbb
Z}\sum_{k\in\mathbb Z}a_{jk}\Delta_jf\Delta_kg \\
&= \sum_{r\in\mathbb Z}\sum_{s\in\mathbb Z}\sum_{j\in\mathbb
Z}\sum_{k\in\mathbb Z}a_{jk}\widetilde\Delta_{j-r}\Delta_jf
\widetilde\Delta_{k-s}\Delta_kg\\ &= \sum_{r\in\mathbb Z}\sum_{s\in\mathbb
Z}a_{j+r,k+s}\widetilde\Delta_j\Delta_{j+r}f\widetilde\Delta_k\Delta_{k+s}g\\
&= \sum_{r\in\mathbb Z}\sum_{s\in\mathbb
Z}W_{A^{[r,s]}}(\sum_{j\in\mathbb Z}
\widetilde\Delta_j\Delta_{j+r}f,\sum_{k\in\mathbb
Z}\widetilde\Delta_k\Delta_{k+s}g)\\
&= \sum_{r\in\mathbb Z}\sum_{s\in\mathbb
Z}W_{A^{[r,s]}}(V_{-r}^*f,V_{-s}^*g),\end{aligned}$$ where $V_r$ is defined in the proof of Theorem \[equivalence\]. Using Proposition \[LP-martingale\] we obtain $$\|V_A\|_{L_2\times L_2\to L_1}\le C
\sum_{r\in\mathbb Z}\sum_{s\in\mathbb
Z}2^{-|r|-|s|}\|W_{A^{[r,s]}}\|_{L_2\times L_2\to L_1}.$$ (All these quantities are finite since $A$ has only finitely many non-zero entries, and so there is a uniform bound on $W_{A^{[r,s]}}.$)
It follows that we have an estimate (for a suitable $C_0,$) $$\label{est1}
h(A)\le C_0 \sum_{r\in\mathbb Z}\sum_{s\in\mathbb
Z} 2^{-|r|-|s|}\|W_{A^{[r,s]}}\|_{L_2\times L_2\to L_1}.$$
Next we estimate $H(A^{[r,s]}).$ If $s\ge r$ it is clear that $A$ remains lower-triangular and the invariance properties of $h(A)$ imply that $H(A^{[r,s]})\le H(A).$ If $s<r$ then it is easy to estimate $$h(A^{[r,s]}_L) \le h(A_L)+(r-s)\|A\|_{\infty}$$ and $$h((A^{[r,s]}_U)^t) \le (r-s)\|A\|_{\infty}.$$ We deduce that $$H(A^{[r,s]})\le h(A)+|r-s|\|A\|_{\infty}$$ for all $r,s.$ Thus we have for a suitable constant $C_0$ $$\label{est2}
\|W_{A^{[r,s]}}\|_{L_2\times L_2\to L_1}\le C_1(1+|r-s|)h(A).$$
Now we may pick an integer $N$ large enough so that $$C_1C_0\sum_{|r|>N}\sum_{|s|>N}(1+|r-s|)2^{-|r|-|s|}\le \frac12.$$ Then we can combine (\[est1\]) and (\[est2\]) to obtain $$\label{est3}
h(A) \le C_2\sum_{|r|\le N}\sum_{|s|\le
N}\|W_{A^{[r,s]}}\|_{L_2\times L_2\to L_1}.$$ At this point Lemma \[translation2\] gives the conclusion that $$h(A)\le C\|W_A\|_{L_2\times L_2\to L_1}.$$ Now suppose $A$ is arbitrary. If we let $W_{jk}$ be the bilinear operator with symbol $a_{jk}\widehat{\phi_j}(\xi)\widehat{\phi_k}(\eta),$ Lemma \[translation2\] (ii) implies that we can use Proposition \[subdiagonal\] (ii) to deduce that $\|W_{A_L}\|_{L_2\times L_2\to L_1}\le
C\|W_A\|_{L_2\times L_2\to L_1}$ for some absolute constant $C.$ Thus the above argument yields $h(A_L)\le C\|W_A\|_{L_2\times L_2\to L_1}.$ Similarly $h(A_U^t)\le C\|W_A\|_{L_2\times L_2\to L_1}$ and Lemma \[bounded2\] is enough to show that $\|A\|_{\infty}\le C\|W_A\|_{L_2\times L_2\to L_1}.$ Combining these we have the estimate $$H(A)\le C\|W_A\|_{L_2\times L_2\to L_1}.$$
The proof is completed by a simple interpolation technique. We will argue first that an estimate of the type $$\label{teydr}
H(A)\le C(p_1,p_2)\|\sigma_A\|_{\mathcal M_{p_1,p_2}}$$ for some fixed $1<p_1,p_2<\infty$ implies the estimate $$\label{huy}
H(A)\le C(q,p_2)\|\sigma_A\|_{\mathcal M_{p_1,q}^w }$$ for every $1<q<\infty.$ We only need to consider the first case and $q\neq p_2$ (when $q=p_2$ one repeats the step). Then we may find $1<r<\infty$ and $0<\theta<1$ so that $$\frac1p_2=\frac{1-\theta}{q}+\frac{\theta}{r}.$$ The Marcinkiewicz interpolation theorem yields $$\label{765rt}
\|\sigma_A\|_{\mathcal M_{p_1,p_2}}\le C(p_1,p_2,\theta)
(\|\sigma_A\|_{\mathcal M_{p_1,q}^w })^{1-\theta}(\|\sigma_A\|_{
\mathcal M_{p_2,r} })^{\theta} .$$ Since $\|\sigma_A\|_{\mathcal M_{p_2,r}}\le C(p_2,r)H(A)$, using (\[765rt\]), and (\[teydr\]) we obtain estimate (\[huy\]) as required (recall that we assume $A$ is a $c_{00}$-matrix so that all these quantities are finite).
Repeated use of this argument starting from $p_1=p_2=2$ gives the theorem in the cases $1<p_1,p_2<\infty$.
Finally in the case where either $p_1\le 1$ or $p_2\le 1$ (or both) one can use complex interpolation to deduce $$\|\sigma_A\|_{\mathcal M^w_{q_1,q_2}}\le C
(\|\sigma_A\|_{\mathcal M^w_{p_1,p_2}})^{1-\theta}(\|\sigma_A\|_{
\mathcal M_{2,2}})^{\theta}$$ where $q_1,q_2>1$ and $$\frac1q_1=\frac{1-\theta}{p_1}+\frac\theta2,\qquad
\frac1q_2=\frac{1-\theta}{p_2} +\frac\theta2.$$ This clearly extends the lower estimate to the cases $p_1,p_2\le 1.$
Applications to bilinear multipliers {#s-applications}
====================================
We will now consider the boundedness of the bilinear operator $W_{\sigma}$ under conditions of Marcinkiewicz type on the symbol $\sigma$. We will say that a symbol $\sigma$ is $C^N$ if it is $C^N$ on the set $\{(\xi,\eta):\
|\xi|,|\eta|>0\}.$ We first give an example to show that conditions (\[hypothesis\]) for a function $\sigma$ on $\mathbb R^{2n}$ do not imply boundedness for the corresponding bilinear map on $\mathbb R^{n} \times \mathbb R^{n}$.
[**Example.**]{} There is a $C^{\infty}-$symbol $\sigma$ so that for every pair of multi-indices $(\alpha,\beta)$ there is a constant $C_{\alpha,\beta}$ so that $$\label{Marcinkiewicz}
|\xi|^{|\alpha|}|\eta|^{|\beta|}|\partial_{\xi}^{\alpha}
\partial_{\eta}^{\beta} \sigma (\xi,\eta)|\le
C_{\alpha,\beta}$$ but $W_{\sigma}$ is not of weak type $(p_1,p_2)$ for any $0<p_1,p_2<\infty.$
Indeed if we let $A$ be a bounded infinite matrix and $\sigma(\xi,\eta)=\sigma_A(\xi,\eta)$, then $\sigma$ satisfies the condition (\[Marcinkiewicz\]). However $W_A$ is of weak type $(p_1,p_2)$ if and only if $H(A)<\infty$ by theorem \[matrixnorms\]. At the end of Section \[estimates\] we showed that there are examples (with $A$ lower-triangular) where $H(A)=\infty.$
In fact more is true. It is shown that the condition $0<\theta<\frac12$ in (\[logcondition\]) is insufficient to give a bound on $h(A)$ or $H(A)$ when $A$ is lower-triangular. This means that if $0<\theta<\frac12$ we can construct a symbol $\sigma$ which is $C^{\infty}$, with $W_{\sigma}$ not of weak type $(p_1,p_2)$ for any $0<p_1,p_2<\infty$ and such that for each pair of multi-indices $(\alpha,\beta)$ there is a constant $C_{\alpha,\beta}$ with $$\label{Marcinkiewicz2}
|\xi|^{|\alpha|}|\eta|^{|\beta|}|\partial_{\xi}^{\alpha}
\partial_{\eta}^{\beta} \sigma (\xi,\eta)|\le
C_{\alpha,\beta}\big(\log(1+|\log\tfrac{|\xi|}{|\eta|}|
)\,\big)^{-\theta}$$ but $W_{\sigma}$ is not of weak type $(p_1,p_2)$ for any $p_1,p_2>0.$
These examples indicate that the Marcinkiewicz-type conditions (\[Marcinkiewicz\]) need to be modified if they are to imply boundedness for bilinear operators on $\mathbb R^{n} \times \mathbb R^{n}$.
In order to formulate some general results, let us introduce the following notation. For $\sigma\in L_{\infty}$ we define $$\label{normH} \|\sigma\|_H=\sup_{1\le |\xi|\le
2}\sup_{1\le |\eta|\le 2}H((\sigma(2^j\xi,2^k\eta)_{j,k}).$$ If $\sigma$ is of class $C^N$ we define $$\label{normH2}
\|\sigma\|_{H}^{(N)}=\sum_{|\alpha|\le
N}\||\xi|^{|\alpha|}\partial_{\xi}^{\alpha}\sigma\|_H
+\sum_{|\beta|\le N}\||\eta|^{|\beta|}\partial_{\eta}^{\beta}\sigma\|_H.$$ It will also be useful to define in this case $$\label{multiplierderiv}
\|\sigma\|_{\mathcal M_{p_1,p_2}}^{(N)}=
\sum_{|\alpha|\le
N}\||\xi|^{|\alpha|}\partial_{\xi}^{\alpha}\sigma\|_{\mathcal M_{p_1,p_2}}+
\sum_{|\beta|\le
N}\||\eta|^{|\beta|}\partial_{\eta}^{\beta}\sigma\|_{\mathcal M_{p_1,p_2}}.$$
Now consider an arbitrary $L^{\infty}$ symbol $\sigma $ of class $C^{n+1}$. Let $$\label{breakup}
\sigma_{jk}(\xi,\eta)=\sigma(\xi,\eta)\widehat\phi(2^{-j}\xi)
\widehat\phi(2^{-k}\eta).$$ Set $\widehat{\zeta}(\xi)= \widehat{\phi_{-2}}(\xi)+
\widehat{\phi_{-3}}(\xi)+\widehat{\phi_{-4}}(\xi)$. Then $\widehat{\zeta}$ is equal to $1$ on the annulus $1/16 \le |\xi|\le 1/4 $ and vanishes off the annulus $1/32 \le |\xi|\le 1/2 $. Thus the function $\widehat{\zeta}(\xi)\widehat{\zeta}(\eta)$ is supported in the unit cube $[0,1]^{2n}$ and is equal to one on the support of $$(\xi,\eta) \to \sigma_{jk} (2^{j+3}\xi,2^{k+3}\eta)$$ which is also contained in $[0,1]^{2n}$. Inspired by [@CM2], we expand the function above in Fourier series on $[0,1]^{2n}$. We have $$\sigma_{jk} (2^{j+3}\xi,2^{k+3}\eta) =
\sum_{\nu\in \mathbb Z^n} \sum_{\rho\in \mathbb Z^n}
a_{jk}(\nu,\rho) e^{2\pi i (\langle \xi , \nu \rangle +
\langle \eta , \rho \rangle )} \widehat\zeta(\xi) \widehat\zeta(\eta),$$ where for $(\nu,\rho)\in\mathbb Z^n\times \mathbb Z^n $ we set $$\label{defmatrix}
a_{jk}(\nu,\rho) = \int_{\mathbb R^n} \int_{\mathbb R^n}
\sigma(2^{j+3}t,2^{k+3}s)\widehat{\phi}(8 t)\widehat{\phi}(8s)
e^{-2\pi i( \langle t,\nu\rangle+
\langle s,\rho\rangle)} dt\,ds.$$ We will denote by $A(\nu, \rho)$ the matrix with entries $a_{jk}(\nu,\rho)$. Now setting $$\label{deftau}
\tau^{\nu,\rho}(\xi,\eta)=
\bigg(\sum_{j\in\mathbb Z}\sum_{k\in\mathbb
Z}a_{jk}(\nu,\rho)e^{ \frac{\pi i}{4} (2^{-j}\langle \xi,\nu\rangle+
2^{-k}\langle\eta,\rho\rangle)} \bigg)
\widehat{\zeta}(2^{-j-3}\xi) \widehat{\zeta }(2^{-k-3}\eta),$$ we can write a symbol $\sigma$ of class $C^{n+1}$ as $$\label{pointwise}
\sigma(\xi,\eta)=\sum_{\nu\in\mathbb
Z^n}\sum_{\rho\in\mathbb Z^n}
\tau^{\nu,\rho}(\xi,\eta).$$
In the next lemma we obtain some elementary estimates based on this expansion.
\[fourierestimates\] Suppose $0<p_1,p_2<\infty$ and $\frac1p_0=\frac1p_1+\frac1p_2.$ Then:(i) There is a constant $C=C(p_1,p_2)$ so that for any $(\nu,\rho)$ $$\|\tau^{\nu,\rho}\|_{\mathcal M_{p_1,p_2}}\le
C (1+|\nu|+|\rho|)^{2m}H(A(\nu,\rho))$$ where $m=[(n+1)/2].$(ii) There is a constant $C=C(N,p_1,p_2)$ such that if $\sigma$ is of class $C^N,$ and $|\nu|+|\rho|>0$, then $$H(A(\nu,\rho))\le C(1+|\nu|+|\rho|)^{-N}\|\sigma\|_H^{(N)},$$ while $$H(A(0,0)) \le C\|\sigma\|_H.$$ (iii) If $p_0\ge 1$ and $\sigma$ is of class $C^N$ then there is a constant $C=C(N,p_1,p_2)$ such that $$H(A(\nu,\rho))\le
C(1+|\nu|+|\rho)^{2m-N}\|\sigma\|_{\mathcal M_{p_1,p_2}}^{(N)}.$$
Observe that $\widehat{\zeta}(2^{-j-3}\xi)= \widehat{\phi}(2^{-j-1}\xi) +
\widehat{\phi}(2^{-j }\xi) +\widehat{\phi}(2^{-j+1}\xi) $ and therefore $\tau^{\nu,\rho}(\xi, \eta) $ is the sum of nine terms of the form $$\sum_{j\in\mathbb Z}\sum_{k\in\mathbb
Z}a_{j ,k }(\nu,\rho) \big( e^{ \frac{\pi i}4 \langle 2^{-j}\xi,\nu\rangle}
\widehat \phi(
2^{-j-r} \xi) \big)
\big(e^{ \frac{\pi i}4 \langle 2^{-k}\eta,\rho\rangle}\widehat \phi
(2^{-k-s} \eta) \big)$$ where $r,s\in \{-1,0,+1\}$. We now use Lemma \[factors\], Lemma \[translation2\] (ii), and Lemma \[upper\] in that order to obtain $$\|\tau^{\nu,\rho} \|_{\mathcal M_{p_1,p_2}}\le
C(1+|\nu|)^m(1+|\rho|)^m H(A (\nu,\rho))$$ where $m=[(n+1)/2].$ This proves (i).
For (ii) note that if $|\alpha|,|\beta|\le N$ integration by parts gives $$\begin{aligned}
\label{fourier1}
a_{jk}(\nu,\rho)& \!=\!
\int_{\mathbb R^n}\!\int_{\mathbb R^n}\!\!\!
\partial_{\xi}^{\alpha} \big(\sigma(2^{j+3}\xi,2^{k+3}\eta)
\widehat{\phi}(8\xi)\widehat{\phi}(8\eta)\big)
\frac{e^{-2\pi i( \langle \xi,\nu\rangle+
\langle\eta,\rho\rangle)}}{(-2\pi i \nu)^{\alpha}}
d\xi d\eta , \\ \label{fourier2}
a_{jk}(\nu,\rho)& \!=\!
\int_{\mathbb R^n} \!\int_{\mathbb R^n}\!\!\!
\partial_{\eta}^{\beta}\big(\sigma(2^{j+3}\xi,2^{k+3}\eta)\widehat{\phi}(8\xi)
\widehat{\phi}(8\eta)\big)
\frac{e^{-2\pi i( \langle \xi,\nu\rangle+
\langle\eta,\rho\rangle)}}{(-2\pi i\rho)^{\beta}}
d\xi d\eta ,\end{aligned}$$ provided $\nu_1^{\alpha_1}\dots \nu_n^{\alpha_n}$ and $\rho^{\beta_1}\dots
\rho_n^{\beta_n}$ are nonzero.
Now using the fact that $H$ is a norm it is easy to see that by choosing an appropriate $\alpha$ or $\beta$ for each pair $(\nu,\rho)\neq (0,0)$ one obtains the estimate $$H(A(\nu,\rho)) \le C(N,p_1,p_2)(1+|\nu|+|\rho|)^{-N}\|\sigma\|_H^{(N)}.$$ If $(\nu,\rho)=(0,0) $ the same estimate follows directly from (\[defmatrix\]).
Finally we turn to (iii). For fixed $\delta_j,\delta_k'$ with $\sup|\delta_j|,\sup|\delta_k'|\le 1$ let us define $\mu(\xi)=\sum_{j\in\mathbb Z}\delta_j\widehat{\phi_j}(\xi)$ and $\upsilon(\eta)=\sum_{k\in\mathbb Z}\delta_k'\widehat{\phi_j}(\eta).$ Then it follows from Lemma \[factors\] that for any multi-indices $\alpha,\alpha'$ we have $$\||\xi|^{|\alpha|+|\alpha'|}\partial_{\xi}^{\alpha}\mu(\xi)
\partial_{\xi}^{\alpha'} \sigma(\xi,\eta)\upsilon(\eta)\|_{\mathcal
M_{p_1,p_2}}\le C(\alpha,\alpha')\|\sigma\|^{(|\alpha'|)}_{\mathcal
M_{p_1,p_2}}$$ This implies that for fixed $N$ and any $\alpha$ with $|\alpha|=N$ we have $$\label{a1}
\sup_{|\delta_j|\le 1}\sup_{|\delta_k'|\le 1}\||\xi|^{N}\sum_{j\in\mathbb
Z}\sum_{k\in\mathbb Z}
\delta_j\delta_k'\partial_{\xi}^{\alpha}\sigma_{jk}(\xi,\eta)
\|_{\mathcal
M_{p_1,p_2}}\le C(N)\|\sigma\|^{(N)}_{\mathcal
M_{p_1,p_2}}.$$
We now use either (\[defmatrix\]) if $(\nu,\rho)=(0,0)$ or we refer back to Proposition \[averaging\] (\[fourier1\]) or (\[fourier2\]) according to the values of $\nu$ or $\rho$, when $(\nu,\rho) \neq (0,0)$. For example when $N=|\nu |\ge |\rho |$ and the $l$th entry of $\nu$ has maximal size $N$, then $$\begin{aligned}
&\|\sum_{j\in\mathbb Z}\sum_{k\in\mathbb
Z}a_{jk}(\nu,\rho)\widehat{\phi_j}(\xi)\widehat{\phi_k}(\eta)\|_{\mathcal
M_{p_1,p_2}} \\
\le &C\sup_{|\delta_j|\le 1}\sup_{|\delta'_k|\le
1}\|\sum_{j\in \mathbb Z}\sum_{k\in\mathbb
Z}\delta_j\delta_k'2^{jN}\frac{\partial^{N}}{\partial\xi_l^N}
\sigma_{jk}(\xi,\eta) \frac{e^{-2\pi i (\langle 2^{-j}
\xi,\nu\rangle+\langle 2^{-k} \eta,\rho\rangle)}}{(-2\pi i \nu_l)^N}\|_{\mathcal
M_{p_1,p_2}}.\end{aligned}$$ Now by Lemma \[factors\] we can estimate the last expression side above by $$C(1+|\nu|+|\rho|)^{2m-N}\sup_{|\delta_j|\le1}\sup_{|\delta_k'|\le 1}
\|\sum_{j\in\mathbb Z}\sum_{k\in\mathbb Z}\delta_j\delta_k'|\xi|^N
\frac{\partial^{N}}{\partial\xi_l^N}
\sigma_{jk}(\xi,\eta)\|_{\mathcal M_{p_1,p_2}}.$$ Using (\[a1\]) we obtain (iii).
Let us state the main result of this section.
\[main1\] Suppose $0<p_1,p_2<\infty$ and $\frac1p_0=\frac1p_1+\frac1p_2.$ Let $N=2n+1$ if $p_0\ge 1$ and $N=n+2+[\frac{n}{p_0}]$ if $p_0<1.$ Then for any $\sigma$ $C^N-$symbol such that $\|\sigma\|_H^{(N)}<\infty$ we have $\|\sigma\|_{\mathcal
M_{p_1,p_2}}<\infty$. Furthermore, there is a constant $C=C(p_1,p_2)$ so that $\|\sigma\|_{\mathcal M_{p_1,p_2}}\le C\|\sigma\|_H^{(N)}.$
This follows directly from Lemma \[fourierestimates\] and (\[pointwise\]). Indeed, we have $$\|\tau^{\nu,\rho}\|_{\mathcal M_{p_1,p_2}}\le
C(1+|\nu|+|\rho|)^{2m-N}.$$ If $t=\min(p_0,1)$ we have $$\|\sigma\|_{\mathcal M_{p_1,p_2}}\le C (\sum_{\nu\in\mathbb
Z}\sum_{\rho\in\mathbb Z}
(1+|\nu|+|\rho|)^{(2m-N)t})^{\frac1t}\|\sigma\|_H^{(N)}.$$ Since $(N-2m)t>n$ this gives the result.
We next show that in a certain sense the preceding theorem is best possible.
\[bestposs\] Suppose $1<p_1,p_2<\infty$ and $\frac1p_0=\frac1p_1+\frac1p_2\le 1.$ Suppose $\sigma$ is a $C^{\infty}-$symbol. Then the following are equivalent:(i) $
\|\sigma\|_{\mathcal M_{p_1,p_2}}^{(N)}<\infty$ for every $N\ge
0.$(ii)$ \|\sigma\|_H^{(N)}<\infty$ for every $N\ge 0.$
Assume (i); then it follows from Lemma \[fourierestimates\] that for any $N>0$ we have an estimate $H(A(\nu,\rho))\le C_N (1+|\nu|+|\rho|)^{-N}.$ Now it is clear from the definition and from Theorem \[matrixnorms\] and Lemma \[translation2\] that we have an estimate $$\||\xi|^{|\alpha|}\partial_{\xi}^{\alpha}\tau^{\nu,\rho}\|_H \le
C_{\alpha}(1+|\nu|)^{|\alpha|}H(A(\nu,\rho)).$$ Hence we can deduce easily that $$\||\xi|^{\alpha}\partial_{\xi}^{\alpha}\sigma\|_H
<\infty$$ for each multi-index $\alpha.$ Repeating the same reasoning with the second variable $\eta$ gives (ii).
Now assume (ii). Then for any multi-index $\alpha$ one can see easily by differentiation that for any pair of multi-indices $\alpha,\beta$ we have that (ii) is satisfied by the symbols $|\xi|^{|\alpha|}\partial_{\xi}^{\alpha}\sigma$ and $|\eta|^{|\beta|}\partial_{\eta}^{\beta}\sigma$ in place of $\sigma.$ Applying Theorem \[main1\] gives (i).
Now let us recast Theorem \[main1\] in terms of estimates on the symbol $\sigma$ using the results of Section \[estimates\].
\[main2\] Suppose $0<p_1,p_2<\infty$ and $\frac1p_0=\frac1p_1+\frac1p_2.$ Let $N=2n+1$ if $p_0\ge 1$ and $N=n+2+[\frac{n}{p_0}]$ if $p_0<1.$ Suppose $\theta>1$ Suppose $\sigma$ is a $C^N-$symbol such that for any pair of multi-indices $\alpha,\beta$ with $0\le
|\alpha|\le N$ and $0\le |\beta|\le N$ there exist constants $C_{\alpha},
C_{\beta},$ with $$\begin{aligned}
\label{Marcinkiewicz3}
|\xi|^{|\alpha|} |\partial_{\xi}^{\alpha}\sigma(\xi,\eta)| &\le
C_{\alpha}
(\log(1+|\log\tfrac{|\xi|}{|\eta|}|))^{-\theta} \\
\label{Marcinkiewicz4}
|\eta|^{|\beta|} |\partial_{\eta}^{\beta}\sigma(\xi,\eta)| &\le
C_{\beta}
(\log(1+|\log\tfrac{|\xi|}{|\eta|}|))^{-\theta}.\end{aligned}$$ Then $\|\sigma\|_{\mathcal M_{p_1,p_2}}<\infty.$
[**Remark.**]{} We have already seen that in (\[Marcinkiewicz2\]) that this is false when $0<\theta<\frac12.$ However the arguments of Section \[estimates\] shows that we can improve (\[Marcinkiewicz3\]) and (\[Marcinkiewicz4\]) somewhat. For example we can replace $
\big(\log(1+|\log\frac{|\xi|}{|\eta|}|)\big)^{-\theta} $ where $\theta>1$ by $
\big( \log(1+|\log\frac{|\xi|}{|\eta|}|)\big)^{-1}\big(
\log(1+\log(1+|\log\frac{|\xi|}{|\eta|})\big)^{-\gamma}$ where $\gamma>1.$
This follows immediately from Theorem \[main1\] and Theorem \[lorentz\] which yields the estimate $$H(A) \le C\sup_{j,k}\frac{|a_{jk}|}{w_{|j-k|+1}}$$ with $w_k=\log
(1+k)^{-\theta}.$
It is possible to “mix and match” the estimates in Section \[estimates\]: for example, in the following theorem we remove the conditions for $|\alpha|,|\beta|=0$ but insist on a stronger condition for $|\alpha|=|\beta|=1$:
\[main3\] Suppose $0<p_1,p_2<\infty$ and $\frac1p_0=\frac1p_1+\frac1p_2.$ Let $N=2n+1$ if $p_0\ge 1$ and $N=n+2+[\frac{n}{p_0}]$ if $p_0<1.$ Suppose $\theta>1$ Suppose $\sigma$ is a $C^N-$symbol which satisfies conditions (\[Marcinkiewicz3\]) and (\[Marcinkiewicz4\]) for $2\le |\alpha|,|\beta|\le N$ and if $|\alpha|=|\beta|=1$ $$\begin{align}\label{Marcinkiewicz5}
|\xi|^{|\alpha|} |\partial_{\xi}^{\alpha}\sigma(\xi,\eta)| &\le
C_{\alpha}
(1+|\log\tfrac{|\xi|}{|\eta|}|)^{-\theta} \\
\label{Marcinkiewicz6}
|\eta|^{|\beta|} |\partial_{\eta}^{\beta}\sigma(\xi,\eta)| &\le
C_{\beta}
(1+|\log\tfrac{|\xi|}{|\eta|}|)^{-\theta}.\end{align}$$ Then $\|\sigma\|_{\mathcal M_{p_1,p_2}}<\infty.$
It is only necessary to show that $\|\sigma\|_H<\infty.$ Note first that Proposition \[bv\] can be used to give the estimate for any infinite matrix: $$H(A) \le C\big(\|A\|_{\infty}+\sup_j\sum_{k<j}|a_{j,k}-a_{j,k+1}|+
\sup_k\sum_{j<k}|a_{j,k}-a_{j+1,k}|\big).$$ Now suppose $1\le |\xi|,|\eta|\le 2$. Then if $k<j,$ $$|\sigma(2^j\xi, 2^k\eta)-\sigma(2^j\xi,2^{k+1}\eta)|\le Ck^{-\theta}$$ by (\[Marcinkiewicz6\]). Combining with a similar estimate from (\[Marcinkiewicz5\]) gives the theorem.
We conclude this section with a theorem of the type of Theorem \[main1\] for operators on $L_1.$
\[L1\] Suppose $N=2n+3$ and that $\sigma$ is a $C^N$-symbol with $\|\sigma\|_H^{(N)}<\infty$; then $W_{\sigma}:L_1\times L_1\to L_{\frac12,\infty}$ is bounded.
Let $Q$ be the cube $\{x: \max_k|x_k|\le 1\}$ and consider the bilinear operator $W_{\sigma,Q}(f,g)=\chi_{Q}W_{\sigma}(f,g).$ We will show that if $r<\frac12$ is such that $n+2+[\frac{n}{2r}]=N $, then $W_{\sigma,Q}:L_1(2Q)\times L_1(2Q)
\to L_r(Q)$ is bounded and $\|W_{\sigma,Q}\|\le C\|\sigma\|_H^{(N)}$ where $C$ is a constant depending only on dimension.
Suppose that $f,g\in\mathcal S$ are functions with support contained in $2Q$ and such that $\int f(x)\,dx=\int g(x)\,dx=0.$ Then $f,g\in H_{2r}$ with $\|f\|_{H_{2r}}\le C\|f\|_{L_1}$ and $ \|g\|_{H_{2r}}\le C\|g\|_{L_1}.$ Applying Theorem \[main1\] we obtain that $$\label{req}
\|W_{\sigma}(f,g)\|_{L_r} \le
C\|\sigma\|_H^{(N)}\|f\|_{L_1}\|g\|_{L_1}$$ where $C$ is an absolute constant. It follows that $W_{\sigma}$ extends unambiguously to any $f,g\in L_1(2Q)$ with $\int f(x)\,dx=\int
g(x)\,dx=0$ and (\[req\]) holds.
Next fix $\psi\in \mathcal S$ so that $\int \psi(x)\,dx=1$ and $\psi$ has support contained in $Q.$ Now for any $f,g\in L_1(3Q)$ let $f_0=f-(\int f(x)\,dx)\psi$ and $g_0=g-(\int g(x)\,dx)\psi.$ Then (\[req\]) gives $$\|W_{\sigma,Q}(f_0,g_0)\|_{L_r} \le C\|\sigma\|_H^{(N)}\|f\|_{L_1}\|g\|_{L_1}.$$ We also note that $\|W_{\sigma,Q}(\psi,\psi)\|_{L_r} \le
C\|\sigma\|_H^{(N)}.$ Now consider the linear map $Tf= W_{\sigma}(f,\psi).$ Since $\psi\in
L_2$ we have that, if $\frac1s=\frac1{2r}+\frac12$, $T:H_{2r}\to L_s$ is bounded with norm controlled by $C\|\sigma\|_H^{(N)}$ (again using Theorem \[main1\].) Hence since $r<s,$ $$\|W_{\sigma,Q}(f_0,\psi)\|_{L_r} \le C\|\sigma\|_H^{(N)}\|f\|_{L_1}.$$ Similarly $$\|W_{\sigma,Q}(\psi,g_0)\|_{L_r} \le C\|\sigma\|_H^{(N)}\|g\|_{L_1}.$$ Combining these estimates gives $$\label{start} \|W_{\sigma,Q}(f,g)\|_{L_r} \le
C\|\sigma\|_H^{(N)}\|f\|_{L_1}\|g\|_{L_1}.$$
We now use a Nikishin type argument as earlier in Lemma \[Lem1.3\]. Suppose $(f_j)_{j=1}^J$ and $(g_j)_{j=1}^J$ satisfy $
\|f_j\|_{L_1},\|g_j\|_{L_1}\le 1$ and that $\sum_{j=1}^J|b_j|^{1/2}=1.$ Then if $(\epsilon_j)_{j=1}^J$ and $(\epsilon'_j)_{j=1}^J$ are two independent sequences of Bernoulli random variables we have $$\big(\mathbb E(\|\sum_{j=1}^J\sum_{k=1}^J
\epsilon_j\epsilon'_k|b_j|^{\frac12}|b_k|^{\frac12}W_{\sigma,Q}
(f_j,g_k)\|_{L_r}^r) \big)^{\frac1r} \le C\|\sigma\|_H^{(N)}.$$ Again by using the result of Bonami [@bonami], we obtain an estimate $$\|(\sum_{j=1}^J\sum_{k=1}^J |b_j||b_k|
|W_{\sigma,Q}(f_j,g_k)|^2)^{1/2}\|_{L_r} \le C\|\sigma\|_H^{(N)}.$$ Extracting the diagonal gives $$\big\|\max_{1\le j\le J}|b_j||W_{\sigma,Q}(f_j,g_j)|\big\|_{L_r} \le
C \|\sigma\|_H^{(N)}.$$ We now use [@Pisier] as before. There is a weight function $w\in
L_1(Q)$ with $w\ge 0$ a.e. and $\int w(x)\,dx=1$ so that for any $f,g\in
L_1(3Q)
$ with $\|f\|_{L_1},\|g\|_{L_1}\le 1$ and any measurable $E\subset Q$ we have $$\left (\int_E |W_{\sigma}(f,g)|^rdx\right)^{\frac1r} \le
C\|\sigma\|_H^{(N)} \left(\int_Ew(x)\,dx\right)^{\frac1r-2}.$$ Now suppose $f,g$ are supported in $Q$ and $\lambda>0.$ Let $E=\{x\in Q:
|W_{\sigma}(f,g)|>\lambda.$ Then the above equation yields $$\label{mmmooo}
\lambda |E|^{\frac1r} \le
C\|\sigma\|_H^{(N)}\left(\int_E w(x)\,dx\right)^{\frac1r-2}.$$ On the other hand if we apply (\[mmmooo\]) to $f_t(x)=f(x-t)$ where $t\in
Q$ and note that $W_{\sigma}(f_t,g)=(W_{\sigma}(f,g))_t$ we also obtain that $$\lambda |E\cap (Q+t)|^{\frac1r} \le C\|\sigma\|_H^{(N)}
\left(\int_Ew(x-t)\,dx\right)^{\frac1r-2}.$$ Raising to the power $(\frac1r-2)^{-1}$ and averaging gives: $$\lambda |E|^{\frac1r} \le C\|\sigma\|_H^{(N)} |E|^{\frac1r-2}.$$ Thus $W_{\sigma,Q}$ maps $L_1(2Q)\times L_1(2Q)$ into $L_{\frac12,\infty}(Q)$ with norm at most $C\|\sigma\|_H^{(N)}.$
Now let $\lambda>1$. If we define $\sigma_{\lambda}(\xi,\eta)=\sigma(\lambda^{-1}\xi,\lambda^{-1}\eta),$ then we have $\|\sigma_{\lambda}\|_H^{(N)}=\|\sigma_{\lambda}\|_H^{(N)}$ and we can apply this result to $\sigma_{\lambda}$. Notice that $W_{\sigma_{\lambda}}(f,g)(x)=W_{\sigma}(f_{\lambda},g_{\lambda})(\lambda
x)$ where $f_{\lambda}(x)=f(\lambda x)$ and $g_{\lambda}(x)= g(\lambda x).$ This implies that for any $\lambda>0$ we have the estimate $$\|\chi_{\lambda Q}W_{\sigma}(f,g)\|_{L_{\frac12,\infty}}\le
C\|\sigma\|_H^{(N)}\|f\|_{L_1}\|g\|_{L_1}$$ for $f,g$ supported in $\lambda Q$. Letting $\lambda\to\infty$ gives the result.
Discussion on paraproducts {#s-paraproducts}
==========================
0
Paraproducts are bilinear operators of the type $\sigma_A$ for some specific upper (or lower) triangular matrices $A$ of zeros and ones. Paraproducts are important tools which have been used in several occasions in harmonic analysis, such as in the proof of the $T1$ theorem of David and Journé [@DJ]. We define the lower and upper paraproducts as the bilinear operators $\Pi_L$ and $\Pi_U$ with symbols $$\tau_L(\xi,\eta)=\sum_{j\in\mathbb Z}
\sum_{k\le j-3}\widehat{\phi_j}(\xi)\widehat{\phi_k}(\eta)$$ and $$\tau_U(\xi,\eta)=\sum_{k\in\mathbb
Z}\sum_{j\le k-3}\widehat{\phi_j}(\xi)\widehat{\phi_k}(\eta)$$ respectively. It is easy to see that $\|\tau_L\|_{\mathcal M_{p_1,p_2}},\|\tau_U\|_{\mathcal M_{p_1,p_2}}<\infty$ for all $0<p_1,p_2<\infty$. This can be deduced in several ways, e.g. from Proposition \[subdiagonal\] using Lemma \[translation2\] or directly from Theorem \[main1\] and Proposition \[bv\]. We conclude that for all $0<p,q<\infty$ $\Pi_L$ maps $H_{p_1}\times H_{p_2}\to H_{p_0}$ when $1/p_1+1/p_2=1/p_0$ and $H_q=L_q$ when $1<q<\infty$. We now turn to some endpoint cases regarding the paraproduct operator $\Pi_L$.
Let $0<q<\infty$. Then the paraproduct operator $\Pi_L$ is bounded on the following products of spaces.
1. $ BMO\times H_q(\mathbb R^n)\to H_q(\mathbb R^n)$, where $H_q=L_q$ when $1<q<\infty$.
2. $ BMO\times H_1(\mathbb R^n)\to L_1(\mathbb R^n)$.
3. $ BMO\times L_{\infty}(\mathbb R^n)\to BMO$.
4. $ H_q(\mathbb R^n)\times L_{\infty}(\mathbb R^n)\to H_q(\mathbb
R^n)$, where $H_q=L_q$ when $1<q<\infty$.
5. $ L_1(\mathbb R^n)\times L_{\infty}(\mathbb R^n)\to
L_{1,\infty}(\mathbb R^n)$.
6. $ BMO\times L_1(\mathbb R^n)\to L_{1,\infty}(\mathbb R^n)$.
7. $ L_1(\mathbb R^n)\times L_1(\mathbb R^n)\to L_{1/2,\infty}(\mathbb
R^n)$.
Statement (1) is a classical result on paraproducts when $1<q<\infty$ and we refer the reader to [@stein-new] p. 303 for a proof. Note that for a fixed $f\in BMO$, the map $g\to \Pi_L(f,g)$ is a Calderón-Zygmund singular integral. The extension of (1) to $H_q$ for $q\le 1$, is consequence of the that if a convolution type singular integral operator maps $L_2\to L_2$ with bound a multiple of $\|f \|_{BMO}$, then it also maps $H_q$ into itself with bound a multiple of this constant. (2) follows from a similar observation while (3) is a dual statement to (2). To prove (4) set $\widetilde S_jg= \sum_{k\le j-3} \widetilde \Delta_k g$. We have that $\Pi_L(f,g)= \sum_{j \in \mathbb Z} \widetilde \Delta_j f
\widetilde S_jg$ and the Fourier transform of $\widetilde \Delta_j f
\widetilde S_jg$ is supported in the annulus $2^{j-2}\le |\xi| \le 2^{j+2}$. It follows that $$\| \Pi_L(f,g) \|_{H_q}\le C \big\| \big(\sum_{j \in \mathbb Z}
| \widetilde \Delta_j f
\widetilde S_jg |^2 \big)^{1/2}\big\|_{H_q} \le \|f\|_{H_q}\|Mg\|_{L_\infty},$$ where $M$ is the Hardy-Littlewood maximal operator which is certainly bounded on $L_\infty$. To prove (5) we freeze $g$ and look at the linear operator $f\to \Pi_L(f,g)$ whose kernel is $K(x,y)= \sum\limits_{j\in \mathbb Z} \phi_j(x-y) S_j(g)(x)$. It is easy to see that $$|\nabla_{y} K(x,y)|\le C\|g\|_{L_\infty} |x-y|^{-n-1}.$$ This estimate together with the fact that the linear operator $f\to \Pi_L(f,g)$ maps $L_2\to L_2$ gives that $f\to \Pi_L(f,g)$ maps $L_1\to L_{1,\infty}$ using the Calderón-Zygmund decomposition. This proves (5). To obtain (6) we use (1) (with $q=2$) and we apply to the Calderón-Zygmund decomposition to the operator $g\to \Pi_L(f,g)$ for fixed $f\in BMO$. Finally (7) is a consequence of Theorem \[L1\].
[WW]{}
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[^1]: The research of both authors was supported by the NSF
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{
"pile_set_name": "ArXiv"
}
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abstract: 'We study the relational variant of the categorical compositional distributional (DisCoCat) models of Coecke et al. [@ClarkEtAl08], where we replace vector spaces and linear maps by sets and relations. We show that RelCoCat models factorise through Cartesian bicategories, as a corollary we get logspace reductions from semantics and entailment to evaluation and containment of conjunctive queries respectively. Finally, we define question answering as an $\tt{NP-complete}$ problem.'
author:
- 'Giovanni de Felice$^\dagger$, Konstantinos Meichanetzidis$^{\dagger\star}$, Alexis Toumi$^\dagger$'
bibliography:
- 'discocat-complexity.bib'
title: Functorial Question Answering
---
Introduction {#introduction .unnumbered}
============
Lambek Pregroups and Free Rigid Categories
==========================================
Conjunctive Queries and Free Cartesian Bicategories
===================================================
RelCoCat Semantics and Natural Language Entailment
==================================================
Question Answering as an NP-complete Problem
============================================
We conclude with related work and potential directions for future work:
- text summarisation through conjunctive query minimisation [@ChekuriRajaraman00],
- semantics of “How many?” questions and counting problems [@StefanoniEtAl18],
- many-sorted RelCoCat models with graphical regular logic [@FongSpivak18a],
- from Boolean semantics to generalised relations in arbitrairy topoi [@CoeckeEtAl18],
- from regular logic to description logics in bicategories of relations [@Patterson17],
- comonadic semantics for bounded short-term memory [@AbramskyShah18],
- quantum speedup for question answering via Grover’s search [@ZengCoecke16].
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors would like to thank Bob Coecke, Dan Marsden, Antonin Delpeuch, Vincent Wang, Jacob Leygonie, Dusko Pavlovic, Rui Soares Barbosa and Samson Abramsky for inspiring discussions in the process of writing this article, as well as anonymous Reviewers of ACT2019 for constructive comments that improved the presentation of this work. K.M. is supported by the EPSRC NQIT Hub and Cambridge Quantum Computing Ltd. A.T. acknowledges Simon Harrison for financial DPhil support.
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{
"pile_set_name": "ArXiv"
}
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---
abstract: 'Two proof-of-principle experiments towards $T_1$-limited magnetic resonance imaging with NV centers in diamond are demonstrated. First, a large number of Rabi oscillations is measured and it is demonstrated that the hyperfine interaction due to the NV’s $^{14}$N can be extracted from the beating oscillations. Second, the Rabi beats under V-type microwave excitation of the three hyperfine manifolds is studied experimentally and described theoretically.'
author:
- 'H. Fedder'
- 'F. Dolde'
- 'F. Rempp'
- 'T. Wolf'
- 'P. Hemmer'
- 'F. Jelezko'
- 'J. Wrachtrup'
date: '2010-09-03'
title: 'Towards $T_1$–limited magnetic resonance imaging using Rabi beats'
---
Introduction
============
The ever increasing need for high resolution imaging in the life sciences, material science, and more recently quantum information processing, has led over the past decade to the development of a variety of methods that try to overcome the limits set by optical diffraction [@pertsinidis2010subnanometre]. Exploiting the non-linear behaviour of fluorescent labels, a spatial resolution down to few nanometers was demonstrated using optical techniques such as STED [@hell1994breaking], PALM [@betzig2006science-PALM], and STORM [@rust2006natmeth-STORM]. A technique whose resolution is independent of the wavelength, is magnetic resonance imaging. Magnetic resonance imaging –and nano scale magnetic- and electric field sensing– are key technologies in various areas of science. Recently, magnetic resonance imaging as well as scanning probe magnetic imaging on scales relevant to molecular biological processes was demonstrated [@balasubramanian2008nanoscale; @maze2008nanoscale]. A spin label that is suitable for high resolution measurements are nitrogen vacancy centers embedded in diamond nano crystals [@tisler2009fluorescence]. Owing to their optical adressability, photostability and long spin coherence time recently sub 10nm spatial resolution as well as the detection of magnetic fields close to individual electron spins has been demonstrated using NV nano sensors[@balasubramanian2008nanoscale; @maze2008nanoscale]. The spatial resolution of a magnetic resonance measurement is determined by the strength of the field gradient and the effective linewidth of the electron spin transition. Pulsed imaging modes, such as the Hahn-Echo and CPMG sequence can be used to decrease the effective linewidth. The continuous driving of Rabi oscillations has been proposed to decrease the effective linewidth down to the limit given by the spin decay time [@shin2009sub]. In this paper we perform a proof-of-principle study of this imaging mode. The feasibility of the method is demonstrated by resolving the hyperfine interaction of the NV center due to its $^{14}N$ nuclear spin.
In the simplest magnetic resonance imaging mode, a CW electron spin resonance (ESR) measurement is performed, and the effective linewidth is given by the spin dephasing time $T_2^*$, which is typically short and only of the order of few micro seconds in case of the NV center. In the Hahn-Echo sequence, $\pi/2-\pi-\pi/2$ pulses are used to refocus precessing spins, such that noise generated by slowly fluctuating spins in the environment is canceled. For the Hahn-Echo sequence, the effective linewidth is given by the dephasing time $T_2$ of the spin system, which is larger than $T_2^*$, and can reach several hundred microseconds in the case of the NV center [@balasubramanian2009ultralong]. More advanced pulse sequences, such as the CPMG sequence [@meiboom1958modified], hold promise to decrease the effective linewidth down to the limit set by $T_1^\rho$ [@PrinciplesOfPulsedElectronParamagneticResonance; @deLange2010single]. Another elegant imaging mode is based on the continuous driving of Rabi oscillations. The decay time of Rabi oscillations is given precisely by $T_1^\rho$ and depends on the microwave power. When the microwave power is high and the Rabi frequency approaches the transition frequency – corresponding to the limit of the rotating wave approximation– $T_1^\rho$ approaches $T_1$. Among the various magnetic resonance imaging modes, this mode may potentially achieve the longest decay time and highest resolution. Over the past years there has been a continued effort to apply this magnetic resonance imaging mode to the NV center [@shin2009sub]. However, this poses great challenges. In this mode, spatial information is extracted from beat frequencies observed in the Rabi oscillations. A high spatial resolution requires the measurement of a large number of Rabi oscillations. Here we demonstrate the feasibility of this imaging mode by realising the measurement of a large number ($>500$) of Rabi oscillations, and resolve the hyperfine splitting due to the NV’s $^{14}N$ nuclear spin from the Rabi beats. Moreover, we investigate experimentally and describe theoretically the observed Rabi beats in case of a V-type energy level scheme, which is the desired excitation mode in a high resolution experiment.
![Principle of magnetic resonance imaging based on Rabi beats.[]{data-label="fig:principle"}](RabiBeatImaging.pdf)
The principle of magnetic resonance imaging using Rabi beats is illustrated in Fig. \[fig:principle\]. We envision a sample, such as a living cell, that is tagged with nano diamonds that contain single NV-centers. The NV electron spins are inititalized and read out optically using a confocal microscope. An inhomogeneous microwave field – such as the one generated by a coplanar waveguide – is applied to the sample. The microwaves are matched to the $m_s=0\rightarrow \pm 1$ electron spin transition and drive Rabi oscillations between the groundstate spin sublevels. The Rabi frequency depends on the microwave power, thus encoding the position and angle within the microwave field. From an a priori knowledge of the microwave field (or from a reference measurement), the position and angle of the individual NV-centers with respect to the electrodes can be determined. To obtain the NV position in two spatial directions, two successive measurements are performed with microwave gradients applied in different directions using a two dimensional waveguide structure.
In this magnetic resonance imaging mode, the resolution is given by the decay time $T_1^\rho$ of the Rabi oscillations, the Rabi frequency $\Omega$, and the strength of the microwave gradient. The number $N$ of observable Rabi oscillations is linked to $T_1^\rho$ and the Rabi frequency as $$\frac{\Omega T_1^\rho}{2\pi}=N,$$\[eq:N\] For the present coplanar waveguide, the field gradient is determined by the width $G$ of the gap, and the resolution $\delta x$ follows approximately as $$\delta x=\frac{G}{N}.$$ The maximum achievable Rabi frequency is of the order of the Lamor frequency of the spin [@chiorescu2003coherent]. In this case, the decay time $T_1^\rho$ of the Rabi signal becomes $T_1$ and the resolution approaches $$\delta x=\frac{2\pi G}{T_1 \Omega},$$ where $\Omega=\omega=D\approx 2.88$GHz is equal to the electron spin transition. For NV centers, the $T_1$ time is typically few ms at room temperature [@redman1991spin], such that a ratio $T_1/\omega=10^{-6}$ could in principle be achievable. With $G=10\mu$m a spatial resolution of $0.01$nm would be reached. However, this poses stringent requirements on the stability of the applied microwave field. Both the power stability and the spatial stability of the microwave field must be at least as good as the ratio $T_1^\rho/\Omega$, and to achieve sub Angstrom resolution, the microwave wire should not drift by more than a fraction of an atomic diameter.
Results and Discussion
======================
![ESR spectrum of the NV center showing the hyperfine splitting of the $m_s=0\rightarrow -1$ spin level due to $^{14}N$. The spectrum was recorded with an applied DC magnetic field of about 150G. Detuning is denoted relative to the lowest hyperfine transition corresponding to 2.4524GHz.[]{data-label="fig:esr"}](NonDegenerateODMRannotated.pdf)
To demonstrate the observation of Rabi beats, we study a single NV-center in a fixed microwave field. Rabi beat measurements with a single NV center are possible owing to hyperfine interaction with the NV’s $^{14}N$ nuclear spin, that results in a splitting of the groundstate spin manifold into several hyperfine sublevels. Each hyperfine transition has a slightly different Rabi frequency, which results in a beat signal in the measured Rabi oscillations. These hyperfine beats will also be seen in high resolution Rabi beat imaging data as a modulation, and it is imortant to identify and assign them correctly. In here we are interested in the hyperfine interaction caused by the $^{14}N$ nucleus (nuclear spin $J=1$) of the NV center, which splits the $m_s=\pm 1$ ground state spin levels each into three hyperfine sublevels with an energy splitting of $\delta\approx 2.18$MHz. This energy splitting is seen in the CW electron spin resonance (ESR) measurement shown in Fig.\[fig:esr\]. In here, the ESR measurement is performed on the $m_s=0\rightarrow -1$ electron spin transition. Zero detuning corresponds to a driving microwave field of 2.4524GHz. Note, that a permanent magnetic field with a component along the NV axis of about 150G has been applied to shift the $m_s=0\rightarrow -1$ transition away from its zero field value (2.88GHz) owing to the Zeeman effect, and thus to separate it from the $m_s=0\rightarrow +1$ spin transition (which is shifted to higher frequency of 3.3GHz). This separation between the two electron spin transitions is necessary in order to ensure that the strong microwave field does not simultaneously drive both spin manifolds.
We now consider the Rabi beats expected for the given hyperfine splitting. For a driving microwave field that is detuned by a small frequency $\delta$ from the resonant transition, the Rabi frequency $\Omega$ is increased as compared to the resonant Rabi frequency $\Omega_0$, following [@scully01quantum_optics] $$\Omega=\sqrt{\Omega_0^2+\delta^2}.$$\[eq:rabi\] For small detuning ($\delta\ll\Omega_0$), the shift of the Rabi frequency $\delta\Omega$ becomes $$\delta\Omega=\frac{\delta^2}{2\Omega_0}.$$\[eq:detuning\] Suppose the microwave frequency corresponds to the lowest hyperfine transition (as in the present experiments). In this case, the lowest hyperfine transition is driven with Rabi frequency $\Omega_0$, while the two other hyperfine transitions are driven with faster Rabi frequencies corresponding to detunings $\delta_0=2.18$MHz and $\delta_{-1}=4.36$MHz. Using Eq.\[eq:rabi\] we can evaluate the expected Rabi frequencies. The resulting signal is the sum of three oscillations, whose frequencies can be extracted by Fourier transformation. Also, the three frequencies result in several beat frequencies in the Rabi signal.
![Electron spin Rabi oscillations. Solid blue line: experimental data, solid magenta line: result from Fourier transform (Fig.\[fig:fft\]). Three beating cosine with frequencies corresponding to the three local maxima in the FFT are plotted.[]{data-label="fig:rabi"}](NonDegenerateRabiLong.pdf)
![Fourier transform of the Rabi signal. The data shows three distinct Rabi frequencies corresponding to the individual hyperfine levels.[]{data-label="fig:fft"}](NonDegenerateFFTannotated.pdf)
To measure Rabi beats, a coplanar waveguide (gap=$10\mu$s, width=$10\mu$s) was fabricated directly onto a type IIa diamond sample with $\langle 100\rangle$ crystallographic orientation and less than 4ppb nitrogen concentration (element6, electronic grade). A DCmagnetic field was applied with a permanent magnet. A microwave field resonant to the lowest of the hyperfine transitions (2.4524GHz) was generated with a synthesizer (Rhode&Schwarz SMIQ) and applied to the sample through a fast switch (minicircuits ZASW-2-50DR). A single NV center was identified in the coplanar gap using a home built confocal fluorescence microscope. Rabi oscillations were measured using a standard laser- and microwave pulse sequence [@jelezko2004observation]. Figure \[fig:rabi\] shows the corresponding Rabi oscillations. The data shows a base oscillation of about $42$MHz (see inset). Due to the beating this is approximately twice the base Rabi frequency. The two beat frequencies $\delta\Omega_{0}=100$kHz and $\delta\Omega_{+1}=400$kHz correspond to the $m_s=0$ and $m_s=+1$ hyperfine transition. Thus in total three Rabi oscillations are observed. Their individual frequencies are seen in the Fourier transform (Figure \[fig:fft\]), that shows $\Omega_{-1}=22.2$MHz, $\Omega_0=22.3$MHz, and $\Omega_{+1}=22.6 $MHz. Note that the FFT data is less pronounced due to the relatively sparse sampling of less then 10 points per period (see also zoom in Fig. \[fig:rabi\]) and power drift during the measurement. The sparse sampling was necessary to keep the total measurement time (several days for Fig. \[fig:rabi\]) within reasonable bounds. To complete our measurment of the hyperfine splitting from the Rabi beats, we use Eq.\[eq:detuning\] and convert the measured beats into energy level shifts. We find $\delta_{0}=2.1$MHz and $\delta_{-1}=4.2$MHz, which is in good agreement with the ESR spectrum (Fig.\[fig:esr\]).
The Rabi oscillations show a decay $T_1^*\approx 25\mu$s. This decay is much much smaller than the expected $T_1^\rho\approx 1$ms and is limited by microwave power drift as we show below. With the present decay $T_1^*$, the resolution of the hyperfine measurement is as follows. The resolution of a hyperfine measurement from the Rabi beats is determined by the Rabi frequency and the number $N$ of visible Rabi oscillations. To measure a small change in the Rabi period, we need to measure a number of oscillations. Combining Eq.(\[eq:N\]) with Eq.(\[eq:detuning\]), we have $$\delta=\frac{\Omega}{\sqrt{N}}=\frac{2\pi}{\sqrt{TT_1^*}}.$$ In our case we find $\delta\approx 6$MHz, which is in good qualitative agreement with our observations, since our data, in particular the Fourier transform, Fig.\[fig:fft\], implies that we can indeed resolve the 4.32MHz detuning related to the farther detuned hyperfine level (22.6MHz peak in the FFT), however, we can just barely resolve the 2.16MHz detuning related to the lesser detuned hyperfine level (22.3MHz peak in the FFT). In the present measurements the total number of Rabi oscillations is about 500, which translates into an equivalent spatial resolution of about 10nm. We expect that an improvement of the power stability to a level better than $10^{-4}$ should be achievable without a large technical effort, and that spatial drifts of the microwave field should still be negligable in this case. Thus a spatial resolution of about 1nm should be achievable in a realistic measurement.
Any fluctuation or drift of the applied microwave power result in a small shift of the Rabi frequency over time, that washes out the oscillations. To proof that the observed decay is caused by power drift, we monitor the power transmitted through the sample and compare it to the Rabi period. This measurement is realized by successively performing short Rabi measurements in a small window ranging from $25.0\mu$s to $25.1\mu$s. The result is shown in Figure \[fig:power\]. Since the Rabi period is inversely proportional to the frequency and the frequency is proportional to the square root of the power, we can express the relation between the relative change of the Rabi period as $$\frac{\delta T}{T}=-\frac{1}{2}\frac{\delta P}{P}.$$ The data confirms that the decay time of the Rabi oscillations is limited by drift of the microwave power, which is slightly better than $10^{-3}$ over 24h in our case. Note that we also observed a change of the transmitted microwave power when the microscope objective was moved (about $10^{-3}/\mu$m). This is explained by a change of the parasitic capacitance of the coplanar waveguide, which plays a role owing to the close proximity of the high N.A. microscope objective to the sample (working distance $200\mu$m).
![Power drift. The relative change of the Rabi period is compared to the relative change of the microwave power.[]{data-label="fig:power"}](PowerDriftAnnotated.pdf)
With the fast Rabi frequency required for a high spatial resolution, it is no longer feasible to shift the $m_s=\pm 1$ spin levels sufficiently far apart from each other such that a single spin transitions is driven selectively. This is because the separation between the $m_s=\pm 1$ levels needed to be larger than the Rabi frequency. This would in turn require a large magnetic field. However at large magnetic field, level mixing causes the optical ESR contrast to vanish, unless the magnetic field is aligned parallel to the NV axis [@neumann2010single]. Alignment of the magnetic field parallel to the NV axis is however not possible in a sample with many randomly distributed NV centers. Thus, in a high resolution measurement, we will always measure with a small DC bias field and hence we will simulstaneously drive the $m_s=\pm 1$ spin transitions. In this case, six hyperfine levels are present whose Rabi oscillations beat with each other. This situation differs from the previous experiments. While in the previous case, the Rabi signal was an incoherent sum of three oscillations (during a measurement, the nuclear spin switched randomly and slowly between the different spin states), now each nuclear spin manifold forms a V-type energy level scheme that is driven simultaneously. In the following, we evaluate the corresponding Rabi beat signal.
The NV’s spin Hamiltonian with external magnetic field reads $$H={{\bf H}}_\text{ZFS}+{{\bf H}}_\text{Z}$$ with $$\begin{aligned}
{{\bf H}}_\text{ZFS}&=D\,({{\bf S}}_z^2-\frac{2}{3}\,{\mathrm{1}})+E\,({{\bf S}}_x^2-{{\bf S}}_y^2)\\
{{\bf H}}_\text{Z}&=g\,\beta\,B\,{{\bf S}}_z.\end{aligned}$$ After transforming into the energy Eigenbasis where we name the $m_s=0$ state as ${\left| 0 \right\rangle}$ and the two state that correspond to the $m_s=\pm 1$ states as ${\left| 1 \right\rangle}$ and ${\left| 2 \right\rangle}$. Now the semiclassical microwave field is applied $${{\bf H}}_\text{MW}=\lambda\,{\textup{e}^{{\textup{i}}\,\omega\,t}}\,{{\bf S}}_x.$$ here we assume, that both transitions have the same transition matrix elements, which breaks down with rising $E$. Now do the rotating wave approximation (RWA) and transform into the rotating frame which leaves us with a Hamiltonian of the form $${{\bf H}}_\text{rot}=\left(\begin{array}{ccc}
0 & \lambda & \lambda\\
\lambda & \Delta-\delta & 0\\
\lambda & 0 & \Delta+\delta
\end{array}\right).$$ where $\delta=\Delta E_{{\left| 1 \right\rangle}\,{\left| 2 \right\rangle}}/2$ and $\Delta$ the detuning of the microwave from ${\left| 1 \right\rangle}+\delta$.
To obtain an analytical solution we set $\Delta=0$. Calculation of the Eigenvalues leads to the Rabi frequency $\sqrt{2\,V^2+\delta^2}$ which, were the two levels detuned is $\sqrt{2}$ times the Rabi frequency of driving a single level. We now take a look at the time evolution of the system with initial state ${\left| 0 \right\rangle}$. The dynamics of ${\left| 0 \right\rangle}$ are $$\rho_{{\left| 0 \right\rangle}}=\frac{\left(\delta^2+2\,\lambda^2\,\cos{\left[\sqrt{2\,\lambda^2+\delta^2}\,t\right]}\right)^2}{(2\,\lambda^2+\delta^2)^2}.$$ from here we see, that there are actually two oscillation frequencies involved, the Rabi frequency and double the Rabi frequency corresponding to the $\cos{}^2$ term. The latter will be the base Rabi frequency observed in experiments, and we find the identity for the Rabi beat as $$\delta\Omega=\frac{2\delta^2}{\Omega_0},$$\[eq:DetuningV\] which is different from the previous expression, Eq.(\[eq:detuning\]).
![ESR spectrum of the second NV center without DC bias field. The two hyperfine triplets corresponding to the $m_s=-1$ and $m_s=+1$ spin levels overlap in the central dip. Detuning is denoted relative to the central hyperfine transition corresponding to 2.8706GHz.[]{data-label="fig:DegenerateESR"}](DegenerateESRannotated.pdf)
To measure Rabi beats with V-type microwave excitation, we remove the DC bias field. Moreover we pick a different NV center that is closer to the central conductor and feels a higher microwave power. Figure \[fig:DegenerateESR\] shows the ESR spectrum, with a total of five dips. The central dip is about twice larger as compare to the other dips. In this experiment, the Zeeman splitting due to a residual magnetic field (earth magnetic field as well as magnetized parts of the setup) closely matches the hyperfine splitting. As a consequence, the two hyperfine triplets corresponding to the $m_s=-1$ and $m_s=+1$ spin transition are shifted in such a way that the highest peak of the lower triplet overlaps with the lowest peak of the higher triplet, resulting in five dips with a pronounced central dip. Rabi oscillations are now driven with the microwave frequency tuned in resonance with the central dip corresponding to 2.8706GHz. This case corresponds to the theoretical case $\Delta=0$ treated above, with $\delta_{0}=2.18$MHz and $\delta_{+1}=4.36$MHz for the driving of the $m_j=0$ and $m_j=+1$ manifold, respectively.
![Rabi oscillations under V-type microwave excitation. Solid blue line: experimental data. Red and green markers denote the extracted beat frequencies. Solid magenta line: result of the extracted beat frequencies. Three beating cosine with the corresponding frequency shifts are plotted.[]{data-label="fig:DegenerateRabi"}](DegenerateRabiAnnotated.pdf)
![Fourier transform of the Rabi oscillations under V-type microwave excitation. Peaks around 42MHz and 21MHz correspond to the base Rabi oscillations and a modulation at half the frequency. The inset shows the beating Rabi frequencies around 42MHz. Labels mark the three frequencies as extracted from the beat analysis of the raw data. The FFT resolves the fast beat, while it does not resolve the slow beat due to sparse sampling.[]{data-label="fig:DegenerateFFT"}](DegenerateFFTannotated.pdf)
The Rabi oscillations and their Fourier transform are shown in Figure \[fig:DegenerateRabi\] and \[fig:DegenerateFFT\]. The first observation is that the Rabi frequency is increased by about a factor of two as compared to the previous case. Two effects contribute here. First, with lambda type driving the Rabi frequency is enhanced by a factor of $\sqrt{2}$ as compared to driving a single transition, and second, the Rabi frequency is additionally increased due to the higher microwave power in the close vicinity of the central conductor. The second observation is a fast and a slow beat with frequency of about 185kHz and 812kHz, respectively. These beats are seen most clearly in the Rabi oscillations. Note that the slower beat is not resolved in the FFT (Fig.\[fig:DegenerateFFT\]) due to power drift and sparse sampling. We can now convert the observed beatings to energy level shifts according to Eq.(\[eq:DetuningV\]). We find $\delta_{0}=2.0$MHz and $\delta_{+1}=4.1$MHz, which is in good agreement with the ESR spectrum.
Summary and Conclusions
=======================
We have performed two proof-of-principle experiments towards $T_1$ limited magnetic resonance imaging with NV centers in diamond. First, we have demonstrated the measurement of a large number ($>500$) of Rabi flops, and we have shown that the hyperfine interaction due to $^{14}$N can be resolved from such a measurement. Second, we have studied the Rabi beats without a large DC bias field, where the nuclear spin manifolds form V-type energy level schemes. The base Rabi frequency is increased by $\sqrt{2}$ and the time evolution of the state population is modulated by an oscillation with half the base Rabi frequency. While the present experiments were performed in bulk, in the future, it will be an important step to demonstrate Rabi beat imaging with NV centers embedded in diamond nano crystals. The present experiments are relevant to quantum information processing in diamond. The ability to drive a large number of Rabi flops could allow to precisely control the spin state of several (detuned) NV centers simultaneously. We envision a microwave pulse with precisely adjusted length that performs independent unitary transformations on a quantum register, such as rotate an NV A to $|1\rangle$ while simultaneously rotating an NV B to the superposition $\sqrt{2}(|0\rangle+|1\rangle$.
Acknowledgements
================
This work was supported by the European Union, Deutsche Forschungsgemeinschaft (SFB/TR21 and FOR1482), Bundesministerium für Bildung und Forschung, and the Landesstiftung Bandenwürttemberg.
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{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We determine the phase diagram of $N$ identical three-level systems interacting with a single photonic mode in the thermodynamical limit ($N \to \infty$) by accounting for the so-called diamagnetic term and the inequalities imposed by the Thomas-Reich-Kuhn (TRK) oscillator strength sum rule. The key role of transitions between excited levels and the occurrence of first-order phase transitions is discussed. We show that, in contrast to two-level systems, in the three-level case the TRK inequalities do not always prevent a superradiant phase transition in presence of a diamagnetic term.'
author:
- Alexandre Baksic
- Pierre Nataf
- Cristiano Ciuti
title: 'Superradiant phase transitions with three-level systems'
---
The collective superradiant coupling of a large number of atoms (or artificial atoms) has attracted a considerable interest since the pioneering work of Dicke[@Dicke] and is now the focus of many recent studies in cavity[@Dimer; @esslinger; @Nagy; @Baumann; @Bhaseen] and circuit[@NatafPRL1; @NatafNAT; @NatafPRL2] quantum electrodynamics. The well-known Dicke model describes the coupling between a collection of two-level systems and a single photon mode. Remarkably, for increasing light-matter coupling such a model predicts a superradiant phase transition [@Lieb; @Carmichael; @Brandes], with a doubly degenerate ground state above a critical vacuum Rabi coupling. The so-called superradiant phase is characterized by a spontaneous polarization of the atoms and a spontaneous coherence of the cavity field in the ground state. In the case of time-independent Hamiltonians, photons in the ground state are ’virtual’ (i.e., bound in the cavity) and they cannot be radiated out of the cavity[@CiutiPRA2006] unless some non-adiabatic modulation of the Hamiltonian is applied[@DeLiberatoPRL2007] (in analogy with the dynamical Casimir effect[@Casimir]). In this sense, the term ’superradiant’ currently used for the Hepp-Lieb[@Lieb] ground state is unfortunate, because it was originally introduced by Dicke[@Dicke] for collective excited state radiative decay and not for ground state properties. Even in spite of no extracavity emission from the ground state, occurrence of a superradiant critical point can be in principle monitored by measuring the dispersion of the collective excitations via standard optical techniques (e.g., through transmission spectroscopy). In the case of non-equilibrium phase transitions for pumped open systems, there is not a true ground state[@Dimer; @esslinger; @Nagy; @Baumann; @Bhaseen], but instead a stationary state, which is accompanied by the emission of real photons.
However, for the case of electric dipole transitions, the Dicke model does not include the so-called diamagnetic term, which is proportional to the squared electromagnetic vector potential present in the minimal coupling Hamiltonian describing light-matter interaction in the non-relativistic regime. It is known that in the case of two-level real atoms, such a diamagnetic term is crucial, because it forbids the phase transition as a result of the TRK oscillator strength sum rule[@pol1; @NatafNAT]. We point out that such no-go theorem cannot be necessarily applied to time-dependent Hamiltonians with applied dressing fields[@esslinger], to the case of magnetic dipole coupling[@Knight] and to more complex effective systems simulating the Dicke Hamiltonian with different degrees of freedom[@esslinger; @Nagy]. Moreover, the no-go theorem[@pol1; @NatafNAT] is formulated for two-level systems, while the study of the multilevel case has been initiated only recently[@Viehmann; @Ciuti; @Hayn]
A generalized model including three-level atoms in the lambda configuration coupled to two photon modes has been theoretically investigated in a recent paper[@Hayn] showing a very rich phase diagram with superradiant transitions of both second and first order (in the case of two-level atoms the superradiant transition is of second order). In such an interesting work[@Hayn], the diamagnetic term has not been included and the particular two-color lambda configuration has been considered, so the general case of an arbitrary three-level system needs to be explored. In a recent letter[@Viehmann], it has been claimed that for multilevel atoms coupled to a single photon mode the no-go theorem can be generalized using again the TRK oscillator strength sum rule: in the proof reported in such a work[@Viehmann], through a perturbative argument, it is assumed that transitions between excited states can be always neglected in the thermodynamical limit: this is rather surprising[@Ciuti], since in the work by Hayn et al.[@Hayn], transitions between excited states play instead a crucial role in the thermodynamical limit and are responsible for the first-order transition boundaries. It is therefore a fundamental problem to explore superradiant phase transitions with arbitrary three-level systems.
In this letter, we investigate the existence of superradiant phase transitions of a system consisting of $N$ three-level systems coupled to a single photon mode including the diamagnetic term in the Hamiltonian and TRK inequalities. We show the rich phase diagram in the thermodynamical limit ($N \to \infty$), obtained via a multilevel Holstein-Primakoff approach. We have studied particular configurations (ladder, lambda, V-type) and also the general three-level coupling configuration. We find that for three-level systems TRK inequalities do not always prevent superradiant phase transitions and that excited state transitions play a crucial role in the thermodynamical limit.
![A sketch of the considered system consisting of $N$ identical three-level atoms identically coupled to a single cavity mode. Each three-level system is represented by the transition frequencies $\omega_{10} = \omega_1-\omega_0 > 0$, $\omega_{21} = \omega_2-\omega_1 > 0$, $\omega_{20} = \omega_{21}+\omega_{10} > 0$ and by the oscillator strengths $f_{01} \geq 0$, $f_{12} \geq 0$, $f_{02} \geq 0$ (see definition (\[f\]) in the text). \[shapes\]](Configurations2){width="150pt"}
![\[ladder\] Results for the [*ladder*]{} configuration ($f_{02}=0$). Photonic order parameter $x = \alpha/\sqrt{N} = \langle a \rangle/\sqrt{N}$ in the ground state as a function of the oscillator strengths $f_{01}$ and $f_{12}$ for each three-level atom. The normal phase (N) is characterized by $x = 0$ (black color). In the superradiant phase (SR), there is a spontaneous coherence $x \neq 0$. Note that a discontinuous jump of $x$ denotes a first-order superradiant phase transition, otherwise the transition is of second order. The area below the red-dashed lines indicate the region described by the TRK inequalities $0 \leq f_{01} \leq 1$ and $0 \leq f_{10} + f_{12} \leq 1$ where $f_{10} = - f_{01}$. Parameters: $D = 3 \omega_{cav}$, $\omega_{10} = 0.1 \omega_{cav}$, $\omega_{21} = \omega_{cav}$. The collective vacuum Rabi frequency $\Omega_{01}$ and $\Omega_{12}$ are obtained through the relationship in Eq. (\[D\]). ](ladder.eps){width="200pt"}
![\[V\] V-type configuration ($f_{12} =0$). Photonic order parameter as a function of the oscillator strengths $f_{01}$ and $f_{02}$. The red-dashed lines indicate the boundaries imposed by the TRK sum rule, namely $0 \leq f_{01} + f_{02} \leq 1$. Parameters: $D = \omega_{cav}$, $\omega_{10} = \omega_{cav}$, $\omega_{21} = 0.1 \omega_{cav}$. In this configuration, the superradiant phase is always incompatible with the TRK boundaries. ](V.eps){width="200pt"}
![\[Lambda\] Results for the [*lambda*]{} configuration ($f_{01} =0$). Photonic order parameter versus $f_{12}$ and $f_{02}$. The area below the red-dashed lines is compatible with the TRK inequalities, namely $0 \leq f_{12} \leq 1$ and $0 \leq f_{02} \leq 1$. Parameters: $D = 3 \omega_{cav}$, $\omega_{10} = 0.1 \omega_{cav}$, $\omega_{21} = 0.9 \omega_{cav}$. As in the ladder case, the superradiant part of the diagram has an overlap with the region compatible with the TRK inequalities. ](Lambda.eps){width="200pt"}
As sketched in Fig. 1, let us consider $N$ identical three-level systems, whose states are $\left \{ \vert 0_k\rangle \hbox{,} \vert 1_k\rangle \hbox{,} \vert2_k\rangle \right \}\hbox{ , }( k=1,2,..,N) $ where $k$ is the atomic index. Each atom is assumed to be independent from the others and to interact identically with a single bosonic, photonic mode. The energies of the three levels are $\hbar \omega_0< \hbar \omega_1< \hbar \omega_2$.
![ Top panel: the filled red volume in the $(f_{01},f_{02},f_{12})$ space represents the superradiant part of the diagram compatible with the TRK inequalities $0\leq f_{01} + f_{02} \leq 1$ and $0\leq f_{10} + f_{12} \leq 1$. Parameters: $D = 5 \omega_{cav}$, $\omega_{10} = 0.17 \omega_{cav}$, $\omega_{21} = \omega_{cav}$. Middle panel: the photonic order parameter $x$ as a function of $f_{01}$ and $f_{02}$ for a fixed value $f_{12} = 0.735$. The area below the red-dashed line corresponds to the region satisfying the TRK inequalities. Bottom panel: the black line represents the spatial profile of an illustrative artificial one-dimensional potential with squared wells (as those realizable with semiconductor heterostructures) in units of the energy $E_c = \frac{\hbar^2}{2 m L^2}$ where the spatial coordinate is expressed in units of the length $L$. The horizontal lines depict the energies of the three bound states with their corresponding wavefunctions (other states are in the continuum). For this potential shape, one obtains the oscillator strengths $(f_{01} = 0.3995 , f_{02} = 0.4069, f_{12}=0.735)$ and anharmonicity ratio $\omega_{10}/\omega_{21}= 0.1709 $ corresponding to the point depicted by the blue cross in the middle panel. \[SRVol\]](SRvol "fig:"){width="200pt"}\
![ Top panel: the filled red volume in the $(f_{01},f_{02},f_{12})$ space represents the superradiant part of the diagram compatible with the TRK inequalities $0\leq f_{01} + f_{02} \leq 1$ and $0\leq f_{10} + f_{12} \leq 1$. Parameters: $D = 5 \omega_{cav}$, $\omega_{10} = 0.17 \omega_{cav}$, $\omega_{21} = \omega_{cav}$. Middle panel: the photonic order parameter $x$ as a function of $f_{01}$ and $f_{02}$ for a fixed value $f_{12} = 0.735$. The area below the red-dashed line corresponds to the region satisfying the TRK inequalities. Bottom panel: the black line represents the spatial profile of an illustrative artificial one-dimensional potential with squared wells (as those realizable with semiconductor heterostructures) in units of the energy $E_c = \frac{\hbar^2}{2 m L^2}$ where the spatial coordinate is expressed in units of the length $L$. The horizontal lines depict the energies of the three bound states with their corresponding wavefunctions (other states are in the continuum). For this potential shape, one obtains the oscillator strengths $(f_{01} = 0.3995 , f_{02} = 0.4069, f_{12}=0.735)$ and anharmonicity ratio $\omega_{10}/\omega_{21}= 0.1709 $ corresponding to the point depicted by the blue cross in the middle panel. \[SRVol\]](general "fig:"){width="180pt"}\
![ Top panel: the filled red volume in the $(f_{01},f_{02},f_{12})$ space represents the superradiant part of the diagram compatible with the TRK inequalities $0\leq f_{01} + f_{02} \leq 1$ and $0\leq f_{10} + f_{12} \leq 1$. Parameters: $D = 5 \omega_{cav}$, $\omega_{10} = 0.17 \omega_{cav}$, $\omega_{21} = \omega_{cav}$. Middle panel: the photonic order parameter $x$ as a function of $f_{01}$ and $f_{02}$ for a fixed value $f_{12} = 0.735$. The area below the red-dashed line corresponds to the region satisfying the TRK inequalities. Bottom panel: the black line represents the spatial profile of an illustrative artificial one-dimensional potential with squared wells (as those realizable with semiconductor heterostructures) in units of the energy $E_c = \frac{\hbar^2}{2 m L^2}$ where the spatial coordinate is expressed in units of the length $L$. The horizontal lines depict the energies of the three bound states with their corresponding wavefunctions (other states are in the continuum). For this potential shape, one obtains the oscillator strengths $(f_{01} = 0.3995 , f_{02} = 0.4069, f_{12}=0.735)$ and anharmonicity ratio $\omega_{10}/\omega_{21}= 0.1709 $ corresponding to the point depicted by the blue cross in the middle panel. \[SRVol\]](potential "fig:"){width="170pt"}
By introducing the collective operators $
\hat{\Sigma}_{i j}=\sum_{k =1}^{N} \vert i_{k} \rangle\langle j_{k} \vert $, the light-matter Hamiltonian reads $$\begin{array}{cc}
\label{H}
\mathcal{H}/\hbar= & \omega_{cav} \hat{a}^{\dagger}\hat{a}+ \sum_{j=0}^2 \omega_j \hat{\Sigma}_{jj} + D(\hat{a}+\hat{a}^{\dagger})^2 \\
& +\sum_{\substack{i,j=0\\(i \neq j)}}^2 \Omega_{ij} \frac{1}{\sqrt{N}} (\hat{\Sigma}_{ij}+\hat{\Sigma}_{ji})(\hat{a}+\hat{a}^{\dagger}).
\end{array}$$ The cavity mode is described by the frequency $\omega_{cav}$ and by its creation (annihilation) bosonic operator $a^{\dagger}$ ($a$). The coupling between the cavity mode and the atomic $i \to j$ transition is quantified by the collective vacuum Rabi frequency $\Omega_{ij}$. $\Omega_{ij}/\sqrt{N}$ is the light-matter coupling per atom. The term proportional to $D$ and quadratic in the photon operators is the so-called diamagnetic term. It originates from the $\frac{(\mathbf{\hat{p}} -q \mathbf{\hat{A}})^2}{2m} $ form of the non-relativistic electron-light interaction in the so-called $\mathbf{\hat{p}}\cdot \mathbf{\hat{A}}$ gauge, where $\mathbf{\hat{p}}$ is the electron momentum operator, $q$ the electron charge and $\mathbf{\hat{A}}$ is the electromagnetic vector potential operator. As it can be deduced by the treatment in Ref. [@NatafNAT], the value of the vacuum Rabi frequencies $\Omega_{ij}$ are linked to the diamagnetic term amplitude via the oscillator strengths as follows: $$\label{D}
\frac{\Omega_{ij}^2}{\omega_{ji}} = f_{ij} D$$ where $\omega_{ji} = \omega_{j}-\omega_i$ is the transition frequency and $$\label{f} f_{ij} = \frac{2 m}{\hbar} (\omega_j -\omega_i) \vert d_{ij} \vert^2$$ is the transition oscillator strength with $d_{ij}$ the corresponding atomic electric dipole matrix element (along the direction of the photon mode polarization). Indeed, as shown in Ref. [@NatafNAT] , $\hbar D = \frac{q^2}{2 m} n_{el} N \mathbf{A}^2_0$ where $q$ is the electron charge, $n_{el}$ the number of electrons per atom and $\mathbf {\hat{A}} = {\mathbf A}_0 (a + a^{\dagger})$ is the electromagnetic vector potential in the position where the atoms are assumed to be coupled (the spatial variation of the cavity field is neglected in the region occupied by the atoms, i.e., the electric dipole approximation has been considered). For a given collection of atoms and for a given cavity, $D$ is a constant depending on the density of atoms in the cavity volume via the factor $N \mathbf{A}^2_0$. A related similar model can be obtained by considering a two-dimensional electron gas in a semiconductor quantum well heterostructure[@Ciuti2005; @Anappara2; @Anappara; @Todorov] (replacing the bare electron mass with the semiconductor conduction band effective mass).
The TRK oscillator strength reads $\sum_{j} f_{i j} = 1$. Note that by definition the oscillator strengths $f_{ij} > 0$ if $\omega_{j} > \omega_i$, while it is negative if $\omega_{j} < \omega_i$. Hence, for a transition from the ground level $\vert 0\rangle$ to the first excited level $\vert 1 \rangle$, we have always $0 \leq f_{01} \leq 1 $ as $f_{0j} \geq 0$. For two-level system models, the presence of the diamagnetic term and the constraint $0 \leq f_{01} \leq 1$ is sufficient to prevent the superradiant phase transition (no-go theorem)[@pol1; @NatafNAT]. Here, we will analyze what happens with three-level systems. By applying the method detailed in Ref. [@Molmer] , we can express the generalized collective transition operators by using a multilevel Holstein-Primakoff transformation in terms of bosonic annihilation (creation) operators $\hat{c}_k$ ( $\hat{c}^{\dagger}_k$). When we use this procedure for the three-level case, we have to choose a state of reference that we will call $r$. The multilevel boson mapping reads: $$\begin{aligned}
& \hat{\Sigma}_{kk}=\hat{c}^{\dagger}_k\hat{c}_k\phantom{...}(k\neq r) \hspace{0.5cm} \hbox{,} \hspace{0.5cm}\hat{\Sigma}_{rr}=N-\sum_{k\neq r} \hat{c}^{\dagger}_k\hat{c}_k \nonumber \\
& \hat{\Sigma}_{ir}=\hat{c}^{\dagger}_i\sqrt{N-\sum_{k\neq r} \hat{c}^{\dagger}_k\hat{c}_k} \hspace{0.5cm} \hbox{,} \hspace{0.5cm} \hat{\Sigma}_{ij}=\hat{c}^{\dagger}_i\hat{c}_j\phantom{...}(i,j\neq r) \,.\end{aligned}$$ The collective transition operators $\hat{\Sigma}_{ij}$ defined above do not commute and are such that $\left[ \hat{\Sigma}_{ij},\hat{\Sigma}_{kl}\right]=\delta_{jk}\hat{\Sigma}_{il}-\delta_{il}\hat{\Sigma}_{kj}$. The Hamiltonian can be rewritten as $$\begin{aligned}
\mathcal{H}/\hbar = & \omega_{cav} \hat{a}^{\dagger}\hat{a}+ \sum_{k\neq r}(\omega_k-\omega_r) \hat{c}_k^{\dagger}\hat{c}_k+ \omega_r N \nonumber \\
&+ \Bigg[\sum_{k \neq r} \frac{\Omega_{kr}}{\sqrt{N}} \left(\hat{c}_k^{\dagger}\sqrt{N-\sum_l \hat{c}^{\dagger}_l\hat{c}_l} +\sqrt{N-\sum_l \hat{c}^{\dagger}_l\hat{c}_l \phantom{..}} \hat{c}_k \right) \nonumber \\
& + \sum_{\substack{i>j\\(i,j \neq r)}}\frac{\Omega_{ij}}{\sqrt{N}} (\hat{c}_i^{\dagger}\hat{c}_j + \hat{c}_j^{\dagger}\hat{c}_i) \Bigg](\hat{a}+\hat{a}^{\dagger}) + D(\hat{a}+\hat{a}^{\dagger})^2 \label{general}\end{aligned}$$
In the following, we choose $r = 1$ as reference state (choosing another reference state leads to the same results). In order to determine the phase diagram in the thermodynamical limit, we can use the mean-field approach as in Refs. [@Brandes; @Hayn], by the replacement $\langle a \rangle=\langle a^{\dagger}\rangle=\alpha$ , $\langle c_j \rangle=\langle c_j^{\dagger}\rangle=\beta_j$ with $j \in \{0,2 \} $. It can be shown that due to the form of Eq. (\[H\]) the solutions for $\alpha$ and $\beta_j$ are necessarily real numbers. Therefore, we obtain the mean-field expression of the ground state energy (\[general\]) $$\begin{aligned}
E_G/\hbar= & (\omega_{cav}+4D)\alpha^2+\omega_{01}\beta_0^2+\omega_{21}\beta_2^2+\omega_1 N \nonumber \\
& +4\alpha\left[(\Omega_{10}\beta_0+\Omega_{21}\beta_2)\sqrt{N-\beta_0^2-\beta_2^2}+\Omega_{20}\beta_0\beta_2\right],\end{aligned}$$ whose global minimization will give the values of $\alpha$ (photonic coherence), $\beta_0$ and $\beta_2$ (collective atomic coherences) in the ground state. A photonic order parameter $\alpha = 0$ implies that the system in the Normal (N) phase. If $\alpha \neq 0$, a SuperRadiant (SR) phase occurs[@Brandes]. In order to minimize the ground state energy in the thermodynamical limit ($N \to \infty $) it is convenient to introduce the rescaled quantities $x=\frac{\alpha}{\sqrt{N}}$ , $y=\frac{\beta_0}{\sqrt{N}}$ and $z=\frac{\beta_2}{\sqrt{N}}$. The ground state energy can therefore be rewritten as $$\begin{aligned}
\frac{E_G/\hbar}{N}= & (\omega_{cav}+4D)x^2+\omega_{01}y^2+\omega_{21}z^2+\omega_1\nonumber \\
& +4x\left[(\Omega_{10}y+\Omega_{21}z)\sqrt{1-y^2-z^2}+\Omega_{20}yz\right] \label{EGxyz} \, .\end{aligned}$$ As $\frac{\partial^2 E_G}{\partial x^2} = 2 (\omega_{cav} + 4 D) > 0 $, the minimization with respect to $x$ is therefore trivial, giving $x$ as a function of $y$ and $z$. In order to minimize the ground state energy, we therefore need to minimize with respect to $y$ and $z$ the following energy function (obtained from Eq. (\[EGxyz\])):
$$\begin{aligned}
\mathcal{E}_G= & \omega_{01}y^2+\omega_{21}z^2+\omega_1\nonumber \\
& -\frac{4}{\omega_{cav}+4D}\left[(\Omega_{10}y+\Omega_{21}z)\sqrt{1-y^2-z^2}+\Omega_{20}yz\right]^2\label{EGyz}\,.\end{aligned}$$
Note that $\beta_0^2+\beta_2^2\leq N$ (which is equivalent to $y^2+z^2\leq 1$) must be fulfilled within the Holstein-Primakoff framework. Hence, the global minimum of $\mathcal{E}_G$ has to be looked for in a disk with unity radius centered in the origin of the $0yz$ plane, which can be done with a straightforward numerical calculation.
We start by considering the [*ladder*]{} configuration ($f_{02} = 0$) for the three-level systems. In Fig. \[ladder\], we have plotted results for the photonic coherence $ x = \alpha/\sqrt{N} = \langle a \rangle/\sqrt{N}$ as a function of the oscillator strengths $f_{01}$ and $f_{12}$ of the ladder coupling for $ D = 3 \omega_{cav}$, $\omega_{21} = \omega_{cav}$ and $\omega_{10}/\omega_{21} = 0.1$. We point out that these parameters correspond to a strong anharmonicity with the photon mode in resonance with the excited transition. The value of the collective vacuum Rabi frequencies is given by Eq. (\[D\]). The normal phase occurs when $\alpha = 0$ (black region). The results in Fig. \[ladder\] show that in the considered system a superradiant phase ($\alpha \neq 0$) do occur. The phase transition boundary can be of both of first and second order. The first-order phase transition occurs when $\alpha$ has a discontinuous jump from $0$ to a finite value (upper part of the frontier). The first-order transition boundary is due to the excited transition $1\to 2$ as in the case recently studied by Hayn [*et al.*]{}[@Hayn]: at the transition there is a macroscopic occupation of the intermediate state $\vert 1 \rangle$. The second-order phase transition occurs when $\alpha$ is not discontinuous, but its gradient is. The important point to consider here is the fact that due to the TRK sum rule the oscillator strengths are subject to constraints. For the ladder configuration, we have $0 \leq f_{01} \leq 1$ and $0 \leq f_{10} + f_{12} \leq 1$, where $f_{10} = - f_{01} \leq 0$. In Fig. \[ladder\] the TRK boundaries imposed by such inequalities have been indicated: the area compatible with the TRK sum rule is below the red-dashed line. It is therefore apparent that there is an overlap between the superradiant part of the diagram and the region compatible with the TRK sum rule. Note that the relevant superradiant phase region compatible with the TRK inequalities contains a first-order transition boundary.
In Fig. \[V\], we consider instead the V-type configuration, where $f_{12} = 0$, i.e., there is no coupling between the excited states. In such a configuration, the photonic order parameter $x = \alpha/\sqrt{N} $ is plotted as a function of $f_{01}$ and $f_{02}$ (parameters in the caption). The TRK sum rule imposes the inequality $0 \leq f_{01}+f_{02} \leq 1$, again indicated by the red-dashed line. In such a V-type configuration, there is a superradiant phase boundary, which is always of second order. In fact, the transition between excited states is by definition inactive in such a configuration. Importantly, we notice that here there is no overlap between the superradiant part of the diagram and the area compatible with the TRK sum rule. Indeed, the TRK sum rule always prevents the superradiant phase transition in the V-type configuration as in the two-level case[@NatafNAT].
In Fig. \[Lambda\], we show results for the [*lambda*]{} configuration ($f_{01} = 0$): the photonic coherence is shown versus $f_{12}$ and $f_{02}$ (parameters in the caption). Here, the TRK inequalities $0 \leq f_{12} \leq 1$ and $0 \leq f_{02} \leq 1$, whose boundaries are once again delimited by the red-dashed line. As in the ladder case, in the lambda configuration there is a superradiant phase in the region compatible with the TRK inequalities (are below the red-dashed line) with the transition boundaries being of the first order.
So far, we have shown the simplest three-level configurations (ladder, V-type and lambda). In Fig. \[SRVol\], we show results for a generic three level system where all the three oscillator strengths are finite (detailed parameters in the caption). The superradiant part satisfying the TRK inequalities is depicted by the red filled volume in the $(f_{01},f_{02},f_{12})$ space (top panel). The middle panel shows the photonic order parameter on a planar section of such oscillator strength three-dimensional space. We point out that in the generic three-level case the interesting superradiant part compatible with the TRK inequalities is widened due to the additional freedom associated to the third oscillator strengths . In the bottom panel of Fig. \[SRVol\], we give an illustrative example of a spatial artificial potential providing a three-level system with oscillator strengths $(f_{01},f_{02},f_{12})$ and anharmonic spectrum corresponding to the cross in the middle panel, for which a superradiant transition is possible while satisfying the TRK constraints. As a perspective, it will be interesting in the future to address more complex multilevel structures and to investigate also the role of direct Coulomb interactions[@Keeling].
In conclusion, we have determined the phase diagram for a model system consisting of $N$ three-level systems coupled to a single photonic boson mode, by including the diamagnetic contribution in the light-matter coupling. We have demonstrated that in the considered model system superradiant phase transitions can occur while preserving the TRK inequalities, in stark contrast to the case of two-level systems. We have found that the transition between excited levels have a key role in such superradiant phase transition in contrast to what assumed in Ref. [@Viehmann]. Our results show that the physics of superradiant phase transitions with multilevel systems might be achievable in a broad range of physical systems, especially those where it is possible to engineer the spectra and oscillator strengths. We would like to thank T. Brandes, I. Carusotto, C. Emary and J. Keeling for stimulating discussions. C. C. is member of [*Institut Universitaire de France*]{} (IUF). We acknowledge support from the ANR project QPOL.
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{
"pile_set_name": "ArXiv"
}
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abstract: |
The non-associative exponential series $exp(x)$ is a power series with monomials from the magma $M$ of finite, planar rooted trees. The coefficient $a(t)$ of $exp(x)$ relative to a tree $t$ of degree $n$ is a rational number and it is shown that
$$\hat{a}(t) := \frac{a(t)}{2^{n-1}\cdot \prod^{n-1}_{i=1}(2^i -
1)}$$
is an integer which is a product of Mersenne binomials. One obtains summation formulas
$$\sum \hat{a}(t) = \omega(n)$$
where the sum is extended over all trees $t$ in $M$ of degree $n$ and $$\omega(n) = \frac{2^{n-1}}{n!} \prod^{n-1}_{i=1} (2^i -
1).$$
The prime factorization of $\omega(n)$ is described. The sequence $(\omega(n))_{n \ge 1}$ seems to be of interest.
author:
- 'by L. Gerritzen'
title: 'Mersenne Binomials and the Coefficients of the non-associative Exponential'
---
[****]{}
For a natural number $n,$ we denote by $M_n$ the n-th Mersenne number $2^n-1.$
The Mersenne factorial $n!_M$ is defined to be the product $\displaystyle \prod^n_{i=1} M_i$ of all Mersenne number $M_1,...,M_n$ while the Mersenne binomial ${n \choose r}_M$ is defined to be $${n!_M}\over {r!_M(n-r)!_M}$$ for natural numbers $r$ between $0$ and $n.$
In quantum calculus, see for instance \[KC\], there are the notions $[n]$ for the $q$ - analoque of $n \in {\mathbb N},[n]!$ for the $q$ - factorial and $\big[{n \atop r}\big] $ for the $q$ - binomial coefficients which are polynomials in the variable $q$. If $f(2)$ denotes the value of a polynomial $f$ after substituting 2 for $q$, then $[n](2) = M_n, [n]!(2) = n!_M$ and $\big[{n \atop r}\big](2)
= {n \choose r}_M.$ Let $M$ be the magma with unit $1_M$ freely generated by a single element $x$. It can be identified with the set of all finite planar binary rooted trees, see \[DG\].
Let ${\mathbb Q}\{\{x\}\}$ the ${\mathbb Q}$- algebra of power series with monomials from $M$. It was proved in \[DG\] that there is a unique series $exp(x)\in{\mathbb Q}\{\{x\}\}$ such that $exp(x) = 1 + x + $higher terms and $$exp(x) \cdot exp(x) = exp(2x)$$ see also Proposition (3.2). Let $a(t)$ be the coefficient of the non-associative exponential $exp$ relative to the planar binary rooted tree $t$ and put $$\hat{a}(t) = n! \cdot \omega(n)\cdot a(t)$$ where $$\omega(n):=
{{2^{n-1}(n-1)!_M}\over {n!}}.$$
A main result of this article states that all $\hat{a}(t)$ are integers which are obtained as products of Mersenne binomials. The projection of the non-associative exponential to the classical one leads to the decomposition $$\sum \hat{a}(t) = \omega(n)$$ if the sum is extended over all trees of degree $n.$
In section 1 the notions of Mersenne order and Wieferich exponents at odd primes is defined. One gets results about the prime factorization of Mersenne numbers.
In section 2 we show that all $\omega(n)$ are integers by computing the order $ord_p \omega(n)$ for all primes $p.$
The applications about the coefficients $a(t)$ of $exp(x)$ are deduced in section 3.
I would like to thank Doron Zeilberger for helpful hints.
[Mersenne orders]{}
We call $M_n = 2^n-1$ the n-th Mersenne number for $n \in {\mathbb N}, n
\ge 1.$
This is in contrast to the definition by D. Shanks, see \[S\], Chap. 1, which is also widely used in which $M_n$ is called Mersenne number only if $n$ is a prime.
Let $p$ be an odd prime.
The Mersenne order $v(p)$ at $p$ is the smallest number $r \in {\mathbb N},
r \ge 1,$ such that $p$ divides $M_r.$
As $2^{v(p)} \equiv 1 \ mod \ p$ and $v(p)$ is the order of the class of $2$ in $({\mathbb Z}/p{\mathbb Z})^*,$ we get that $v(p)$ divides the order of the multiplicative group $({\mathbb Z}/p{\mathbb Z})^*$ which is $(p-1).$
Also $p$ divides $M_n,$ if and only if $n$ is a multiple of $v(p).$
Also $p < 2^{v(p)}$ from which follows that $log_2(p) < v(p).$
It has the consequence that $\displaystyle lim_{p \to \infty}
v(p)= \infty.$
Let $\varepsilon(p) := ord_p(M_{v(p)})$ for any odd prime; it is called the Wieferich exponent at $p.$
Always $\varepsilon(p) \ge 1$ and $p$ is called a Wieferich prime if $\varepsilon (p) \ge 2.$
Up to now only two Wieferich primes are known, namely 1093 and 3511 and it was checked that these are the only Wieferich primes $
< 4.10^{12}$ in \[CDP\].
The Mersenne order at 1093 is 364 = ${{1092} \over {3}} = 2^2
\cdot 7 \cdot 13$ and the Mersenne order at 3511 is 1755 = ${{3510}\over {2}} = 3^3 \cdot 5 \cdot 13.$
The Wieferich exponent for both primes is equal to 2.
Let $p$ be an odd prime, $n \in {\mathbb N}, n \ge 1.$
- $ord_p(M_n) = 0,$ if $n$ is not a multiple of $v(p).$
- If $n = v(p)\cdot p^r \cdot k$ and $p$ does not divide $k$, the $ord_p(n) = r$ and $ord_p(M_n) = \varepsilon (p) + r$
Statement (i) is well-known.
Let now $n = v(p) \cdot p^r \cdot k$ as in (ii).
Let $m$ be an integer and consider the group $({\mathbb Z}/p^m{\mathbb Z})^*$ of units of the ring ${\mathbb Z}/p^m{\mathbb Z}.$ Let $U_m$ be the subgroup of $({\mathbb Z}/p^m{\mathbb Z})^*$ of class congruent to 1 modulo $p.$
Then $U_m$ has order $p^{m-1}$ because any element $\alpha$ in $U_m$ has a unique representation by a number $a = 1 +
\displaystyle \sum ^{m-1}_{i=1} a_i p_i$ with $a_i \in
\{0,...,p-1\}.$ The order of $\alpha$ is $m - min(\{i: a_i \ne
0\}, m)$ because if $a = 1 + p^\varepsilon \cdot x, x $ any number not divisible by $p$, then $$a^p = 1 + p^{\varepsilon + 1}\cdot
y$$ where $$y = x + \displaystyle \sum^p_{i=2}(^p_i) x^i p ^{i
\varepsilon - \varepsilon - 1}$$ which shows that $y \equiv x \
mod \ p$ as $p > 2.$
Let now $\alpha = \bar {2}^{v(p)}$ where $\bar {2}$ is the class of 2 in ${\mathbb Z}/p^m{\mathbb Z}.$
Then $\alpha \in U_m$ and $\alpha$ is represented by a number $a =
1 + p^{\varepsilon(p)} \cdot x,$ where $x$ is not divisible by $p.$ Then $\alpha^{p^r}$ is represented by a number $$1 +
p^{\varepsilon(p) + r} \cdot y$$ with $y$ not divisible by $p.$
It follows that $ord_p M_n = \varepsilon(p) + r$ because the order of $\alpha^{p^r}$ is equal to the order of $\alpha^{p^rk}.$
$$M_n = \prod_n p^{\varepsilon(p) + ord_p(n)}$$
where the product is extended over all odd primes $p$ for which the Mersenne order $v(p$) divides $n.$
[Factorial Mersenne quotients]{}
$F(n) := \prod^n_{i=1} M_i$ is called the Mersenne factorial of $n
\in {\mathbb N};$ it is also denoted by $n!_M.$
Let $$\omega(n) := {{2^{n-1} \cdot F(n-1)}\over {n!}}$$ for $n \ge 1;$ it is called the factorial Mersenne quotient at $n.$ Let $m \in {\mathbb N}, m \ge 1,$ and $p$ be a prime. There is a unique p-adic expansion $$m = \sum^r_{i=0}m_i p^i$$ with $m_i \in
\{0,1,...,p-1\}.$ Let $$d_p(m) := \sum^r_{i=0}m_i.$$
- $ord_2\omega(n) = d_2(n) - 1$
- If p is an odd prime, then $$ord_p(\omega(n)) = \varepsilon(p)\cdot m - {{(n-d_p(n)) - (m-d_p(m))}\over
{(p-1)}}$$ where $v(p)$ is the Mersenne order at $p$, $m$ is the largest integer $ \le (n-1)/v(p)$
and $\varepsilon(p)$ is the Wieferich exponent at $p$.
- $m
\equiv d_p(m) \ mod \ (p-1)$ because $m \equiv \displaystyle \sum
^r_{i=0} m_i \ mod \ (p-1).$ It is well known that $$ord_p(n!) =
{{n-d_p(n)} \over {(p-1)}} = \sum ^{< \infty}_{i=1} \ [{{n} \over
{p^i}}]$$
<!-- -->
- $ord_2(\omega(n)) = (n -1) - ord_2(n!)$ because $F(n)$ is odd. Thus $$ord_2(\omega(n)) = d_2(n) - 1 \ge 0$$ for $n \ge
1.$
- Let now $p$ be an odd prime. Then $$ord_pF(n-1) =
\sum^{n-1}_{i=1} ord_p M_i.$$ If $i= k \cdot p^r v(p)$ and $k$ is not divisible by $p,$ then $$ord_pM_i = \varepsilon(p) + r)$$ and $ord_p M_i = 0$ if $p$ is not divisible by $p.$ $$ord_p(F(n-1)) =
\sum^m_{j=1} ord_p M_{jv(p)}$$ where $m_p(n) =
m:=[{{n-1}\over {v(p)}}]$ is the largest integer $\le (n-1)/v(p)$. It follows from Proposition 1.3 that $$ord_p
M_{j\sigma(p)} = \varepsilon(p) + ord_p(j)$$ as $ord_p(j) =
ord_p(jv(p)).$ As $$\displaystyle \sum^m_{j=1} ord_p(j) = ord_p(m!)
= {{m-d_p(m)}\over {p-1}}$$ we get the formulas $$ord_p F(n-1) =
\varepsilon(p) \cdot m + {{m-d_p(m)}\over {p-1}}$$and $$ord_p(\omega(n) = \varepsilon(p)m - {{(n-d_p(n)) - (m-d_p
(m))}\over {p-1}}$$
$\omega(n) \in {\mathbb N}.$
We will show now that $ord_p \omega(n) \ge 0.$ From the formula in 3) we get $$ord_p(\omega(n)) \ge m - {{(n-d_p(n)) -
(m-d_p(m))}\over{p-1}}$$ $$\ge m - {{n-d_p(n)}\over {(p-1)}}\ge
[{{n-1}\over {p-1}}] - {{n-d_p(n)} \over {p-1}} \ge 0$$ because $m
- d_p(m) \ge 0, m \ge {{n-1} \over {p-1}},$ $n - d_p(n) \le n-1$ and ${{n-d_p(n)} \over {p-1}} \in {\mathbb N}.$
From 1) and 2) we get that $ord_p(\omega(n-1)) \ge 0$ for all primes $p.$ As $\omega(n-1)$ is a rational number, it follows that $\omega(n-1)$ is an integer.
- $ord_7 \omega(13)= 3,$ because $n= 13, m = {{13-1}\over {v(7)}} = 4, d_7(4) = 4, d_7(13) = 7,
\varepsilon(7) = 1, n - d_7(n) = 6, m - d_7(m) = 0$ $ord_7(\omega(13)) = 4 - {6 \over 6} = 3$
- $ord_7 \omega(100)= 21,$ because $v(7) = 3, n = 100, m =[ {{n-1}\over {3}}]
= 33, d_7 = 100) = 4, d_7(33) = 9$ $ord_7(\omega(100)) = 33 -
{{96-24} \over {6}} = 21,$ because $log(\omega(n)) \approx {n+1
\choose 2} log(2) + n - (n + 1/2) log(n).$
Let $\pi_M(x)$ be defined to be number of odd primes $p$ such that the Mersenne order $v(p)$ is $\le x-1.$
It seems interesting to determine the asymptotic behaviour of $\pi_M.$
One can expect some results about $\pi_M$ from the prime factorization of $n!_M,$ because $\pi_M(n)$ is the number of primes dividing $n!_M.$
$\pi_M(16) = 15$ because the primes dividing $16!_M$ are the odd primes $\le 16$ and the primes 17, 23, 31, 43, 73, 89, 127, 151, 257, 8191.
[Coefficients of the non-associative exponential]{}
Let $M$ be the magma with neutral element $1_M$ freely generated by a set consisting of one element $x$. The multiplication in $M$ is a map $\cdot : M \times M \to M$ and the restriction of this map onto $(M - \{1_M\} ) \times (M x \{1_M\})$ is injective and its image is $M - \{1_M, x\}.$ It means that for any $t \in M -
\{1_M, x \}$ there is a unique pair $(t_1, t_2) \in M \times M, t_1
\neq 1_M \neq t_2$, such that $t = t_1 \cdot t_2.$
There is a unique homomorphism $\deg : M \rightarrow {\mathbb N}$ such that $deg(1_M) = 0, deg(x) = 1.$
It is important to realize that any $t \in M, t \neq 1_M,$ gives rise to a unique planar binary rooted tree with $deg(t)$ leaves, see \[1\], section 1, p. 163. The grafting of such trees corresponds to the multiplication map of $M$.
Let $P = {\mathbb Q}\{\{x \}\}$ be the ${\mathbb Q}$-algebra of power series with monomials from $M$. Thus any $f \in P$ has a unique expansion $$f
= \displaystyle \sum_{t \in M} c(t) \cdot t$$ with $c(t) \in {\mathbb Q}.$ It will also be called the algebra of tree power series in $x$ or the algebra of power series over ${\mathbb Q}$ in a nonassociative and noncommutative variable $x$.
Define the order $ord(f)$ of $f$ to be $min \{deg(t) : c(t) \neq 0
\},$ if $f \neq 0$ and $ord(0) = + \infty.$ The following proposition is obtained by standard calculus methods.
1. There is a unique ${\mathbb Q}$-linear map ${d \over {dx}} :P \rightarrow P$ such that: $${d \over {dx}} (x) =
1$$ and $${d \over {dx}} (f \cdot g) = {d \over {dx}}(f) \cdot g + f \cdot
{d \over {dx}}(g)$$ for all $f, g \in P$.
$f' := {d \over {dx}}(f)$ is called the derivative of $f$ with respect to $x$.
2. Let $g \in P$ with $ord(g) \ge 1$. Then there is a unique ${\mathbb Q}$-algebra homomorphism $\eta_g : P \to P$ such that $\eta_g(x)
= g$. One denotes $\eta_g(x)$ also by $f \circ g$ or $f(g)$ and calls $\eta_g$ the substitution homomorphism induced by $g$.
3. There is a canonical ${\mathbb Q}$-algebra homomorphism $[ \ ] :
P \rightarrow {\mathbb Q}[[x]],$ where ${\mathbb Q}[[x]]$ denotes the classical ${\mathbb Q}$-algebra of power series in an associative (and commutative) variable $x$ such that $[x]=x$.
One calls $[f]$ the classical power series associated to $f \in P$.
There is a unique $f \in {\mathbb Q}\{\{x\}\}$ such that $$f'(0) = 1$$ $$f(x) \cdot f (x) = f (2x)$$
Moreover $f'(x) = f(x)$ where $f'(x)$ denotes the derivative of $f$ relative to $x$. The tree power series $f(x)$ is called the nonassociative, noncommutative exponential series and will be denoted in this article by $e^x$ or $exp(x)$.
Inductively we define a map $a: M \rightarrow {\mathbb Q}$ by putting $a(t)
:= 1,$ if $t=1_M$ or $t=x$ and otherwise $$a(t):= {a(t_1) \cdot
a(t_2) \over 2^n-2}$$ if $n = deg(t)$ and $t= t_1 \cdot t_2$ with $t_i \in M.$
Put $f := \displaystyle \sum_{t \in M} a(t) \cdot t \in P = {\mathbb Q}\{\{x\}\}.$ It is easy to check that $f(x) \cdot f(x) = f(2x):$
$$\begin{split}
f(2x) &= \sum_{t \in M} a(t) \cdot 2^{deg(t)}\cdot t \\ f(x) \cdot
f(x) &= \sum_{(t_1,t_2) \in M \times M} a(t_1) a (t_2) t_1 \cdot
t_2 \\ & = 1+2a(x) + \\& \ \ \ + \sum_{t \in M, deg(t) \geq
2, t= (t_1,t_2)} (a(t) \cdot a(x^0)+a(x^0)a(t) + a(t_1) a(t_2))
\cdot t.
\end{split}$$
As $2a(t) + a(t_1) \cdot a(t_2) = 2^n a(t)$ for any $t \in M$ with $deg(t) \geq 2, t = t_1 \cdot t_2,$ we obtain the desired functional equation.
For the proof that $f'(x) = f(x)$ we refer to \[DG\].
From the proof we get that the coefficients $a(t)$ of $exp$ satisfy: $a(1) = a(x) = 1$ and $$a(t_1 \cdot t_2)= {{a(t_1)a(t_2)}
\over {2^n - 2}}$$ where $n = deg(t_1 \cdot t_2).$
Let $\hat a(t) := n \cdot \omega(n) \cdot a(t).$
Let $t_1,t_2 \in M, n_i = deg(t_i) \ge 1$ and $t = t_1 \cdot
t_2, n = deg(t) = n_1 + n_2.$ Then $$\hat{a}(t) = ({{n-2}\over
{n_1 -1}})_M \hat{a}(t_1) \hat{a}(t_2)$$
$$\begin{split}
\hat{a}(t)&= n!\cdot \omega(n) \cdot {{a(t_1)a(t_2)}\over{2^n - 2}}\\
&= n!\cdot {{\omega(n)}\over {2M_{n-1}}}
{{\hat{a}(t_1)}\over {n_1! \omega(n_1)}}{{\hat{a}(t_2)}\over {n_2! \omega(n_2)}}\\
&= {{2^{n-1} F(n-1)}\over {2M_{n-1}}}
{{\hat{a}(t_1)\hat{a}(t_2)}\over{2^{n_1-1}
F(n_1-1)2^{n_2-1}F(n_2-1)}}\\
&={{F(n-2)}\over {F(n_1-1)F(n_2-1)}} \hat{a}(t_1)
\hat{a}(t_2)\\ &={n-2 \choose n_1-1}_M \hat{a}(t_1)
\hat{a}(t_2).\\
\end{split}$$
It follows from (3.3) that $\hat{a}(t)$ is a product of Mersenne binomials. More precisely:
$\hat{a}(t) = \displaystyle \prod_{a \in I(t)} {n(a) - 2 \choose
n_1(a) - 1)}_M \in {\mathbb N}$ where $I(t)$ is the set of inner nodes of $t$ and $n(a)$ is the degree of $t_{\le a}$ where $t_{\le a}$ is the tree below a which is defined to consist of all nodes $b$ for which the simple path from $b$ to the root of $t$ is passing through $a$. Also $n_1(a)$ is the degree of the left factor $s_1$ of $t_{\le a}$ (such that $t_{\le a} = s_1 \cdot
s_2).$
Let $T_n$ be the set of trees in M of degree $n$ defined as follows: $T_1 = \{x\}.$
If $T_{n-1}$ is already defined, then $T_n = x \cdot T_{n-1}\cup
T_{n-1}\cdot x.$
Then $\sharp T_n = 2^{n-2}$ for $n \ge 2.$ One can show that $\hat{a}(t) = 1$ iff $t \in T_n.$
$$\displaystyle \sum_{deg(t) = n} \hat{a}(t) = \omega(n)$$
We consider the canonical algebra homomorphism
$$[\ ] : {\mathbb Q}\{\{ x \}\} \to {\mathbb Q}[[x]].$$
Then $[exp]$ is the classical series $\displaystyle
\sum^\infty_{n=0} {{x^n}\over{n!}}.$ As $[t] = x^n$ if $t \in M$ has degree $n$, it follows that $\displaystyle \sum_{deg(t) = n}
a(t) = {{1}\over{n!}}.$
For $1 \le k \le n-1$ let
$$S_k(n) = \sum_{deg(t_1) = k \atop deg(t_2) = n-k}\hat{a}(t_1
\cdot t_2).$$
Then $$S_k(n) = {n-2 \choose k-1}_M \displaystyle \sum_{deg(t_1) =
k \atop deg(t_2) = n-k}\hat{a}(t_1) \hat{a}(t_2)$$ $$S_k(n)={n-2
\choose k-1}_M \cdot \omega(k) \cdot \omega(n-k)$$ It follows from Corollary 3.6 that $\omega(n) = \displaystyle \sum^{n-1}_{k = 1}
S_k(n)$. Thus
$\displaystyle \sum^{n-1}_{k=1}{n-2 \choose k-1}_M \omega(k)\cdot
\omega(n-k) = \omega(n).$
$\omega(1) = \omega(2) = 1, \omega(3)= 2, \omega(4)=7, \omega(5) =
42 = 2 \cdot 3 \cdot 7$
Thus $$\begin{aligned}
\omega(6) &=& \displaystyle \sum^5_{k=1} {4 \choose k-1}_M
\omega(k) \omega(6-k)\\ &=& 1 \cdot \omega(5) + M_4 \cdot 1 \cdot
\omega(4) + {{M_4 \cdot M_3} \over{M_2}} \omega(3)^2 + M_4 \cdot
\omega(4) \cdot 1 + 1 \cdot \omega(5)\\ &=& 2 \cdot 42 + 30 \cdot
7 + 5 \cdot 7 \cdot 4 = 434 = 2 \cdot 7 \cdot 31\end{aligned}$$
How to understand from this partition of $\omega(6)$ that 31 is a factor of $\omega(6)?$
[XXXX]{} *Crandall, R. - Dilcher, K. - Pomerance, C.* [: A search for Wieferich and Wilson primes, Math. Comp. 66 (1997), no. 217, p. 433 - 449]{} *Drensky, V.- Gerritzen, L.*[: On non-associative exponential and logarithm, (2002), to appear in Journal of Algebra]{}
*Kac, V. - Cheung, P.*[: Quantum Calculus, Springer-Verlag, Universitext, 2002]{} *Shanks, D.*[: Solved and unsolved problems in number theory, Third edition, Chelsea Publishing Co., New York 1985]{}
[****]{}
Binary rooted trees, non-associative power series, exponential series, $q$-binomials, Mersenne binomials
[****]{} Prof. Dr. Lothar Gerritzen
Ruhr-Universität Bochum
Fakultä für Mathematik
D 44780 Bochum
Germany
Tel.-number: 0049-0234-32-28304
FAX: 0049-0234-32-14025
Email: [email protected]
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We report on the development of a sensitive dilatometer based upon a AFM piezocantilever. This dilatometer has been tested at temperatures down to 25 mK and in magnetic fields up to 16 T. The layered heavy fermion superconductor $CeCoIn_5$ and its non-magnetic analog $LaRhIn_5$ have been measured to demonstrate its use in detecting phase transitions and quantum oscillations. In addition, using this dilatometer, a simultaneous multi-axis dilation measurement has been done. This compact dilatometer has many advantages such as its ability to measure very small samples with lengths at sub-mm levels, low temperature and field dependence, ability to rotate, and works well irrespective of being in a changing liquid or gas environment (i.e. within a flow cryostat or mixing chamber).'
author:
- 'L. Wang'
- 'G. M. Schmiedeshoff'
- 'D. Graf'
- 'J. -H. Park'
- 'T. P. Murphy'
- 'S. W. Tozer'
- 'J. L. Sarrao'
- 'J. C. Cooley'
- 'E. C. Palm'
title: Miniature Dilatometer Based upon an AFM Piezocantilever
---
Introduction
============
Thermal expansion (TE) and magnetostricton (MS) can provide accurate information on changes in the dimensions of materials as the temperature and magnetic field are varied. These quantities are of considerable interest because of the fundamental importance of the structure of materials and their intimate relation to the specific heat of materials[@Barron1999; @Grueneisen]. They provide a way to understand the pressure dependencies of the respective ordering phenomena, thermodynamic properties, relevant energy changes, and magnetic field-induced structural changes, which are very important for the study of high temperature superconductors [@Lortz2003; @Meingast2001], heavy fermion systems[@Schmiedeshoff2011; @AdeVisser1990] and many other novel materials. Unfortunately, many samples of interest are frequently small with a correspondingly small TE or MS, so a dilatometer with a high resolution is needed for these measurements.
Currently, the dilatometers based upon the capacitance method are one of the most sensitive methods for precise TE and MS measurements[@White1993; @White1974; @Green1963; @Pott1983; @Swenson1998; @Rotter1998]. Dilatometers of this design have operated successfully in a wide variety of different cryostats and measurement systems. Their main advantages are that the sample is only under very weak uniaxial stress (50-500 mN)[@Rotter1998] and that a dilation limit of less than 0.1$\AA$ can be reached[@Schmiedeshoff2006]. However, when placed into a low temperature environment that changes from liquid to gas (as inside a He-3 cryostat) or even in a dilution refrigerator mixing chamber where the ratio of $^3$He to $^4$He changes with temperature or field, the capacitance dilatometer has large background effects due to the temperature dependent dielectric constant of the medium. In addition, in a pulsed magnetic field, where the field is changing extremely rapidly, eddy currents in the metallic body of the device can have deleterious effects. Finally, many capacitance dilatometers will change their capacitance upon rotation of the device due to the effect of gravity even without the sample length having changed, making their use in rotators difficult[@Brown1983; @Johansen1986; @Neumeier2008].
To solve these problems, several solutions have been proposed. One is an optical fiber strain gauge using Fiber Bragg Gratings (FBGs). This method has the significant advantage of immunity to electromagnetic interference and works very well in a wide variety of cryostats and large pulsed magnetic fields[@2010RScI81c3909D]. However, the FBG has to be glued to the surface of the sample and this requires that the sample has a flat surface at least a few millimeters in length which is not possible with many samples. In addition, it is difficult to rotate the device and bend the optical fiber.
The solution proposed here uses an atomic force microscope piezo-resistive cantilever as the sensing element. The prototype construction was described earlier[@Park2009] and is here developed into a more robust device. A significant advantage of this device is its immunity to the effects of the cryogenic media it is immersed in. In addition, the cantilever is small and easy to mount in a relatively small space and can be easily rotated. Another advantage is the ability to perform simultaneous mulit-axis measurements. For previous dilatometer forms (capacitive and optical), dilations of the sample can only be measured one axis at a time. With proper construction, more than one cantilever can be used at the same time and measured separately in order to detect the simultaneous changes of different axes of one sample; $x$, $y$ and even $z$ axis.
This work concentrates on the following aspects: finding suitable construction materials and design, exploring low noise and high-quality measurements, and discovering applications and limitations for this method.
Basic measurement set-up
========================
The dimensions of the commercial PRC400 [@Seiko] device measure 3.5$\times$1.6$\times$0.2 mm$^3$ (L$\times$W$\times$H) overall and the sample lever arm has dimensions of 0.4$\times$0.05$\times$0.005 mm$^3$ (L$\times$W$\times$H). $R_{s}$ indicates the resistance of piezoelement of the long-tip lever arm and $R_{r}$ is the resistance of a short-tip reference lever which is synthesized together with the signal piezo element. The typical room temperature resistance of each piezo (both $R_{s}$ and $R_{r}$) is about 600-700 $\Omega$. As shown in Fig.\[1\] (b), the basic measurement circuit of is based on a Wheatstone bridge configuration. Resistance changes in the piezoelement $R_s$ that are induced by dimensional changes from the sample are recorded as a voltage signal in a lock-in amplifier. $V_{bias}$ is the input excitation voltage and $V_{ab}$ is the recorded output signal. $R_{1}$ and $R_{2}$ are the potentiometers used to balance the bridge. The equation is: $$V_{ab}=V_{bias}(\frac{R_{2}}{R_{s}-R_{2}}-\frac{R_{1}}{R_{r}-R_{1}})$$
A top-loading $^3$He/$^4$He dilution refrigerator and superconducting magnet were used to provide temperatures as low as $\sim$25 mK and magnetic fields up to 16 T. The dilatometer is immersed directly in the $^3$He-$^4$He mixture providing excellent thermal contact to the sample.
![\[1\] Construction for a 2-axis measurement. (a) The silicon stage with two cantilever tips touching the sample surface. The cantilever arm and tip, piezo element $R_s$ and reference piezo $R_r$ are indicated. (b) The construction the Wheatstone bridge circuit. $V_{bias}$ is the output voltage, and $V_{ab}$ is the recorded signal. The circuit within the red box has been synthesized by a low noise circuit bridge, designed and built by the NHMFL electronic shop. (c) Image of the measurement setup as two cantilever tips touch the two surfaces of $CeCoIn_5$ separately as illustrated by (a).](Fig1){width="1\linewidth"}
For designing this dilatometer, there are two basic rules: First, the design must be very simple, since any strain in the construction can transfer to the output signal, and the sample signal will normally be hidden by this huge effect. Second, it should be made of the same material as the cantilever body. If the materials are different, the differential contraction between the arm and the base can cause the cantilever tip to “walk” over the surface of the sample and produce erroneous signals. The final design of sample stage is made of silicon and the height of the stage where the sample sits is machined according to the thickness of the sample. The cantilever and sample are both glued to the silicon base with superglue and the sensory tip of the cantilever gently touches the sample surface. This simple design is especially suitable for thin samples with very limited length, $L_{sample}$ $\sim$ 0.1 mm or smaller. For a bulk sample with larger size, by using multiple cantilevers, the dilations of more than one sample direction can be measured simultaneously. The 2-axis measurement setup is shown as an example in Fig.\[1\] (a) and with proper construction, measuring a 3$^{rd}$ axis is also possible.
Fig.\[1\] (b) shows the Wheatstone bridge circuit construction. This part is synthesized in a custom manufactured low noise breakout box. This approach dramatically reduced the connections and exposed wires that can introduce unwanted noise. Signal Recovery (SR) 7280 lock-in amplifiers are used here. Typical settings were $V_{bias}$ = 0.05 V with time constant 500 ms and excitation frequency f = 17 Hz. With the improved circuit, the peak-to-peak noise level can be reduced down to 20 nV.
In capacitance dilation measurements, $L$ represents the sample length and $\Delta L$ is used to quantify the change. In the dilatometer measurements described here, results are recorded in the form of electronic signal, Voltage. The signal $V_{ab}$ can reflect the deviation of the cantilever tip $\Delta L$. At room temperature, the calibration from $V_{ab}$ to $\Delta L$ can be made by using a micrometer to push the cantilever tip (record $\Delta L$) and recording the electrical signal ($V_{ab}$) at the same time. However the calibration relation is not as easily obtained for low temperature environments. There is an important factor which will dramatically influence the calibration, that can best be described in terms of the “cell effect".
In a capacitive dilatometer, the cell effect is caused by the thermal expansion and magnetostriciton of the material(s) making up the dilatometer. In the cantilever dilatometer, the cell effect originates from two piezoresistors that do not match. The two resistors were presumably engineered to be identical at room temperature and in zero fields. What has been observed, though, is that small differences between them increase at low temperatures (slightly different temperature-dependent resistivities) and high fields (slightly different magnetoresistances). To create a new term, this might be called a “resistor mismatch cell effect". Based on the tests of several cantilevers, this kind of cell effect can clearly vary from cantilever to cantilever. So for low temperature calibration, the conversion form $V_{ab}$ to $\Delta L$ cannot be a universal value for all cantilevers.
In spite of this, for repeated measurements with one single cantilever, the cell effect is rather reproducible. Most of the time, the cell effect is shown as a very broad bump or nearly linear background signal throughout the whole measurement range. Most of the dilation changes that are associated with phase transitions can be easily resolved as anomalies in the broad background signal. A rough estimation was made measuring one sample with both capacitance and piezo dilatometers. Under the same conditions (pressure, temperature and magnetic field), the dilation change appeared to be comparable. Here we use the same $CeCoIn_5$ sample which is used in Correa’s magnetostriction measurements[@VectorPRL]. According to their capacitance dilatometer measurements, at approximately 25 mK, for B//$ab$-plane and original sample length $L=0.9144$ mm, the dilation change $\Delta L_{sample}$ at the transition point $H_c$ is $\sim 5$ nm. The voltage signal $\Delta V$ at the same transition point is $\sim 0.7$ $\mu$V. So the calibration relation can then be estimated as $$\Delta V/ \Delta L\approx 140 \mu V/\mu m
\label{eq2}$$
This estimation agrees with the room temperature value found in the reference[@Takahashi2002] within a factor of 2 (PRC400, with $V_{bias}$=0.05 V). In fact, because the different cantilevers have different “cell effect", the signal at the transition point is also affected. We performed measurements on the same $CeCoIn_5$ sample using different cantilevers. At the transition point, the largest $\Delta V$ signal could be up to twice large as the smallest one. The $\Delta V$ shown above is an average value. Therefore, this calibration relation can be used for some rough estimations, but cannot be used to determine absolute sample length changes.
With equation \[eq2\], we can estimate our low temperature measurement resolution. As previously mentioned, the lowest peak to peak noise level is $\sim$ 20 nV, so the best resolution we can achieve is $\sim$2 $\AA$. With this calibration, the temperature and field dependence of the cell effect also can be roughly estimated to be $\sim$3 nm/Kelvin and $\sim$1.5 nm/Tesla respectively. These are average values obtained by measuring 3 different cantilevers. Note these estimated values only apply for low temperature measurements in the millikelvin range. This cell effect includes the result of resistor mismatch, piezoresistor magnetoresistance, the expansion of silicon, etc. But in fact, for many low temperature measurements, one is interested in measuring a phase transition or seeing quantum oscillations, and this cell effect could be subtracted as the background signal. So despite the “cell effect”, this technique is still very useful for detecting dilation changes caused by phase transitions for samples with limited size. Moreover, for samples that are large enough, a very useful simultaneous multi-axis measurement can be performed with the setup shown in Fig.\[1\] (a).
Phase transition measurement
============================
![Graph (a) shows the field dependence results of $CeCoIn_5$ at different temperatures (top to bottom curves are taken from 1.26 K to base temperature). The field is applied in the $ab$-plane ($\theta$ = 90$^{\circ}$). Insert graph is the phase diagram. Graphs (b) and (c) show the results of the field applied to $c$-axis ($\theta$ = 0$^{\circ}$). Graph (b) shows the temperature dependence at different fields (top to bottom data are taken at field from 4.5 T to 6 T). Graph (c) shows the field dependence at different temperatures (top to bottom data are taken from base temperature to $\sim$1 K). The phase diagram based on these two graphs is placed in the middle. Graph (d) shows the field dependence at the different angle (top to bottom data are taken from $\theta$ = 90$^{\circ}$ to 0$^{\circ}$). The bottom graph (e) is the 3D view of a summary of all three phase diagrams.[]{data-label="CeCoIn5"}](Fig2){width="0.9\linewidth"}
The heavy-fermion superconductor 115 family, $REMIn_5$ ($RE = La$ or $Ce$; $M$ = $Co, Rh$ or $Ir$), has been extensively investigated in the last decade owing to several unusual properties of their superconducting (SC) states[@0953-8984-13-17-103; @PhysRevLett.86.5152]. The tetragonal crystal structure alternates magnetic $REIn_3$ and non-magnetic $MIn_2$ layers along the $c$-axis. One of the most interesting members of this family is $CeCoIn_5$, not only because it demonstrates a sharp, clear first-order phase transition from the superconducting state to the normal state at high magnetic fields, but also because it is the first material to exhibit a Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superconducting state[@PhysRevLett.91.187004; @VectorPRL].
The $CeCoIn_5$ sample we used for this measurement is 1$\times$1$\times$1.5 (a$\times$b$\times$c) mm$^3$ in size. The sketch at the right bottom of Fig.\[CeCoIn5\] shows the orientation between the sample and applied magnetic field. The cantilever tip gently rested on top of the sample and the signal shown here reflected only the changes of the $c$-axis. Here $\theta$ is defined as the angle between the magnetic field and the $c$-axis.
As shown in Fig.\[CeCoIn5\], TE and MS measurements were made at different directions and temperatures, which provided a complete 3D-phase diagram. Fig.\[CeCoIn5\] (a) shows the field dependence results of $CeCoIn_5$ at different temperatures. From top to bottom, the data were taken at 1260 mK, 1100 mK, 900 mK, 600 mK, 450 mK, 300 mK, 200 mK, 150 mK, 100 mK and 25 mK respectively. For this graph, a magnetic field was applied along the $ab$-plane ($\theta$ = 90$^{\circ}$). The inset to graph (a) is the resulting phase diagram. Fig.\[CeCoIn5\] (b) and (c) are the results of the field applied to $c$-axis ($\theta$ = 0$^{\circ}$). Fig.\[CeCoIn5\] (b) is the temperature dependence at different fields (4.5 T, 4.8 T, 4.9 T, 4.95 T, 5 T, 5.5 T, 6 T), while graph (c) shows the field dependence at different temperatures (25 mK, 100 mK, 200 mK, 380 mK, 420 mK, 750 mK, 1000 mK, 1200 mK). The resulting phase diagram is shown in the inset of Fig.\[CeCoIn5\] (b) and (c). Graph (d) shows the field dependence at different angles ($\theta$ = 0$^{\circ}$, 4$^{\circ}$, 9$^{\circ}$, 13$^{\circ}$, 17$^{\circ}$, 22$^{\circ}$, 30$^{\circ}$, 50$^{\circ}$, 90$^{\circ}$ respectively).
These overall magnetic field versus temperature phase diagrams are in excellent agreement with previous work[@VectorPRL; @PhysRevB.71.020503; @PhysRevB.71.020503; @PhysRevB.70.134513; @PhysRevB.70.020506]. Most importantly, a complete 3D phase diagram can be made. (See the last graph (e) in Fig.\[CeCoIn5\]). By decreasing the temperature from 1.2 K to the base temperature ($\sim$25 mK), the transition point $H_c$ increased from $\sim$9 T to 11.65 T. While at base temperature, the sample was rotated 90 degrees from the B//$ab$-plane position to the B//$c$-axis position. The transition critical field $H_c$ decreased from 11.65 T to $\sim$4.98 T (see the $\theta$-$H$ plane at $T = 25$ mK). Finally, at position B//$c$-axis, the temperature was increased from 25mK to 1.2 K. The transition point continued to decrease and finally vanished around 4.1 T (see the $H$-$T$ plane at $\theta$ = 0$^{\circ}$).
Simultaneous Measurement of two sample axes
===========================================
![\[2axisCeCoIn\] Angular dependence data for both $c$ and $a$($b$)- axis taken at the same time. Note that the data collected along the $a$($b$)-axis is decreased by a factor of 10 to be shown on a similar scale.](Fig3){width="1\linewidth"}
The same $CeCoIn_5$ sample used in the previous section was also used for these measurements. The orientation between the sample and applied magnetic field is shown in the lower right of Fig.\[CeCoIn5\]. This time two cantilevers were used to measure the two different axis of the sample simultaneously. As can be seen in Fig.\[1\] (c), $c$-axis is perpendicular to the sample surface which is directly facing out of the page (in Fig.\[1\] (a), the surface faces up), and $a$($b$)-axis is perpendicular to the side surface (in Fig.\[1\] (a), the surface faces aside). The signals will reflect the changes of $c$ and $a$($b$)-axis at the same time.
Fig.\[2axisCeCoIn\] shows two groups of angular dependent data, one group (upper in the graph) is the measurement along the a(b)-axis and the other is the measurement along the $c$-axis (lower part in the graph). Both groups contains four curves that are taken at $\theta$ = 5$^{\circ}$, 10$^{\circ}$, 20$^{\circ}$, 90$^{\circ}$ respectively. As can be seen, the transitions for these two groups of data correspond to each other. At the transition point, the signal, $\Delta V$ for the $a$($b$)-axis is more than 10 times larger than the signal for $c$-axis. For clarity, the $y$-axis of the graph was split and data for $a$($b$)-axis are decreased by a factor of 10 in order to get a similar scale with the data for $c$-axis. For the measurements performed on the $c$-axis, even though a different cantilever was used than in section III, the signal amplitude for both measurements are still comparable, despite the different background caused by the cell effect. Thus for the measurements along the $a$($b$)-axis, the large $\Delta V$ at the transition point indicated the dilation change along $a$($b$)-axis is much bigger than that along $c$-axis given the crystal dimensions. This is very important information which can help to estimate the volume change and understand the whole lattice.
Measurement of quantum oscillations
===================================
 Background subtracted trace showing quantum oscillations and (b) the resulting FFT spectrum in $LaRhIn_5$](Fig4){width="1\linewidth"}
In addition to the application for simultaneous multi-axis measurement, the advantages of our cantilever dilatometer manifest in other aspects. The use of cantilevers in pulsed magnets and the oscillatory magnetic torque measurements were successfully studied[@PhysRevB.80.241101; @adhikari:013903; @PhysRevB.81.184506]. The dilation measurements on $LaRhIn_5$ proved the cantilever can detect quantum oscillations via magnetostricton and the signal is significant.
The $LaRhIn_5$ sample we used here is much thinner, only 1$\times$1$\times$0.1 mm$^3$(a$\times$b$\times$c) in size. The sample and field position are similar to the $CeCoIn_5$ sample shown in Fig.\[CeCoIn5\] bottom. The cantilever also touches the top of the sample, which measures the dilation change of $c$-axis.
In Fig.\[LaRhIn5-1\], data from $LaRhIn_5$ with the field parallel to the $c$-axis is shown. In the lower part of the graph, a fast Fourier transform (FFT) spectrum for the $LaRhIn_5$ analysis for fields between 4 T and 12 T is shown. The upper graph shows the clear oscillation data up to 12 T. The quality of this data is typical for all of the investigated trace curves in Fig.\[LaRhIn5-2\]. Multiple frequencies are found in the FFT spectrum; main fundamental branches are $\alpha_1$, $\alpha_2$, $\beta_1$, $\beta_2$, $\varepsilon_1$ and $\varepsilon_2$.
In addition to high resolution, this cantilever dilatometer can be easily rotated in the cryostat. For quantum oscillation measurements, the data taken at different angles will reflect the Fermi surface at that direction, so obtaining the angular dependent data is important for understanding the full Fermi surface. Because of the confined space and liquid/gas environment at field center, sample rotation in other dilation measurements is rather limited. However, this piezo cantilever dilatometer does not have these restrictions, making it the perfect choice to perform angular dependent measurements at low temperature and high field.
Figures \[LaRhIn5-2\] (a) and (b) show the angular dependence of the oscillation frequencies of $LaRhIn_5$, both in the FFT spectra and selected peaks from those spectra. The branches $\alpha_i$ ($i$=1, 2) and $\beta_2$ roughly follow the 1/$cos\theta$ dependence, where $\theta$ indicates a field angle tilted from \[001\] to \[100\]. In comparison to the theoretical calculation in Shishido’s paper[@JPSJ.71.162], this angular dependence suggests that the corresponding Fermi surface is nearly cylindrical. Because of the theoretical calculation, branch $\alpha_2$ is due to a band 15-electron Fermi surface, which is nearly cylindrical, but corrugated, allowing maximum and minimum cross-sections. Branch $\beta_i$ is due to a highly corrugated band 14-electron Fermi surface. A feature of $\beta_i$ is a minimum in its frequency at $\sim$30$^{\circ}$ from the $c$-axis. This feature is shown in our data in Fig.\[LaRhIn5-2\] (b). Branch $\varepsilon_i$ is due to a band 13-hole Fermi surface whose topology is similar to a lattice. In previous studies[@JPSJ.70.2248; @Onuki200213; @PhysRevB.79.033106], these low frequency signals have never been clearly shown. However, by using the cantilever dilatometer, the low frequency features are well resolved in the oscillatory magnetostriction measurements.
The angle-dependent quantum oscillations in $LaRhIn_5$ at $\sim$25 mK for a wide angle region have been successfully observed and shown in Fig.\[LaRhIn5-2\]. It is clearly found that these main frequencies which are corresponding to certain Fermi surfaces, systematically shifted with rotation angle $\theta$. The significant signals at low frequency range may need further investigation.
![\[LaRhIn5-2\]Angular dependence of the frequency in $LaRhIn_5$. (a)FFT spectra at different angle and (b) selected peaks vs. angle. ](Fig5){width="1\linewidth"}
Conclusion
==========
Advancing our previous work, we improved the design implementation of the miniature size AFM cantilever based dilatometer. The successful applications of our new design on heavy Fermion SC samples $REMIn_5$ show that the cantilever dilatometer is a very useful tool for magnetoelastic and quantum oscillation investigations. Compared to the traditional capacitance dilatometer and FBGs, although this dilatometer can not provide accurate length changes of a sample, it still has unique merits. First, the sample used for this dilatometer can be very small (smaller than 0.1 mm in length). This is already beyond the limitations of many other dilatometer forms. Also, the rotation measurement is usually considered to be a big challenge for capacitance and FBGs dilatometers. Since the volume of the piezo dilatometer can be rather small, it can be easily rotated in very constrained spaces, like field center in a dilution refrigerator, without gravity effects, providing another advantage for this technique. Its versatility makes it suitable for measurements under the many conditions imposed by low temperatures (like liquid or gaseous helium) and high magnetic fields. Moreover, this technique shows suitability of application in oscillatory magnetostricton measurements, measuring even high frequency signals easily. The final advantage of our dilatometer is shown by the successful application on simultaneous multi-axis dilation measurements. The dilation measurement for more than one direction at the same time is difficult for other dilatometer techniques, which makes our cantilever dilatometer a unique tool for this type of measurement.
We would like to thank Vaughan Williams for mechanical construction work and James Andrew Powell for electrical design and construction. Support for this work was provided by the DOE/NNSA under Grant No. DEFG52-06NA26193. Work at Occidental College was supported by the National Science Foundation under DMR-1006118. This work was performed at the National High Magnetic Field Laboratory, which is supported by National Science Foundation Cooperative Agreement No. DMR-0654118, the State of Florida, and the U.S. Department of Energy.
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---
abstract: 'This paper considers the topological degree of $G$-shifts of finite type for the case where $G$ is a nonabelian monoid. Whenever the Cayley graph of $G$ has a finite representation and the relationships among the generators of $G$ are determined by a matrix $A$, the coefficients of the characteristic polynomial of $A$ are revealed as the number of children of the graph. After introducing an algorithm for the computation of the degree, the degree spectrum, which is finite, relates to a collection of matrices in which the sum of each row of every matrix is bounded by the number of children of the graph. Furthermore, the algorithm extends to $G$ of finite free-followers.'
address:
- 'Department of Applied Mathematics, National Dong Hwa University, Hualien 97401, Taiwan, ROC.'
- 'Department of Applied Mathematics, National University of Kaohsiung, Kaohsiung 81148, Taiwan, ROC.'
author:
- 'Jung-Chao Ban'
- 'Chih-Hung Chang\*'
- 'Nai-Zhu Huang'
bibliography:
- '../../grece.bib'
date: 'June 28, 2018'
title: Topological Degree of Shift Spaces on Monoids
---
[^1]
=1.5
Introduction
============
Let $\mathcal{A}$ be a finite alphabet. Given $d \in \mathbb{N}$, a *configuration* is a function from $\mathbb{Z}^d$ to $\mathcal{A}$, and a pattern is a function from a finite subset of $\mathbb{Z}^d$ to $\mathcal{A}$. A subset $X \subseteq \mathcal{A}^{\mathbb{Z}^d}$ is called a *shift space* if $X$, denoted by $X = \mathsf{X}_{\mathcal{F}}$, consists of configurations which avoid patterns from some set $\mathcal{F}$ of patterns. A shift space is called a *shift of finite type* (SFT) if $\mathcal{F}$ is a finite set; $\mathbb{Z}^d$ acts on $X$ by translation of configurations making $X$ a symbolic dynamical system. One of the many motivations in studying symbolic dynamical systems is it helps for the investigation of hyperbolic topological dynamical systems. The interested reader can consult standard literature such as [@Bow-1975; @Rue-1978].
While almost all properties of SFTs are decidable for $d=1$ (cf. [@LM-1995]), the investigation of SFTs is rife for $d \geq 2$ since many undecidability issues company with it. It is even undecidable if an SFT is nonempty [@Berger-MAMS1966]. Different kinds of mixing properties have been introduced for examining the existence and denseness of periodic configurations [@BPS-TAMS2010]. A straightforward generalization is considering SFTs on $G$ which is associated with some algebraic structure. Whenever $G$ is a free monoid, it has been demonstrated that many such issues do not arise. For instance, the conjugacy between two irreducible $G$-SFTs is decidable [@AB-TCS2012]; furthermore, the nonemptiness, extensibility, and the existence of periodic configurations are decidable for $G$-SFTs [@BC-JMP2017; @BC-TAMS2017]. Aside from the qualitative behavior of $G$-SFTs, the phenomena from the computational perspective are also fruitful [@BC-2017a; @BC-2017b; @PS-TCS2018; @Piantadosi-DCDS2008].
For the case where $G = \mathbb{Z}^1$, the topological entropy of an SFT relates to the spectral radius of an integral matrix, and the entropy spectrum (i.e., the set of entropies) of SFTs is the set of logarithms of Perron numbers [@Lind-ETDS1984; @LM-1995]. When $G = \mathbb{Z}^d$ for $d \geq 2$, the entropy of an SFT is a right recursively enumerable number which may not be algebraic and is not computable [@HM-AoM2010; @MP-ETDS2013; @PS-TAMS2015]. However, the story is quite different when $G$ is a free monoid.
Suppose that $G$ is a finitely generated free monoid. Let $\Sigma$ be a finite set which generates $G$. An element $g \in G$ is called an *$i$-word* provided the minimal expresion of $g$ is $g = g_1 g_2 \cdots g_i$ for some $g_1, \ldots, g_i \in \Sigma$. For $n \in \mathbb{N}$, let $\Gamma_n$ denote the set of $n$-blocks in a $G$-SFT; an $n$-block is a pattern whose support consists of all $i$-words in $G$ for $i \leq n$. Suppose the cardinality of $\Gamma_n$ behaves approximately like $c(n) \lambda^{\kappa^n}$ for some $\lambda, \kappa \in \mathbb{R}$ and $c(n) = o(\kappa^n)$. It follows that such a $G$-SFT carries the topological entropy $\ln \lambda$ if $\kappa = d$ is the number of generators of $G$ and $0$ otherwise, and the topological degree (defined in ) is $\ln \kappa$ [@BC-N2017]. The degree spectrum (i.e., the set of degrees) of $G$-SFTs relates to the set of Perron numbers [@BC-2017], and Petersen and Salama reveal an algorithm to estimate the entropy of a hom-shift [@PS-TCS2018]. (A hom-shift, roughly speaking, is a $G$-SFT which is isotropic and symmetric; alternatively, a hom-shift is determined by one rule in each direction. For instance, a $d$-dimensional golden mean shift is a hom-shift. The interested reader is referred to [@CM-PJM2018].) Furthermore, there is an infinite series expression for the entropy provided $|\mathcal{A}| = d = 2$ [@BC-2017b].
This paper considers the topological degree of $G$-SFTs for the case where $G$ is a finitely generated nonabelian monoid. The topological degree of a $G$-shift reflects the idea of entropy dimension. More specifically, the topological degree of a $G$-shift having positive topological entropy is $\ln d$, where $d$ is the number of generators of $G$ and $G$ is a free monoid (cf. [@BC-2017; @BC-N2017; @BC-TAMS2017]). In other words, the investigation of topological degree relates to discovering zero entropy systems. The importance of zero entropy systems has been revealed recently; many $\mathbb{Z}^d$-actions with zero entropy exhibit diverse complexities. See [@Carvalho-PM1997; @CL-JSP2010; @DHP-TAMS2011; @DHK-2018] and the references therein for more details. This elucidation extends the computation of topological degree of $G$-SFT to the case where $G$ is a monoid with finite representation (Theorem \[thm:degree-essential-case\]) or $G$ has finite *free-followers* (defined in , see Section \[sec:finite-follower-case\]). Notably, a finitely generated free monoid has finite representation, and a monoid with finite representation has finite free-followers.
On the other hand, Ban *et al.* [@BCH-2017] reveal that, if $G$ is a free monoid with $d$ generators, the degree spectrum of $G$-SFTs is a finite subset of Perron numbers less than or equal to $d$. Theorems \[thm:degree-general-case-2symbol\] and \[thm:degree-general-case-ksymbol\] elaborate that the topological degree of a $G$-SFT relates to the maximal spectral radius of a collection of integral matrices which are constrained by the structure of the Cayley graph of $G$. Meanwhile, the necessary and sufficient conditions for a $G$-SFT having full topological degree are also addressed, which provide a criterion for determining whether a $G$-SFT has zero entropy.
The introduction ends with a summary of the remainder of the paper. Whenever $G$ is a monoid such that a matrix $A$ determines the relationships among the generators of $G$ and $G$ has *finite representation* (see Section \[sec:definition\]), the coefficients of the characteristic polynomial of $A$ relate to the number of children of the Cayley graph of $G$ (Theorem \[thm:char-poly-A-xi\]). After revealing an algorithm for the computation of the topological degree (Theorem \[thm:degree-algorithm\]), the degree spectrum (Theorems \[thm:degree-general-case-2symbol\] and \[thm:degree-general-case-ksymbol\]) extends the previous result under the hypothesis that $G$ is free (cf. [@BCH-2017]). Furthermore, Section \[sec:finite-follower-case\] extends the algorithm to the case where $G$ has finite free-followers.
Definition and Notation {#sec:definition}
=======================
Although most results in this investigation extend to groups with finite representation, the present paper focuses on shift spaces on monoids for clarity. Let $d$ be a positive integer. A *semigroup* is a set $G = \langle \Sigma | R \rangle$ together with a binary operation which is closed and associative, where $\Sigma = \{s_1, \ldots, s_d\}$ is the set of generators and $R$ is a set of equivalences which describe the relationships among the generators. A *monoid* is a semigroup with an identity element $e$.
Given a finite set of generators $\Sigma = \{s_1, s_2, \ldots, s_d\}$ and a $d \times d$ binary matrix $A$. A monoid $G$ of the form $G = \langle \Sigma | R_A \rangle$ means that $$s_i s_j = s_i \quad \text{if and only if} \quad A(i, j) = 0.$$ Alternatively, $s_i$ is a right (resp. left) free generator if and only if $A(i, j) = 1$ ($A(j, i) = 1$) for $1 \leq j \leq d$. Let $\Sigma_R$ (resp. $\Sigma_L$) denote the set of right (resp. left) free generators of $G$. For each $g \in G$, the *length* $|g|$ indicates the number of generators used in its minimal presentation; that is, $$|g| = \min \{j: g = g_1 g_2 \cdots g_j, g_i \in \Sigma \text{ for } 1 \leq i \leq j\}.$$ Suppose that $C = (V, E)$ is the Cayley graph of $G$. Define a subgraph $F = (V_F, E_F) \subseteq C$ as follows.
1. $g = g_1 \cdots g_n \in V_F$ if $g_n$ is the unique right free generator in $g$;
2. if $g_1 \cdots g_n \in V_F$, then $g_1 \cdots g_j \in V_F$ for each $j \leq n$;
3. $(g, g') \in E_F$ if and only if $(g, g') \in E$ and $g, g' \in V_F$.
A monoid $G$ has a *finite representation* if $F$ is a finite graph. For the rest of this paper, $G = \langle \Sigma | R_A \rangle$ denotes a monoid with a finite representation unless otherwise stated. See Example \[eg:G-A3x3-semigroup\].
![The Cayley graph of monoid $G$ in Example \[eg:G-A3x3-semigroup\] has a finite representation. The generators $s_1, s_2, s_3$ satisfy the equivalences $s_1^2 = s_1$ and $s_2 s_1 = s_2^2 = s_2$. A pseudo-identity $e$ makes the Cayley graph of $G$ strongly connected.[]{data-label="fig:G-A3x3-semigroup"}](DegreeFiniteMonoid-20180628-pics-arxiv)
![The Cayley graph of monoid $G$ in Example \[eg:G-A3x3-semigroup\] has a finite representation. The generators $s_1, s_2, s_3$ satisfy the equivalences $s_1^2 = s_1$ and $s_2 s_1 = s_2^2 = s_2$. A pseudo-identity $e$ makes the Cayley graph of $G$ strongly connected.[]{data-label="fig:G-A3x3-semigroup"}](DegreeFiniteMonoid-20180628-pics-arxiv)
\[eg:G-A3x3-semigroup\] Let $d = 3$ and let $$A =
\begin{pmatrix}
0 &1 &1 \\
0 &0 &1 \\
1 &1 &1
\end{pmatrix}.$$ The generators $s_1, s_2, s_3$ satisfy the equivalences determined by $A$; that is, $$s_1^2 = s_1, s_2 s_1 = s_2, \text{ and } s_2^2 = s_2.$$ It follows that $G$ has a finite representation, sees Figure \[fig:G-A3x3-semigroup\] for the Cayley graph of $G$ together with a pseudo-identity and the graph of its finite representation.
Let $\mathcal{A} = \{1, 2, \ldots, k\}$ be a finite alphabet, where $k$ is a natural number greater than one. A *labeled tree* is a function $t: G \to \mathcal{A}$. For each $g \in G$, $t_g = t(g)$ denotes the label attached to the vertex $g$ of the Cayley graph of $G$. The *full shift* $\mathcal{A}^G$ collects the labeled trees, and the *shift map* $\sigma: G \times \mathcal{A}^G \to \mathcal{A}^G$ is defined as $(\sigma_g t)_{g'} = t_{gg'}$ for $g, g' \in G$.
For each $n \geq 0$, let $\Delta_n = \{g \in G: |g| \leq n\}$ denote the initial $n$-subgraph of the Cayley graph. An *$n$-block* is a function $\tau: \Delta_n \to \mathcal{A}$. A labeled tree $t$ *accepts* an $n$-block $\tau$ if there exists $g \in G$ such that $t_{g g'} = \tau_{g'}$ for all $g' \in \Delta_n$; otherwise, $\tau$ is a *forbidden block* of $t$ (or $t$ *avoids* $\tau$). A *$G$-shift space* is a set $X \subseteq \mathcal{A}^G$ of all labeled trees which avoid all of a certain set of forbidden blocks.
Characterization of Finite Representation {#sec:finite-representation-characteristic-polynomial}
=========================================
For each $n \in \mathbb{N}$, let $$\xi_n = \# \{g \in G: |g| = n, g_n \text{ is the only right free generator}\}$$ be the number of $n$-words which contain exactly one right free generator and end in it. Theorem \[thm:char-poly-A-xi\] reveals $\{\xi_n\}$ plays an important role in the characteristic polynomial of $A$.
\[thm:char-poly-A-xi\] Suppose $G = \langle \Sigma | R_A \rangle$ is a monoid determined by a binary matrix $A$. Then the characteristic polynomial of $A$ is $$\label{eq:char-poly-A-xi}
f(\lambda) = \lambda^d - \sum_{i=1}^d \xi_i \lambda^{d-i}.$$
Before proving Theorem \[thm:char-poly-A-xi\], it is essential to characterize the structure of the Cayley graph of $G$. Let $$P_n = \{g \in G: |g| = n+1 \text{ and } g_1 = g_{n+1}\}$$ and $$\Xi _n = \{g \in P_n: g_n \text{ is the only right free generator}\}$$ be the sets of periodic $(n+1)$-words and periodic $(n+1)$-words whose second last symbol is the one and only right free generator, respectively. It follows immediately that $|P_n| = \mathrm{tr}(A^n)$ and $|\Xi _n| = \xi_n$.
\[lem:periodic-word-contain-free-generator\] For each $(n+1)$-word $g = g_1 g_2 \cdots g_n g_1 \in P_n$, there exists $1 \leq i \leq n$ such that $g_i$ is a right free generator.
Suppose not, it comes immediately that $(g_1 \cdots g_n)^m \in G$ is a vertex of $F$ for all $m \in \mathbb{N}$, which contradicts to $|V_F| < \infty$. The proof is complete.
For each positive integer $n > d$, $\xi_n = 0$. That is, every $(n+1)$-word contains at least one right free generator.
Suppose there exists $n > d$ and $g = g_1 \cdots g_n \in G$ such that $g_n$ is the only free generator. The pigeonhole principle asserts that $g_i = g_j$ for some $1 \leq i < j \leq d+1$. Lemma \[lem:periodic-word-contain-free-generator\] demonstrates that there exists $i \leq \ell \leq j$ such that $g_{\ell}$ is a right free generator, which is a contradiction. This derives the desired result.
Let $$T(\Xi _n) = \{ u_i \cdots u_n u_1 \cdots u_i: i = 1, \ldots, n, u \in \Xi_n \}$$ collect the translation of all elements of $\Xi_n$. Observe that $|T(\Xi_n)| = n\xi_n$. Let $$L(P_m, \Xi_n) = \{u_1 \cdots u_s v_1 \cdots v_n u_{s+1} \cdots u_{m+1}: s = \min_i \{i: u_i \in S_R\}, u \in P_m, v \in \Xi_n \},$$ where $S_R$ is the set of right free generators. That is, $L(P_m, \Xi_n)$ consists of words obtained by inserting the initial $n$-subword of every $v = v_1 \cdots v_{n+1} \in \Xi_n$ in a periodic $(m+1)$-word $u$ right after the first right free generator of $u$. Obviously, $L(P_m, \Xi_n) \subseteq P_{m+n}$.
\[lem:1st2nd-generator-distance\] Suppose $x = x_1 \cdots x_{n+1} \in P_n$ contains at least two right free generators. Let $x_s$ and $x_r$ be the first and second free generators, respectively. Then $$r - s = l \quad \text{if and only if} \quad x \in L(P_{n-l}, \Xi_l)$$
If $x \in L(P_{n-l}, \Xi_l)$, then $x = u_1 \cdots u_s v_1 \cdots v_l u_{s+1} \cdots u_{n-l+1}$ for some $u \in P_{n-l}, v \in \Xi_l$, where $s = \min\{i: u_i \in S_R\}$. Since $v \in \Xi_l$, $x_{s + l} = v_l$ is the second right free generator in $x$. This concludes that $r - s = ( s + l ) - s = l$.
For each $x = x_1 \cdots x_{n+1} \in P_n$ which contains at least two right free generators, let $u = x_1 \cdots x_s x_{s+l+1} \cdots x_{n+1}$ and let $v = x_{s+1} \cdots x_{s+l} x_{s+1}$. Since $u_s = x_s \in S_R $, $x_s x_{s+l-1}$ is a two-word. Furthermore, $x \in P_n$ and $x_1 = x_{n+1}$ indicates that $u_1 = u_{n-l+1}$. Alternatively, $u = u_1 \cdots u_{n-l+1} \in P_{n-l}$. Similarly, $v_l = x_{s+l} \in S_R$ shows that $x_{s+l} x_{s+1}$ is also a two-word. The fact of $x_r = x_{s + l}$ being the second free generator elaborates that $v_1, \ldots, v_{l+1} \notin S_R$, $v_l\in S_R$, and $v_{l+1}= v_1$. Hence, $v \in \Xi_l$. Conclusively, $x \in L(P_{n-l}, \Xi_l)$. This completes the proof.
Lemma \[lem:1st2nd-generator-distance\] illustrates $L(P_{n-l}, \Xi_l) \bigcap L(P_{n-m}, \Xi_m) = \varnothing$ if and only if $l \neq m$. Proposition \[prop:T-L-partition-Pn\], additionally, reveals that the translation and insertion of $\Xi_n$ form a partition of periodic words.
\[prop:T-L-partition-Pn\] For each $n \in \mathbb{N}$, $\{L(P_{n-i}, \Xi_i)\}_{i=1}^{n-1} \bigcup \{T(\Xi_n)\}$ forms a partition of $P_n$.
Obviously, $L(P_{n-i}, \Xi_i) \bigcap T(\Xi_n) = \varnothing$ for $1 \leq i \leq n-1$ since every element of $T(\Xi_n)$ accepts exactly one generator while $L(P_{n-i}, \Xi_i)$ consists of words which contain at least two free generators. The desired result comes immediately from $$P_n = \bigcup_{i=1}^{n-1} L(P_{n-i}, \Xi_i) \bigcup T(\Xi_n).$$ Indeed, the definitions of $L(P_{n-i}, \Xi_i)$ and $T(\Xi_n)$ indicate that $$\bigcup_{i=1}^{n-1} L(P_{n-i}, \Xi_i) \bigcup T(\Xi_n) \subseteq P_n.$$ For each $x \in P_n$, $x \in T(\Xi_n)$ if $x$ has exactly one free generator. Otherwise, $x$ has $x_s$ and $x_r$ as its first two free generators for some $s < r$. Let $l = r - s$. Lemma \[lem:1st2nd-generator-distance\] shows that $x \in L(P_{n-l}, \Xi_l)$. The proof is complete.
Continue with Example \[eg:G-A3x3-semigroup\], recall that $$A=
\begin{pmatrix}
0 &1 &1 \\
0 &0 &1 \\
1 &1 &1
\end{pmatrix}$$ and $G$ has only one right free generator $s_3$. Then $\Xi_1 = \{s_3 s_3\} = T(\Xi_1) = P_1$. Since $\Xi_2$ consists of words of the form $u_1 s_3 u_1$, $$\Xi_2 = \{s_1 s_3 s_1, s_2 s_3 s_2\} \quad \text{and} \quad T(\Xi_2) = \{s_1 s_3 s_1, s_3 s_1 s_3, s_2 s_3 s_2, s_3 s_2 s_3\}.$$ As defined above, $L(P_1, \Xi_1) = \{s_3 s_3 s_3\}$ collects the words obtained by inserting the first word of $\Xi_1$ in each word of $P_1$ right after the first right free generator. It follows that $$P_2 = \{s_1 s_3 s_1, s_2 s_3 s_2, s_3 s_1 s_3, s_3 s_2 s_3, s_3 s_3 s_3\} = T(\Xi_2) \bigcup L(P_1, \Xi_1).$$
Similarly, $\Xi_3 = \{s_1 s_2 s_3 s_1\}$ and $T(\Xi_3) = \{s_1 s_2 s_3 s_1, s_2 s_3 s_1 s_2, s_3 s_1 s_2 s_3\}$. $$\begin{aligned}
L(P_2, \Xi_1) &= \{s_1 s_3 s_3 s_1, s_2 s_3 s_3 s_2, s_3 s_3 s_1 s_3, s_3 s_3 s_2 s_3, s_3 s_3 s_3 s_3\}, \\
L(P_1,\Xi_2) &= \{s_3 s_1 s_3 s_3, s_2 s_3 s_3 s_3\}.\end{aligned}$$ Then $$\begin{aligned}
P_3 &= \{s_1 s_2 s_3 s_1, s_1 s_3 s_3 s_1, s_2 s_3 s_1 s_2, s_2 s_3 s_3 s_2, s_3 s_1 s_2 s_3, \\
& \qquad s_3 s_1 s_3 s_3, s_3 s_2 s_3 s_3, s_3 s_3 s_1 s_3, s_3 s_3 s_2 s_3, s_3 s_3 s_3 s_3\} \\
&= T(\Xi_3) \bigcup L(P_2, \Xi_1) \bigcup L(P_1,\Xi_2).\end{aligned}$$ Furthermore, $\Xi_n = \varnothing$ for $n \geq 4$.
For each real $n \times n$ matrix $A$, there is a recursive formula for the coefficients of the characteristic polynomial of $A$; more explicitly, $f(\lambda) = \det (A - \lambda I) = \sum_{i=0}^n b_i \lambda^{n-i}$, where $$\begin{aligned}
b_0 &= (-1)^n, \quad b_1 = - (-1)^n A_1, \quad b_2 = -\dfrac{1}{2} (b_1 A_1 + (-1)^n A_2), \\
b_3 &= -\dfrac{1}{3} (b_2 A_1 + b_1 A_2 + (-1)^n A_3), \ldots \\
b_n &= -\dfrac{1}{n} (b_{n-1} A_1 + b_{n-2} A_2 + \cdots + b_1 A_{n-1} + (-1)^n A_n),\end{aligned}$$ and $A_i$ is the trace of $A^i$ for $1 \leq i \leq n$ (cf. [@ZD-1963 p.303-305]).
Proposition \[prop:T-L-partition-Pn\] shows that, for $n \in \mathbb{N}$, $$|P_n| = |T(\Xi_n)| + \sum_{i=1}^{n-1} |L(P_i, \Xi_{n-i})|;$$ that is, $A_1 = \xi_1$ and $A_n = n \xi_n + \sum_{i=1}^{n-1} A_i \xi_{n-i}$ for $n \geq 2$. Since $\xi_n = 0$ for $n > d$, follows from $$\xi_n= \frac{1}{n}(A_n-\sum_{i=1}^{n-1}A_i\xi_{n-i}), \quad 1 \leq n \leq d,$$ and the recursive formula of the coefficients of the characteristic polynomial of $A$.
Topological Degree of Shift Spaces on Monoids {#sec:degree-essential-case}
=============================================
Suppose that $X$ is a $G$-shift space. Let $\Gamma_n^{[g]}(X)$ denote the set of $n$-blocks of $X$ rooted at $g$; that is, the support of each block of $\Gamma_n^{[g]}(X)$ is $g\Delta_n$. Let $\gamma_n^{[g]}$ denote the cardinality of $\Gamma_n^{[g]}(X)$. The *topological degree* of $X$ is defined as $$\label{eq:defn-degree}
\mathrm{deg}(X) = \limsup_{n \to \infty} \dfrac{\ln \ln \gamma_n(X)}{n},$$ where $\gamma_n(X) = \gamma_n^{[e]}(X)$. The rest of this paper omits the notation $X$ when it causes no confusion.
For each $a \in \mathcal{A}$, let $\Gamma_{a, n}^{[g]} \subseteq \Gamma_n^{[g]}$ consist of $n$-blocks rooted at $g$ and labeled $a$ at root. A symbol $a$ is *essential*[^2] if $\gamma_{a, n} = |\Gamma_{a, n}| \geq 2$ for some $n \in \mathbb{N}$; otherwise, $a$ is an *inessential* symbol. Proposition \[prop:degree-exists-related-essential\] indicates that the limit in exists provided $X$ is a $G$-SFT, and only the essential symbols matter for calculating the topological degree.
\[prop:degree-exists-related-essential\] Suppose that $X$ is a $G$-SFT. Then the limit exists and $$\mathrm{deg}(X) = \lim_{n \to \infty} \dfrac{\ln \sum_{i=1}^k \ln \gamma_{i, n}}{n} = \lim_{n \to \infty} \dfrac{\ln \sum_{i \in \mathcal{E}} \ln \gamma_{i, n}}{n},$$ where $\mathcal{E} \subseteq \mathcal{A}$ denotes the set of essential symbols.
Theorem \[thm:degree-essential-case\] reveals that, whenever every symbol is essential, the degree of $G$-shift of finite type ($G$-SFT) is the logarithm of the spectral radius of $A$ (recall that $G = \langle \Sigma | R_A \rangle$ is determined by a $d \times d$ matrix $A$, see Section \[sec:definition\]).
\[thm:degree-essential-case\] Suppose that $X$ is a $G$-SFT and every symbol is essential. Then $\mathrm{deg}(X) = \ln \rho_A$, where $\rho_A$ is the spectral radius of $A$.
Ban and Chang [@BC-2017a] reveal an algorithm for computing the degree of $G$-SFTs, where $G$ is a Fibonacci set ($G = \langle \Sigma | R_A \rangle$ with $\Sigma = \{s_1, s_2\}$ and $A = \begin{pmatrix}
1 & 1 \\
1 & 0
\end{pmatrix}$). The algorithm extends to general $G$ via analogous argument. For the compactness and self-containment of the present paper, this section rephrases main ideas and propositions of the algorithm without detailed proofs, and Example \[eg:hom-shift-essential-case\] shows how the algorithm works. For more details about the algorithm, see [@BC-2017a] and the references therein.
Since $G$ has a finite representation, each $G$-SFT relates to a *recurrence representation* (or *system of nonlinear recurrence equations*, SNRE) of the form $$\begin{aligned}
\gamma_{i, n} = \sum c_\mathbf{j} \gamma_{1,n-1}^{j_{1,1}} \cdots \gamma_{k, n-1}^{j_{k,1}} \gamma_{1,n-2}^{j_{1,2}} \cdots \gamma_{k, n-2}^{j_{k,2}} \cdots \gamma_{1,n-l}^{j_{1,l}} \cdots \gamma_{k, n-l}^{j_{k,l}}\end{aligned}$$ for some $l \in \mathbb{N}$, where $c_\mathbf{j} \in \mathbb{N}$, $\mathbf{j} = (j_{1,1}, \ldots, j_{k,1}, \ldots, j_{1,l}, \ldots, j_{k,l})$, and $1 \leq i \leq k$. A *simple subsystem* of $X$ is of the form $$\begin{aligned}
\gamma_{i, n} = \gamma_{1,n-1}^{j_{1,1}} \cdots \gamma_{k, n-1}^{j_{k,1}} \gamma_{1,n-2}^{j_{1,2}} \cdots \gamma_{k, n-2}^{j_{k,2}} \cdots \gamma_{1,n-l}^{j_{1,l}} \cdots \gamma_{k, n-l}^{j_{k,l}}\end{aligned}$$ for some $j_{1,1}, \ldots, j_{k,1}, \ldots, j_{1,l}, \ldots, j_{k,l}$ and $1 \leq i \leq k$. Take logarithm on the above equation and let $$\theta_n = ( \ln \gamma_{1,n}, \ldots, \ln \gamma_{k,n}, \ln \gamma_{1;n-1}, \ldots, \ln \gamma_{k,n-1}, \ldots, \ln \gamma_{1, n-l+1}, \ldots, \ln \gamma_{k, n-l+1})',$$ where $v'$ refers to the transpose of $v$. Then there exists a $kl \times kl$ matrix $M$ called *adjacency matrix* (of the simple subsystem) such that $\theta_n = M \theta_{n-1}$ for $n \geq l+1$. Theorem \[thm:degree-algorithm\] reveals that the degree of $X$ relates to the maximal spectral radius among the adjacency matrices of simple subsystems of $X$.
\[thm:degree-algorithm\] Suppose that $X$ is a $G$-SFT. Then $$\mathrm{deg}(X) = \max\{\ln \rho_M: M \text{ is the adjacency matrix of a simple subsystem of } X\}.$$
\[eg:hom-shift-essential-case\] Suppose that $X$ is a hom-shift on $G$ determined by a $k \times k$ binary matrix $T$; that is, for each labeled tree $t \in X$ and $g \in G$, a pattern $(t_g, t_{g s_i})$ is allowable if and only if $T(t_g, t_{g s_i}) = 1$. For instance, consider the case where $k = 2$ and $T = \begin{pmatrix}
1 & 1 \\
1 & 1
\end{pmatrix}$. A hom-shift defined by $T$ is a full $G$-shift and $\mathrm{deg}(X) = \ln \rho_A$. This example shows that the above algorithm derives the desired result.
It follows from $s_3$ being a free generator that, for $i = 1, 2$, $\gamma_{i, n}^{[g]} = \gamma_{i, n}$ if $g = g' s_3$ for some $g, g' \in G$. Hence, for $i=1, 2$, $$\begin{aligned}
\gamma_{i,n} &= (\gamma_{1,n-1}^{[s_1]}+\gamma_{2,n-1}^{[s_1]})(\gamma_{1,n-1}^{[s_2]}+\gamma_{2,n-1}^{[s_2]})(\gamma_{1,n-1}^{[s_3]}+\gamma_{2,n-1}^{[s_3]}) \\
&= (\gamma_{1,n-1}^{[s_1]}+\gamma_{2,n-1}^{[s_1]})(\gamma_{1,n-1}^{[s_2]}+\gamma_{2,n-1}^{[s_2]})(\gamma_{1,n-1}+\gamma_{2,n-1})\end{aligned}$$ Combining $$\begin{aligned}
\gamma_{i,n-1}^{[s_1]} &= (\gamma_{1,n-2}^{[s_1 s_2]}+\gamma_{2,n-2}^{[s_1 s_2]})(\gamma_{1,n-2}^{[s_1 s_3]}+\gamma_{2,n-2}^{[s_1 s_3]}) = (\gamma_{1,n-2}^{[s_1 s_2]}+\gamma_{2,n-2}^{[s_1 s_2]})(\gamma_{1,n-2} + \gamma_{2,n-2}) \\
\gamma_{i,n-1}^{[s_2]} &= \gamma_{1,n-2}^{[s_2 s_3]}+\gamma_{2,n-2}^{[s_2 s_3]} = \gamma_{1,n-2} + \gamma_{2,n-2}\end{aligned}$$with $$\begin{aligned}
\gamma_{i,n-1}^{[s_1 s_2]} = \gamma_{1,n-3}^{[s_1 s_2 s_3]} + \gamma_{2,n-3}^{[s_1 s_2 s_3]} = \gamma_{1,n-3} + \gamma_{2,n-3}\end{aligned}$$ derives that $$\begin{aligned}
\gamma_{i,n} &=
(4\gamma_{1,n-3}\gamma_{1,n-2}+4\gamma_{2,n-3}\gamma_{1,n-2}+4\gamma_{1,n-3}\gamma_{2,n-2}+4\gamma_{2,n-3}\gamma_{2,n-2}) \cdot \\
& \qquad (2\gamma_{1,n-2}+2\gamma_{2,n-2}) (\gamma_{1,n-1}+\gamma_{2,n-1})\end{aligned}$$ Let $$\theta_n = (\ln \gamma_{1,n},\ln \gamma_{2,n},\ln \gamma_{1,n-1},\ln \gamma_{2,n-1},\ln \gamma_{1,n-2},\ln \gamma_{2,n-2})'.$$ For every simple subsystem of $X$, the corresponding adjacency matrix is of the form $$M=
\begin{pmatrix}
B_{1}&B_{2}&B_{3}\\
I&0&0\\
0&I&0%
\end{pmatrix},$$ where $B_l$ is a $2 \times 2$ matrix satisfies $\sum_{q = 1}^2 B_l(p, q) = \xi_l$ for all $l = 1,2,3$, $p = 1, 2$. That is, $\theta_n = M \theta_{n-1}$ for $n \geq 3$. Let $$v = (\rho_A^2, \rho_A^2, \rho_A, \rho_A, 1, 1)'.$$ Observe that $Mv = \rho_A v$. Perron-Frobenius Theorem demonstrates that $\rho_A$ is also the spectral radius of $M$. In other words, $\mathrm{deg}(X) = \ln \rho_A$.
The proof focuses on the case where $X$ is a $G$-SFT determined by $k \times k$ binary matrices $A_1, \ldots, A_d$ for clarification, the demonstration of the general case is analogous. In this case, for each labeled tree $t \in X$ and $g \in G$, $(t_g, t_{g s_l})$ is allowable if and only if $A_l (t_g, t_{g s_l}) = 1$ for $1 \leq l \leq d$.
Write $A_l = (a_{l; i_1, i_2})$ for $1 \leq l \leq d, 1 \leq i_1, i_2 \leq k$. Since $\gamma_{j, n}^{[g s_l]} = \gamma_{j, n}$ for all $1 \leq j \leq k, n \in \mathbb{N}, g \in G$ provided $s_l$ is a free generator, for $1 \leq i \leq k$, $$\begin{aligned}
\gamma_{i,n} &= \prod_{s_l \in \Sigma} (\sum_{j_1=1}^k a_{s_l;i,j_1}\gamma_{{j_1},n-1}^{[s_l]}) \\
&= \prod_{s_l \notin S_R} (\sum_{j_1=1}^k a_{s_l;i,j_1}\gamma_{{j_1},n-1}^{[s_l]})\prod_{s_l\in S_R} (\sum_{j_1=1}^k a_{s_l;i,j_1} \gamma_{{j_1},n-1}^{[s_l]}) \\
&= \prod_{s_l \notin S_R} (\sum_{j_1=1}^k a_{s_l;i,j_1}\gamma_{{j_1},n-1}^{[s_l]})\prod_{s_l\in S_R} (\sum_{j_1=1}^k a_{s_l;i,j_1} \gamma_{{j_1},n-1})\end{aligned}$$ Observe that $f_1 = \prod_{s_l\in S_R} (\sum_{j_1=1}^k a_{s_l;i,j_1} \gamma_{{j_1},n-1})$ is a polynomial of degree $\xi_1$ over $\gamma_{1,n-1}, \ldots,\gamma_{k,n-1}$.
Similarly, for each $s_l$ which is not a free generator, $$\begin{aligned}
\gamma_{j_1,n-1}^{[s_l]} = \prod_{s_l s_m \in G} (\sum_{j_2=1}^k a_{s_m; j_1, j_2} \gamma_{{j_2},n-2}^{[s_l s_m]})\end{aligned}$$infers that $$\begin{aligned}
\gamma_{i,n} = f_1 \cdot \prod_{s_l s_m \in G, s_l \notin S_R} (\sum_{j_1,j_2=1}^k a_{s_l;i,j_1} a_{s_m;j_1,j_2}\gamma_{j_2,n-2}^{[s_l s_m]}).\end{aligned}$$ Let $$\begin{aligned}
f_2 &= \prod_{s_l s_m \in G, s_l \notin S_R, s_m \in S_R} (\sum_{j_1,j_2=1}^k a_{s_l;i,j_1} a_{s_m;j_1,j_2}\gamma_{j_2,n-2}^{[s_l s_m]}) \\
&= \prod_{s_l s_m \in G, s_l \notin S_R, s_m \in S_R} (\sum_{j_1,j_2=1}^k a_{s_l;i,j_1} a_{s_m;j_1,j_2}\gamma_{j_2,n-2})\end{aligned}$$ Then $f_2$ is a polynomial of degree $\xi_2$. Repeating the same process decompose $\gamma_{i,n} = f_1 f_2 \cdots f_{\ell}$, where $\ell = \max\{j: \xi_j \neq 0\} \leq d$, and $f_j$ is a polynomial of degree $\xi_j$ over $\gamma_{1,n-j}, \ldots, \gamma_{k,n-j}$ for $1 \leq j \leq \ell$.
Let $$\theta_n= (\ln \gamma_{1,n},\cdots,\ln \gamma_{k,n},\ln \gamma_{1,n-1},\cdots,\ln \gamma_{k,n-1},\cdots,\ln \gamma_{1,n-d+1},\cdots,\ln \gamma_{k,n-d+1})'.$$ For each simple subsystem of $X$, there exists $$M=
\begin{pmatrix}
B_1& B_2& B_3& \cdots & B_{\ell} \\
I& 0 & \cdots & \cdots &0 \\
0&I & 0 & \cdots & 0\\
\vdots && \ddots& & \vdots \\
0 & \cdots & 0 &I&0%
\end{pmatrix},$$ where $B_j$ is a $k \times k$ nonnegative integral matrix satisfies $\sum_{q=1}^k B_j (p, q) = \xi_i$ for all $1 \leq j \leq \ell, 1 \leq p \leq k$, such that $M$ is the corresponding adjacency matrix (note that $\xi_j = 0$ for $j > \ell$). That is, $\theta_n = M \theta_{n-1}$ is the designated simple subsystem. Let $v = (\rho_A^{\ell-1} \cdots \rho_A 1)' \otimes \mathbf{1}_k$, where $\otimes$ is the Kronecker product and $\mathbf{1}_k \in \mathbb{R}^k$ is the vector consisting of $1$’s. It comes immediately that $M v = \rho_A v$. Perron-Frobenius Theorem infers that $\rho_A$ is also the spectral radius of $M$. Hence, $\mathrm{deg}(X) = \ln\rho_A$.
This completes the proof.
For the general cases, Proposition \[prop:degree-exists-related-essential\] demonstrates that Theorem \[thm:degree-algorithm\] holds if the rows and columns of matrix $M$ indexed by inessential symbols are eliminated.
Degree Spectrum of $G$-SFTs {#sec:degree-spectrum}
===========================
Theorem \[thm:degree-essential-case\] reveals that the degree of $G$-SFTs is $\ln \rho_A$ whenever every symbol is essential. This section extends to the general case and gives the complete characterization of degree spectrum of $G$-SFTs.
Let $\mathbb{Z}_+$ be the set of nonnegative integers. For $\mathbf{m}, \mathbf{n} \in \mathbb{Z}_+^d$, define $\mathbf{m} \preceq \mathbf{n}$ if $m_i \leq n_i$ for $1 \leq i \leq d$, and $\mathbf{m} \prec \mathbf{n}$ if $\mathbf{m} \preceq \mathbf{n}$ and $\mathbf{m} \neq \mathbf{n}$. Theorem \[thm:degree-general-case-2symbol\] characterizes the degree spectrum of $G$-SFTs for the case where $k = 2$.
\[thm:degree-general-case-2symbol\] Suppose that $k = 2$. Let $\xi = (\xi_1, \ldots, \xi_d)$. The degree spectrum of $G$-SFTs is $$\begin{aligned}
H = \{\ln \lambda: \lambda = \max \{x: x^d - \sum_{i=1}^d \alpha_i x^{d-i} = 0\} \text{ for some } \alpha \in \mathbb{Z}_+^d, \alpha \preceq \xi \}.\end{aligned}$$
Obviously, two inessential symbols infers that the degree is $0$; Theorem \[thm:degree-essential-case\] indicates the degree is $\ln \rho_A$ and $\rho_A = \max \{x: x^d - \sum_{i=1}^d \beta_i x^{d-i} = 0\}$ if every symbol is essential. It suffices to consider the case where $1 \in \mathcal{A}$ is essential and $2 \in \mathcal{A}$ is inessential.
Without loss of generality, assume that $\xi_i > 0$ for $1 \leq i \leq d$. Similar to the discussion in Example \[eg:hom-shift-essential-case\], write $\gamma_{1,n} = f_1 f_2 \cdots f_d$, where $$f_1 = \prod_{u_1 \in S_R} (\sum_{j_1=1}^2 a_{u_1;1,j_1} \gamma_{j_1,n-1})$$ and $$f_i = \prod_{u_1\cdots u_i \in G, u_1, \ldots, u_{i-1} \notin S_R, u_i \in S_R} (\sum_{j_1, \ldots, j_i =1}^2 a_{u_1;1,j_1} a_{u_2;j_1,j_2}\cdots a_{u_i; j_{i-1}, j_i} \gamma_{{j_i}, n-i})$$ for $2 \leq i \leq d$, and $f_i$ is a polynomial of degree $\xi_i$. Hence, every simple subsystem of $X$ is of the form $$\begin{aligned}
\gamma_{1,n} &= \gamma_{1,n-1}^{\eta_1}\gamma_{2,n-1}^{\tau_1}\gamma_{1,n-2}^{\eta_2}\gamma_{2,n-2}^{\tau_2}\cdots\gamma_{1,n-d}^{\eta_d}\gamma_{2,n-d}^{\tau_d}, \\
\gamma_{2,n} &= \gamma_{2,n-1}^{\xi_1} \gamma_{2,n-2}^{\xi_2} \cdots \gamma_{2,n-d}^{\xi_d},\end{aligned}$$where $\eta_i+\tau_i = \xi_i$ for $1 \leq i \leq d$.
Let $\theta_n= (\ln \gamma_{1,n},\ln \gamma_{2,n},\ln \gamma_{1,n-1},\ln \gamma_{2,n-1},\cdots,\ln \gamma_{1,n-d+1},\ln \gamma_{2,n-d+1})'$, and let $$M=
\begin{pmatrix}
\eta_1& \tau_1& \eta_2 & \tau_2 &\cdots & \eta_d & \tau_d \\
0 & \xi_1 & 0 & \xi_2 &\cdots &0 &\xi_d \\
1 & & & & & 0 &0 \\
& 1 & & & &\vdots & \vdots\\
& &\ddots & & &\vdots &\vdots\\
& & & \ddots& &\vdots &\vdots \\
& & & & 1 &0 &0%
\end{pmatrix}.$$ Then the simple subsystem is $\theta_n = M \theta_{n-1}$. Since $2$ is inessential, the degree of such a simple subsystem is $\ln \lambda$, where $\lambda$ is the spectral radius of $$\overline{M} =
\begin{pmatrix}
\eta_1& \eta_2 &\cdots& \eta_d \\
1 & & &0 \\
&\ddots& &\vdots\\
& & 1 &0%
\end{pmatrix}.$$ A straightforward examination elaborates that $\lambda = \max \{x: x^d - \sum_{i=1}^d \eta_i x^{d-i} = 0\}$. This derives $$H \subseteq \{\ln \lambda: \lambda = \max \{x: x^d - \sum_{i=1}^d \alpha_i x^{d-i} = 0\} \text{ for some } \alpha \in \mathbb{Z}_+^d, \alpha \preceq \xi \}.$$
To show that, for each $\alpha \in \mathbb{Z}_+^d$ satisfying $\alpha \preceq \xi$, there exists a $G$-SFT such that $\mathrm{deg}{X} = \ln \lambda$ with $\lambda = \max \{x: x^d - \sum_{i=1}^d \alpha_i x^{d-i} = 0\}$, construct a one-step $G$-SFT as follows. Without loss of generality, assume that $\xi_i > 0$ for $i \leq d$. The symbol $2$ is inessential in the following construction, thus it suffices to mention where to label $1$.
For $n \in \mathbb{N}$, let $S_1 = \{e\}$ and, for $n \geq 2$, let $$S_n = \{g = g_1 \cdots g_{n-1}: g s \in G \text{ for some } s \in S_R, g_i \notin S_R \text{ for } 1 \leq i \leq n-1\}.$$ Observe that $S_n = \varnothing$ if and only if $n > d$ (under the assumption that $\xi_n = 0$ if and only if $n > d$). Let $$\overline{S}_n = \{g s: g \in S_n, s \in \Sigma, g s \in G\}.$$ Then $\bigcup_{i=1}^d \overline{S}_n$ is the set of supports of two-blocks of $X$ up to shift. For $n = 1$, let $B_1 \subseteq \overline{S}_1^{\mathcal{A}}$ consists of $1$-blocks $\phi$ which satisfy $\phi_g = 1$ if and only if $$g \in S_1 \bigcup \Sigma \setminus S_R \quad \text{and} \quad |\{g \in S_R: \phi_g = 1\}| = \alpha_1.$$ In other words, each pattern of $B_1$ labels $1$ at, except from the root and non-free generators, arbitrary $\alpha_1$ free generators. This makes $\max\{p: \gamma_{1,n-1}^p | \gamma_{1, n}\} = \alpha_1$.
Analogously, let $B_2 \subseteq \overline{S}_2^{\mathcal{A}}$ consists of $1$-blocks $\phi$ which satisfy $\phi_g = 1$ if and only if $$g \in S_2 \bigcup \{g' s \in \overline{S}_2: s \notin S_R\} \quad \text{and} \quad |\{g' s \in \overline{S}_2: s \in S_R, \phi_{g's} = 1\}| = \alpha_2.$$ Then $\max\{p: \gamma_{1,n-2}^p | \gamma_{1, n}\} = \alpha_2$. Repeating the same process to construct $B_i$ for $i \leq d$ makes $$\max\{p: \gamma_{1,n-i}^p | \gamma_{1, n}\} = \alpha_i \quad \text{for} \quad 1 \leq i \leq d.$$ For each subset $H \subseteq G$ such that $H$ forms the support of a one-block, observe that there exists $g \in G$ ended in free generator and $1 \leq i \leq d$ such that $H = g H'$ for some $H' \subseteq \overline{S}_i$. Then each labeled pattern of support $H$ follows the same rule as determined in $\overline{S}_i$. Notably, Such a pattern is still in $B_i$.
Therefore, every simple subsystem of $X$ generated by $B = \bigcup_{i=1}^d B_i$ is of the form $$\gamma_{1, n} = c \cdot \gamma_{1, n-1}^{\alpha_1} \gamma_{2, n-1}^{\beta_1} \gamma_{1, n-2}^{\alpha_2} \gamma_{2, n-2}^{\beta_2} \cdots \gamma_{1, n-d}^{\alpha_d} \gamma_{2, n-d}^{\beta_d}, \quad \gamma_{2, n} = \gamma_{2, n-1}^d,$$ where $c$ is a constant, and $\alpha_i + \beta_i = \xi_i$ for all $i$. A straightforward examination indicates that $\mathrm{deg}{X} = \ln \lambda$ with $\lambda = \max \{x: x^d - \sum_{i=1}^d \alpha_i x^{d-i} = 0\}$.
The proof is complete.
Notably, $\xi_n = 0$ for $n \geq 2$ if and only if $G$ is a free monoid. In this case, $H = \{0, \ln 2, \ldots, \ln d\}$ is revealed in [@BCH-2017].
Theorem \[thm:degree-general-case-ksymbol\] extends Theorem \[thm:degree-general-case-2symbol\] to the general case. The proof is similar, thus it is omitted.
\[thm:degree-general-case-ksymbol\] Let $\mathcal{M}$ be the set consisting of $$M=
\begin{pmatrix}
C_1& C_2& C_3& \cdots & C_d \\
I& 0 & \cdots & \cdots &0 \\
0&I & 0 & \cdots & 0\\
\vdots && \ddots& & \vdots \\
0 & \cdots & 0 &I&0%
\end{pmatrix}$$for some $l \times l$ matrices $C_i$, $l \leq k$, satisfying $\sum_{q=1}^l C_i (p, q) \leq \xi_i$ for all $1 \leq i \leq d$, $1 \leq p \leq l$. The degree spectrum of $G$-SFTs is $$\begin{aligned}
H = \{\ln \lambda: \lambda \text{ is the spectral radius of } M \in \mathcal{M} \}.\end{aligned}$$
Corollary \[cor:iff-cond-for-full-degree\], follows from the proof of Theorem \[thm:degree-general-case-2symbol\], elaborates a necessary and sufficient condition of a $G$-SFT achieved full degree.
\[cor:iff-cond-for-full-degree\] Suppose that $X$ is a $G$-SFT. Then $\mathrm{deg}(X) = \ln \rho_A$ if and only if the essential symbols form a subshift on right free generators; that is, for each $s \in S_R$ and $\phi$ is a one-block with support $\mathrm{supp}(\phi) = s \Sigma$, $\phi_g$ is essential for $g \in \mathrm{supp}(\phi)$.
It suffices to consider the case where $k = 2$ since the demonstration of the general case is analogous but more complicated. Recall that, in the proof of Theorem \[thm:degree-general-case-2symbol\], every simple subsystem of $X$ is of the form $$\begin{aligned}
\gamma_{1,n} &= \gamma_{1,n-1}^{\eta_1}\gamma_{2,n-1}^{\tau_1}\gamma_{1,n-2}^{\eta_2}\gamma_{2,n-2}^{\tau_2}\cdots\gamma_{1,n-d}^{\eta_d}\gamma_{2,n-d}^{\tau_d}, \\
\gamma_{2,n} &= \gamma_{1,n-1}^{\delta_1}\gamma_{2,n-1}^{\iota_1}\gamma_{1,n-2}^{\delta_2}\gamma_{2,n-2}^{\iota_2}\cdots\gamma_{1,n-d}^{\delta_d}\gamma_{2,n-d}^{\iota_d},\end{aligned}$$ where $\eta_i + \tau_i = \delta_i + \iota_i = \xi_i$ for $1 \leq i \leq d$. In other words, $\mathrm{deg}(X) = \ln \lambda$, where $\lambda$ is the spectral radius of one of the following matrix, which depends on the essential of symbols. $$\begin{aligned}
M_1 &=
\begin{pmatrix}
\eta_1& \tau_1& \eta_2 & \tau_2 &\cdots & \eta_d & \tau_d \\
\delta_1 & \iota_1 & \delta_2 & \iota_2 &\cdots &\delta_d &\iota_d \\
1 & & & & & 0 &0 \\
& 1 & & & &\vdots & \vdots\\
& &\ddots & & &\vdots &\vdots\\
& & & \ddots& &\vdots &\vdots \\
& & & & 1 &0 &0%
\end{pmatrix}, \\
M_2 &=
\begin{pmatrix}
\eta_1& \eta_2 &\cdots& \eta_d \\
1 & & &0 \\
&\ddots& &\vdots\\
& & 1 &0
\end{pmatrix}, \quad M_3 =
\begin{pmatrix}
\iota_1& \iota_2 &\cdots& \iota_d \\
1 & & &0 \\
&\ddots& &\vdots\\
& & 1 &0
\end{pmatrix}.\end{aligned}$$ It follows that $\lambda = \rho_A$ if and only if exactly one of the following three conditions holds.
1. (Case $M_1$) Two symbols are essential.
2. (Case $M_2$) Symbol $1$ is essential and $\eta_i = \xi_i$ for $1 \leq i \leq d$.
3. (Case $M_3$) Symbol $2$ is essential and $\iota_i = \xi_i$ for $1 \leq i \leq d$.
This completes the proof.
Groups with Finite Free-Followers {#sec:finite-follower-case}
=================================
Suppose that $G$ is a monoid. For each $g \in G$, defined the *free-follower set* (free-follower for short) of $g$ as $$\label{eq:follower-set-definition}
F_g = \{g' \in G: |gg'| = |g| + |g'| \}.$$ Set $F = \{F_g: g \in G\}$. Then $G$ *has finite free-followers* if $F$ is finite. It is easily seen that every finitely generated free monoid $G$ has finite -free-followers since $F_g = G$ for each $g \in G$. The investigation in Sections \[sec:degree-essential-case\] and \[sec:degree-spectrum\] extends to the case where $G$ has finite free-followers via analogous elaboration. This section, rather than rephrases every result in the previous two sections, presents an example to address how to compute the degree of a $G$-SFT ($G$ has finite free-followers herein) for the compactness of the paper.
Suppose that $d = k = 2$. In this case, $\Sigma = \{s_1, s_2\}$ and $\mathcal{A} = \{1, 2\}$. Let $G = \langle \Sigma | R \rangle$ be the monoid with $R = \{s_2 s_1^{2i+1} s_2 = s_2\}_{i \geq 0}$. It follows that $G$ has finite free-followers. Indeed, let $$\begin{aligned}
&F_{s_1} = \{s_1, s_2, s_1^2, s_1 s_2, s_2 s_1, s_2^2, \ldots\} = G, \\
&F_{s_2} = \{s_1, s_2, s_1^2, s_2 s_1, s_2^2, s_1^3, s_1^2 s_2, \ldots\} = \{s_1^n\}_{n \geq 1} \bigcup \{s_1^{2i} s_2 g: g = s_1^n, s_1^{2j} s_2^n, i, j, n \geq 0 \}, \\
&F_{s_2 s_1} = \{s_1, s_1^2, s_1 s_2, s_1^3, \ldots\} = \{s_1^n\}_{n \geq 1} \bigcup \{s_1^{2i+1} s_2 g: g = s_1^n, s_1^{2j} s_2^n, i, j, n \geq 0 \}.\end{aligned}$$ An examination indicates that, for each $g \in G$, $$F_g = \left\{ \begin{aligned}
&F_{s_1}, && g = s_1^n; \\
&F_{s_2}, && g \text{ ends in } s_2 s_1^{2i}, i \geq 0; \\
&F_{s_2 s_1}, && g \text{ ends in } s_2 s_1^{2i+1}, i \geq 0.
\end{aligned}\right.$$ A straightforward examination elaborates that there is a one-to-one correspondence between the monoid $G$ and the set of finite words of one-dimensional even-shift.
Let $X$ be a hom-shift on $G$ determined by $T = \begin{pmatrix}
1 & 1 \\
1 & 1
\end{pmatrix}$. Alternatively, $X$ is a full $G$-shift; it follows immediately that $\mathrm{deg}(X) = \ln \lambda$, where $\lambda = \frac{1 + \sqrt{5}}{2}$ satisfies $\lambda^2 - \lambda - 1 = 0$. The following shows that the algorithm in Section \[sec:degree-essential-case\] derives the same result.
Observe that $\gamma_{i, n}^{[g]} = \gamma_{i, n}$ for $i = 1, 2$ since $F_{s_1^j} = G$ for $j \in \mathbb{N}$. For $i = 1, 2$, $$\begin{aligned}
\gamma_{i, n} &= (\gamma_{1, n-1}^{[s_1]} + \gamma_{2, n-1}^{[s_1]}) (\gamma_{1, n-1}^{[s_2]} + \gamma_{2, n-1}^{[s_2]}) \\
&= (\gamma_{1, n-1} + \gamma_{2, n-1}) (\gamma_{1, n-1}^{[s_2]} + \gamma_{2, n-1}^{[s_2]})\end{aligned}$$ Also, $F_{s_2^2} = F_{s_2}$ and $F_{s_2 s_1^2} = F_{s_2}$ infer that $$\begin{aligned}
\gamma_{i, n-1}^{[s_2]} &= (\gamma_{1, n-2}^{[s_2 s_1]} + \gamma_{2, n-2}^{[s_2 s_1]}) (\gamma_{1, n-2}^{[s_2 s_2]} + \gamma_{2, n-2}^{[s_2 s_2]}) \\
&= (\gamma_{1, n-2}^{[s_2 s_1]} + \gamma_{2, n-2}^{[s_2 s_1]}) (\gamma_{1, n-2}^{[s_2]} + \gamma_{2, n-2}^{[s_2]}) \\
\gamma_{i, n-2}^{[s_2 s_1]} &= \gamma_{1, n-3}^{[s_2 s_1^2]} + \gamma_{2, n-3}^{[s_2 s_1^2]} = \gamma_{1, n-3}^{[s_2]} + \gamma_{2, n-3}^{[s_2]}\end{aligned}$$ Hence, the SNRE of $X$ is $$\begin{aligned}
\gamma_{i, n} &= 2 (\gamma_{1, n-1} + \gamma_{2, n-1}) (\gamma_{1, n-2}^{[s_2]} + \gamma_{2, n-2}^{[s_2]}) (\gamma_{1, n-3}^{[s_2]} + \gamma_{2, n-3}^{[s_2]})\end{aligned}$$ for $i = 1, 2$. Let $\theta_n = (\ln \gamma_{1, n}^{[s_2]}, \ln \gamma_{2, n}^{[s_2]}, \ln \gamma_{1, n-1}^{[s_2]}, \ln \gamma_{2, n-1}^{[s_2]})'$ and let $$M = \begin{pmatrix}
\eta_1 & \tau_1 & \eta_2 & \tau_2 \\
\delta_1 & \iota_1 & \delta_2 & \iota_2 \\
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0
\end{pmatrix}.$$ Then, every simple subsystem of the invariant system $\ln \gamma_{1, n}^{[s_2]}, \ln \gamma_{2, n}^{[s_2]}$ is of the form $\theta_n = M \theta_{n-1}$ with $\eta_j + \tau_j = \delta_j + \iota_j = 1$ for $1 \leq j \leq 2$. It follows that $\ln \gamma_{i, n}^{[s_2]} \approx e^{\lambda n}$ for $i = 1, 2$ and $n$ large enough.
Furthermore, every simple subsystem of $X$ is of the form $$\begin{aligned}
\ln \gamma_{1, n} &\approx \eta \ln \gamma_{1, n-1} + \tau \ln \gamma_{2, n-1} + e^{(n-2) \lambda} + e^{(n-3) \lambda} \\
\ln \gamma_{2, n} &\approx \delta \ln \gamma_{1, n-1} + \iota \ln \gamma_{2, n-1} + e^{(n-2) \lambda} + e^{(n-3) \lambda}\end{aligned}$$ where $\eta + \tau = \delta + \iota = 1$. A straightforward examination shows that $$\mathrm{deg}(X) = \lim_{n \to \infty} \dfrac{\ln (\ln \gamma_{1, n} + \ln \gamma_{2, n})}{n} = \ln \lambda.$$ This concludes the desired result.
[^1]: \*Author to whom any correspondence should be addressed.
[^2]: In one-dimensional symbolic dynamical systems, a graph presentation of an SFT is called essential if there is no stranded vertex [@LM-1995]. In other words, every vertex has its contribution in the corresponding SFT. This paper extends the idea to the alphabet of $G$-SFTs.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Hyperbolicity measures, in terms of (distance) metrics, how close a given graph is to being a tree. Due to its relevance in modeling real-world networks, hyperbolicity has seen intensive research over the last years. Unfortunately, the best known algorithms for computing the hyperbolicity number of a graph (the smaller, the more tree-like) have running time $O(n^4)$, where $n$ is the number of graph vertices. Exploiting the framework of parameterized complexity analysis, we explore possibilities for “linear-time FPT” algorithms to compute hyperbolicity. For instance, we show that hyperbolicity can be computed in time $O(2^{O(k)} + n +m)$ ($m$ being the number of graph edges) while at the same time, unless the SETH fails, there is no $2^{o(k)}n^2$-time algorithm.'
author:
- 'Till Fluschnik[^1]'
- 'Christian Komusiewicz[^2]'
- 'George B. Mertzios'
- 'André Nichterlein[^3]'
- Rolf Niedermeier
- 'Nimrod Talmon[^4]'
bibliography:
- 'hyper-arxiv2017-short.bib'
title: 'When can Graph Hyperbolicity be computed in Linear Time?[^5]'
---
Introduction
============
(Gromov) hyperbolicity [@Gro87] of a graph is a popular attempt to capture and measure how *metrically* close a graph is to being a tree. The study of hyperbolicity is motivated by the fact that many real-world graphs are tree-like from a distance metric point of view [@AD16; @BCCM15]. This is due to the fact that many of these graphs (including Internet application networks or social networks) possess certain geometric and topological characteristics. Hence, for many applications, including the design of (more) efficient algorithms, it is useful to know the hyperbolicity of a graph. The hyperbolicity of a graph is a nonnegative number $\delta$; the smaller $\delta$ is, the more tree-like the graph is; in particular, $\delta =0$ means that the graph metric indeed is a tree metric. Typical hyperbolicity values for real-world graphs are below 5 [@AD16].
Hyperbolicity can be defined via a four-point condition: Considering all size-four subsets $\{ a, b, c, d \}$ of the vertex set of the graph, one takes the (nonnegative) difference between the biggest two of the three sums ${\ensuremath{\overline{a b}}}+{\ensuremath{\overline{c d}}}$, ${\ensuremath{\overline{a c}}}+{\ensuremath{\overline{b d}}}$, and ${\ensuremath{\overline{a d}}}+{\ensuremath{\overline{b c}}}$, where, e.g., ${\ensuremath{\overline{a b}}}$ denotes the length of the shortest path between vertices $a$ and $b$ in the given graph. For an $n$-vertex graph, this characterization of hyperbolicity directly implies a simple (brute-force) $O(n^4)$-time algorithm to compute its hyperbolicity. It has been observed that this polynomial running time is too slow for computing the hyperbolicity of big graphs as occurring in applications [@AD16; @BCCM15; @BCH16; @FIV15]. On the theoretical side, it was shown that relying on some (rather impractical) matrix multiplication results, one can improve the upper bound to $O(n^{3.69})$ [@FIV15]. Moreover, roughly quadratic lower bounds are known [@BCH16; @FIV15]. In practice, however, the best known algorithm still has an $O(n^4)$-time worst-case bound but uses several clever tricks when compared to the straightforward brute-force algorithm [@BCCM15]. Indeed, based on empirical studies an $O(mn)$ running time is claimed, where $m$ is the number of edges in the graph.
To explore the possibility of faster algorithms for hyperbolicity in relevant special cases is the guiding principle of this work. More specifically, introducing some graph parameters, we investigate whether one can compute hyperbolicity in linear time when these parameters take small values. In other words, we employ the framework of parameterized complexity analysis (so far mainly used for studying [$\mathsf{NP}$]{}-hard problems) applied to the polynomial-time solvable hyperbolicity problem. In this sense, we follow the recent trend of studying “FPT in P” [@GMN15]. Indeed, other than for [$\mathsf{NP}$]{}-hard problems, for some parameters we achieve not only exponential dependence on the parameter but also polynomial ones.
#### Our contributions.
\[tab:results\] summarizes our main results. On the positive side, for a number of natural graph parameters we can attain “linear FPT” running times. Our “positive” graph parameters here are the following:
the covering path number, that is, the minimum number of paths where only the endpoints have degree greater than two and which cover all vertices;
the feedback edge number, that is, the minimum number of edges to delete to obtain a forest;
the number of graph vertices of degree at least three;
the vertex cover number, that is the minimum number of vertices needed to cover all edges in the graph;
the minimum vertex deletion number to cographs, that is, the minimum number of vertices to delete to obtain a cograph.[^6]
On the negative side, we prove that that with respect to the parameter vertex cover number $k$, we cannot hope for any $2^{o(k)} n^{2-\epsilon}$ algorithm unless the SETH fails. We also obtain a “quadratic-time FPT” lower bound with respect to the parameter maximum vertex degree, again assuming SETH. Finally, we show that computing the hyperbolicity is at least as hard as computing a size-four independent set of a graph. It is conjectured that computing size-four independent sets needs $\Omega(n^3)$ time.
Parameter
------------------------------------ --------------------------------- -----------------------------
covering path number $O(k^4(n+m))$ \[\[thm:maximal-paths\]\]
feedback edge number $O(k^4(n+m))$ \[\[thm:fes\]\]
number of $\geq 3$-degree vertices $O(k^8(n+m))$ \[\[thm:degree3vertices\]\]
vertex cover number $2^{O(k)} + O(n+m)$ \[\[theorem:vc\]\]
distance to cographs $O(4^{4k}\cdot k^7\cdot (n+m))$ \[\[thm:cograph-dist\]\]
: Summary of our algorithmic results. Herein, $k$ denotes the parameter and $n$ and $m$ denote the number of vertices and edges, respectively.
\[tab:results\]
Preliminaries and Basic Observations
====================================
We write $[n]:=\{1,\ldots,n\}$ for every $n\in\mathbb N$. For a function $f:X\to Y$ and $X'\subseteq X$ we set $f(X'):=\{y\in Y\mid\exists x\in X':f(x)=y\}$.
#### Graph theory.
Let $G=(V,E)$ be a graph. We define $|G|=|V|+|E|$. For $W\subseteq V$, we denote by $G[W]$ the graph *induced* by $W$. We use $G-W:=G[V\setminus W]$ to denote the graph obtained from $G$ by deleting the vertices of $W \subseteq V$. A *path* $P=(v_1,\ldots,v_k)$ in $G$ is a tuple of distinct vertices in $V$ such that $\{v_i,v_{i+1}\}\in E$ for all $i\in[k-1]$; we say that such a path $P$ has endpoints $v_1$ and $v_k$, we call the other vertices of $P$ (i.e., $P \setminus \{v_1, v_k\}$) as inner nodes, and we say that $P$ is a $v_1$-$v_k$ path. We denote by ${\ensuremath{\overline{a b}}}$ the length of a shortest $a$-$b$ path if such a path exists; otherwise, that is, if $a$ and $b$ are in different connected components, ${\ensuremath{\overline{a b}}}:=\infty$. Let $P=(v_1,\ldots,v_k)$ be a path and $v_i,v_j$ two vertices on $P$. We denote by ${\ensuremath{\overline{v_i v_j}|_{P}}}$ the distance of $v_i$ to $v_j$ on $P$, that is, ${\ensuremath{\overline{v_i v_j}|_{P}}} = |j-i|$. For a graph $G$ we denote with $V^{\ge 3}_{G}$ the set of vertices of $G$ that have degree at least three.
#### Hyperbolicity.
Let $G=(V,E)$ be graph and $a,b,c,d\in V$. We denote the distance between two vertices $a$ and $b$ by [$\overline{a b}$]{}. We define $D_1:={\ensuremath{\overline{a b}}}+{\ensuremath{\overline{c d}}}$, $D_2:={\ensuremath{\overline{a c}}}+{\ensuremath{\overline{b d}}}$, and $D_3:={\ensuremath{\overline{a d}}}+{\ensuremath{\overline{b c}}}$ (referred to as *distance sums*). Moreover, we define $\delta(a,b,c,d) := |D_i-D_j|$ if $D_k\le\min\{D_i,D_j\}$, for pairwise distinct $i,j,k\in\{1,2,3\}$. The *hyperbolicity* of a graph is defined as $\delta(G) = \max_{a,b,c,d \in V}\{\delta(a,b,c,d)\}$. We say that the graph is *$\delta$-hyperbolic* for some $\delta \in \mathbb N$ if it has hyperbolicity at most $\delta$. That is, a graph is $\delta$-hyperbolic if for each 4-tuple $a,b,c,d \in V$ we have $${\ensuremath{\overline{a b}}} + {\ensuremath{\overline{c d}}} \le \max\{{\ensuremath{\overline{a c}}}+{\ensuremath{\overline{b d}}},{\ensuremath{\overline{a d}}}+{\ensuremath{\overline{b c}}}\} + \delta.$$ Formally, the [<span style="font-variant:small-caps;">Hyperbolicity</span>]{}problem is defined as follows.
<span style="font-variant:small-caps;">[<span style="font-variant:small-caps;">Hyperbolicity</span>]{}</span>\
[@lX@]{} **Input:** & An undirected graph $G=(V,E)$ and a positive integer $\delta$.\
**Question:** & Is $G$ $\delta$-hyperbolic?
The following lemmas would be useful later. For any quadruple $\{a,b,c,d\}$, \[lem:hyp-distance-bounded\] upper bounds $\delta(a, b, c, d)$ by twice the distance between any pair of vertices of the quadruple. \[lem:diamequalshyp\] discusses graphs for which the hyperbolicity equals the diameter. \[lemma:1seps\] is used in the proof of \[rrule:degree1\].
\[lem:hyp-distance-bounded\] $\delta(a,b,c,d) \le 2 \cdot \min_{u \neq v \in \{a,b,c,d\}} \{{\ensuremath{\overline{u v}}}\}$
\[lem:diamequalshyp\] Let $G$ be a graph with diameter $h$ and $\delta(G)=h$. Then for each quadruple $a,b,c,d\in V(G)$ with $\delta(a,b,c,d)=h$, it holds that exactly two disjoint pairs are at distance $h$ and all the other pairs are at distance $h/2$.
Let $a,b,c,d\in V(G)$ be an arbitrary but fixed quadruple with $\delta(a,b,c,d)=h$. By \[lem:hyp-distance-bounded\], $\min_{u \neq v \in \{a,b,c,d\}} \{{\ensuremath{\overline{u v}}}\}\geq h/2$. Let w.l.o.g. be $S_1={\ensuremath{\overline{a b}}}+{\ensuremath{\overline{c d}}}$ and $S_1\geq \max\{S_2,S_3\}$. Then $h=S_1-\max\{S_2,S_3\}\leq S_1-h$. It follows that $S_1\geq 2h$ and since $G$ is of diameter $h$, it follows that ${\ensuremath{\overline{a b}}}={\ensuremath{\overline{c d}}}=h$. Moreover, it follows that $\max\{S_2,S_3\}=h$ and together with $\min_{u \neq v \in \{a,b,c,d\}} \{{\ensuremath{\overline{u v}}}\}\geq h/2$, we obtain that each other distance equals $h/2$.
\[lemma:1seps\] Given a graph $G=(V,E)$ with $|V|>4$ and $v\in V$ being a 1-separator in $G$. Let $A_1,\ldots,A_\ell$ be the components in $G-\{v\}$. Then there is an $i\in[\ell]$ such that $\delta(G)=\delta(G-V(A_i))$.
Let $A_i$ be one of the components in $G-\{v\}$ with $\delta(G[A_i\cup\{v\}])$ being minimum if $|V(A_j)\cup\{v\}|\geq 4$ for all $j\in[\ell]$, or with $|V(A_i)|$ being minimum otherwise. We distinguish three cases. Let $a,b,c,d\in V$ such that (we assume $|V(A_1)|\geq 3$) either
(i) $a,b,c\in V\backslash (V(A_i)\cup\{v\})$ and $d\in V(A_i)$, or
(ii) $a,b\in V\backslash (V(A_i)\cup\{v\})$ and $c,d\in V(A_i)$ (assuming $|V(A_i)|\geq 2$), or
(iii) $a,b\in V\backslash (V(A_i)\cup\{v\})$, $c=v$, and $d\in V(A_i)$.
#### Case (i):
In this case, every shortest path from $d$ to any of $a,b,c$ contains $v$. Hence, we obtain $$\begin{aligned}
{\ensuremath{\overline{a b}}}+{\ensuremath{\overline{c d}}} &= {\ensuremath{\overline{a b}}}+{\ensuremath{\overline{v c}}}+{\ensuremath{\overline{v d}}}, \\
{\ensuremath{\overline{a c}}}+{\ensuremath{\overline{b d}}} &= {\ensuremath{\overline{a c}}}+{\ensuremath{\overline{v b}}}+{\ensuremath{\overline{v d}}}, \\
{\ensuremath{\overline{a d}}}+{\ensuremath{\overline{b c}}} &= {\ensuremath{\overline{b c}}}+{\ensuremath{\overline{v a}}}+{\ensuremath{\overline{v d}}},
\end{aligned}$$ and thus $\delta(a,b,c,d)=\delta(a,b,c,v)$.
#### Case (ii):
In this case, every shortest path between $a,b$ and $c,d$ contains $v$. Hence, we obtain $$\begin{aligned}
{\ensuremath{\overline{a b}}}+{\ensuremath{\overline{c d}}} &= {\ensuremath{\overline{a b}}}+{\ensuremath{\overline{c d}}}, \\
{\ensuremath{\overline{a c}}}+{\ensuremath{\overline{b d}}} &= {\ensuremath{\overline{a v}}}+{\ensuremath{\overline{v c}}}+{\ensuremath{\overline{v b}}}+{\ensuremath{\overline{v d}}}, \\
{\ensuremath{\overline{a d}}}+{\ensuremath{\overline{b c}}} &= {\ensuremath{\overline{a v}}}+{\ensuremath{\overline{v c}}}+{\ensuremath{\overline{v b}}}+{\ensuremath{\overline{v d}}}.
\end{aligned}$$ Since ${\ensuremath{\overline{a b}}}\leq {\ensuremath{\overline{a v}}}+{\ensuremath{\overline{v b}}}$ on the one hand, and ${\ensuremath{\overline{c d}}}\leq {\ensuremath{\overline{c v}}}+{\ensuremath{\overline{v d}}}$ on the other hand, it follows that $\delta(a,b,c,d)=0$.
#### Case (iii):
In this case, $c$ is contained in every shortest path. Hence, we obtain $$\begin{aligned}
{\ensuremath{\overline{a b}}}+{\ensuremath{\overline{c d}}} &= {\ensuremath{\overline{a b}}}+{\ensuremath{\overline{c d}}}, \\
{\ensuremath{\overline{a c}}}+{\ensuremath{\overline{b d}}} &= {\ensuremath{\overline{a c}}}+{\ensuremath{\overline{b c}}}+{\ensuremath{\overline{c d}}}, \\
{\ensuremath{\overline{a d}}}+{\ensuremath{\overline{b c}}} &= {\ensuremath{\overline{a c}}}+{\ensuremath{\overline{c d}}}+{\ensuremath{\overline{b c}}}.
\end{aligned}$$ Since ${\ensuremath{\overline{a b}}}\leq {\ensuremath{\overline{a c}}}+{\ensuremath{\overline{c b}}}$, it follows that $\delta(a,b,c,d)=0$.
Observe that the case where $a\in V\backslash (V(A_i)\cup\{v\})$, $c=v$, and $b,d\in V(A_i)$ reduces to (iii). Since $A_i$ was chosen as $\delta(G[A_i\cup\{v\}])$ being minimum if $|V(A_j)\cup\{v\}|\geq 4$ for all $j\in[\ell]$, or with $|V(A_i)|$ being minimum otherwise, it follows that $\delta(G)=\delta(G-V(A_i))$.
\[rrule:degree1\] As long as there are more than four vertices, remove vertices of degree one.
\[lem:rrule-1-correct-lin-time\] \[rrule:degree1\] is correct and can be applied exhaustively in linear time.
The soundness of \[rrule:degree1\] follows immediately from \[lemma:1seps\]. To apply \[rrule:degree1\] in linear time do the following. First, collect all degree one vertices in linear time in a list $L$. Then, iteratively delete degree-one vertices and put their neighbor in $L$ if it has degree one after the deletion. Each iteration can be applied in constant time. Thus, \[rrule:degree1\] can be applied in linear time.
Polynomial Linear-Time Parameterized Algorithms
===============================================
In this section, we provide polynomial linear-time parameterized algorithms with respect to the parameters feedback edge number and number of vertices with degree at least three; that is, algorithms with a linear-time dependence on the input size times a polynomial-time dependence on the parameter value.
To this end, we first introduce an auxiliary parameter, the *minimum maximal paths cover number*, which we formally define below and also describe a polynomial linear-time paramaterized algorithm for it.
Building upon this result, for the parameter feedback edge number we then show that, after applying \[rrule:degree1\], the number of maximal paths can be upper bounded by a polynomial of the feedback edge number. This implies a polynomial linear-time parameterized algorithm for the feedback edge number as well. For the parameter number of vertices with degree at least three, we introduce an additional reduction rule to achieve that the number of maximal paths is bounded in a polynomial of this parameter. Again, this implies a polynomial linear-time algorithm.
#### Minimum maximal paths cover number.
Consider the following definition.
Let $G$ be a graph and $P$ be a path in $G$. Then, $P$ is a *maximal path* if the following hold: (1) it contains at least two vertices; (2) all its inner nodes have degree two in $G$; and (3) either both its endpoints have degree at least three in $G$, or one of its endpoints has degree at least three in $G$ while the other endpoint is of degree two in $G$; and (4) $P$ is size-wise maximal with respect to these properties.
We will be interested in the minimum number of maximal paths needed to cover the vertices of a given graph; we call this number the *minimum maximal paths cover number*. While not all graphs can be covered by maximal paths (e.g., edgeless graphs), graphs which have minimum degree two and contain no isolated cycles can be covered by maximal paths (it follows by, e.g., a greedy algorithm which iteratively selects an arbitrary uncovered vertex and exhaustively extend it arbitrarily; since there are no isolated cycles and the minimum degree is two, we are bound to eventually hit at least one vertex of degree three). In the following lemma we show how to approximate the minimum maximal paths cover number, for graphs which have minimum degree two and contain no isolated cycles.
\[lem:maximal-paths\] There is a linear time algorithm which approximates the minimum maximal paths cover number for graphs which have minimum degree two and contain no isolated cycles.
The algorithm operates in two phases. In the first phase, we greedily cover all vertices of degree two. Specifically, we arbitrarily select a vertex of degree two, view it as a path of length one, and arbitrarily try to extend it in both directions (it has degree two, so, pictorially, has two possible directions for extension). We stop extending it in each direction whenever we hit a vertex of degree at least three; if it is the same vertex in both directions then we extend it only in one direction (since a path cannot contain the same vertex more than once).
The second phase begins when all vertices of degree two are already covered. In the second phase, ideally we would find a matching between those uncovered vertices of degree at least three. To get a $2$-approximation we arbitrarily select a vertex of degree at least three, view it as a path of length one, and arbitrarily extend it until it is maximal. This finishes the description of the linear-time algorithm.
For correctness of the first phase, the crucial observation is that each vertex of degree two has two be covered by at least one path. For the second phase, $2$-approximation follows since each maximal path can cover at most two vertices of degree at least three.
Now we are ready to design a polynomial linear-time parameterized algorithm for [<span style="font-variant:small-caps;">Hyperbolicity</span>]{}with respect to the minimum maximal paths cover number.
\[thm:maximal-paths\] Let $G = (V,E)$ be a graph and $k$ be its minimum maximal paths cover number. Then, [<span style="font-variant:small-caps;">Hyperbolicity</span>]{}can be solved in $O(k^4 (n+m))$ time.
We begin with some preprocessing. First, we apply \[rrule:degree1\] to have a graph with no vertices of degree one. Second, we check whether there are any isolated cycles; if there are, then we consider the largest isolated cycle, and compute its hyperbolicity. If its hyperbolicity is at least $\delta$ then we have a yes-instance and we halt; otherwise, we remove all isolated cycles and continue.
Now we use \[lem:maximal-paths\] to get a set of at most $2k$ maximal paths which cover $G$. By initiating a breadth-first search from each of the endpoints of those maximal paths, we can compute the pairwise distances between those endpoints in $O(k (n+m))$ time. Thus, for the rest of the algorithm we assume that we can access the distances between any two vertices which are endpoints of those maximal paths in constant time. Let $(a,b,c,d)$ be a quadruple such that $\delta(a,b,c,d)=\delta(G)$. Since the set ${\cal P}$ covers all vertices of $G$, each vertex of $a$, $b$, $c$, and $d$ belongs to some path $P\in {\cal P}$. Since $|\mathcal{P}| = k$, there are $O(k^4)$ possibilities to assign the vertices $a$, $b$, $c$, and $d$ to paths in $\mathcal{P}$. For each possibility we compute the maximum hyperbolicity respecting the assignment in linear time, that is, we compute the positions of the vertices on their respective paths that maximize $\delta(a,b,c,d)$. We achieve the running time by formulating an integer linear program (ILP) with a constant number of variables and constraints whose coeffecients have value at most $n$.
To this end, denote with $P_a, P_b, P_c, P_d \in \mathcal{P}$ the paths containing $a,b,c,d$, respectively. We assume for now that these paths are different and deal later with the case that one path contains at least two vertices from $a,b,c,d$. Let $a_1$ and $a_2$ ($b_1,b_2,c_1,c_2,d_1,d_2$) be the endpoints of $P_a$ ($P_b, P_c, P_d$, respectively). Furthermore, denote by $\ell(P)$ the length of a path $P \in \mathcal{P}$, that is, the number of its edges. Without loss of generality assume that $D_1 \le D_2 \le D_3$. We now compute the positions of the vertices on their respective paths that maximize $D_1 - D_2$ by solving an ILP. Recall that ${\ensuremath{\overline{v_1 v}|_{P_v}}}$ denotes the distance of $v$ to $v_1$ on $P_v$. Thus, ${\ensuremath{\overline{v_1 v}|_{P_v}}} + {\ensuremath{\overline{v v_2}|_{P_v}}} = \ell(P_v)$ and ${\ensuremath{\overline{v_1 v}|_{P_v}}} \ge 0$ and ${\ensuremath{\overline{v v_2}|_{P_v}}} \ge 0$. The following is a compressed description of the ILP containing the minimum function. We describe below how to remove it. $$\begin{aligned}
\text{maximize:} && & D_1 - D_2 \\
\text{subject to:} && D_1 & = {\ensuremath{\overline{a b}}}+{\ensuremath{\overline{c d}}} \\
&& D_2 & = {\ensuremath{\overline{a c}}}+{\ensuremath{\overline{b d}}} \\
&& D_3 & = {\ensuremath{\overline{a d}}}+{\ensuremath{\overline{b c}}} \\
&& D_1 &\le D_2 \le D_3 \\
\forall x \in \{a,b,c,d\}: && \ell(P_x) & = {\ensuremath{\overline{x_1 x}|_{P_x}}} + {\ensuremath{\overline{x x_2}|_{P_x}}} \label{line:vars} \\
\forall x,y \in \{a,b,c,d\}: && {\ensuremath{\overline{x y}}} & = \min \left\{
\begin{array}{c}
{\ensuremath{\overline{x_1 x}|_{P_x}}} + {\ensuremath{\overline{x_1 y_1}}} + {\ensuremath{\overline{y_1 y}|_{P_y}}}, \\
{\ensuremath{\overline{x_1 x}|_{P_x}}} + {\ensuremath{\overline{x_1 y_2}}} + {\ensuremath{\overline{y y_2}|_{P_y}}}, \\
{\ensuremath{\overline{x x_2}|_{P_x}}} + {\ensuremath{\overline{x_2 y_1}}} + {\ensuremath{\overline{y_1 y}|_{P_y}}}, \\
{\ensuremath{\overline{x x_2}|_{P_x}}} + {\ensuremath{\overline{x_2 y_2}}} + {\ensuremath{\overline{y y_2}|_{P_y}}}
\end{array} \right\}\label{line:min}
\end{aligned}$$ First, observe that the ILP obviously has a constant number of variables. The only constant coefficients are ${\ensuremath{\overline{x_i y_j}}}$ for $x,y \in \{a,b,c,d\}$ and $i,j \in \{1,2\}$ and obviously have value at most $n-1$. To remove the minimization function in \[line:min\], we use another case distinction: We simply try all possibilities of which value is the smallest one and adjust the ILP accordingly. For example, for the case that the minimum in \[line:min\] is ${\ensuremath{\overline{x x_1}|_{P_x}}} + {\ensuremath{\overline{x_1 y_1}}} + {\ensuremath{\overline{y_1 y}|_{P_y}}}$, we replace this equation by the following: $$\begin{aligned}
{\ensuremath{\overline{x y}}} = {\ensuremath{\overline{x_1 x}|_{P_x}}} + {\ensuremath{\overline{x_1 y_1}}} + {\ensuremath{\overline{y_1 y}|_{P_y}}} \\
{\ensuremath{\overline{x y}}} \le {\ensuremath{\overline{x_1 x}|_{P_x}}} + {\ensuremath{\overline{x_1 y_2}}} + {\ensuremath{\overline{y y_2}|_{P_y}}} \\
{\ensuremath{\overline{x y}}} \le {\ensuremath{\overline{x x_2}|_{P_x}}} + {\ensuremath{\overline{x_2 y_1}}} + {\ensuremath{\overline{y_1 y}|_{P_y}}} \\
{\ensuremath{\overline{x y}}} \le {\ensuremath{\overline{x x_2}|_{P_x}}} + {\ensuremath{\overline{x_2 y_2}}} + {\ensuremath{\overline{y y_2}|_{P_y}}}
\end{aligned}$$ There are four possibilities of which value is the smallest one, and we have to consider each of them independently for each of the $\binom{4}{2} = 6$ pairs. Hence, for each assignment of the vertices $a$, $b$, $c$, and $d$ to paths in $\mathcal{P}$, we need to solve $4 \cdot 6 = 24$ different ILPs in order to remove the minimization function. Since each ILP has a constant number of variables and constraints, this takes $L^{O(1)}$ time where $L=O(\log n)$ is the total size of the ILP instance (for example by using the algorithm of Lenstra [@Len83]).
It remains to discuss the case that at least two vertices of $a$, $b$, $c$, and $d$ are assigned to the same path $P \in \mathcal{P}$. We show the changes in case that $a$, $b$, and $c$ are mapped to $P_{a} \in \mathcal{P}$. We assume without loss of generality that the vertices $a_1,a,b,c,a_2$ appear in this order in $P$ (allowing $a=a_1$ and $c=a_2$). The adjustments for the other cases can be done in a similar fashion. The objective function as well as the first four lines of the ILP remain unchanged. \[line:vars\] is replaced with the following: $$\begin{aligned}
\ell(P_a) & = {\ensuremath{\overline{a_1 a}|_{P_a}}} + {\ensuremath{\overline{a b}|_{P_a}}} + {\ensuremath{\overline{b c}|_{P_a}}} + {\ensuremath{\overline{c a_2}|_{P_a}}} \\
\ell(P_d) & = {\ensuremath{\overline{d_1 d}|_{P_d}}} + {\ensuremath{\overline{d d_2}|_{P_a}}}
\end{aligned}$$ To ensure that \[line:min\] works as before, we add the following: $$\begin{aligned}
{\ensuremath{\overline{a a_2}|_{P_a}}} & = {\ensuremath{\overline{a b}|_{P_a}}} + {\ensuremath{\overline{b c}|_{P_a}}} + {\ensuremath{\overline{c a_2}|_{P_a}}} \\
{\ensuremath{\overline{b_1 b}|_{P_b}}} & = {\ensuremath{\overline{a_1 a}|_{P_a}}} + {\ensuremath{\overline{a b}|_{P_a}}} \\
{\ensuremath{\overline{b b_2}|_{P_b}}} & = {\ensuremath{\overline{b c}|_{P_a}}} + {\ensuremath{\overline{c a_2}|_{P_a}}} \\
{\ensuremath{\overline{c_1 c}|_{P_c}}} & = {\ensuremath{\overline{a_1 a}|_{P_a}}} + {\ensuremath{\overline{a b}|_{P_a}}} + {\ensuremath{\overline{b c}|_{P_a}}} \\
{\ensuremath{\overline{c c_2}|_{P_c}}} & = {\ensuremath{\overline{c a_2}|_{P_a}}}
\end{aligned}$$
#### Feedback edge number.
We next show a polynomial linear-time parameterized algorithm with respect to the parameter feedback edge number $k$. The idea is to show that a graph that is reduced with respect to \[rrule:degree1\] contains $O(k)$ maximal paths.
\[thm:fes\] [<span style="font-variant:small-caps;">Hyperbolicity</span>]{}can be computed in $O(k^4(n+m))$ time, where $k$ is the feedback edge number.
The first step of the algorithm is to reduce the input graph exhaustively with respect to \[rrule:degree1\]. By \[lem:rrule-1-correct-lin-time\] we can exhaustively apply \[rrule:degree1\] in linear time.
Denote by $X \subseteq E$ a minimum feedback edge set for the reduced graph $G = (V,E)$ and observe that $|X| = k$. We will show that the minimum maximal paths cover number of $G$ is $O(k)$. More precisely, we show the slightly stronger claim that the number of maximal paths in $G$ is $O(k)$.
Observe that all vertices in $G$ have degree at least two since $G$ is reduced with respect to \[rrule:degree1\]. Thus, every leaf of $G-X$ is incident with at least one feedback edge which implies that there are at most $2k$ leaves in $G-X$. Moreover, since $G-X$ is a forest, the number of vertices with degree at least three in $G-X$ is at most the number of leaves in $G-X$ and thus at most $2k$. This implies that the number of maximal paths in $G-X$ is at most $2k$ (each maximal path corresponds to an edge in the forest obtained from $G-X$ by contracting all degree-two vertices).
We now show the bound for $G$ by showing that an insertion of an edge into any graph $H$ increases the number of maximal paths by at most five. Hence, consider a graph $H$ and let $\{u,v\}$ be an edge that is inserted into $H$; denote the resulting graph by $H'$. First, each edge can be part of at most one maximal path in any graph. Therefore, there is at most one maximal path $P$ in $H'$ that contains $\{u,v\}$. The only vertices of $P$ that can be in further in maximal paths of $H'$ are the endpoints of $P$. If an endpoint $w$ of $P$ has degree at least three, in $H$ then each maximal path of $H$ containing this endpoint is also maximal path in $H'$. Otherwise, that is, if $w$ has degree two in $H$, then there can be at most two new maximal paths containing $w$, one for each edge that is incident with $w$ in $H$. Thus, the number of maximal paths containing $w$ and different from $P$ increases by at most two. Therefore, the insertion of the $k$ edges of $X$ in $G-X$ increases the number of maximal paths by at most $5k$. Thus $G$ contains at most $7k$ maximal paths. The statement of the theorem now follows from \[thm:maximal-paths\].
#### Number of vertices with degree at least three.
We finally show a polynomial-linear time parameterized algorithm with respect to the number $k$ of vertices with degree three or more. To this end, we use the following data reduction rule to bound the number of maximal paths in the graph by $O(k^2)$ (in order to make use of \[thm:maximal-paths\]).
\[rrule:paths-between-high-degree-vertices\] Let $G=(V,E)$ be a graph, $u,v \in V^{\ge 3}_{G}$ be two vertices of degree at least three, and $\mathcal{P}_{uv}$ be the set of maximal paths in $G$ with endpoints $u$ and $v$. Let $\mathcal{P}_{uv}^9 \subseteq \mathcal{P}_{uv}$ be the set containing the shortest path, the four longest even-length paths, and the four longest odd-length paths in $\mathcal{P}_{uv}$. If $\mathcal{P}_{uv} \setminus \mathcal{P}_{uv}^9 \neq \emptyset$, then delete in $G$ all inner vertices of the paths in $\mathcal{P}_{uv} \setminus \mathcal{P}_{uv}^9$.
\[lem:paths-rule-correct\] \[rrule:paths-between-high-degree-vertices\] is correct and can be exhaustively applied in linear time.
We first prove the running time. We compute in linear time the set $V^{\ge 3}_{G}$ of all vertices with degree at least three. Then for each $v \in V^{\ge 3}_{G}$ we do the following. Starting from $v$, we perform a modified breadth-first search that stops at vertices in $V^{\ge 3}_{G}$. Let $R(V^{\ge 3}_{G},v)$ denote the visited vertices and edges. Observe that $R(V^{\ge 3}_{G},v)$ consists of $v$, some degree-two vertices, and all vertices of $V^{\ge 3}_{G}$ that can be reached from $v$ via maximal paths in $G$. Furthermore, with the breadth-first search approach we can also compute for all $u \in R(V^{\ge 3}_{G},v) \cap V^{\ge 3}_{G}$ with $u \neq v$ the number of maximal paths between $u$ and $v$ and their respective lengths. Then, in time linear in $|R(V^{\ge 3}_{G},v)|$, we remove the paths in $\mathcal{P}_{uv} \setminus \mathcal{P}_{uv}^9$ for all $u \in R(V^{\ge 3}_{G},v) \cap V^{\ge 3}_{G}$. Thus, we can apply \[rrule:paths-between-high-degree-vertices\] for each $v \in V^{\ge 3}_{G}$ in $O(|R(V^{\ge 3}_{G},v)|)$ time. Altogether, the running time is $$O(\sum_{v\in V^{\ge 3}_{G}} |R(V^{\ge 3}_{G},v)|) = O(n+m)$$ where the equality follows from the fact each edge and each maximal path in $G$ is visited twice by the modified breadth-first search.
We now prove the correctness of the data reduction rule. To this end, let $G = (V,E)$ be the input graph, let $P \in \mathcal{P}_{uv} \setminus \mathcal{P}_{uv}^9$ be a maximal path from $u$ to $v$ whose inner vertices are removed by the application of the data reduction rule, and let $G' = (V', E')$ be the resulting graph. We show that $\delta(G) = \delta(G')$. The correctness of \[rrule:paths-between-high-degree-vertices\] follows then from iteratively applying this argument. First, observe that since $\mathcal{P}_{uv}^9$ contains the shortest maximal path of $\mathcal{P}_{uv}$, it follows that $u$ and $v$ have the same distance in $G$ and $G'$. Furthermore, it is easy to see that each pair of vertices $w,w' \in V'$ has the same distance in $G$ and $G'$ (\[rrule:paths-between-high-degree-vertices\] removes only paths and does not introduce degree-one vertices). Hence, we have that $\delta(G) \ge \delta(G')$ and it remains to show that $\delta(G) \le \delta(G')$
Towards showing that $\delta(G) \le \delta(G')$, let $a,b,c,d \in V$ be the four vertices defining the hyperbolicity of $G$, that is, $\delta(G) = \delta(a,b,c,d)$. If $P$ does not contain any of these four vertices, then we are done. Thus, assume that $P$ contains at least one vertex from $\{a,b,c,d\}$. (For convenience, we say in this proof that a path $Q$ contains a vertex $v$ if $v$ is an inner vertex of $Q$ because \[rrule:paths-between-high-degree-vertices\] does neither delete $u$ nor $v$.) We next make a case distinction on the number of vertices of $\{a,b,c,d\}$ that are contained in $P$.
*Case (I): $P$ contains one vertex of $\{a,b,c,d\}$.* Without loss of generality assume $P$ contains $a$. We show that we can replace $a$ by another vertex $a'$ in a path $P' \in \mathcal{P}_{uv}^9$ such that $\delta(a,b,c,d) = \delta(a',b,c,d)$. Since $P$ contains $a$, we can chose $P'$ as one of the four (odd/even)-length longest paths in $\mathcal{P}_{uv}^9$ such that
- $\ell(P') - \ell(P)$ is nonnegative and even (either both lengths are even or both are odd) and
- $P'$ contains no vertex of $\{b,c,d\}$.
Since $P$ is removed by \[rrule:paths-between-high-degree-vertices\], it follows that $\ell(P) \le \ell(P')$. We chose $a'$ on $P'$ such that ${\ensuremath{\overline{u a'}|_{P'}}} = {\ensuremath{\overline{u a}|_{P}}} + (\ell(P') - \ell(P))/2$. Observe that this implies that ${\ensuremath{\overline{a' v}|_{P'}}} = {\ensuremath{\overline{a v}|_{P}}} + (\ell(P') - \ell(P))/2$ and thus $${\ensuremath{\overline{u a}|_{P}}} - {\ensuremath{\overline{a v}|_{P}}} = {\ensuremath{\overline{u a'}|_{P'}}} - {\ensuremath{\overline{a' v}|_{P'}}}.$$ Recall that $$\begin{aligned}
D_1 :={\ensuremath{\overline{a b}}}+{\ensuremath{\overline{c d}}}, && D_2 :={\ensuremath{\overline{a c}}}+{\ensuremath{\overline{b d}}}, \text{ and } && D_3 :={\ensuremath{\overline{a d}}}+{\ensuremath{\overline{b c}}}.
\end{aligned}$$ Denote with $D'_1$, $D'_2$, and $D'_3$ the respective distance sums resulting from replacing $a$ with $a'$, for example $D'_1 = {\ensuremath{\overline{a' b}}} + {\ensuremath{\overline{c d}}}$. Observe that by the choice of $a'$ we increased all distance sums by the same amount, that is, for all $i\in\{1,2,3\}$ we have $D'_i = D_i + (\ell(P') - \ell(P))/2$. Since $\delta(a,b,c,d) = D_i - D_j$ for some $i,j \in \{1,2,3\}$, we have that $$\delta(G') = \delta(a',b,c,d) = D'_i - D'_j = \delta(a,b,c,d) = \delta(G).$$
*Case (II): $P$ contains two vertices of $\{a,b,c,d\}$.* Without loss of generality, assume that $P$ contains $a$ and $b$ but not $c$ and $d$. We follow a similar pattern as in the previous case and again use the same notation. Let $P', P'' \in \mathcal{P}_{uv}^9$ be the two longest paths such that both $P'$ and $P''$ do neither contain $c$ nor $d$ and both $\ell(P') - \ell(P)$ and $\ell(P'') - \ell(P)$ are even. We distinguish two subcases:
*Case (II-1): $D_1$ is not the largest sum ($D_1 < D_2$ or $D_1 < D_3$).* We replace $a$ and $b$ with $a'$ and $b'$ on $P'$ such that ${\ensuremath{\overline{u a'}|_{P'}}} = {\ensuremath{\overline{u a}|_{P}}} + (\ell(P') - \ell(P))/2$ and ${\ensuremath{\overline{u b'}|_{P'}}} = {\ensuremath{\overline{u b}|_{P}}} + (\ell(P') - \ell(P))/2$. Thus, $D'_1 = D_1$ since ${\ensuremath{\overline{a b}}} = {\ensuremath{\overline{a' b'}}}$. However, for $i \in \{2,3\}$ we have $D'_i = D_i + (\ell(P') - \ell(P))/2$. Since either $D_2$ or $D_3$ was the largest distance sum, we obtain $$\delta(G) = \delta(a,b,c,d) = D_i - D_j \le D'_i - D'_{j'} = \delta(a',b',c,d) = \delta(G')$$ for some $i \in \{2,3\}$, $j,j' \in \{1,2,3\}$, $i \neq j$, and $i \neq j'$.
*Case (II-2): $D_1$ is the largest sum ($D_1 \ge D_2$ and $D_1 \ge D_3$).* We need another replacement strategy since we did not increase $D_1$ in case (II-1). In fact, we replace $a$ and $b$ with two vertices on different paths $P'$ and $P''$. We replace $a$ with $a'$ on $P'$ and $b$ with $b'$ on $P''$ such that ${\ensuremath{\overline{u a'}|_{P'}}} = {\ensuremath{\overline{u a}|_{P}}} - (\ell(P') - \ell(P))/2$ and ${\ensuremath{\overline{u b'}|_{P''}}} = {\ensuremath{\overline{u b}|_{P}}} - (\ell(P'') - \ell(P))/2$. Observe that for $i \in \{2,3\}$ it holds that $$D'_i = D_i + (\ell(P') - \ell(P))/2 + (\ell(P'') - \ell(P))/2.$$ Moreover, since $a'$ and $b'$ are on different maximal paths, we also have $$\begin{aligned}
{\ensuremath{\overline{a b}}} & \le \min_{x \in \{u,v\}}\{{\ensuremath{\overline{x a}|_{P}}}+{\ensuremath{\overline{x b}|_{P}}}\} \\
& = \min_{x \in \{u,v\}}\{{\ensuremath{\overline{x a'}|_{P'}}}+{\ensuremath{\overline{x b'}|_{P''}}}\} - \frac{\ell(P') - \ell(P)}{2} - \frac{\ell(P'') - \ell(P)}{2} = {\ensuremath{\overline{a' b'}}}
\end{aligned}$$ and thus $D'_1 \ge D_1 + (\ell(P') - \ell(P))/2 + (\ell(P'') - \ell(P))/2$. Hence, we have $$\delta(G) = \delta(a,b,c,d) = D_1 - D_j \le D'_1 - D'_{j} = \delta(a,b,c,d) = \delta(G')$$ for some $j \in \{2,3\}$.
*Case (III): $P$ contains all four vertices of $\{a,b,c,d\}$.* We consider two subcases.
*Case (III-1): the union of the shortest paths between these four vertices induces a path.* In this case, we have $\delta(G) = 0$ and thus trivially $\delta(G) \le \delta(G')$.
*Case (III-2): the union of the shortest paths between these four vertices induces a cycle.* From \[lem:hyp-distance-bounded\] we derive $\delta(G) \le \ell(P)/2$ since at least two of the four vertices $a,b,c,d$ have distance at most $\ell(P)/4$. We can replace the four vertices with four vertices on a path $P' \in \mathcal{P}_{uv}^9$ such that $\ell(P') - \ell(P)$ is nonnegative and even. Observe that if $\ell(P') = \ell(P)$, then taking the vertices on the same positions as $a,b,c,d$ gives a 4-tuple with the same distances. Hence, assume $\ell(P')>\ell(P)$. Consider the union of the vertices on $P'$ and the shortest path between $u$ and $v$. The union of the shortest paths of all vertices in this set is a cycle of length at least $\ell(P')+1\ge \ell(P)+2$. By known results of @KM02 there is a 4-tuple of cycle vertices $A',b',c',d'$ such that $\delta(a,b,c,d)\ge \lfloor (\ell(P')+1)/2\rfloor > \ell(P)/2$. Thus, we have $$\delta(G') \ge \delta(a',b',c',d') > \ell(P)/2 \ge \delta(G).$$
*Case (IV): $P$ contains three vertices of $\{a,b,c,d\}$.* Without loss of generality, assume that $P$ contains $a$, $b$, and $c$ but not $d$ and that $a$ is the closest vertex to $u$ on $P$ and $c$ is the closest vertex to $v$ on $P$ (that is, $a,b,c$ appear in this order on $P$). We distinguish two subcases.
*Case (IV-1): ${\ensuremath{\overline{a c}|_{P}}} = {\ensuremath{\overline{a c}}}$.* We follow a similar pattern as in case (I) and use the same notation. Again, there is a $P' \in \mathcal{P}_{uv}^9$ such that $\ell(P') - \ell(P)$ is even (either both lengths are even or both are odd) and $P'$ does not contain $d$. We replace each vertex $a,b,c$ as in case (I), that is, for each $x \in \{a,b,c\}$ we chose $x'$ on $P'$ such that ${\ensuremath{\overline{u x'}|_{P'}}} = {\ensuremath{\overline{u x}|_{P}}} + (\ell(P') - \ell(P))/2$. Observe that only the distances between $d$ and the other three vertices change. Thus, we have again for all $i\in\{1,2,3\}$ that $D'_i = D_i + (\ell(P') - \ell(P))/2$ and hence $\delta(G) = \delta(G')$.
*Case (IV-2): ${\ensuremath{\overline{a c}|_{P}}} > {\ensuremath{\overline{a c}}}$.* We use again a similar strategy as in case (I) and use the same notation. Again, there is a $P' \in \mathcal{P}_{uv}^9$ such that $\ell(P') - \ell(P)$ is even (either both lengths are even or both are odd) and $P'$ does not contain $d$. We replace the vertices $a,b,c$ with $a',b',c'$ on $P'$ such that
- ${\ensuremath{\overline{a u}}} = {\ensuremath{\overline{a u}|_{P}}} = {\ensuremath{\overline{a' u}|_{P'}}} = {\ensuremath{\overline{a' u}}}$,
- ${\ensuremath{\overline{c v}}} = {\ensuremath{\overline{c v}|_{P}}} = {\ensuremath{\overline{c' v}|_{P'}}} = {\ensuremath{\overline{c' v}}}$,
- ${\ensuremath{\overline{b u}|_{P}}} = {\ensuremath{\overline{b' u}|_{P'}}} - (\ell(P') - \ell(P))/2$, and
- ${\ensuremath{\overline{b v}|_{P}}} = {\ensuremath{\overline{b' v}|_{P'}}} - (\ell(P') - \ell(P))/2$.
Note that since ${\ensuremath{\overline{a c}|_{P}}} > {\ensuremath{\overline{a c}}}$, it follows that the distances not involving $b$ remain unchanged, that is, ${\ensuremath{\overline{a b}}} = {\ensuremath{\overline{a' b'}}}$, and for $x \in \{a,b\}$ we have ${\ensuremath{\overline{x d}}} = {\ensuremath{\overline{x' d}}}$. Furthermore, all distances involving $b$ increase by $(\ell(P') - \ell(P))/2$, that is, ${\ensuremath{\overline{b d}}} = {\ensuremath{\overline{b' d}}} - (\ell(P') - \ell(P))/2$ and for $x \in \{a,b\}$ we have ${\ensuremath{\overline{b x}}} = {\ensuremath{\overline{b' x'}}}$. Thus, we have again for all $i\in\{1,2,3\}$ that $D'_i = D_i + (\ell(P') - \ell(P))/2$ and hence $\delta(G) = \delta(G')$.
Observe that if the graph $G$ is reduced with respect to \[rrule:paths-between-high-degree-vertices\], then there exist for each pair $u,v \in V^{\ge 3}_G$ at most nine maximal paths with endpoints $u$ and $v$. Thus, $G$ contains at most $O(k^2)$ maximal paths and using \[thm:maximal-paths\] we arrive at the following.
\[thm:degree3vertices\] [<span style="font-variant:small-caps;">Hyperbolicity</span>]{}can be solved in $O(k^8 (n+m))$ time, where $k$ is the number of vertices with degree at least three.
Parameter Vertex Cover
======================
A *vertex cover* of a graph $G = (V, E)$ is a subset $W \subseteq V$ of vertices of $G$ such that each edge in $G$ is incident to at least one vertex in $W$. Deciding whether a graph $G$ has a vertex cover of size at most $k$ is [$\mathsf{NP}$]{}-complete in general [@GJ79]. There is, however, a simple linear-time factor-$2$ approximation (see, e.g., [@PapadimitriouS82]). In this section, we consider the size $k$ of a vertex cover as the parameter. We show that we can solve [<span style="font-variant:small-caps;">Hyperbolicity</span>]{}in time linear in $|G|$, but exponential in $k$; further, we show that, unless SETH fails, we cannot do asymptotically better.
#### A Linear-Time Algorithm Parameterized by the Vertex Cover Number. {#section:vcub}
We prove that [<span style="font-variant:small-caps;">Hyperbolicity</span>]{}can be solved in time linear in the size of the graph and exponential in the size $k$ of a vertex cover. This result is based on a linear-time computable kernel of size $O(2^k)$, that can be obtained by exhaustively applying the following reduction rule.
\[rrule:twins\] If there are at least five vertices $v_1, v_2, \ldots, v_\ell \in V$, $\ell > 4$, with the same (open) neighborhood $N(v_1) = N(v_2) = \ldots = N(v_\ell)$, then delete $v_5, \ldots, v_\ell$.
We next show that the above rule is correct, can be applied in linear time, and leads to a kernel for the parameter vertex cover number.
\[lemma:vcredrules\] \[rrule:twins\] is correct and can be applied exhaustively in linear time. Furthermore, if \[rrule:twins\] is not applicable, then the graph contains at most $k + 4 \cdot 2^k$ vertices and $O(k \cdot 2^k)$ edges, where $k$ is the vertex cover number.
Let $G=(V,E)$ be the input graph with a vertex cover $W \subseteq V$ of size $k$ and let $v_1, v_2, \ldots, v_\ell \in V$, $\ell > 4$, be vertices with the same open neighborhood.
First, we show that \[rrule:twins\] is correct, that is, $\delta(G[V \setminus \{v_5, \ldots, v_\ell\}])=\delta(G)$. To see this, consider two vertices $v_i$, $v_j$ with the same open neighborhood, and consider any other vertex $u$. The crucial observation is that ${\ensuremath{\overline{u v_i}}} = {\ensuremath{\overline{u v_j}}}$. This means that the two vertices are interchangeable with respect to the hyperbolicity. In particular, if $v_i,v_j \in V$ have the same open neighborhood, then $\delta(v_i,x,y,z)=\delta(v_j,x,y,z)$ for every $x,y,z\in V\setminus \{v_i,v_j\}$. As the hyperbolicity is obtained from a quadruple, it is sufficient to consider at most four vertices with the same open neighborhood. We conclude that $\delta(G[V \setminus \{v_5, \ldots, v_\ell\}])=\delta(G)$.
Next we show how to exhaustively apply \[rrule:twins\] in linear time. To this end, we apply in linear time a *partition refinement* [@HabibP10] to compute a partition of the vertices into twin classes. Then, for each twin class we remove all but 4 (arbitrary) vertices. Overall, this can be done in linear time. Since $|W|\leq k$, it follows that there are at most $2^{k}$ pairwise-different neighborhoods (and thus twin classes) in $V \setminus W$. Thus, if \[rrule:twins\] is not applicable, then the graph consists of the vertex cover $W$ of size $k$ plus at most $4 \cdot 2^{k}$ vertices in $V \setminus W$. Furthermore, since $W$ is a vertex cover, it follows that the graph contains at most $ 4k \cdot 2^k$ edges.
With \[rrule:degree1\] we can compute in linear time an equivalent instance having a bounded number of vertices. Applying on this instance the trivial $O(n^4)$-time algorithm yields the following.
\[theorem:vc\] [<span style="font-variant:small-caps;">Hyperbolicity</span>]{}can be computed in $O(2^{4k} + n + m)$ time, where $k$ denotes the size of a vertex cover of the input graph.
#### SETH-based Lower bounds. {#ssec:vclowerbound}
We show that, unless SETH breaks, the $2^{O(k)} + O(n+m)$-time algorithm obtained in the previous subsection cannot be improved to an algorithm even with running time $2^{o(k)}\cdot (n^{2-\epsilon})$. This also implies, that, assuming SETH, there is no kernel with $2^{o(k)}$ vertices computable in $O(n^{2-\epsilon})$ time, i.e. the kernel obtained by applying \[rrule:twins\] cannot be improved significantly. The proof follows by a reduction from the following problem.
<span style="font-variant:small-caps;">Orthogonal Vectors</span>\
[@lX@]{} **Input:** & Two sets ${\overrightarrow{A}}$ and ${\overrightarrow{B}}$ each containing $n$ binary vectors of length $\ell=O(\log n)$.\
**Question:** & Are there two vectors ${\overrightarrow{a}}\in {\overrightarrow{A}}$ and ${\overrightarrow{b}}\in {\overrightarrow{B}}$ such that ${\overrightarrow{a}}$ and ${\overrightarrow{b}}$ are orthogonal, that is, such that there is no position $i$ for which ${\overrightarrow{a}}[i]={\overrightarrow{b}}[i]=1$?
Williams and Yu [@WY14] proved that, if <span style="font-variant:small-caps;">Orthogonal Vectors</span> can be solved in $O(n^{2-\epsilon})$ time, then SETH breaks. We provide a linear-time reduction from <span style="font-variant:small-caps;">Orthogonal Vectors</span> to [<span style="font-variant:small-caps;">Hyperbolicity</span>]{}where the graph $G$ constructed in the reduction contains $O(n)$ vertices and admits a vertex cover of size $O(\log(n))$ (and thus contains $O(n\cdot \log n)$ edges). The reduction then implies that, unless SETH breaks, there is no algorithm solving [<span style="font-variant:small-caps;">Hyperbolicity</span>]{}in time polynomial in the size of the vertex cover and linear in the size of the graph. We mention that Borassi et al. [@BCH16] showed that under the SETH [<span style="font-variant:small-caps;">Hyperbolicity</span>]{}cannot be solved in $O(n^{2-\epsilon})$. However, the instances constructed in their reduction have a minimum vertex cover of size $\Omega(n)$. Note that our reduction is based on ideas from the reduction of Abboud et al. [@AWW16] for the <span style="font-variant:small-caps;">Diameter</span> problem.
\[thm:SETH-lowerbound\] Assuming SETH, [<span style="font-variant:small-caps;">Hyperbolicity</span>]{}cannot be solved in $2^{o(k)}\cdot (n^{2-\epsilon})$ time, even on graphs with $O(n \log n)$ edges, diameter four, and domination number three. Here, $k$ denotes the vertex cover number of the input graph.
We reduce any instance $({\overrightarrow{A}},{\overrightarrow{B}})$ of <span style="font-variant:small-caps;">Orthogonal Vectors</span> to an instance $(G,\delta)$ of [<span style="font-variant:small-caps;">Hyperbolicity</span>]{}, where we construct the graph $G$ as follows (we refer to \[fig:orthogvec\] for a sketch of the construction).
Make each ${\overrightarrow{a}}\in {\overrightarrow{A}}$ a vertex $a$ and each ${\overrightarrow{b}}\in {\overrightarrow{B}}$ a vertex $b$ of $G$, and denote these vertex sets by $A$ and $B$, respectively. Add two vertices for each of the $\ell$ dimensions, that is, add the vertex set $C:=\{c_1,\ldots, c_\ell\}$ and the vertex set $D=\{d_1,\ldots, d_\ell\}$ to $G$ and make each of $C$ and $D$ a clique. Next, connect each $a\in A$ to the vertices of $C$ in the natural way, that is, add an edge between $a$ and $c_i$ if and only if ${\overrightarrow{a}}[i]=1$. Similarly, add an edge between $b\in B$ and $d_i\in D$ if and only if ${\overrightarrow{b}}[i]=1$. Moreover, add the edge set $\{\{c_i,d_i\}\mid i\in [\ell]\}$. This part will constitute the central gadget of our construction.
Our aim is to ensure that the maximum hyperbolicity is reached for 4-tuples $(a,b,c,d)$ such that $a\in A$, $b\in B$, and $a$ and $b$ are orthogonal vectors. The construction of $G$ is completed by adding two paths $(u_A,u,u_B)$ and $(v_A,v,v_B)$, and making $u_A$ and $v_A$ adjacent to all vertices in $A\cup C$ and $u_B$ and $v_B$ adjacent to all vertices in $B\cup D$.
=\[circle, fill, scale=1/2, draw\] (c1) at (-1.5,0-)\[node\]; (c2) at (-1.5,-0.5-)\[node\]; (ci) at (-1.5,-1-)\[\][$\vdots$]{}; (cell) at (-1.5,-1.5-)\[node\];
(d1) at (1.5,0-)\[node\]; (d2) at (1.5,-0.5-)\[node\]; (ci) at (1.5,-1-)\[\][$\vdots$]{}; (dell) at (1.5,-1.5-)\[node\];
\(c) at (-1.5,-0.75-)\[ellipse, minimum width=30pt, minimum height=80pt,draw,label=80:[$C$]{}\]; (d) at (1.5,-0.75-)\[ellipse, minimum width=30pt, minimum height=80pt,draw,label=100:[$D$]{}\];
(c1) – (d1); (c2) – (d2); (cell) – (dell);
(a1) at (-4,0)\[node,label=180:[$a_1$]{}\]; (adots1) at (-4,-0.5)\[\][$\vdots$]{}; (ai) at (-4,-1)\[node,label=180:[$a_i$]{}\]; (adots2) at (-4,-1.5)\[\][$\vdots$]{}; (an) at (-4,-2)\[node,label=180:[$a_n$]{}\];
(b1) at (4,0)\[node,label=0:[$b_1$]{}\]; (bdots1) at (4,-0.5)\[\][$\vdots$]{}; (bi) at (4,-1)\[node,label=0:[$b_j$]{}\]; (bdots2) at (4,-1.5)\[\][$\vdots$]{}; (bn) at (4,-2)\[node,label=0:[$b_n$]{}\];
(-4.75,-2.5) rectangle (-3.5,0.5); at (-5,0.5)\[\][$A$]{}; (4.75,-2.5) rectangle (3.5,0.5); at (5,0.5)\[\][$B$]{};
at (-2.75,-1)\[\][$\vdots$]{}; at (2.75,-1.1)\[\][$\vdots$]{};
(ai) – node\[above,midway,scale=3/4\][iff $a_i[1]=1$]{}(c1); (bi) – node\[above,midway,scale=3/4\][iff $b_j[2]=1$]{}(d2);
(ua) at (-2.5,1.75)\[node, label=90:[$u_A$]{}\]; (u) at (0,1.75)\[node, label=90:[$u$]{}\]; (ub) at (2.5,1.75)\[node, label=90:[$u_B$]{}\];
(va) at (-2.5,-3.75)\[node, label=-90:[$v_A$]{}\]; (v) at (0,-3.75)\[node, label=-90:[$v$]{}\]; (vb) at (2.5,-3.75)\[node, label=-90:[$v_B$]{}\];
(va) – (v) – (vb); (ua) – (u) – (ub);
in [1,2,...,9]{}[ (ua) –(-3.5-0.1\*,0.5); (ua) –(\[xshift=2\*-10 pt\] c.center); ]{}
in [1,2,...,9]{}[ (va) –(-3.5-0.1\*,-2.5); (va) – (\[xshift=2\*-10 pt\] c.center); ]{}
in [1,2,...,9]{}[ (ub) –(3.5+0.1\*,0.5); (ub) –(\[xshift=2\*-10 pt\] d.center); ]{}
in [1,2,...,9]{}[ (vb) –(3.5+0.1\*,-2.5); (vb) –(\[xshift=2\*-10 pt\] d.center); ]{}
\(c) at (-1.5,-0.75-)\[ellipse, minimum width=30pt, minimum height=80pt,draw,fill=white\]; (d) at (1.5,-0.75-)\[ellipse, minimum width=30pt, minimum height=80pt,draw,fill=white\];
(c1) at (-1.5,0-)\[node,label=90:[$c_1$]{}\]; (c2) at (-1.5,-0.5-)\[node\]; (ci) at (-1.5,-1-)\[scale=3/4\][$\vdots$]{}; (cell) at (-1.5,-1.5-)\[node,label=-90:[$c_\ell$]{}\]; (c1) to \[out=-45,in=45\](c2); (c1) to \[out=-45,in=45\](ci); (c2) to \[out=-45,in=45\](cell); (c2) to \[out=-135,in=135\](ci); (c1) to \[out=-135,in=135\](cell);
(d1) at (1.5,0-)\[node,label=90:[$d_1$]{}\]; (d2) at (1.5,-0.5-)\[node\]; (di) at (1.5,-1-)\[scale=3/4\][$\vdots$]{}; (dell) at (1.5,-1.5-)\[node,label=-90:[$d_\ell$]{}\]; (d1) to \[out=-45,in=45\](d2); (d1) to \[out=-135,in=135\](di); (d2) to \[out=-135,in=135\](dell); (d2) to \[out=-45,in=45\](di); (d1) to \[out=-45,in=45\](dell);
(ai) – node\[above,midway,scale=3/4\][iff $a_i[1]=1$]{}(c1); (bi) – node\[above,midway,scale=3/4\][iff $b_j[2]=1$]{}(d2);
Observe that $G$ contains $O(n)$ vertices, $O(n \cdot \log n)$ edges, and that the set $V\setminus (A\cup B)$ forms a vertex cover in $G$ of size $O(\log n)$. Moreover, observe that $G$ has diameter four. Note that each vertex in $A\cup B\cup C\cup D$ is at distance two to each of $u$ and $v$. Moreover, $v_A$ and $v_B$ are at distance three to $u$. Analogously, $u_A$, $u_B$ are at distance three to $v$. Furthermore $u$ and $v$ are at distance four. Finally, observe that $\{u_A,u_B,v\}$ forms a dominating set in $G$.
We complete the proof by showing that $({\overrightarrow{A}},{\overrightarrow{B}})$ is a yes-instance of <span style="font-variant:small-caps;">Orthogonal Vectors</span> if and only if $G$ has hyperbolicity at least $\delta=4$.
[($\Rightarrow$)]{} Let $({\overrightarrow{A}},{\overrightarrow{B}})$ be a yes-instance, and let ${\overrightarrow{a}}\in {\overrightarrow{A}}$ and ${\overrightarrow{b}}\in {\overrightarrow{B}}$ be a pair of orthogonal vectors. We claim that $\delta(a,b,u,v)=4$. Since ${\overrightarrow{a}}$ and ${\overrightarrow{b}}$ are orthogonal, there is no $i\in[\ell]$ with ${\overrightarrow{a}}[i]={\overrightarrow{b}}[i]=1$ and, hence, there is no path connecting $a$ and $b$ only containing two vertices in $C\cup D$, and it holds that ${\ensuremath{\overline{a b}}}=4$. Moreover, we know that ${\ensuremath{\overline{u v}}}=4$ as that ${\ensuremath{\overline{a u}}}={\ensuremath{\overline{b u}}}={\ensuremath{\overline{a v}}}={\ensuremath{\overline{a v}}}=2$. Thus, $\delta(a,b,u,v)=8-4=4$, and $G$ is 4-hyperbolic.
[($\Leftarrow$)]{} Let $S=\{a,b,c,d\}$ be a set of vertices such that $\delta(a,b,c,d)\ge4$. By \[lem:hyp-distance-bounded\], it follows that no two vertices of $S$ are adjacent. Hence, we assume without loss of generality that ${\ensuremath{\overline{a b}}}={\ensuremath{\overline{c d}}}=4$. Observe that all vertices of $C$ and $D$ have distance at most three to all other vertices. Similarly, each vertex of $\{u_A,v_A,u_B,v_B\}$ has distance at most three to all other vertices. (Consider for example $u_A$. By construction, $u_A$ is a neighbor of all vertices in $A\cup C\cup \{u\}$ and, hence, $u_A$ has distance at most two to $v_A$ and to all vertices in $D$. Thus, $u_A$ has distance at most three to $v$, $B$, $u_B$ and $v_B$ and therefore to all vertices of $G$. The arguments for $v_A$, $u_B$, and $v_B$ are symmetric).
It follows that $S\subseteq A\cup B\cup \{u,v\}$, and therefore at least two vertices in $S$ are from $A\cup B$. Thus, assume without loss of generality that $a$ is contained in $A$. By the previous assumption, we have that ${\ensuremath{\overline{a b}}}=4$. This implies that $b\in B$ and ${\overrightarrow{a}}$ and ${\overrightarrow{b}}$ are orthogonal vectors, as every other vertex in $V\setminus B$ is at distance three to $a$ and each $b'\in B$ with ${\overrightarrow{b'}}$ being non-orthogonal to ${\overrightarrow{a}}$ is at distance three to $a$. Hence, $({\overrightarrow{A}},{\overrightarrow{B}})$ is a yes-instance.
We remark that, with the above reduction, the hardness also holds for the variants in which we fix one vertex ($u$) or two vertices ($u$ and $w$). The reduction also shows that approximating the hyperbolicity of a graph within a factor of $4/3-\epsilon$ cannot be done in strongly subquadratic time or with a PL-FPT running time.
Next, we adapt the above reduction to obtain the following hardness result on graphs of bounded maximum degree.
\[thm:SETH-lowerbound2\] Assuming SETH, [<span style="font-variant:small-caps;">Hyperbolicity</span>]{}cannot be solved in $f(\Delta)\cdot (n^{2-\epsilon})$ time, where $\Delta$ denotes the maximum degree of the input graph.
We reduce any instance $({\overrightarrow{A}},{\overrightarrow{B}})$ of <span style="font-variant:small-caps;">Orthogonal Vectors</span> to an instance $(G,\delta)$ of [<span style="font-variant:small-caps;">Hyperbolicity</span>]{}as follows.
We use the following notation. For two sets of vertices $X$ and $Y$ with $|X|=|Y|$, we say that we introduce *matching paths* if we connect the vertices in $X$ with the vertices in $Y$ with paths with no inner vertices from $X\cup Y$ such that for each $x\in X$, $x$ is connected to exactly one $y\in Y$ via one path and for each $y\in Y$, $y$ is connected to exactly one $x\in X$ via one path.
Let $G'$ be the graph obtained from the graph constructed in the proof of \[thm:SETH-lowerbound\] after deleting all edges. For each $x_A$, $x\in \{u,v\}$, add two binary trees, $T_{x_A}^A$ with $n$ leaves and height at most ${\lceil\log n\rceil}$, and $T_{x_A}^C$ with $\ell$ leaves and height at most ${\lceil\log \ell\rceil}$. Connect each tree root by an edge with $x_A$. Next introduce matching paths between $A$ and the leaves of $T_{x_A}^A$ such that each shortest path connecting a vertex in $A$ with $x_A$ is of length $h:=2({\lceil\log(n)\rceil}+1)+1$. Similarly, introduce matching paths between $C$ and the leaves of $T_{x_A}^C$ such that each shortest path connecting a vertex in $C$ with $x_A$ is of length $h$. Apply the same construction for $x_B$, $x\in \{u,v\}$, $B$, and $D$.
For $x\in A\cup B$, we denote by $|x|_1$ the number of 1’s in the corresponding binary vector ${\overrightarrow{x}}$. Moreover, for $c_i\in C$, we denote by $|c_i|$ the number of vectors in $A$ with a 1 as its $i$th entry. For $d_i\in D$, we denote by $|d_i|$ the number of vectors in $B$ with a 1 as its $i$th entry.
For each vertex $a\in A$, add a binary tree with $|a|_1$ leaves and height at most ${\lceil\log |a|_1\rceil}$ and connect its root by an edge with $a$. For each $i\in[\ell]$, add a binary tree with $|c_i|$ leaves and height at most ${\lceil\log |c_i|\rceil}$ and connect its root by an edge with $c_i$. Next, construct matching paths between the leaves of all binary trees introduced for the vertices in $A$ on the one hand, and the leaves of all binary trees introduced for the vertices in $C$ on the other hand, such that the following holds: (i) for each $a\in A$ and $c_i\in C$, there is a path only containing the vertices of the corresponding binary trees if and only if ${\overrightarrow{a}}[i]=1$, and (ii) each of these paths is of length exactly $h$. Apply the same construction for $B$ and $D$.
Next, for each $i\in[\ell]$, add a binary tree with $\ell-1$ leaves and height at most ${\lceil\log (\ell-1)\rceil}$ and connect its root by an edge with $c_i$. Finally, add paths between the leaves of all binary trees introduced in this step such that (i) each leaf is incident to exactly one path, (ii) for each $i,j\in[\ell]$, $i\neq j$, there is a path only containing the vertices of the corresponding binary trees, and (iii) each of these paths is of length exactly $h$. Apply the same construction for $D$.
Finally, for each $i\in[\ell]$, connect $c_i$ with $d_i$ via a path of length $h$. Moreover, for $x\in \{u,v\}$, connect $x_A$ with $x$ and $x$ with $x_B$ each via a path of length $h$. This completes the construction of $G$. Observe that the number of vertices in $G$ is at most the number of vertices in the graph obtained from $G'$ by replacing each edge with paths of length $h$. As $G'$ contains $O(n\log n)$ edges, the number of vertices in $G$ is in $O(n\log^2 n)$. Finally, observe that the vertices in $C\cup D$ are the vertices of maximum degree which is five.
Next, we discuss the distances of several vertices in the constructed graph. Observe that $u$ and $v$ are at distance $4h$. For $x\in \{u,v\}$, the distance between $x$ and $x_A$ or $x_B$ is $h$, and the distance between $x_A$ and $x_B$ is $2h$. The distance from any $c\in C$ to any $d\in D$ is at least $h$ and at most $2h$. Moreover, the distance between any $a\in A$ and $b\in B$ is at least $3h$ and at most $4h$.
\[claim:orthog4h\] For any $a\in A$ and $b\in B$, ${\ensuremath{\overline{a b}}}=4h$ if and only if ${\overrightarrow{a}}$ and ${\overrightarrow{b}}$ are orthogonal.
[($\Leftarrow$)]{} Let ${\overrightarrow{a}}$ and ${\overrightarrow{b}}$ be orthogonal. Suppose that there is a shortest path $P$ between $a$ and $b$ of length smaller than $4h$. Observe that any shortest path between $a$ and $b$ containing $u$ or $v$ is of length $4h$. Hence, $P$ contains vertices in $C\cup D$. As the shortest paths from $a$ to $C$, $C$ to $B$, and $B$ to $b$ are each of length $h$, the only shortest path containing vertices in $C\cup B$ of length smaller than $4h$ is of the form $(a,c_i,d_i,b)$ for some $c_i\in C$ and $d_i\in D$ (recall that the shortest path between any two vertices in $C$ or $D$ is of length $h$). Hence, ${\overrightarrow{a}}$ and ${\overrightarrow{b}}$ have both a 1 as their $i$th entry, and thus are not orthogonal. This contradicts the fact that ${\overrightarrow{a}}$ and ${\overrightarrow{b}}$ form a solution. It follows that ${\ensuremath{\overline{a b}}}=4h$.
[($\Rightarrow$)]{} Let ${\overrightarrow{a}}$ and ${\overrightarrow{b}}$ be not orthogonal. Then there is an $i\in[\ell]$ such that $a[i]=b[i]=1$. Hence, there is a path $(a,c_i,d_i,b)$ of length $3h<4h$.
Let $M:=A\cup B\cup C\cup D\cup \{x,x_A,x_B\mid x\in\{u,v\}\}$. So far, we know that the only vertices that can be at distance $4h$ are those in $A\cup B\cup \{u,v\}$.
Consider any vertex $p\in V(G)\setminus M$. Then $p$ is contained in a shortest between two vertices $x$ and $y$ in $M$ at distance $h$. Moreover, $\max\{{\ensuremath{\overline{p x}}},{\ensuremath{\overline{p y}}}\}=:h'<h$. Let $P_x^Y$ denote the set of inner vertices of the shortest path connecting $x$ and $x_A$, for $x\in\{u,v\}$, $Y\in\{A,B\}$. Moreover, let $M^*:=\{p\in P_x^Y\mid x\in\{u,v\}, Y\in\{A,B\}\}$. We first discuss the case where $p\in M^*$. By symmetry, let $p\in P_u^A$. Observe that for $q\in P_v^B$ with ${\ensuremath{\overline{v q}}}={\ensuremath{\overline{u p}}}$ holds ${\ensuremath{\overline{p q}}}=4h$.
Let $p\not\in M\cup M^*$. Then, we claim that for all vertices $q\in V(G)$ it holds that ${\ensuremath{\overline{p q}}}<4h$. Suppose not, so that there is some $q\in V(G)$ with ${\ensuremath{\overline{p q}}}\geq 4h$. Observe that $q$ is not contained in a shortest path between $x$ and $y$. It follows that ${\ensuremath{\overline{x q}}}\geq 4h-h'>3h$ or ${\ensuremath{\overline{y q}}}\geq 4h-h'>3h$. Let $z\in\{x,y\}$ denote the vertex of minimal distance among the two, and let $\bar{z}$ denote the other one. Note that since $h$ is odd, the distances to $z$ and $\bar{z}$ are different.
*Case 1*: $q\in M$. Then $z,q\in A\cup B$, where $z$ and $q$ are not both contained in $A$ or $B$. Recall that $p\not\in M\cup M^*$ and, hence, the case $z,q\in \{u,v\}$ is not possible. By symmetry, assume $z\in A$ and $q\in B$. As ${\ensuremath{\overline{z q}}}>3h$, it follows that $\bar{z}=c_i\in C$ for some $i\in[\ell]$ with $1={\overrightarrow{z}}[i]\neq {\overrightarrow{q}}[i]$, or $\bar{z}\in\{u_A,v_A\}$. Hence, the distance of $\bar{z}$ to $q$ is at most the distance of $z$ to $q$, contradicting the choice of $z$.
*Case 2*: $q\not \in M$. Then $q$ is contained in a shortest path between two vertices $x',y'\in M$ of length $h$. Moreover, $\max\{{\ensuremath{\overline{q x'}}},{\ensuremath{\overline{q y'}}}\}=:h''<h$. Consider a shortest path between $p$ and $q$ and notice that it must contain $z$ and $z'\in\{x',y'\}$. It holds that ${\ensuremath{\overline{z z'}}}\geq 4h-h'-h''>2h$. By symmetry, assume $z\in A$, and $z'\in D\cup\{u_B,v_B\}$ (recall that $p\not\in M\cup M^*$). Then $\bar{z}$ is in $C\cup \{u_A,v_A\}$, and hence of shorter distance to $q$, contradicting the choice of $z$.
We proved that ${\ensuremath{\overline{p q}}}<4h$ for all $p\in V(G)\setminus (M\cup M^*)$, $q\in V(G)$. We conclude that the vertex set $A\cup B\cup \{u,v\}\cup M^*$ is the only set containing vertices at distance $4h$. Moreover, $G$ is of diameter $4h$.
We claim that $({\overrightarrow{A}},{\overrightarrow{B}})$ is a yes-instance of <span style="font-variant:small-caps;">Orthogonal Vectors</span> if and only if $G$ has hyperbolicity at least $\delta=4h$.
[($\Rightarrow$)]{} Let ${\overrightarrow{a}}\in {\overrightarrow{A}}$ and ${\overrightarrow{b}}\in {\overrightarrow{B}}$ be orthogonal. We claim that $\delta(a,b,u,v)=4h$. Observe that ${\ensuremath{\overline{u v}}}=4h$, and that ${\ensuremath{\overline{a b}}}=4h$ by \[claim:orthog4h\]. The remaining distances are $2h$ by construction, and hence $\delta(G)=\delta(a,b,u,v)=4h$.
[($\Leftarrow$)]{} Let $\delta(G)=4h$ and let $w,x,y,z$ be a quadruple with $\delta(w,x,y,z)=4h$. By \[lem:diamequalshyp\], we know that there are exactly two pairs of distance $4h$ and, hence, $\{w,x,y,z\}\subseteq A\cup B\cup\{u,v\}\cup M^*$. We claim that $|\{w,x,y,z\}\cap (M^*\cup \{u,v\})|\leq 2$. By \[lem:diamequalshyp\], we know that, out of $w,x,y,z$, there are exactly two pairs at distance $4h$ and all other pairs have distance $2h$. Assume that $|\{w,x,y,z\}\cap M^*\cup \{u,v\}|\geq 3$. Then, at least two vertices are in $P_v^A\cup P_v^B\cup \{v\}$ or in $P_u^A\cup P_u^B\cup \{u\}$. Observe that any two vertices in $P_v^A\cup P_v^B\cup \{v\}$ or in $P_u^A\cup P_u^B\cup \{u\}$ are at distance smaller than $2h$, but this contradicts the choice of the quadruple. It follows that $|\{w,x,y,z\}\cap M^*\cup \{u,v\}|\leq 2$, and w.l.o.g. let $w,x\in A\cup B$. As each vertex in $A$ is at distance smaller than $3h$ to any vertex in $A\cup \{u,v\}\cup M^*$, it follows that the other vertex is in $B$. Applying \[claim:orthog4h\], we have that $w$ and $x$ are at distance $4h$ if and only if ${\overrightarrow{w}}$ and ${\overrightarrow{x}}$ are orthogonal; hence, the statement of the lemma follows.
Parameter Distance to Cographs
==============================
We now describe a fixed-parameter linear-time algorithm for <span style="font-variant:small-caps;">Hyperbolicity</span> parameterized by the vertex deletion distance $k$ to cographs. A graph is a cograph if and only if it is $P_4$-free. Given a graph $G$ we can determine in linear time whether it is a cograph and return an induced $P_4$ if this is not the case. This implies that in $O(k\cdot (m+n))$ time we can compute a set $X\subseteq V$ of size at most $4k$ such that $G-X$ is a cograph.
A further characterization is that a cograph can be obtained from graphs consisting of one single vertex via unions and joins [@BLS99].
A *union* of two graphs $G_1=(V_1,E_1)$ and $G_2=(V_2,E_2)$ is the graph $(V_1\cup V_2,E_1\cup E_2)$.
A *join* of two graphs $G_1=(V_1,E_1)$ and $G_2=(V_2,E_2)$ is the graph $(V_1\cup V_2,E_1\cup E_2\cup \{\{v_1,v_2\}|v_1\in V_1,v_2\in V_2\})$.
The union of $t$ graphs and the join of $t$ graphs are defined by taking successive unions or joins, respectively, of the $t$ graphs in an arbitrary order. Each cograph $G$ can be associated with a rooted cotree $T_G$. The leaves of $T_G$ are the vertices of $V$. Each internal node of $T_G$ is labeled either as a union or join node. For node $v$ in $T_G$, let $L(v)$ denote the leaves of the subtree rooted at $v$. For a union node $v$ with children $u_1,\ldots, u_t$, the graph $G[L(v)]$ is the union of the graphs $G[L(u_i)]$, $1\le i\le t$. For a join node $v$ with children $u_1,\ldots, u_t$, the graph $G[L(v)]$ is the join of the graphs $G[L(u_i)]$, $1\le i\le t$.
The cotree of a cograph can be computed in linear time [@CPS85]. In a subroutine in our algorithm for <span style="font-variant:small-caps;">Hyperbolicity</span> we need to solve the following variant of <span style="font-variant:small-caps;">Subgraph Isomorphism</span>.
<span style="font-variant:small-caps;"><span style="font-variant:small-caps;">Colored Induced Subgraph Isomorphism</span></span>\
[@lX@]{} **Input:** & An undirected graph $G=(V,E)$ with a vertex-coloring $\gamma:V\to \mathbb{N}$ and an undirected graph $H=(W,F)$, where $|W|=k$, with a vertex-coloring $\chi:W\to \mathbb{N}$.\
**Question:** & Is there a vertex set $S\subseteq V$ such that there is an isomorphism $f$ from $G[S]$ to $H$ such that $\gamma(v)=\chi(f(v))$ for all $v\in S$?
Informally, the condition that $\gamma(v)=\chi(f(v))$ means that every vertex is mapped to a vertex of the same color. We say that such an isomorphism *respects the colorings*. As shown by Damaschke [@Dam90], <span style="font-variant:small-caps;">Induced Subgraph Isomorphism</span> on cographs is [$\mathsf{NP}$]{}-complete. Since this is the special case of <span style="font-variant:small-caps;">Colored Induced Subgraph Isomorphism</span> where all vertices in $G$ and $H$ have the same color, <span style="font-variant:small-caps;">Colored Induced Subgraph Isomorphism</span> is also [$\mathsf{NP}$]{}-complete (containment in [$\mathsf{NP}$]{}is obvious). In the following, we show that on cographs <span style="font-variant:small-caps;">Colored Induced Subgraph Isomorphism</span> can be solved by a linear-time fixed-parameter algorithm when the parameter is the order $k$ of $H$.
\[lemma:cisifpt\] <span style="font-variant:small-caps;">Colored Induced Subgraph Isomorphism</span> can be solved in $O(3^k(n+m))$ time in cographs.
We use dynamic programming on the cotree. Herein, we assume that for each internal node $v$ there is an arbitrary (but fixed) ordering of its children; the $i$th child of $v$ is denoted $c_i(v)$ and the set of leaves in the subtrees rooted at the first $i$ children of $v$ is denoted $L_i(v)$. We fill a three-dimensional table $D$ with entries of the type $D[v,i,X]$ where $v$ is a node of the cotree with at least $i$ children and $X\subseteq W$ is a subset of the vertices of the pattern $H$. The entry $D[v,i,X]$ has value 1 if $(G[L_i(v)], \gamma|_{L_{i}(v)}, H[X], \chi|_{X})$ is a yes-instance of <span style="font-variant:small-caps;">Colored Induced Subgraph Isomorphism</span>, that is, there is a subgraph isomorphism from $G[L_i(v)]$ to $H[X]$ that respects the coloring. Otherwise, the entry has value 0. Thus, $D[v,\deg(v)-1,X]$ has value 1 if and only if there is an induced subgraph isomorphism from $G[L_{i}(v)]$ to $H[X]$. After the table is completely filled, the instance is a yes-instance if and only if $D[r,\deg(r),W]$ has value 1 where $r$ is the root of the cotree. We initialize the table for leaf vertices $v$, by setting $D[v,0,X]=1$ if either $X=\emptyset$ or $X = \{u\}$ with $\gamma(v) = \chi(u)$; otherwise $D[v,0,X]=0$.
For union nodes, the table $D$ is filled by the following recurrence $$D[v,i,X]=
\begin{cases}
1 & \exists X'\subseteq X: D[v,i-1,X']= D[c_{i}(v),\deg(c_{i}(v))-1,X\setminus X']=1\\
& \text{$\wedge$ there are no edges between~$X'$ and~$X\setminus X'$ in~$H$}\\
0 & \text{otherwise}.
\end{cases}$$ For join nodes, the table $D$ is filled by the following recurrence $$D[v,i,X]=
\begin{cases}
1 & \exists X'\subseteq X: D[v,i-1,X']= D[c_{i}(v),\deg(c_{i}(v))-1,X\setminus X']=1\\
& \text{$\wedge$ every~$q\in X'$ is adjacent to every~$p\in X\setminus X'$ in~$H$}\\
0 & \text{otherwise}.
\end{cases}$$ The correctness of the recurrence can be seen as follows for the union nodes. First assume there is a color-respecting induced subgraph isomorphism from $G[L_i(v)]$ to $H[X]$. Then there is a set $S\subseteq L_i(v)$ such that there is a color-respecting isomorphism $f$ from $G[S]$ to $H[X]$. Let $S':=S\cap L_{i-1}(v)$ be the set of vertices that are from $S$ and from $L_{i-1}(v)$ which implies that $S\setminus S'=S\cap L(c_{i}(v))$. Since $v$ is a union node, there are no edges between $S'$ and $S\setminus S'$ in $G$. Let $X':=f(S')$ and $X\setminus X'=f(S\setminus
S')$ denote the image of $S'$ and $S\setminus S'$, respectively. Since $f$ is an isomorphism there are no edges between between $X'$ and $X\setminus X'$. Moreover, since restricting a color-respecting isomorphism $f\colon G[S]\to H[X]$ to a subset $S'$ gives a color-respecting isomorphism from $f\colon G[S']\to H[f(S')]$ we have that $D[v,i-1,X']$ and $D[c_{i}(v),\deg(c_{i}(v))-1,X\setminus X']$ have value 1. Therefore, there is a case such that the recurrence evaluates correctly to 1.
Conversely, if the recurrence evaluates to 1, then the conditions in the recurrence (about the existence of $X'$) imply a color-respecting induced subgraph isomorphism from $G[L_i(V)]$ to $H[X]$. Therefore the table is filled correctly for union nodes. The correctness of the recurrence for join nodes follows by symmetric arguments.
The running time is bounded as follows. The cotree has size $O(n+m)$ and thus, there are $O((n+m)\cdot 2^k)$ entries in the table. For each $X\subseteq W$, filling the entries of a particular table entry is done by considering all subsets of $X$, thus the overall number of evaluations is $O(3^k\cdot (n+m))$.
We now turn to the algorithm for [<span style="font-variant:small-caps;">Hyperbolicity</span>]{} on graphs that can be made into cographs by at most $k$ vertex deletions.
<span style="font-variant:small-caps;"><span style="font-variant:small-caps;">Distance-Constrained 4-Tuple</span></span>\
[@lX@]{} **Input:** & An undirected graph $G=(V,E)$ and six integers $d_{\{a,b\}}$, $d_{\{a,c\}}$, $d_{\{a,d\}}$, $d_{\{b,c\}}$, $d_{\{b,d\}}$, and $d_{\{c,d\}}$.\
**Question:** & Is there a set $S\subseteq V$ of four vertices and a bijection $f\colon S\to \{a,b,c,d\}$ such that for each $x,y\in S$ we have ${\ensuremath{\overline{x y}}} =d_{\{f(x),f(y)\}}$?
\[lem:cograph-dc-4-tuple\] <span style="font-variant:small-caps;">Distance-Constrained 4-Tuple</span> can be solved in $O(4^{4k}\cdot k\cdot (n+m))$ time if $G-X$ is a cograph for some $X\subseteq V$ of size $k$.
Let $G = (V,E)$ be the input graph and $X \subseteq V$, $|X| \le k$, such that $G-X$ is a cograph. Without loss of generality, let $X=\{x_1,\ldots ,x_k\}$.
In a preprocessing step, we classify the vertices of each connected component in $G[V
\setminus X]$ according to the length of shortest paths to vertices in $X$ such that all internal vertices of the shortest path are in $V\setminus X$.
More precisely, for a vertex $v \in V \setminus X$ in a connected component $C_v$ of $G-X$, the *type* $t_v$ of $v$ is a length-$k$ vector containing the distance of $v$ to each vertex $x_i$ of $X$ in $G[C_v \cup \{x_i\}]$. That is, $t_v[i]$ equals the distance from $v$ to $x_i \in X$ within the graph $G[C_v \cup \{x_i\}]$. Since the diameter of $G[C_v]$ is at most two, $t_v[i] \in \{1,2,3,\infty\}$. Therefore, the number of distinct types in $G$ is at most $4^k$. For simplicity of notation, for every type $t$ we denote by $v_t$ an arbitrary vertex such that $t_v=t$.
\[obs:dist-type\] Let $u$ and $v$ be two vertices of the same type, that is, $t_u=t_v$. Then for each vertex $w$ in $G-(C_u\cup C_v)$, we have ${\ensuremath{\overline{u w}}}={\ensuremath{\overline{v w}}}$.
Given two vertices $u$ and $v$ such that $C_u\neq C_v$, we can compute ${\ensuremath{\overline{u v}}}$ in $O(k)$ time when the distance between each $u\in X$ and each $v\in V\setminus X$ can be retrieved in $O(1)$ time.
The dominating part of the running time of the preprocessing is the computation of the vertex types which can be performed in $O(k\cdot (n+m))$ time as follows. Create a length-$k$ vector for each of the at most $n$ vertices of $V\setminus X$ and initially set all entries to $\infty$. Then compute for each $x_i\in X$, the graph $G-(X\setminus
\{x_i\})$ in $O(n+m)$ time. In this graph perform a breadth-first search from $x_i$ to compute the distances between $x_i$ and each vertex $v\in V\setminus X$ that is in the same connected component as $x_i$. This distance is exactly the one in the graph $G[C_v
\cup \{x_i\}]$. Thus, for all vertices that reach $x_i$, the $i$th entry in their type vector is updated. Afterwards, each vertex has the correct type vector.
After this preprocessing, the algorithm proceeds as follows by restricting the choice of vertices for the 4-tuple.
- First, branch into all $O(k^4)$ cases of taking a subset $X'\subseteq X$ of size at most four. (We will assume that $X'=S\cap X$.)
- For each such $X'\subseteq X$, branch into the different cases for the types of vertices in $S\setminus X'$. That is, consider all multisets $M_T$ of size $|S\setminus X'| = 4 - |X'|$ over the universe of all types. (There are $4^k$ types and thus at most $4^{4k}$ cases for each $X'\subseteq X$.)
- For each such $X'\subseteq X$ and multiset $M_T$, branch into all cases of matching the vertices in $\{a,b,c,d\}$ to the vertices in $X$ and types in $M_T$ (branch into all “bijections” $f$ between $X' \cup M_T$ and $\{a,b,c,d\}$). (There are at most $4!$ cases.)
- For each such branch, branch into the different possibilities to assign the types in $M_T$ to connected components of $G-X$. That is, create one branch for each partition of the multiset $M_T$ and assume in this branch that two types are in the same connected component if and only if they are in the same set of the partition of $M_T$. The current partition is called the *component partition* of the branch.
We now check whether there is a solution to the <span style="font-variant:small-caps;">Distance-Constrained 4-Tuple</span> instance that fulfills the additional assumptions made in the above branches. To this end, for each pair of vertices $x,y\in X'$, check whether ${\ensuremath{\overline{x y}}}=d_{\{f(x),f(y)\}}$. Now, for each vertex $x\in X'$ and each type $t\in M_T$, check whether ${\ensuremath{\overline{x v_t}}}=d_{\{f(x),f(v_t)\}}$, where $v_t$ is an arbitrary vertex of type $t$. Observe that this is possible since, by \[obs:dist-type\] the distance between $X$ and any vertex of type $t$ is the same in $G$. Next, for pair of types $t,t'\in M_T$ such that the branch assumes that $t$ and $t'$ do not lie in the same connected component of $G-X$, check whether ${\ensuremath{\overline{v_{t} v_{t'}}}}=d_{\{f(v_{t}),f(v_{t'})\}}$. Again, this is possible due to \[obs:dist-type\].
The remaining problem is thus to determine whether the types of $M_T$ can be assigned to vertices in such a way that
- for each pair of types $t,t'\in M_T$ the assigned vertices are in the same connected component of $G-X$ if and only if it is constrained to be in the same type of connected component in the current branch,
- for each pair of types $t,t'\in M_T$ such that their assigned vertices $v_t$ and $v_{t'}$ are constrained to be in the same connected component, we need to ensure that ${\ensuremath{\overline{v_t v_{t'}}}}=d_{\{f(v_t),f(v_{t'})\}}$.
We solve this problem by a reduction to <span style="font-variant:small-caps;">Colored Induced Subgraph Isomorphism</span>. Observe that, since $G-X$ is a cograph, for each pair $u$ and $v$ of vertices in the same connected component of $G-X$, the distance between $u$ and $v$ is 2 if and only if they are not adjacent. With this observation, the reduction works as follows. Let $j\le 4$ denote the number of distinct connected components of $G-X$ that shall contain at least one type of $M_T$. Now, for each connected component $C$ of $G-X$ add one further vertex $v_C$ by making it adjacent to all vertices of $C$ and call the resulting graph $G'$. Now color the vertices of $G-X$ as follows. The additional vertices of each connected component receive the color $0$. Next, for each vertex type $t$ in $G-X$ introduce one color and assign this color to each vertex of type $t$. Call the vertices with color $0$ the *component-vertices* and all other vertices the *type-vertices*. To complete the construction of the input instance, we build $H$ as follows. Add $j$ vertices of color $0$. Then add a vertex for each type $t$ of $M_T$ and color it with the color corresponding to its type. As in $G'$, call the vertices with color $0$ *component-vertices* and all other vertices *type-vertices*. Add edges between the component-vertices and the type-vertices in such a way that every type-vertex is adjacent to one vertex of color 0 and two type-vertices are adjacent to the same color-0 vertex if and only if they are constrained to be in the same connected component. Finally, if two type-vertices are constrained to be in the same connected component and have distance 1 in $G$, then add an edge between them, otherwise add no edge between them. This completes the construction of $H$. The instance of <span style="font-variant:small-caps;">Colored Induced Subgraph Isomorphism</span> consists of $G'$ and $H$ and of the described coloring. We now claim that this instance is a yes-instance if and only if there is a solution to the <span style="font-variant:small-caps;">Distance-Constrained 4-Tuple</span> instance that fulfills the constraints of the branch.
If the instance has a solution, then the subgraph isomorphism $\phi$ from $G'[S]$ to $H$ corresponds to a selection of types from $j$ connected components since $H$ contains neighbors of $j$ component-vertices. Moreover, in $H$, and thus in $G'[S]$, every type-vertex is adjacent to exactly one component-vertex and thus the component-vertices define a partition of the type-vertices of $G'[S]$ that is, due to the construction of $H$, exactly the component partition of $M_T$. Selecting the type-vertices of $S$ and assigning them to $\{a,b,c,d\}$ as specified by $f$ gives, together with the selected vertices of $X$, a special 4-tuple $Q$ Observe that $Q$ fulfills all constraints of the branch except for the conditions on the distances between the type-vertices of the same component. Now for two vertices $u$ and $v$ of $S$ in the same connected component of $G-X$, the distance is 1 if they are adjacent and 2 otherwise. Due to the construction of $H$, and the fact that $\phi$ is an isomorphism, the distance is thus 1 if $d_{\{f(u),f(v)\}}=1$ and 2 if $d_{\{f(u),f(v)\}}=2$. Thus, if the <span style="font-variant:small-caps;">Colored Induced Subgraph Isomorphism</span> instance is a yes-instance, so is the <span style="font-variant:small-caps;">Distance-Constrained 4-Tuple</span> instance. The converse direction follows by the same arguments.
The running time can be seen as follows. The preprocessing can be performed in $O(k(n+m))$ time, as described above. Then, the number of branches is $O(4^{4k})$: the only time when the number of created branches is not constant is when the types of the vertices in the 4-tuple are constrained or when the 4-tuple vertices are fixed to belong to $X$. In the worst case, we have $X' = \{a,b,c,d\}\cap X=\emptyset$, that is, $S \subseteq V\setminus X$ and for all four vertices of $S$ one has to branch in total into $4^{4k}$ cases to fix the types. In each branch, the algorithm first checks the conditions on all distances except for the distances between vertices of the same parts of the component partition. This can be done in $O(k)$ time for each of these distance. Afterwards, the algorithm builds and solves the <span style="font-variant:small-caps;">Colored Induced Subgraph Isomorphism</span> in $O(n+m)$ time. Altogether, this gives the claimed running time bound.
The final step is to reduce [<span style="font-variant:small-caps;">Hyperbolicity</span>]{} to <span style="font-variant:small-caps;">Distance-Constrained 4-Tuple</span>. This can be done by creating $O(k^4)$ instances of <span style="font-variant:small-caps;">Distance-Constrained 4-Tuple</span> as shown below.
\[thm:cograph-dist\] [<span style="font-variant:small-caps;">Hyperbolicity</span>]{}can be solved in $O(4^{4k}\cdot k^7\cdot (n+m))$ time, where $k$ is the vertex deletion distance of $G$ to cographs.
Let $G = (V,E)$ be the input graph and $X \subseteq V$, $|X| \le k$, such that $G-X$ is a cograph and observe that $X$ can be computed in $O(4^k\cdot (n+m))$ time. Since every connected component of $G-X$ has diameter at most two, the maximum distance between any pair of vertices in the same component of $G$ is at most $4k+2$: any shortest path between two vertices $u$ and $v$ visits at most $k$ vertices in $X$, at most three vertices between every pair of vertices $x$ and $x'$ from $X$ and at most three vertices before encountering the first vertex of $X$ and at most three vertices before encountering the last vertex of $X$.
Consequently, for the 4-tuple $(a,b,c,d)$ that maximizes $\delta(a,b,c,d)$, there are $O(k^6)$ possibilities for the pairwise distances between the four vertices. Thus, we may compute whether there is a 4-tuple such that $\delta(a,b,c,d)=\delta$ by checking for each of the $O(k^6)$ many $6$-tuples of possible pairwise distances of four vertices in $G$ whether there are $4$ vertices in $G$ with these six pairwise distances and whether this implies $\delta(a,b,c,d)\ge \delta$. The latter check can be performed in $O(1)$ time, and the first is equivalent to solving <span style="font-variant:small-caps;">Distance-Constrained 4-Tuple</span> which can be done in $O(4^{4k}\cdot k\cdot (n+m))$ time by \[lem:cograph-dc-4-tuple\]. The overall running time follows.
Reduction from 4-Independent Set {#sec:ind-set}
================================
In this section, we provide a further relative lower bound for <span style="font-variant:small-caps;">Hyperbolicity</span>. Specifically, we prove that, if the running time is measured in terms of $n$, then <span style="font-variant:small-caps;">Hyperbolicity</span> is at least as hard as the problem of finding an independent set of size four in a graph. The currently best running time for this problem is $O(n^{3.257})$ [@EG04; @WWWY15]. Hence, any improvement on the running time of <span style="font-variant:small-caps;">Hyperbolicity</span> which breaks this bound (e.g., an algorithm running in $o(n^3)$ time), would also yield a substantial improvement for the <span style="font-variant:small-caps;">4-Independent Set</span> problem.
To this end, we reduce from a 4-partite (or 4-colored) variant of the <span style="font-variant:small-caps;">Independent Set</span> problem. The standard reduction from <span style="font-variant:small-caps;">Independent Set</span> to <span style="font-variant:small-caps;">Multicolored Independent Set</span> shows that this 4-colored variant has the same asymptotic running time lower bound as <span style="font-variant:small-caps;">4-Independent Set</span>.
\[thm:4is\] Any algorithm solving [<span style="font-variant:small-caps;">Hyperbolicity</span>]{}in $O(n^c)$ time for some constant $c$ yields an $O(n^c)$-time algorithm solving <span style="font-variant:small-caps;">4-Independent Set</span>.
Let $G=(V=V_1\uplus V_2\uplus V_3\uplus V_4,E)$ be an instance of the <span style="font-variant:small-caps;">4-Colored-Independent Set</span> problem. Assume an arbitrary order on the vertices of $V_i$, that is, $V_i=\{v_1^i,\ldots,v_{n_i}^i\}$, where $n_i;=|V_i|$, for each $1\leq i\leq 4$. We construct a graph $G'$, initially being the empty graph, as follows (we refer to \[fig:4isreduc\] for an illustration).
=\[circle,fill=white,draw=black!80,minimum size=25pt,inner sep=0pt\] =\[circle,fill=white,draw=black!80,minimum size=5pt,inner sep=0pt\] =\[circle,fill=white,draw=black!80,minimum size=20pt,inner sep=0pt\] =\[circle,fill=white,draw=black!80,minimum size=35pt,inner sep=0pt\] ; ; / / in [ [(2,10\*)]{}/$X_1$/vone, [(8\*,10\*)]{}/$X_2$/vone, [(8\*,4)]{}/$X_3$/vone, [(2,4)]{}/$X_4$/vone, [(4\*,11\*)]{}//vtwo, [(6\*,11\*)]{}//vtwo, [(9\*,8\*)]{}//vtwo, [(9\*,6\*)]{}//vtwo, [(6\*,3)]{}//vtwo, [(4\*,3)]{}//vtwo, [(1,6\*)]{}//vtwo, [(1,8\*)]{}//vtwo, [(5\*,10\*)]{}/$X_1' $/vthree, [(8\*,7\*)]{}/$X_2'$/vthree, [(5\*,4)]{}/$X_3' $/vthree, [(2,7\*)]{}/$X_4' $/vthree, [(-1\*,8\*)]{}/$Y_1$/vthree, [(6.5\*,1\*)]{}/$ Z_1$/vthree, [(10.75\*,8\*)]{}/$Y_2$/vthree, [(3.5\*,1\*)]{}/$Z_2$/vthree]{} [ (Vi) at ; ]{}
at (V5)\[label=90:[$u_{1,2}$]{}\]; at (V6)\[label=90:[$u_{2,1}$]{}\]; at (V7)\[label=0:[$u_{2,3}$]{}\]; at (V8)\[label=0:[$u_{3,2}$]{}\]; at (V9)\[label=-90:[$u_{3,4}$]{}\]; at (V10)\[label=-90:[$u_{4,3}$]{}\]; at (V11)\[label=180:[$u_{4,1}$]{}\]; at (V12)\[label=180:[$u_{1,4}$]{}\];
in [0.1,0.2,...,0.5]{} [ (V5) to ($(V1)+(0.1*\x+0.2,\x)$); (V6) to ($(V2)+(0.1*\x+0.1,\x)$); (V7) to ($(V2)+(0.75*\x-0.4,1.2*\x)$); (V8) to ($(V3)+(0.75*\x-0.4,-0.5*\x)$); (V9) to ($(V3)+(-0.1*\x-0.2,-\x)$); (V10) to ($(V4)+(-0.4*\x+0.3,-\x+0.1)$); (V11) to ($(V4)+(0.75*\x-0.4,1.2*\x)$); (V12) to ($(V1)+(0.75*\x-0.4,-0.5*\x)$); ]{} i/ in [5/6,7/8,9/10,11/12]{} [ (Vi) edge (V) ; ]{}
i/ in [1/13,2/14,3/15,4/16]{} [ (Vi) edge (V) ; ]{}
(V13) to node\[below,scale=0.66\][$\not\in E$]{}(V2); (V14) to node\[left,scale=0.66\][$\not\in E$]{}(V3); (V15) to node\[above,scale=0.66\][$\not\in E$]{}(V4); (V16) to node\[right,scale=0.66\][$\not\in E$]{}(V1);
i/ in [1/17]{} [ (Vi) edge (V) ; ]{} i/ in [2/19]{} [ (Vi) edge (V) ; ]{} (V19) to \[out=-90,in=-45\](V20); (V17) to \[out=-90,in=225\](V18); (V20) to node\[right,scale=0.66\][$\in E$]{}(V4); (V18) to node\[right,scale=0.66\][$\in E$]{}(V3);
/ / in [ [(2,10\*)]{}/$X_1$/vone, [(8\*,10\*)]{}/$X_2$/vone, [(8\*,4)]{}/$X_3$/vone, [(2,4)]{}/$X_4$/vone, [(4\*,11\*)]{}//vtwo, [(6\*,11\*)]{}//vtwo, [(9\*,8\*)]{}//vtwo, [(9\*,6\*)]{}//vtwo, [(6\*,3)]{}//vtwo, [(4\*,3)]{}//vtwo, [(1,6\*)]{}//vtwo, [(1,8\*)]{}//vtwo, [(5\*,10\*)]{}/$X_1' $/vthree, [(8\*,7\*)]{}/$X_2'$/vthree, [(5\*,4)]{}/$X_3' $/vthree, [(2,7\*)]{}/$X_4' $/vthree, [(-1\*,8\*)]{}/$Y_1$/vthree, [(6.5\*,1\*)]{}/$ Z_1$/vthree, [(10.75\*,8\*)]{}/$Y_2$/vthree, [(3.5\*,1\*)]{}/$Z_2$/vthree]{} [ (V) at ; ]{}
\(w) at (5\*,7\*)\[circle,draw,scale=0.5,label=45:[$w$]{}\]; in [-0.4,-0.3,...,0.4]{}[ (w) to ($(w)+(1.3*\x,1.1*\ysh)$); (w) to ($(w)+(1.1*\xsh,1.3*\x)$); (w) to ($(w)+(1.3*\x,-1.1*\ysh)$); (w) to ($(w)+(-1.1*\xsh,1.3*\x)$); ]{}
Add the vertex sets $X_1$, $X_2$, $X_3$, and $X_4$, where $X_i=\{x_1^i,\ldots,x_{n_i}^i\}$, $1\leq i\leq 4$. We say $x_j^i$ corresponds to the vertex $v_j^i\in V_i$ in $V$, for each $1\leq i\leq 4$, $1\leq j\leq n_i$. Introduce a copy $X'_i$ of each $X_i$ and further copies $Y_1, Z_1$ of $X_1$ and $Y_2, Z_2$ of $X_2$. Make each $X_i$ and each copy of each $X_i$ a clique. We say that the $j$th vertex of some copy of $X_i$ *corresponds to the $j$th vertex* of $X_i$ and hence corresponds to the $j$th vertex in $V_i$.
For each vertex in $X_i$ introduce an edge to its corresponding vertex in $X'_i$.
For $i\in\{1,2,3\}$, introduce an edge between a vertex in $X'_i$ and a vertex in $X_{i+1}$ if their corresponding vertices in $V$ are *not* adjacent in $G$. Introduce edges between vertices in $X_4'$ and $X_1$ analogously.
For $i\in\{1,2\}$, introduce edges for corresponding vertices between $X_i$ and $Y_i$, and between $Y_i$ and $Z_i$.
For $i\in\{1.2\}$. introduce an edge between a vertex in $Z_i$ and a vertex in $X_{i+2}$ if their corresponding vertices in $V$ *are* adjacent in $G$.
Introduce a set $U:=\{u_{1,2}^1,u_{1,2}^2,u_{2,3}^2,u_{2,3}^3,u_{3,4}^3,u_{3,4}^4,u_{4,1}^4,u_{4,1}^1\}$ of eight further vertices and call the vertices in $U$ the *connection vertices*.
Introduce the edges $\{u_{i,j}^i,u_{i,j}^j\}$, and connect each vertex in $X_i$ with $u^i_{i,j}$ and $X_j$ with $u^j_{i,j}$ via an edge.
Finally, add the vertex $w$ and connect $w$ via an edge with all vertices except the vertices in $X_i$, $1\leq i \leq 4$.
This finishes the construction of $G'=(V',E')$. Observe that $|V(G')|=2\cdot|V(G)|+2\cdot(n_1+n_2)+9$. Moreover, observe that the diameter of $G'$ is four. To see this, observe that $w$ has distance at most two to each vertex in $G'$. Assuming that there exist at least one pair of vertices $a \in V_1$ and $c \in V_3$ such that $\{a, c\} \notin E$ (as otherwise $G$ is a trivial no-instance), the distance between $a$ and $c$ is exactly $4$. We prove that $G'$ is 4-hyperbolic if and only if $G$ has an independent set of size 4.
[($\Rightarrow$)]{} Let $\{a,b,c,d\}$ be a colored-independent set of size four in $G$, and let without loss of generality $a\in V_1$, $b\in V_2$, $c\in V_3$, and $d\in V_4$. Let $a',b',c',d'\in V(G')$ with $a'\in X_1$, $b'\in X_2$, $c'\in X_3$, and $d'\in X_4$ be the corresponding vertices in $X:=X_1\cup X_2\cup X_3\cup X_4$. We show that $\delta(a',b',c',d')=4$.
First, we show that ${\ensuremath{\overline{a' b'}}} = 2$. As no vertex in $X_1$ is adjacent to any vertex in $X_2$, we have ${\ensuremath{\overline{a' b'}}} \geq 2$. As $a$ and $b$ are not adjacent in $G$, they have a common neighbor in $X_1'$ by construction of $G'$. It follows that ${\ensuremath{\overline{a' b'}}} = 2$. By a symmetric argument, we conclude that ${\ensuremath{\overline{b' c'}}} = {\ensuremath{\overline{c' d'}}} = {\ensuremath{\overline{d' a'}}} = 2$.
Further, we show that ${\ensuremath{\overline{a' c'}}}=4$. As each vertex in $G'$ is at distance two to vertex $w$, it follows that ${\ensuremath{\overline{a' c'}}}\leq 4$. Moreover, all the neighbors of $a'$ are in $X'_1\cup Y_1\cup X'_4\cup \{u_{1,2}^1,u_{4,1}^1\}$ and all the neighbors of $c'$ are in $Z_1\cup X'_2\cup X'_3\cup \{u_{2,3}^3,u_{3,4}^3\}$. Thus, to have a distance of at most three, a neighbor of $a'$ must be adjacent to a neighbor of $c'$. By construction, this is only possible if the unique neighbor $a'_Y$ of $a'$ in $Y_1$ is adjacent to a neighbor of $c'$ in $Z_1$. The unique neighbor $a'_Z\in Z_1$ of $a'_Y\in Y_1$ is, however, not adjacent to $c'$ since $a$ and $c$ are not adjacent in $G$. It follows that ${\ensuremath{\overline{a' c'}}}=4$. By a symmetric argument, we conclude that ${\ensuremath{\overline{b' d'}}}=4$.
Finally, altogether we have $\delta(a',b',c',d')=(4+4)-(2+2)=4$.
[($\Leftarrow$)]{} Let $S:=\{a',b',c',d'\}\subseteq V(G')$ be a vertex set such that $\delta(a',b',c',d')=4$. We show that these vertices correspond to four vertices in $G$ forming a colored independent set in $G$. By \[lem:hyp-distance-bounded\], we have $4=\delta(a',b',c',d') \le 2 \cdot \min_{u \neq v \in S} \{{\ensuremath{\overline{u v}}}\}$, and hence no two vertices in $S$ are adjacent. Now, assume without loss of generality that ${\ensuremath{\overline{a' c'}}}+{\ensuremath{\overline{b' d'}}}$ is the largest sum among the distances. Since the diameter of $G'$ is four, we have that ${\ensuremath{\overline{a' c'}}}+{\ensuremath{\overline{b' d'}}}\le 8$. Moreover, all other distances are at least two, since $S$ forms an independent set in $G'$. Then, since $G'$ is 4-hyperbolic, this implies that ${\ensuremath{\overline{a' c'}}}={\ensuremath{\overline{b' d'}}}=4$ and ${\ensuremath{\overline{a' b'}}}={\ensuremath{\overline{b' c'}}}={\ensuremath{\overline{c' d'}}}={\ensuremath{\overline{d' a'}}}=2$.
This already implies that $w\notin S$ as $w$ has distance at most two to all other vertices in $G'$. Moreover, since all vertices in $V'\setminus X$ are adjacent to $w$, each of them is at distance at most three to all other vertices in $G'$. Thus, $S\subseteq X$, but as $S$ forms an independent set in $G'$, there are no two vertices of $X_i$, $1\leq i\leq 4$, in $S$ (recall each $X_i$ forms a clique in $G'$). Let $a\in V_1$, $b\in V_2$, $c\in V_3$, and $d\in V_4$ be the vertices in $G$ corresponding to $a'$, $b'$, $d'$, and $d'$, respectively. Assume without loss of generality that $a'\in X_1$, $b'\in X_2$, $c'\in X_3$, and $d'\in X_4$. Finally, by the construction of $G'$ together with ${\ensuremath{\overline{a' b'}}}={\ensuremath{\overline{b' c'}}}={\ensuremath{\overline{c' d'}}}={\ensuremath{\overline{d' a'}}}=2$ and ${\ensuremath{\overline{a' c'}}}={\ensuremath{\overline{b' d'}}}=4$, it follows that $\{a,b,c,d\}$ forms a colored-independent set in $G$.
Conclusion
==========
To efficiently compute the hyperbolicity number, parameterization sometimes may help. In this respect, perhaps our practically most promising results relate to the $O(k^4(n+m))$ running times (for the parameters covering path number and feedback edge number, see \[tab:results\])—note that they clearly improve on the standard algorithm when $k=o(n^{1/4})$. Moreover, the linear-time data reduction rules we presented may be of independent practical interest. On the lower bound side, together with the work of Abboud et al. [@AWW16] our SETH-based lower bound with respect to the parameter vertex cover number is among few known “exponential lower bounds” for a polynomial-time solvable problem.
As to future work, we particularly point to the following open questions. First, we left open whether there is a linear-time FPT algorithm exploiting the parameter feedback vertex number for computing the hyperbolicity number. Second, for parameter vertex cover number we have an SETH-based exponential lower bound for the parameter function in any linear-time FPT algorithm. This does not imply that it is impossible to achieve a polynomial parameter dependence when asking for algorithms with running time factor $O(n^2)$ or $O(n^3)$.
[^1]: Supported by the DFG, project DAMM (NI 369/13-2).
[^2]: Supported by the DFG, project MAGZ (KO 3669/4-1).
[^3]: Supported by a postdoc fellowship of the DAAD while at Durham University.
[^4]: Nimrod Talmon was supported by a postdoctoral fellowship from I-CORE ALGO.
[^5]: This work was initiated at the 2016 research retreat of the Algorithmics and Computational Complexity (AKT) group of TU Berlin.
[^6]: Cographs are the graphs without induced $P_4$s. For instance, distance to cographs is never bigger than the graph parameter cluster graph vertex deletion distance [@DK12]. Moreover, it is also upper-bounded by the vertex cover number.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We present the new TNG50 cosmological, magnetohydrodynamical simulation – the third and final volume of the IllustrisTNG project. This simulation occupies a unique combination of large volume and high resolution, with a 50 Mpc box sampled by $2160^3$ gas cells (baryon mass of $8 \times 10^4$[M$_{\odot}$]{}). The *median* spatial resolution of star-forming ISM gas is $\sim$100$-$140 parsecs. This resolution approaches or exceeds that of modern ‘zoom’ simulations of individual massive galaxies, while the volume contains $\sim$ 20,000 resolved galaxies with $M_\star \ga 10^7$[M$_{\odot}$]{}. Herein we show first results from TNG50, focusing on galactic outflows driven by supernovae as well as supermassive black hole feedback. We find that the outflow mass loading is a non-monotonic function of galaxy stellar mass, turning over and rising rapidly above $10^{10.5}$[M$_{\odot}$]{}due to the action of the central black hole. Outflow velocity increases with stellar mass, and at fixed mass is faster at higher redshift. The TNG model can produce high velocity, multi-phase outflows which include cool, dense components. These outflows reach speeds in excess of 3000 km/s out to 20 kpc with an ejective, BH-driven origin. Critically, we show how the relative simplicity of model inputs (and scalings) at the injection scale produces complex behavior at galactic and halo scales. For example, despite isotropic wind launching, outflows exhibit natural collimation and an emergent bipolarity. Furthermore, galaxies above the star-forming main sequence drive faster outflows, although this correlation inverts at high mass with the onset of quenching, whereby low luminosity, slowly accreting, massive black holes drive the strongest outflows.'
author:
- |
Dylan Nelson$^{1}$[^1], Annalisa Pillepich$^{2}$, Volker Springel$^{1,3,4}$, R[ü]{}diger Pakmor$^{1,3}$, Rainer Weinberger$^{5}$, Shy Genel$^{6}$, Paul Torrey$^{7}$, Mark Vogelsberger$^{8}$, Federico Marinacci$^{9}$, Lars Hernquist$^{5}$\
\
$^{1}$Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Str. 1, 85741 Garching, Germany\
$^{2}$Max-Planck-Institut für Astronomie, Königstuhl 17, 69117 Heidelberg, Germany\
$^{3}$Heidelberg Institute for Theoretical Studies, Schloss-Wolfsbrunnenweg 35, 69118 Heidelberg, Germany\
$^{4}$Zentrum für Astronomie der Universität Heidelberg, ARI, Mönchhofstr. 12-14, 69120 Heidelberg, Germany\
$^{5}$Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA, 02138, USA\
$^{6}$Center for Computational Astrophysics, Flatiron Institute, 162 Fifth Avenue, New York, NY 10010, USA\
$^{7}$University of Florida, Department of Astronomy, 211 Bryant Space Sciences Center, Gainesville, FL 32611, USA\
$^{8}$Kavli Institute for Astrophysics and Space Research, Department of Physics, MIT, Cambridge, MA, 02139, USA\
$^{9}$Department of Physics and Astronomy, University of Bologna, Viale Berti Pichat 6/2, I-40127 Bologna, Italy\
bibliography:
- 'refs.bib'
title: |
First Results from the TNG50 Simulation:\
Galactic outflows driven by supernovae and black hole feedback
---
galaxies: evolution – galaxies: formation – galaxies: outflows and feedback – circumgalactic medium
Introduction
============
Energetic outflows driven out of galaxies signpost the resistance of astrophysical feedback processes against the inevitability of gravitational collapse. By itself, the accretion of gas into dark matter halos and its subsequent cooling onto a centrally forming galaxy [@silk77; @wr78] leads to an excess of baryons across the mass scale [@thoul95]. This is evidenced most clearly by the differential shape of the galaxy stellar mass function with respect to the underlying dark matter halo mass function from which it must emerge [@kauffmann93; @cole94].
To successfully regulate the stellar mass content of galaxies gas must be prevented from joining the star-forming phase, efficiently ejected out of this phase, or both. Preventive feedback alone is not thought to be sufficient [@benson03], while galactic outflows are both theoretically expected [@tomisaka88] and observationally detected [@heckman90]. Physically, they are expected to be driven by the energy released from stars and supernovae [@cc85], as well as from supermassive black holes [@begelman91].
No model for galaxy formation currently succeeds without incorporating both mechanisms [@spr03; @croton06]. Outflows and feedback are therefore a fundamental aspect of galaxy formation. However, the underlying physical mechanisms are complex [@murray05] and operate at extremely small scales – a supermassive black hole accretion disk (10$^{-5}$ pc) is roughly ten orders of magnitude smaller than its host galaxy or the intracluster medium (10$^{5}$ pc) whose thermodynamics it can dominate [@tabor93; @mcnamara00; @fabian03]. As feedback-driven outflows are both difficult to simulate and difficult to observe, the topic invites study [reviewed in @veilleux05; @king15; @heckman17].
Observations of galactic outflows
---------------------------------
Pioneering studies have demonstrated the existence of outflows in the local universe [@heckman90; @lehnert96] and at high redshifts $z>2$ [@pettini01]. Galactic outflows are understood to be ‘ubiquitous’ in star-forming galaxies [@shapley03; @martin05; @weiner09; @zhu15], implying that normal main-sequence galaxies continuously drive winds. Observational probes for the incidence and properties of outflows extend from $z \sim 0$, including analyses of SDSS [@chen10] through $z \sim 0.5-1$ [@rubin10; @rubin11] to the peak epoch of cosmic star formation at $z \sim 2-3$ in both absorption [@steidel10] and emission, focusing on star formation (SF)-driven outflows [@newman12; @genzel14] and blackhole/active galactic nuclei (AGN) sources [@harrison16; @leung17; @circosta18] even up to $z \gtrsim 5$ quasars [e.g. @bischetti18].
Gas which makes up such a galactic-scale wind can be identified in a number of different phase tracers [@cicone18]: cold gas (e.g. $\lesssim 10^4$ K) including molecules such as CO [@fluetsch18] as well as neutral HI and metals such as NaD [@concas17b; @bae18]. Cool gas (e.g. $\sim 10^4$ K) including metal ion tracers such as FeII, MgII and OIII [@martin09; @rubin14], as well as hydrogen in the form of H$\alpha$ [@shapiro09]. Warm gas (e.g. $\sim 10^5$ K - $10^6$ K) can also be traced by ionized metals such as OVI [@kacprzak15; @nielsen17]. Finally, hot gas (e.g. $\gtrsim 10^6$ K) as observed at x-ray wavelengths [@lehnert99; @strickland09].
Except in the most local examples, observations of more than one of these outflow phases in the same system are rare. When available, such data imply that outflows are multi-phase, having several co-spatial, possibly kinematically coherent components with a large range in density and temperature [@heckman17]. Different phases can have significantly different inferred outflow velocities [@grimes09]. Typically, information is only available on one phase, so an inferred outflow rate of mass, energy, or momentum may only be a fraction of the total [@forsterschreiber18b]. This is particularly problematic when comparing data and theoretical models, for example of mass loading factors $\eta = \dot{M}_{\rm out} / \dot{M}_\star$ or outflow velocities, as we explore below.
Only in the past few years have we begun to spatially resolve outflows with integral field units (IFU) surveys [@rupke17]. Recent explorations with MUSE have demonstrated the power of blind, deep fields out to high-z [@finley17b; @feltre18] as well as resolved detail at low-z [@venturi18]. Data from the KCWI instrument has revealed E+A galaxies hosting strong conical outflows, and shown how MgII-traced outflows can be mapped in spatially resolved emission [@baron18]. Studies using SINFONI have shown how AO-assisted ground-based IFU data can connect galaxy and outflow properties on resolved, kpc scales at $z \sim 2$ [@forsterschreiber18a; @davies18; @circosta18]. From the observational point of view, even identifying a given data feature as an outflow can be challenging. For direct ‘down the barrel’ spectral observations of a host galaxy, two orthogonal techniques are at play: absorption and emission. A blue-shifted absorption line can be interpreted as an intervening mass of the given gas tracer, flowing towards the observer and so away from the galaxy, blocking some of its background continuum light. While the line center offset from systemic gives an indication of the velocity of the bulk of the material, the shape (equivalent width) simultaneously encodes the full velocity distribution of the outflowing material, the covering factor of the observed phase, and the optical depth [@chen10]. Optical emission-line profiles are often asymmetric in shape, plausibly consisting of superimposed narrow and broad components, the latter preferentially blueshifted. These can be understood as a galactic-scale outflow, where obscuration by the interstellar medium (ISM) of the host galaxy hides the receding, redshifted gas [@armus89]. Studies in absorption as well as emission must make a large number of assumptions to convert observed quantities to physical values. There are enough caveats that final uncertainties are large [@chisholm17]. Given the breadth of techniques, definitions, galaxy selection functions, mass and redshift coverage, and physical assumptions, even fundamental scaling relations between outflow properties (e.g. velocity, mass/energy loading) and host galaxy properties (e.g. stellar mass, star formation rate, black hole luminosity) are often revised [@rupke05c; @perna17].
As a result, and despite the abundance of evidence for the existence of outflowing gas around galaxies, questions of its origin (launch mechanism, energy source) as well as its eventual fate (ability to escape the galaxy, dark matter halo, or to return as a recycled fountain flow) remain largely open topics. Of particular interest is our ability to discriminate winds with stellar versus black hole feedback origins. In particular, the correlations of outflow properties with the star formation and black hole activity of the galaxy itself can shed light on this dichotomy [@forsterschreiber18b], ultimately providing a crucial view of, and constraint on, astrophysical feedback processes.
Cosmological hydrodynamical simulations
---------------------------------------
One promising avenue to study this interplay of feedback, outflows, and the subsequent cycle of baryons in and around galaxies is through cosmological hydrodynamical simulations [see review in @somerville15]. Recent large-volume efforts have clarified the need for efficient outflows in order to make realistic galaxy populations whose characteristics and integral properties are roughly in agreement with observational constraints [@vog14a; @genel14; @crain15; @schaye15; @dubois16]. These simulations have started to disentangle the dominant physics driving galaxy formation, understanding not only mean relations and ‘typical’ galaxies [@genel18], but also outliers and unique examples [@zhu18] – providing, as a result, interpretation for a diverse array of observations.
The IllustrisTNG project [@pillepich18b; @naiman18; @nelson18a; @marinacci18; @springel18] is a recent addition to this class of simulation. The first two simulations of this effort, TNG100 and TNG300, realize volumes of $\sim 100^3$ and $\sim 300^3$ comoving Mpc$^3$, respectively. This allows them to sample galaxy populations which include hundreds of thousands of objects with $M_\star \geq 10^9$[M$_{\odot}$]{}by $z=0$. However, the unavoidable trade off in such population-level statistical power is – by construction – limited resolution.
With baryon mass resolutions of $\sim$10$^6$[M$_{\odot}$]{}(corresponding roughly to the canonical ‘1 kpc’ spatial resolution) of the current generation of cosmological volumes, we can realistically study the structure of galaxies with $M_\star \gtrsim 10^9$[M$_{\odot}$]{}. This hinders investigation of many interesting scientific questions, particularly those related to the structural properties of the stellar and gaseous components of galaxies, the internal structure of disks and their ISM, the co-evolution of black holes and galactic nuclei, feedback, and the resulting baryon cycle of inflows and outflows.
These small-scale phenomena are frequently investigated with ‘zoom’ simulations of individual galaxies [@hop14fire], with project campaigns of order 10 to 100 galaxies in total [@wang15], often oriented around a focus such as Milky Way mass halos [@grand17], Local Group analogs [@sawala16] or the challenging regime of galaxies in high-density cluster environments [@bahe17b; @tremmel18]. However, zoom simulations immediately sacrifice one of the greatest advantages of cosmological simulations – namely, the ability to make statistically robust statements about large, unbiased populations and thereby comment on galaxy formation and evolution in full generality. Together with the companion paper () we here present the new TNG50 simulation, the third and final volume of the IllustrisTNG project. TNG50 has been designed to overcome the resolution/volume limitation by simulating a large, cosmological volume at a resolution which approaches or even exceeds that of modern ‘zoom’ simulations of individual galaxies. It enables an insightful view into the structure, chemodynamical evolution, and small-scale properties of galaxies and their halos.
Simulation work on outflows
---------------------------
At small scales there are many idealized studies aimed at understanding the details of supernova-driven wind launching mechanisms [@girichidis16; @fielding17b; @kim17b] or detailed phase-interaction physics [@richings18; @schneider18a]. General relativistic (GR)MHD simulations explore black hole outflow phenomena including both winds and jets [@blandford99; @mckinney14], recently reviewed in [@yuan14].
One of the long-term goals of such efforts is to capture and parameterize the emergent behavior at larger scales in order to develop effective models which can be used in full galaxy or cosmological simulations. The sophistication of such effective models is presently limited by several factors, most critically the finite numerical resolution available. As it will be impossible to ever resolve the energy injection scales of critical astrophysical processes, sub-resolution (or sub-grid) approximations are unavoidable.
As a result, cosmological simulations have rarely if ever been compared directly to outflow observations (i.e. the baryon cycle), but rather to indirect properties such as the amount of metals left in the ISM [@torrey18], the ionized metal content of the circumgalactic medium [@nelson18b], or the stellar formation efficiency of the galaxy itself [@pillepich18b].
Notably, [@oppenheimer10] studied SN wind-driving models in early cosmological simulations, emphasizing the importance of recycling and the dependence of recycled accretion on the balance between mass loading and outflow velocity. In the context of the more realistic FIRE model, [@muratov15] presented measurements of the scaling of mass loading (and outflow velocity) which decrease (increase) strongly with $M_\star$, such that lower-mass progenitors at early times would have had much higher mass loadings than their present day descendants. Similarly, [@christensen16] evaluated outflow scalings [and metal content; @christensen18] over a similar mass range (halos $< 10^{12}$[M$_{\odot}$]{}) and zoom sample size ($\sim$10 - 20) using a delayed cooling blastwave SN feedback model, emphasizing the importance of ejective feedback (i.e. high mass loadings) towards low $M_\star$.
Non-cosmological, idealized simulations of dwarf galaxy disks enable the possibility of higher resolution and sophisticated physical modeling, and have started to explore the phase-structure and feedback dependencies of galactic outflows. [@emerick18] highlight how different treatments of the radiative feedback from young stars can modify the mass loading and temperature of outflows. [@hu18] demonstrate how different SN energy injection schemes similarly lead to different outflow properties – in particular, that a terminal momentum injection approach does not produce the same pressure-driven wind as when residual thermal energy is also accounted for [see also @smith18].
None of these works have emphasized any direct observational signatures of their outflows, i.e. forwarding modeling. Recently, [@tescari18] made a first comparison between EAGLE ‘warm gas’ and spatially resolved SAMI/IFU H$\alpha$ emission to relate gas kinematics to wind signatures. The notable exception is [@ceverino16], who generated synthetic H$\alpha$ emission line profiles of $z \sim 2$ zoom galaxies and correlated their information content with intrinsic outflow properties. Any robust comparison between outflow observations and hydrodynamical simulations will require such progress in the future, as we discuss below.
All of these investigations have focused *exclusively on SN-driven outflows*, mostly in models specifically neglecting BH feedback, which is however thought to play a dominant role in outflow production in massive galaxies. For example, the [@muratov15] measurement of $\eta_{\rm M}$ scalings in FIRE fails to capture the onset of the BH mechanism at $M_\star \gtrsim 10^{10.5}$[M$_{\odot}$]{}and therefore misses the inversion of this relation towards higher masses, as we demonstrate below. A notable exception is [@brennan18], which explores a kinetic BH feedback model in a suite of 24 zooms (halos $10^{12} - 10^{13.5}$[M$_{\odot}$]{}), showing how black holes can produce faster outflows (up to 1000 km/s) which travel farther and are thus less likely to be subsequently recycled [see also @nelson15a].
The detailed properties (as opposed to the consequences) of SN and BH-driven outflows, as well as the interplay between these two origin mechanisms, have not yet been studied with cosmological simulations, nor across a full sampling of the galaxy mass range, from $M_\star = 10^7$[M$_{\odot}$]{}dwarfs to $M_\star = 10^{11.5}$[M$_{\odot}$]{}BCGs.
The present work
----------------
We first present the TNG50 simulation, a new class of cosmological volume simulation realized at a resolution of modern ‘zoom’ simulations of individual galaxies. Leveraging this new tool, we describe first results focusing on the properties of galactic outflows in the cosmological setting and with respect to the galaxies from which they arise. In this context we are not trying to understand the launching mechanism or the relevant microphysics, but rather the way in which outflows shape the galaxy population as a whole, modulate galaxy evolution, and generate associated observational signatures. As emphasized above, the feedback models in simulations such as TNG are effective in nature, meaning that the outflow properties of the model *at injection* are an *input* rather than an output of the model. We therefore contrast, throughout this work, our results on the emergent properties of galactic outflows with the assumptions of the underlying physical models.
TNG50 incorporates unchanged a robust scheme for cosmological magnetohydrodynamics coupled to a comprehensive and well-validated model for galaxy formation physics [@weinberger17; @pillepich18a]. This motivates our study of the resulting outflows, as they reflect the net outcome of a model which produces a realistic galaxy population matching many key properties and scaling relations of observed galaxies (see in the context of TNG50 galactic structural properties). The TNG model was designed via calibration against a number of observations, mainly at $z=0$ [see @pillepich18a for a detailed discussion of our calibration/tuning], and parameters related to feedback processes at the smallest scales are arguably the most important ingredient of such a model. We focus here on intermediate and high redshift, namely $z=1$, $z=2$, and above, as particularly interesting and observationally rich epochs of galaxy assembly and galactic-scale outflows.
The paper is organized as follows: in Section \[sec\_sims\] we introduce the TNG50 simulation and the TNG model for galaxy formation, while Section \[sec\_methods\_outflows\] details the methodology and analysis details for our study of outflows. Section \[sec\_results\] presents our results: the high redshift galaxy population of TNG50 (Section \[sec\_results\_tng50\]); measurement of mass outflow rates and mass loading factors (Section \[sec\_results\_rates\]); outflow velocities and the ability of BHs to generate flows (Section \[sec\_results\_vel\]); the multi-phase character of our outflows (Section \[sec\_results\_multiphase\]); the angular dependence of winds and their hydrodynamic collimation (Section \[sec\_results\_angle\]); and correlations between outflows and the properties of their host galaxy and central black hole, in comparison to observed relations (Section \[sec\_results\_vsgal\]). Section \[sec\_discussion\] discusses future directions and comparison with observations, while Section \[sec\_conclusions\] summarizes our conclusions. Finally, Appendix \[sec:appendix\] presents halo-normalized outflow velocity measurements for comparison.
Methods {#sec_methods}
=======
The TNG Simulations {#sec_sims}
-------------------
The IllustrisTNG project[^2] [@nelson18a; @naiman18; @pillepich18b; @marinacci18; @springel18] is a series of three large cosmological volumes, simulated with gravo-magnetohydrodynamics (MHD) and incorporating a comprehensive model for galaxy formation physics [@weinberger17; @pillepich18a]. All aspects of the model, including parameter values and the simulation code, are described in these two methods papers and remain in practice *unchanged* for our production simulations, and we give here only a brief overview.
The TNG project includes three distinct simulation volumes: TNG50, TNG100, and TNG300. The larger two have previously been presented: these are **TNG100**, which includes 2$\times$1820$^3$ resolution elements in a $\sim$100 Mpc (comoving) box, and **TNG300**, which includes 2$\times$2500$^3$ resolution elements in a $\sim$300 Mpc box. Here we present the third, final, and by far most computationally demanding simulation of the IllustrisTNG project: **TNG50**, our high-resolution effort. This run includes 2$\times$2160$^3$ resolution elements in a $\sim$50 Mpc (comoving) box. In TNG50 the baryon mass resolution is $8.5 \times 10^4$[M$_{\odot}$]{}, the gravitational softening lengths are $290$ parsecs at $z$=0 for the stars and dark matter, while the gas softening is adaptive with a minimum of $74$ comoving parsecs. TNG50 has roughly fifteen times better mass resolution, and two and a half times better spatial resolution, than TNG100. This represents an unprecedented combination of resolution and volume for a cosmological hydrodynamical simulation, which approaches or exceeds that of modern ‘zoom’ simulations of individual galaxies. The main parameters of the three TNG volumes are compared in Table \[simTable\].
TNG uses the <span style="font-variant:small-caps;">Arepo</span> code [@spr10] which solves for the coupled evolution under (self-)gravity and ideal, continuum MHD [@pakmor11; @pakmor13]. Gravity employs the Tree-PM approach, and the fluid dynamics use a Godunov type finite-volume scheme where an unstructured, moving, Voronoi tessellation provides the spatial discretization. The simulations include a physical model for the most important processes relevant for the formation and evolution of galaxies. Specifically: (i) gas radiative processes, including primordial/metal-line cooling and heating from the background radiation field, (ii) star formation in the dense ISM, (iii) stellar population evolution and chemical enrichment following supernovae Ia, II, as well as AGB stars, with individual accounting for the nine elements H, He, C, N, O, Ne, Mg, Si, and Fe, plus NS-NS byproducts, (iv) supernova driven galactic-scale outflows or winds, (v) the formation, coalescence, and growth of supermassive black holes, (vi) and dual-mode BH feedback operating in a thermal ‘quasar’ state at high accretion rates and a kinetic ‘wind’ state at low accretion rates. Note that TNG does not yet include an explicit treatment of radiation nor cosmic rays, both of which could plausibly affect the phase structure of outflows.
Black holes are seeded in massive halos and then accrete nearby gas at the Eddington limited Bondi rate. Based on this accretion rate, their feedback mode is determined. When the Eddington ratio exceeds a threshold of $\chi = \rm{min} [ 0.002 (M_{\rm BH}/10^8 \rm{M}_\odot)^2, 0.1]$ thermal energy is injected continuously into the surrounding gas. The rate is $\Delta E_{\rm high} = \epsilon_{\rm f,high} \epsilon_{\rm r} \dot{M}_{\rm BH} c^2$ where $\epsilon_{\rm f,high}$ is the high-state coupling efficiency, $\epsilon_{\rm r}$ is the radiative accretion efficiency, and $\epsilon_{\rm f,high} \epsilon_{\rm r} = 0.02$. Below this threshold, kinetic energy is injected as a time-pulsed, oriented ‘wind’ with a random direction which reorients for each event. The rate is $\Delta E_{\rm low} = \epsilon_{\rm f,low} \dot{M}_{\rm BH} c^2$ with $\epsilon_{\rm f,low} \le 0.2$ (its typical value at onset), the efficiency decreasing at low environmental density [see @weinberger17].
Run Name **TNG50** TNG100 TNG300
--------------------------- --------------- ------------------------ ------------------- -------------------
Volume \[Mpc$^3$\] $\bm{51.7^3}$ $110.7^3$ $302.6^3$
$L_{\rm box}$ \[Mpc/$h$\] $\bm{35}$ 75 205
$N_{\rm GAS}$ - $\bm{2160^3}$ $1820^3$ $2500^3$
$N_{\rm DM}$ - $\bm{2160^3}$ $1820^3$ $2500^3$
$N_{\rm TR}$ - $\bm{2160^3}$ $2 \times 1820^3$ $2500^3$
$m_{\rm baryon}$ \[M$_\odot$\] $\bm{8.5 \times 10^4}$ $1.4 \times 10^6$ $1.1 \times 10^7$
$m_{\rm DM}$ \[M$_\odot$\] $\bm{4.5 \times 10^5}$ $7.5 \times 10^6$ $5.9 \times 10^7$
$\epsilon_{\rm gas,min}$ \[pc\] $\bm{74}$ 185 370
$\epsilon_{\rm DM,stars}$ \[pc\] $\bm{288}$ 740 1480
CPU Time \[Mh\] $\bm{130}$ 18 35
: Key details of the TNG50 simulation in comparison to its larger volume siblings. These are: the simulated volume and box side-length (both comoving), the number of initial gas cells, dark matter particles, and Monte Carlo tracers. The mean baryon and dark matter particle mass resolutions, in solar masses. The minimum allowed adaptive gravitational softening length for gas cells (comoving Plummer equivalent), the redshift zero softening of the collisionless components in physical parsecs, and CPU time.[]{data-label="simTable"}
Galactic-scale outflows generated by stellar feedback are modeled using a kinetic wind approach, whereby the available energy from SNII is used to stochastically eject star-forming gas cells from galaxies. The injection velocity of such wind particles is $v_{\rm w} \propto \sigma_{\rm DM}$ where $\sigma_{\rm DM}$ is the local dark matter velocity dispersion, subject to a minimum $v_{\rm w,min} = 350$ km/s. The mass loading of the winds is then $\eta = \dot{M}_{\rm w} / \dot{M}_{\rm SFR} = 2 (1 - \tau_{\rm w}) e_{\rm w} / v_{\rm w}^2$ where $\tau_{\rm w} = 0.1$ is the thermal energy fraction and $e_{\rm w}$ is a metallicity dependent modulation of the canonical $10^{51}$ erg available per SNII. Wind particles are hydrodynamically decoupled from surrounding gas until they exit the dense, star-forming environment. This occurs when the wind reaches a background density lower than 0.05 times the star-formation threshold, or after 0.025 of the current Hubble time elapses, and typically within a few kiloparsecs. The total energy available to drive winds from a gas cell therefore depends on its instantaneous star formation rate as $\dot{E}_{\rm w} = e_{\rm w} \dot{M}_{\rm SFR}$ which is roughly $\simeq 10^{41} - 10^{42}$ erg/s $\dot{M}_{\rm SFR} / (M_\odot / \rm{yr})$ depending on the local gas metallicity [see @pillepich18a].
In Figure \[fig\_sim\_comparison\] we offer a comparison between recent cosmological hydrodynamical simulations of galaxy formation, putting TNG50 in context given its combination of volume and resolution, which uniquely positions it in this space. TNG50 is by far the most computationally demanding run of the TNG suite, and one of the most ambitious cosmological simulations to date.
We adopt a cosmology consistent with the [@planck2015_xiii] results, namely $\Omega_{\Lambda,0}=0.6911$, $\Omega_{m,0}=0.3089$, $\Omega_{b,0}=0.0486$, $\sigma_8=0.8159$, $n_s=0.9667$ and $h=0.6774$.
Model update for TNG50 {#sec_sims_tng50}
----------------------
![ The TNG50 simulation occupies a unique region of parameter space for typical cosmological hydrodynamical simulations. TNG50 includes 2$\times$2160$^3$ resolution elements, giving a baryon mass resolution of with adaptive gas softening down to $74$ comoving parsecs. This approaches or exceeds that of modern ‘zoom’ simulations of individual galaxies, while maintaining the statistical power and unbiased sampling of the full $\sim$50 cMpc cosmological volume. Here we show TNG50 (dark blue) in comparison to other cosmological volumes (circles) and zoom simulation suites (diamonds) at $z \sim 0$, based on the total number of resolved galaxies (i.e. at least 100 star/gas particles) with $M_\star \geq 10^9$[M$_{\odot}$]{}. For TNG50 we also indicate $N_{\rm gal}$ above $10^7$[M$_{\odot}$]{}(rightmost blue circle), which are still resolved at this level – as in many zooms, but in contrast to other large-volume simulations. The computational difficulty of pushing towards the upper right represents the frontier for next-generation galaxy formation simulations. \[fig\_sim\_comparison\]](figures/sim_comparison_meta.pdf){width="3.3in"}
To execute the TNG50-1 (highest resolution level of the TNG50 volume) simulation a minor change has been made to the fiducial TNG model. With the inclusion of MHD, we found that extremely dense gas cells often have $P_{\rm B} \gg P_{\rm gas}$. The resulting Courant-like constraint on the hydrodynamical timestep, restricted by the magnetic signal velocity instead of the sound speed, requires physical timesteps as small as 10 *years* at the resolution of TNG50-1. Such high-density gas should naturally convert rapidly into stars, but in the fiducial configuration the numerical timestep decreases faster than the star formation probability increases (with increasing gas density), leading to an unfavorable race condition. We instead want that gas cells with extremely short star formation timescales convert into collisionless star particles in a reasonable number of numerical timesteps. We have therefore modified the base [@spr03] star formation model, wherein the dependence of the star formation timescale on density $t_\star(\rho)$ is given by (Eqn. 21)
$$t_\star(\rho) = t_0^\star \left( \frac{\rho}{\rho_{\rm th}} \right)^{-\alpha}$$
where $t_0^\star$ is a model parameter which modulates the global gas consumption time-scale, and $\rho_{\rm th}$ is a model parameter setting the threshold density for star formation. In the fiducial configuration, used for all simulations employing this model, the choice of $\alpha = 1/2$ has been made, corresponding to the canonical assumption that $t_\star$ is proportional to the local dynamical time of the gas.
Instead, for TNG50-1, we postulate a somewhat steeper dependence of the star formation rate on gas density *only* for the densest gas. We take $\alpha = 1$ for gas above the runaway threshold, which is defined as the density where the slope of the effective pressure versus density curve $n_{\rm eff} < 4/3$. At this point the effective pressure can no longer support gas against dynamical instabilities, and this occurs at , which is $\sim$ 230 $\rho_{\rm th}$. Since the star formation rate goes as $\dot{M}_\star \propto \rho^\alpha$, this implies the physical ansatz of more rapid star formation in the densest environments – the nuclear starburst regime of galactic disks, for instance.
In practice, this change was taken for numerical reasons, and we have verified with test simulations at full resolution ($1024^3$ particles in 25 Mpc/h volumes) that this change affects a negligible amount of gas, by mass or by number, and has no impact on galactic structure nor galaxy population statistics, at any redshift.
Measuring Outflow Rates and Velocities {#sec_methods_outflows}
--------------------------------------
To compute mass outflow rates two orthogonal techniques exist: (i) deriving instantaneous fluxes based on gas kinematics at one point in time [e.g. @ocvirk08], and (ii) deriving mass fluxes by tracking the Lagrangian evolution of gas mass across two distinct points in time [e.g. @nelson13]. As only the former has the possibility for direct comparisons to observables, we here focus on these instantaneous rates. We compute the radial mass flux of all the gas cells in a thin shell centered on a galaxy as
$$\dot{M} = \left.\frac{ \partial M}{ \partial t }\right\rvert_{\rm \,rad} = \frac{1}{\Delta r} \sum_{\overset{i=0}{|r_i-r_0| < \Delta r/2}}^{N} \left( \frac{ {\mbox{\boldmath$v_i$}} \cdot {\mbox{\boldmath$r_i$}} }{ |r_i| } m_{i} \right)$$
where the subscript $i$ enumerates all the gas cells with mass $m_i$ in a particular volume of space, taken as a spherical shell with some thickness $\Delta r$ and mean distance $r_0$ from a central galaxy. We calculate $\dot{M}$ for a range of $r_0$ and corresponding $\Delta r$ values, 5 kpc thick in the inner halo coarsening outwards. The cell position ${\mbox{\boldmath$r_i$}}$ is relative to the subhalo center, taken as the position of the particle/cell with the minimum gravitational potential, while the velocity ${\mbox{\boldmath$v_i$}}$ is relative to the subhalo center of mass motion, accounting for local Hubble expansion. $v_{\rm rad} > 0$ denotes outflow, and $v_{\rm rad} < 0$ is inflow.
Herein we frequently decompose the outflowing gas by its instantaneous properties, whereby $\dot{M}$ has parameter dependencies:
$$\dot{M} = \dot{M} \left( z, M_\star; r, v_{\rm rad}, \rho, T, Z \right).$$
To compute this distribution for each galaxy (with a given redshift and stellar mass) we calculate the radial mass flux of each gas cell and take the 5D histogram of this mass flux, or mass outflow rate, binning in the gas properties of interest above: radius, radial velocity, density, temperature, and metallicity. A total mass outflow rate is derived by marginalizing over the state of the gas (i.e. $\rho, T, Z$) and satisfying a threshold in outflow velocity of $v_{\rm rad} > v_{\rm thresh}$. We frequently require only $v_{\rm rad} > 0$ km/s to define outflowing material, though higher values are also considered as conservative limits and/or to mimic observational sensitivity limits.
In measuring total mass outflow rates, mass loading factors, or outflow velocities, we consider actual gas cells as well as wind-phase particles, the latter of which are generated by the TNG wind model [see @pillepich18a] and are hydrodynamically (though not gravitationally) decoupled from their surrounding gas for the duration of their existence, which is typically restricted to small distances away from galaxies. At $r \gtrsim 10$ kpc, where we generally focus our analysis, wind-phase particles are a negligible contribution and outflows are hydrodynamically resolved. Excluding wind particles has only a minor effect on all our measurements, and we include them as a relevant component of our simulations, one currently modeled at a particularly simple level (i.e. with strong sub-grid assumptions about phase structure and interaction). To explore trends of outflow velocity, we must reduce the complex distribution of outflow velocities around a galaxy down to a single number $v_{\rm out}$. We do so by taking mass outflow rate weighted percentiles of $v_{\rm rad}$. In particular, we define the quantity $v_{\rm out,N}$ as the radial velocity above which $(1-N)$% of the outflow is moving. For example, implies that ten percent of outflowing mass flux is moving 500 km/s or faster. The ‘median’ value $v_{\rm out,50}$ therefore gives the speed achieved by at least half of the total outflow mass flux. This quantity can be measured at a particular radial distance from the galaxy, $v_{\rm out,50,r=10 kpc}$ for example.
We note that these theoretical velocity percentiles may be similar, but not directly comparable to, observational measurements of outflow velocity (e.g. ‘$v_{\rm 05}$’ or ‘$v_{\rm 90}$’ in the literature), because we consider 3D $v_{\rm rad}$ instead of 1D projected $v_{\rm LOS}$, our velocities are mass flux (not optical depth or emission) weighted, and we do not yet attempt any modeling of synthetic spectral features.
We define and measure a mass loading factor with respect to the star formation rate of the galaxy in the usual way,
$$\label{eqn_eta_M_SN}
\eta_{\rm M}^{\rm SN} = \frac{ {\vphantom{\int}}\dot{M}_{\rm out} }{ {\vphantom{\int}}\dot{M}_\star } .$$
This measure is typically used in the context of supernovae-driven winds, to diagnose the relative efficiency of stellar feedback produced galactic-scale outflows. It has the same dependencies as the mass outflow rate, namely $\eta_{\rm M} = \eta_{\rm M} \left( z, M_\star; r, v_{\rm rad}, \rho, T, Z \right)$.
We typically drop the superscript ‘SN’, which emphasizes that the normalization (denominator) assumes that stellar feedback is the relevant energetic source. As BH energy injection is typically coexistent and always present, we always interpret its possible role in setting $\eta_{\rm M}$ or contributing to any measured $\dot{M}_{\rm out}$.
Galaxy Identification and General Properties
--------------------------------------------
Subhalos are identified with the <span style="font-variant:small-caps;">Subfind</span> algorithm [@spr01], and we exclusively consider central (non-satellite) galaxies with $7 < \log(M_\star / \rm{M}_\odot) < 11$ at redshifts $z \geq 1$.
Throughout this work we take 30 physical kpc aperture values for the stellar mass $M_\star$ of a galaxy. We compute star formation rates $\dot{M}_\star$ within the same aperture and with a temporal smoothing of 100 Myr, as appropriate for observationally derived SFR indicators based on the infrared or 24$\mu m$ measurements [@calzetti08], using an accounting of recently formed stars, rather than the instantaneous star formation rate of dense gas.
We compute the bolometric luminosities of accreting black holes under the assumption of a $\dot{M}$-dependent radiative efficiency [following @churazov05; @weinberger18] as
$$L_{\rm bol} \, = \,
\left\{\!\begin{aligned}
\, &\epsilon_r\, \dot{M} \, c^2 \quad &; \quad \lambda_{\rm edd} \ge 0.1\\[1ex]
\, &10\, \lambda_{\rm edd}^2\, L_{\rm Edd} \quad &; \quad \lambda_{\rm edd} < 0.1
\end{aligned}\right\} ,$$
where $\lambda_{\rm edd} = \dot{M} / \dot{M}_{\rm edd}$ is the Eddington ratio, is the Eddington luminosity, and the Eddington accretion rate, with the usual fundamental constants. $L_{\rm bol}$ is an intrinsic luminosity – obscuration is not included.
Results {#sec_results}
=======
TNG50: Resolving outflows from resolved galaxies {#sec_results_tng50}
------------------------------------------------
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The fundamental means by which feedback regulates galaxy formation and evolution is by generating non-gravitational motions in the gas. Energy injection gives rise to *outflows*, the subject of this paper, which are particularly evident in the velocity structure of the cosmic gas surrounding galaxies. To set the stage, Figure \[fig\_timeevo1\] therefore visualizes the time evolution of a strong BH-driven outflow in TNG50 originating from a massive $M_\star \simeq 10^{11.4}$[M$_{\odot}$]{}galaxy at $z \sim 2$. We show gas velocity (left column) and gas temperature (right column), with time progressing downwards, each row roughly 100 Myr apart. Energy injection from the BH produces a high-velocity, large-scale, and highly collimated (directional) outflow. The gas reaches velocities $\gtrsim$ 2500 km/s even as it crosses the halo virial radius (shown as the circle in the last row).
This halo-scale outflow results from numerous, high time cadence energy inputs within a spatial region of $\lesssim 150$ pc: in the center of the upper right panel, five distinct events are already visible as loop-like features in temperature which have begun to flow outwards. The most recent (smallest) has already reached a distance of $\sim 4$ kpc from the SMBH, which has a mass of $10^{8.7}$[M$_{\odot}$]{}and is accreting gas at a rate of , such that it is in the ‘low accretion state’ of the TNG model. It has an Eddington ratio of $\lambda_{\rm edd} \simeq 0.002$ and a bolometric luminosity of only $L_{\rm bol} \simeq 10^{42.3}$ erg/s [cf. the system of @may18]. This initial ‘burst’ of energy inputs is followed by continual, though less frequent, feedback from the BH over the next $\sim$500 Myr. The central galaxy is the eighth most massive at this time in TNG50, with a SFR of $\simeq$ 50[M$_{\odot}$]{}yr$^{-1}$, log(sSFR) $\simeq$ -9.7 yr$^{-1}$, and is in the process of quenching. We see that the outflow heats two centrally offset lobes to temperatures $\gtrsim 10^{7.5} - 10^8$ K. A succession of cocoon like shells pile up as they encounter resistance propagating through the halo gas. Although not shown here, energy dissipation due to hydrodynamical shocks traces the outer envelopes (i.e. density edges) of individual bubbles. The outflow geometry is more jet-like in its collimation than bipolar, being quite linear along the horizontal direction. The overall morphology is reminiscent of early simulations by [@suchkov94] of superwind structure, as well as radio observations of [@cecil01] who studied the coherence of the magnetic field inside similarly trailing lobes. That work suggested vortex-like dynamics at the edge of the bubble could loft material outwards, while inversion of the observed rotation measure across the boundary implied a separation of outflow and infall regions – signatures in the magnetic field topology which could be explored in TNG50.
The occasional tilt with respect to the host rotation axis (as seen in the first and last rows) could, if the outflow origin was stellar in nature, reflect asymmetries in the distribution of rapidly star-forming gas and young stars. Here it is instead due to a large momentum injection by the BH in a random direction [e.g. @schmitt02; @liska18] which then propagates along a preferred direction. We discuss this natural collimation in Section \[sec\_results\_angle\].
The structure is not bi-symmetric as is the case in idealized jet/wind models, and asymmetries of the outflow at larger, halo scales are expected in this more realistic setting [@nelson16]. In particular, the left lobe is more pronounced, sustaining higher velocities, than the right lobe in the last two time snapshots. This is plausibly due to density fluctuations in the halo gas of the host galaxy. In particular, we see how an intruding satellite at $\Delta t = 270$ Myr is shredded as it passes through pericenter, populating the inner halo to the right of the central disk with dense gas structures which likely help inhibit outflow propagation in that direction.
Figure \[fig\_timeevo2\] shows the same time sequence, this time in projected gas column density (left) and gas-phase metallicity (right). At the final $\Delta t = 370$ Myr the remnants of this merging satellite are clearly visible as a number of dense gas fragments flowing outwards – a feature which could easily be mistaken as a feedback-driven outflow. The most striking feature in the density projections is the bipolar cavity structure, which inflates from a few kpc to the scale of the virial radius over the span of a few hundred Myr. Along the edges of each cavity accumulated shells of swept up and ejected mass are clearly visible as strong density contrasts relative to the under dense, post-shock regions in their wakes. Even without further energy injection from the central engine these highly over pressurized bubbles will expand outwards as they dissipate their energy into the halo gas over time.
The evolving gas metallicity distribution over this time span directly shows how galactic-scale outflows enrich the circumgalactic and intergalactic medium with heavy elements [@tegmark93]. Although a considerable fraction of the virial volume is already enriched to $0.1 Z_\odot$ by $z \sim 2$, prior to the onset of strong BH activity, the distribution of metals is inhomogeneous and regions of the halo and its immediate environment remain at $\lesssim 0.01 Z_\odot$. Sitting already on the mass-metallicity relation, the central galaxy is, however, highly enriched, and this $\gtrsim Z_\odot$ gas is directly ejected out to scales of 100s of kpc by the BH-driven wind. This high metallicity material is preferentially cospatial with hot gas in the cavities, particularly at early times. A rich turbulent mixing structure develops, producing metallicity sub-structure within the bubbles and possibly across their boundaries. We discuss the multiphase nature and metal content of the outflows in Section \[sec\_results\_multiphase\].
Because the outflow is not isotropic, it allows predominantly metal-poor gas to remain along the major axes of the disk (i.e. in the vertical direction of this projection). Any ongoing accretion will lead to metal dilution over time. Evidence for this fueling of the gas reservoir of the central galaxy, particularly near the end of this strong feedback episode, is also apparent. Cold, overdense gas fragments begin to populate the inner halo at $r \lesssim 0.25 r_{\rm vir}$. They originate predominantly along the major axis of the galaxy, though also appear to form directly in the wake of the outflow. These gas overdensities are clearly infalling, with mixing tails extending radially outward, and often not obviously associated with any substructure in the halo at earlier times. We speculate that they may arise from an in-situ, condensation-like process in the thermally unstable hot halo gas [@sharma12; @mccourt12; @voit15]. Given the ability of TNG50 to resolve detailed structure in the lower density halo gas, a detailed study of precipitation and its ability to feed the central BH and/or regulate the feedback cycle will be an interesting topic for future work.
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Sitting at the center of this outflow, nearly invisible at this scale, is the stellar body and gaseous reservoir of the central galaxy. Figure \[fig\_mosaic\] shows one such massive galaxy at $z \sim 2$ in a face-on gas density projection (upper left), and at later times an example of the stellar structure of a $z \sim 1$ descendant of such a system, traced through a mock of its optical stellar light (upper right). In both cases the edge-on view is shown immediately below. The redshift two system hosts a large, flocculent, clumpy spiral structure, roughly $10 - 20$ kpc in radial extent. This disk is relatively thin in its vertical scale-height at the center, but warped at the outskirts, with gas features coupling the outer disk to ongoing accretion from its larger scale environment. The line-of-sight velocity dispersion of the star-forming (i.e. H$\alpha$-traced) component varies across the disk; between $1-2 r_{\rm 1/2,\star}$ the mean value is $\sim 70$ km/s, reaching as high as $170$ km/s – see . The maximum rotational velocity is $\sim 350$ km/s, giving $V/\sigma \sim 2-5$ depending on measurement methodology. The redshift one descendant hosts a large stellar bulge surrounded by a thin stellar disk composed of irregular spiral-like features. A stellar clump is visible migrating inwards to eventually coalesce with the bulge. Bombardment by minor mergers at $z < 1$ will convert this galaxy into the spheroidal BCG of a massive group by $z=0$.
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By showing the highly resolved structure of an individual galaxy we emphasize that TNG50 enables the study of the connection between small-scale (i.e. few hundred pc) feedback and large-scale (i.e. few hundred kpc) outflows. Furthermore, we clarify that winds launched by stellar feedback originate from the entirety of the star-forming gaseous disk, on a cell-by-cell basis at the scale of the simulation resolution. As the local star formation properties vary substantially within a galaxy, the resulting SN-driven outflows have non-constant, and non-trivial, distributions of outflow velocity, mass outflow rate, and even temperature (all at launch). That is, galaxies of a given mass or circular velocity do not produce outflows at a single constant velocity or mass loading, as is sometimes mistakenly inferred for large-volume feedback models like TNG. Simultaneously in time and possibly coincident in space, the central black hole will also be injecting either thermal or kinetic energy into its immediate surroundings – contributing to, or possibly dominating the production of an outflow.
Figure \[fig\_mosaic\] also highlights the statistical power of the galaxy population realized in TNG50 already by $z=2$ (lower half), visualizing the most massive $\sim$ 750 galaxies present in the volume, in the same face-on gas density (left) and face-on stellar light mocks (right) as above. They are sorted from the most massive, in the upper left corner, to the least massive, in the lower right, ranging from . With few exceptions, galaxies at this mass and redshift are gas rich, with rotationally supported gas reservoirs which are typically much more extended than the stellar component. The diversity present even within this sample highlights the importance of the large cosmological volume. The of TNG50 provides a sampling of rare, high-mass halos, together with the statistics needed to make robust statements about a representative galaxy population.
Outflow rates and mass loading factors vs. M\* and z {#sec_results_rates}
----------------------------------------------------
In Figure \[fig\_massloading1\] we begin to explore the quantitative properties of TNG50 outflows. Here we measure the outflow mass loading [$\eta_{\rm M}$]{}, the ratio of the mass outflow rate and the galaxy star formation rate, as a function of galaxy stellar mass at $z=2$ (top panel). Considering a fixed distance of 10 kpc from the galaxy, we evaluate the mass outflow rates subject to three different minimum [$v_{\rm rad}$]{}values to be classified as an outflow (blue, orange, and green).
Considering all $v_{\rm rad} > 0$ km/s material in the mass outflow rate (blue), [$\eta_{\rm M}$]{}drops rapidly with increasing stellar mass, from $\eta_{\rm M} \simeq 50$ at $M_\star = 10^{7.5}$[M$_{\odot}$]{}to $\eta_{\rm M} \simeq 3-5$ at $M_\star = 10^{10.5}$[M$_{\odot}$]{}. Enforcing a more stringent $v_{\rm rad} > 150$ km/s cut as measurably outflowing material (orange) weakens the dependence and flattens the relation with stellar mass considerably – [$\eta_{\rm M}$]{}decreases by only a factor of two over the same mass range. For even higher velocity thresholds (green), the slope of $\eta_{\rm M} \propto M_\star$ changes sign, because low mass galaxies drive very little mass at such high velocities.
The black line shows the value of $\eta_{\rm M}^{\,i}$ at the injection scale, as prescribed by the TNG model. We derive a single mean value for each galaxy from its ensemble of star-forming gas cells, then present an approximate fit through these mean values as a function of $M_\star$, as well as its scatter (gray band). The model $\eta_{\rm M}^{\,i}$ is nearly flat below $M_\star \lesssim 10^{8.5}$[M$_{\odot}$]{}and then decreases monotonically with increasing mass, dropping to unity by $10^{10.5}$[M$_{\odot}$]{}and $< 1$ thereafter. However, this behavior is fundamentally different than what is seen in the resolved flows which result at 10 kpc scales. Here the trend of [$\eta_{\rm M}$]{}reverses, reaching a minimum at $M_\star \simeq 10^{10.5}$[M$_{\odot}$]{}and before rising rapidly again towards higher masses. This corresponds to the mass scale for the onset of quenching as galaxies begin to transition out of the blue, star-forming population in the TNG model [@nelson18a; @weinberger18]. Star formation rates are suppressed by the action of efficient BH feedback which simultaneously produces a large mass of outflowing gas through direct ejection of the nuclear (i.e. central) ISM, driving $\eta_{\rm M} = \dot{M}_{\rm out} / \dot{M}_\star$ up. As we move beyond $M_\star \gtrsim 10^{11}$[M$_{\odot}$]{}, mass loading factors approach similar values ($> 10$) as in the lowest mass dwarfs. As a result, $\eta_{\rm M}(M_\star)$ has a broken ‘v’ shape: the low-mass behavior regulated by stellar feedback, while the high-mass behavior is set by the energetics and coupling efficiency of BH feedback.
Even where [$\eta_{\rm M}$]{}is established largely by the SN energy budget alone, its value and scaling with $M_\star$ differs from the TNG model input at the injection scale. In the lower left panel we clarify this difference, by showing the dependence on threshold [$v_{\rm rad}$]{}(different line colors) as well as at different distances away from the galaxy (different line styles). The wind mass loading decreases strongly as a function of distance from 10 kpc outwards at $M_\star < 10^{10.5}$[M$_{\odot}$]{}, dropping for low-mass galaxies in the $v_{\rm rad} > 50$ km/s case from $\eta_{\rm M} \simeq 20-50$ at 10 kpc to only $\eta_{\rm M} \simeq 1$ at 50 kpc. Depending on velocity cut and radius, the actual mass outflow rate can exceed that prescribed at injection by the TNG model, presumably due to local entrainment as winds begin to interact in the outskirts of the disk. Declining $\eta_{\rm M}(r)$ is generically expected for a flow which is at least partially fountain, i.e. not entirely escaping [e.g. @sarkar14]. In Appendix \[sec:appendix\] we discuss mass loading trends scaled by the virial velocity of the parent halo.
The radial dependence of [$\eta_{\rm M}$]{}in TNG weakens towards high stellar mass, pointing towards a different launch mechanism. At we see that even for $v_{\rm rad} > 150$ km/s the wind mass loading exceeds that produced by the SF-driven wind model alone. We speculate that the mutual, positive coupling between the two feedback mechanisms is roughly maximal at this mass scale - where black holes are able to contribute energy to drive outflows, but not yet severely impacting galactic star formation and so total available SN energy. As the majority of BHs transition into the low accretion state at $M_\star \gtrsim 10^{10.5}$[M$_{\odot}$]{}the radial dependence is entirely eliminated, even for fast $v_{\rm rad} > 350$ km/s outflows. As we discuss later, the vast majority of gas driven by kinetic BH feedback exceeds this velocity to distances greater than $r > 50$ kpc, leaving [$\eta_{\rm M}$]{}invariant to distance in this regime.
In the lower right panel of Figure \[fig\_massloading1\] we explore the redshift dependence of [$\eta_{\rm M}$]{}in different bins of stellar mass, from $\log(M_\star/\rm{M}_\odot) = 10^8 - 10^{11}$. For low mass galaxies with $M_\star \lesssim 10^{10}$M$_\odot$ the wind mass loading is roughly constant as a function of redshift. This is because, although the actual outflow rates around such galaxies decrease with redshift at fixed mass, by roughly one order of magnitude from $z=6$ to $z=0.2$, their star formation rates do likewise. On the other hand, galaxies at the high-mass end exhibit a rapidly increasing [$\eta_{\rm M}$]{}towards late times. For example, galaxies with $M_\star \sim 10^{10.5}$M$_\odot$ have $\eta_{\rm M} \sim 1$ at $z=6$, increasing to $\eta_{\rm M} \sim 100$ by $z \simeq 0$. In this case mass outflow rates are roughly constant with redshift, while star formation rates begin to decline after some characteristic redshift as a result of the quenching process. At redshift two we see the inversion of [$\eta_{\rm M}$]{}noted above as the crossover of the high-mass curves; at $z \gtrsim 4$ it is clear that [$\eta_{\rm M}$]{}is monotonic with $M_\star$, unlike at low redshift.
Outflow velocities: SN versus BH-driven winds {#sec_results_vel}
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Moving to a second fundamental property beyond mass outflow rates, Figure \[fig\_vout\] measures outflow velocities as a function of stellar mass. As before, we select outflowing material at some distance away from the galaxy. We also condense the full spectrum of the outflowing velocity distribution around a galaxy into a single number - mass outflow rate weighted percentiles $v_{\rm out,50}$ to $v_{\rm out,95}$ (as described in Section \[sec\_methods\_outflows\]).[^3] In the top panel we show the 75th and 95th percentiles of $v_{\rm out}$ at a distance of 10 kpc at $z=2$. Both increase monotonically with $M_\star$ from $100-200$ km/s at stellar masses of $10^{7.5}$[M$_{\odot}$]{}up to $\sim$ 400 km/s ($v_{\rm 75}$) or $\gtrsim$ 800 km/s ($v_{\rm 95}$) for . They are typically below the escape velocity $v_{\rm esc} = [-2 \phi(r = \rm{10 kpc})]^{1/2}$ at this distance. Given that the global potential also traces the large scale environment, particularly for smaller galaxies [@oppenheimer08], we also include a second line for $v_{\rm esc}$ (dashed), computed using the potential difference between 10 kpc and $r_{\rm vir}$. This roughly captures, therefore, the energy required to move between these two locations, against the force of gravity. In this case the conclusion would be that $v_{\rm out}$ can commonly exceed the escape velocity, although we caution that in neither case does this actually mean that an outflow can necessarily escape the halo, as the comparison neglects hydrodynamical interactions, drag, and other similar effects. In Appendix \[sec:appendix\] we discuss velocity trends scaled by the virial velocity of the parent halo.
As before, we include the prescription from the TNG model for the injection (launch) velocity $v_{\rm out}^{\, i}$ of winds generated by stellar feedback as the gray line and shaded band, which represent a fit to the mean value derived for each galaxy. We see that $v_{\rm out}^{\, i}$ represents more an upper envelope to the resulting flow at 10 kpc, rather than its characteristic velocity. We note however that $v_{\rm out,99}$ does exceed the injection velocity even at this distance. The minimum wind velocity of 350 km/s enforced by the TNG wind model leads to the constant $v_{\rm out}^{\, i}$ below $M_\star \lesssim 10^9$[M$_{\odot}$]{}. Interestingly, we see that this does not mean that material flowing away from a galaxy at any resolved scale actually inherits such a velocity (nor even its trend with stellar mass). The $v_{\rm out,75}$ velocity percentile is actually below this ‘minimum’ for all $M_\star \lesssim 10^{10}$[M$_{\odot}$]{}. We interpret this as an indication of halo drag, either gravitational or hydrodynamical [@silich01; @dallavecchia08], an important demonstration of how emergent outflow properties can differ from model inputs due to resolved physical processes.
In the lower left panel of Figure \[fig\_vout\] we explore the dependence of $v_{\rm out}$ on distance (colors) and velocity percentile (linestyles). Outflow velocity notably decreases with distance for all $r > 10$ kpc. At the same time, our velocity percentile ‘summary statistic’ will be influenced by all gas with $v_{\rm rad} > 0$ km/s, i.e. it could also be pulled down by a progressively larger fraction of quasi-equilibrium, slowly moving halo gas relative to an actual, faster outflow component. Outflow velocity scales strongly with the velocity percentile, particularly at the massive end. The distributions of $v_{\rm out}$ clearly have extended tails which are not captured by $v_{\rm 50}$ for instance. At $M_\star = 10^{11}$[M$_{\odot}$]{}, the 95th tail reaches $\sim$ 1000 km/s while $v_{\rm out,50}$ is only $\sim 200$ km/s. By measuring the maximal velocities achieved by gas shortly after being kicked by the low-state BH feedback, we determine a corresponding ‘injection velocity’ of $\gtrsim$ 12,000 km/s, applicable to the high-mass end, which shows how rapidly such nuclear-originating flows must decelerate.
The lower right panel of Figure \[fig\_vout\] shows the redshift dependence of outflow speed at fixed stellar mass, where we focus on $v_{\rm out,90}$ and $r = 10$ kpc for simplicity. Winds are faster at higher redshift for a given $M_\star$, both in the SF-driven and BH-driven regimes. In the former, a plausible cause is that, at fixed stellar mass, the DM velocity dispersion will be higher at earlier times. We note an important new aspect of the TNG wind model is the explicit scaling $v_{\rm out}^{\, i} \propto [H_0 / H(z)]^{1/3}$. This dependence produces a co-evolution of wind velocity and virial halo mass, i.e. redshift-independent velocity at a fixed host halo mass [@pillepich18a]. After convolution with the evolving stellar mass to halo mass relation, we see that in practice the model injection velocities are only higher at a given $M_\star$ for very high redshift ($z \sim 6$), but settle down to a roughly constant set of the overlapping black curves for all $z < 4$. In practice, this scaling is not relevant for $M_\star \lesssim 10^9$[M$_{\odot}$]{}due to the imposition of the minimum wind injection velocity of 350 km/s. However, we see that the resolved wind velocities evolve with redshift even at these low masses. In this regime $v_{\rm out,90} \propto (1+z)^{0.2 - 0.4}$ or $v_{\rm out} \propto H(z)^{0.2 - 0.3}$ depending on $M_\star$. We note that the measured scaling with $H(z)$ has the opposite sign as the model injection scaling, emphasizing that the mapping between model inputs and resolved outputs can be non-trivial. At the high mass end, the redshift scaling is much stronger, $v_{\rm out,90} \propto (1+z)^{0.8 - 1.3}$ for $M_\star \simeq 10^{11}$[M$_{\odot}$]{}, indicating that the velocities of BH-driven outflows evolve differently with redshift.
![ Two dimensional phase space diagram showing high-velocity outflows ($v_{\rm rad} > 0$) in the conditional gas mass distribution on the ($v_{\rm rad}$,$r$) plane for one illustrative halo being strongly affected by BH feedback The bulk of gas mass is confined to roughly $\pm v_{\rm circ}$ of the halo, with accelerating infall visible as features shifting towards more negative velocities in the halo center. At small distances, distinct branches extend to high velocity, corresponding to individual outflow episodes. These arcs flow outward (in radius) and downward (in velocity) with time, although their upward shape implies an instantaneously increasing radial velocity with distance. \[fig\_phase2d\]](figures/phase2d_TNG50-1_59_gas_x-rad_kpc_y-vrad_wt-mass_h-22.pdf){width="3.4in"}
Having presented the population-level view of outflow velocity, we go on to explore the detailed velocity structure of outflows. Figure \[fig\_phase2d\] shows the phase space diagram of gas mass in the plane of around a single halo with total mass $\simeq 10^{12.8}$[M$_{\odot}$]{}, from just outside the direct region of influence of the black hole (1 kpc) to about a third of the host virial radius (100 kpc). The bulk of mass at almost all radii is sitting just below $v_{\rm rad} \sim 0$ indicating consistent inflow through the inner halo. Outflowing winds are evident as a succession of discrete features at positive radial velocities. These outflows launched by black hole feedback form a sawtooth-like pattern in $(v_{\rm rad},r)$, consisting of a series of triangular components – each results from a single energy injection event at a small scale off the left-edge of this panel. There are three such flows evident at $r < 20$ kpc, and $\sim$ 5 visible in the inner halo.
The maximum radial velocity of each outflow component *increases* with distance, i.e $v_{\rm out} \propto r^{\,\alpha} \,(\alpha > 0)$. This shape is a natural consequence if material has a spectrum of launch velocities and the most rapidly outflowing gas reaches, at any given time, the largest distances. The maximal outflow velocity at each radius is established by the upper envelope of the successive peaks, corresponding to the highest velocity tails of each flow. Over time, each individual feature collapses outward, shifting to larger distance and lower velocity. As a result, the envelope of outflowing radial velocities actually *decreases* with distance, i.e $v_{\rm out,max} \propto r^{\,\alpha} \,(\alpha < 0)$. This velocity structure, coming from the superposition of the individual outflow components, is continually reinforced by new injection events at $r \sim 0$ and generically maintains a quasi-steady structure over time. The high velocity tails of the outflowing material lie above the escape velocity curve $v_{\rm esc}(r) = [-2 \phi(r)]^{1/2}$ at all radii (black line).
![ The average distributions of outflow velocity around galaxies in different stellar mass bins at $z=2$. Specifically, we decompose the radial mass outflow rate as a function of radial velocities [$v_{\rm rad}$]{}, focusing on the flow across $r = 20$ kpc, in units of [M$_{\odot}$yr$^{-1}$]{}per 10 km/s bin of [$v_{\rm rad}$]{}. At low mass, the outflow velocity distributions have a similar, centrally symmetric shape originating from SN-driven winds. They peak at velocities $\lesssim$ a few hundred km/s, with a maximal velocity cutoff which scales with $M_\star$. At high masses, these slower outflows are augmented by a long tail to high velocities due to the emergence of BH-driven outflows, which easily reach $\gtrsim$ 3000 km/s. Small solid, dashed, and dotted vertical lines denote $v_{\rm 50}$, $v_{\rm 90}$, and $v_{\rm 95}$ for each mass bin, respectively. \[fig\_vrad\_dist\]](figures/outflowRate_Gas_vrad_mstar_TNG50-1_33_mean_skipzeros-False.pdf){width="3.4in"}
Figure \[fig\_vrad\_dist\] presents the distribution functions of outflow velocity, decomposing the mass outflow rate $\dot{M}_{\rm out}$ as a function of $v_{\rm rad}$. We stack galaxies in bins of stellar mass, from $8.0 < \log(M_\star/\rm{M}_\odot) < 11.5$ (different line colors), and focus on a distance of $r = 20$ kpc at $z=2$ for simplicity. At lower stellar masses the distributions have a similar shape, being peaked at low velocities hundred km/s, which increases slowly with $M_\star$. The amount of outflowing gas with higher velocity drops rapidly beyond this central core, such that there is an effective maximum outflow velocity of $\sim$ 300 km/s for $M_\star = 10^8$[M$_{\odot}$]{}, $\sim$ 400 kms for $M_\star = 10^9$[M$_{\odot}$]{}, and $\sim$ 500 km/s for $M_\star = 10^{10}$[M$_{\odot}$]{}. This is the regime of the stellar feedback driven wind model, with the approximate scaling of $v_{\rm out} \propto M_{\rm halo}^{1/3}$. As a result, typical wind velocities scale up with mass, but have bounded maximal values.
However, at stellar masses $M_\star \gtrsim 10^{10.5}$[M$_{\odot}$]{}a high-velocity tail begins to emerge. By $10^{11}$[M$_{\odot}$]{}this feature is fully developed and results in a roughly linear extension out to high $v_{\rm rad}$ (in this log-linear plane). As a result, we can start to find outflowing mass at $>$ 1000 km/s velocities, where the distributions are then composed of two distinct components: the SF-driven core at low velocity, and the tail at high velocity. In the TNG model, only the kinetic BH feedback mechanism can generate such high velocities, and this mode becomes efficient at $M_\star \gtrsim 10^{10.5}$[M$_{\odot}$]{}as black holes transition below the threshold accretion rate. The mass outflow rate decreases more slowly with $v_{\rm rad}$ than in the SF-driven wind case, such that there is no strong cutoff at some maximal velocity. However, the decline is still precipitous: for the highest mass bin considered here at $M_\star = 10^{11.5}$[M$_{\odot}$]{}, for $v_{\rm out} \sim 1000 \pm 50$ km/s, the total flux is on average $\dot{M}_{\rm out} \simeq 10 \,\rm{M}_\odot \,\rm{yr}^{-1}$. However, for gas moving with $v_{\rm out} \sim 2000 \pm 50$ we have only $\dot{M}_{\rm out} \simeq 1 \,\rm{M}_\odot \,\rm{yr}^{-1}$, and the flux of gas moving at $\sim$ 3000 km/s is a further factor of ten less.
As we discuss further in Section \[sec\_results\_vsgal\], it is important to note that these highest velocity BH-driven outflows in the TNG model do not arise from ultra-luminous or ‘radio-loud’ quasars (e.g. with space densities $n \sim 10^{-7}$ Mpc$^{-3}$ or less, and/or $L_{\rm bol} \sim 10^{45}$ erg/s or greater), which are much too rare to arise in the volume of TNG50. Instead, they are generated from the much more common low luminosity, low-accretion rate ($\lambda_{\rm edd} \sim 10^{-2}$ or less) population of BHs for which we posit a radiatively inefficient flow which converts gravitational binding energy into a non-relativistic wind [@blandford99; @yuan14]. The implication is that – in the TNG model – low luminosity, slowly accreting black holes can drive some of the most powerful outflows. Other models produce strong BH-driven outflows by invoking different physical mechanisms and/or numerical implementations [e.g. @mccarthy11; @henden18], and differentiating between these scenarios will provide useful theoretical constraints.
Multiphase outflows: temperature, density, metallicity {#sec_results_multiphase}
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![ Gas inflow/outflow rates decomposed as a function of gas temperature and outflow velocity at $z=2$. Here we focus on a regime where black holes dominant outflow properties: massive galaxies with $M_\star \simeq 10^{11}$[M$_{\odot}$]{}at a distance of $r = 30$ kpc. We find that different phases attain different outflow speeds (or, equivalently, that different outflow velocity components occupy different regions of $\rho-T$ space). In this case, it is only hot gas at temperatures $\gtrsim 10^6$ K which is able to reach $>$1000 km/s, while the cooler component is confined to around $\sim$500 km/s or less. \[fig\_outflowrate\_temp\_vrad\]](figures/outflowRate2D_Gas_temp-vrad_mstar_TNG50-1_33_mean_skipzeros-False.pdf){width="3.3in"}
We have so far considered only the kinematics of the outflow as a whole, i.e. phase-agnostic. Observationally there exists a broad correlation between outflow velocity and gas phase temperature, such that neutral phases are slower than ionized components [@heckman01; @rupke02], which are themselves slower than a hot wind fluid phase [@veilleux05].
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In Figure \[fig\_outflowrate\_temp\_vrad\] we show that different outflow phases in TNG have different kinematics. We present the galaxy-stacked 2D histogram of mass outflow rate, as a function of temperature (x-axis) and instantaneous velocity (y-axis). We focus on a mass scale ($M_\star = 10^{11}$[M$_{\odot}$]{}) where the action of the central black hole plays an important role in setting the properties of the galactic-scale outflows. For winds which reach an appreciable distance away from the galaxy we find a relative dichotomy of hot/cooler gas phases and rapid/slower outflow velocities. In particular, the only outflowing gas which achieves velocities of $>$ 1000 km/s is hot, with . Within this phase there is a minor correlation between $v_{\rm out}$ and temperature: the hottest gas travels the fastest, with outflows having temperatures of order $10^{6.5}$ K.
On the other hand, cooler gas is slower, and so unlikely to propagate as far. The outflow component which would be visible in ionized H$\alpha$ or metal transition lines, between $10^4$ K and $10^5$ K, has typical velocities of a few hundred km/s and a maximal $v_{\rm out}$ of $\sim$ 600 km/s. The phase separation at high velocity implies that the temperature of a high-velocity component can be clearly deduced. The bulk of the outflowing gas, however, coexists at a common, lower velocity interval, preventing one from easily identifying the phase structure of this dominant, low-velocity component. We note that for lower stellar masses $M_\star \sim 10^{9-10}$[M$_{\odot}$]{}(not shown), where stellar rather than BH-driven winds are most relevant, this correlation of hotter outflow phases moving faster likewise holds.
Figure \[fig\_outflowrate\_phase\] quantifies the multiphase nature of TNG outflows in terms of temperature, density, and metallicity. In the main panel (top) we decompose the mass outflow rate as a function of gas temperature, where each line shows the mean stack within a given stellar mass bin, from $8.0 < \log(M_\star/\rm{M}_\odot) < 11.5$ (different line colors). Two important trends emerge.
First, there exists a warm/hot outflowing component with a well-behaved, roughly gaussian distribution of mass flux weighted log temperature. The peak of this component shifts smoothly upwards with increasing galaxy stellar mass, from $\sim 10^{5.5}$ K at to $\sim 10^6$ K by $10^{9.5}$[M$_{\odot}$]{}, and all the way up to K at $10^{11}$[M$_{\odot}$]{}. The width (in log K) is roughly constant, implying that this constituent of the outflow is set by the temperature structure of the virialized hot halo gas and tracks the increasing host halo virial temperature. Note that bulk (i.e. non-circular) motions in the inner halo gas which exceed the chosen velocity threshold would contribute here, so the hot outflow component could partially reflect increasing amounts of such non-equilibrium kinematics of virialized gas. The peak outflow rate of this component increases by roughly a factor of ten with each decade in stellar mass – i.e., it is a roughly constant ‘specific gas outflow rate’. In the integral it dominates, at all mass scales, the total outflowing mass flux.
Next, a second cool outflow component emerges towards higher stellar masses, resulting in a bimodal outflow temperature distribution. This bimodality is clear at all $M_\star > 10^{10}$[M$_{\odot}$]{}and becomes stronger towards higher mass, where the peak outflow rates at $T \sim 10^4$ K and $T \sim 10^7$ K come into near equality.[^4] The origin is twofold - the hint of this secondary peak already developing even for galaxies as small as $M_\star = 10^9$[M$_{\odot}$]{}, together with the invariance of the peak temperature itself in relation to the shape of our adopted cooling curve [@wiersma09] suggests a cooling process [possibly in the outflow itself, e.g. @thompson16; @schneider18b]. In the future we can use the Monte Carlo tracer particle scheme in TNG [@genel13] to directly track the thermal history of outflowing gas and (dis)prove this point. On the other hand, the strong increase of outflowing mass at this temperature above $M_\star > 10^{10.5}$[M$_{\odot}$]{}indicates a relation to the onset of efficient black hole feedback. We observe that the kinetic BH mechanism in this regime directly evacuates the central ISM reservoir, leading to centrally suppressed gas densities (and star formation rates, as we discuss later). At stellar masses $\gtrsim 10^{11}$[M$_{\odot}$]{}this dichotomy in outflow temperature is therefore a direct consequence of ejective feedback launching formerly dense ISM out into the halo [e.g. @scannapieco15].
Observationally, [@chisholm17] recently measured the mass loading of OVI absorbing gas in a single lensed $z \sim 3$ galaxy, concluding that the fastest outflowing material (with $v_{\rm out} > v_{\rm esc}$) has a mass outflow rate larger than in the cooler, photoionized phase. Similarly, from a sample of local starbursts [@grimes09] find that OVI-traced gas at $\sim 10^{5.5}$K has a higher mean outflow velocity than seen in lower ionization states. If the expectation presented here is roughly correct, then the most important outflow phase for $M_\star \gtrsim 10^{10}$[M$_{\odot}$]{}moves quickly away from available ionized tracers and even past OVI bearing ‘coronal’ gas into a much hotter x-ray traced plasma, where observational constraints are presently challenging.
In the lower left panel of Figure \[fig\_outflowrate\_phase\] we similarly decompose the mass outflow rate now as a function of gas density, in the same stacked distributions as a function of stellar mass. A two component structure is also evident: the low density peak is associated to the high temperature gas, with the high density peak corresponding to the low temperature component which prominently emerges only at $M_\star > 10^{10}$[M$_{\odot}$]{}. As before, the location of the lower density component tracks the increasing halo gas density with higher host mass. On the other hand, the high density ISM component has a roughly constant maximal outflow rate for densities of , although the most massive galaxies above $M_\star > 10^{11}$[M$_{\odot}$]{}can launch outflows to 20 kpc which contain densities of $n > 10$ cm$^{-3}$ and higher. This also indicates a direct ejective origin for this dense material, having originally compressed in the center of the galactic potential. Given our star formation threshold density, some of this ejecta will therefore be star-forming and on the effective equation of state.
The lower right panel of Figure \[fig\_outflowrate\_phase\] measures the metallicity of outflowing gas, in three stellar mass bins (colored lines) at $z=1$, $z=2$, and $z=4$ (solid, dashed, dotted). At all redshifts, the outflow metallicity increases with the stellar mass of the galaxy as expected. At redshift two, the dominant metallicity of wind material is $0.1 Z_\odot$ for $M_\star = 10^8$[M$_{\odot}$]{}, increasing to $0.3 Z_\odot$ at $M_\star = 10^{9.5}$[M$_{\odot}$]{}and reaching $0.7 Z_\odot$ by $M_\star = 10^{10.5}$[M$_{\odot}$]{}. The distributions are broad and gas at $\pm 0.5$ dex of these values is always also present at roughly 10% of the mass outflow rate when compared to the dominant metallicity. At low masses the redshift evolution is roughly independent of stellar mass and fairly weak – the average $Z_{\rm out}$ increases by $\sim$ 0.2 dex from $z=4$ to $z=2$ and by a further $\sim$ 0.1 dex to $z=1$. At high masses the redshift evolution is even shallower due to the flattening of the gas-phase mass metallicity relation. In both cases the evolution of outflow metallicity is approximately similar to the evolving MZR of TNG [@torrey18], although we defer to future work a detailed comparison of wind metallicity relative to the galaxy ISM metallicity itself [e.g. @muratov17].
Angular dependence: emergent bipolarity {#sec_results_angle}
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![ The dependence of mass outflow rate on galactocentric angle at $z=1$, where directions corresponding to the minor axes are $\theta = \pm \pi/2$, marked with horizontal gray lines. Each panel stacks all galaxies in mass bins centered on $M_\star \simeq 10^{10}$[M$_{\odot}$]{}(top; $\sim$ 280 systems) or $M_\star \simeq 10^{11}$[M$_{\odot}$]{}(bottom; $\sim$ 70 systems). These two mass scales are dominated by stellar and black hole driven outflows, respectively. White lines quantify the opening angle bounding half the total mass outflow at each radius. Despite the isotropy of both feedback models at the energy *injection* scale, a hydrodynamic collimation naturally occurs and results in wide opening angle, bipolar outflows which preferentially escape along the minor axes of galaxies. \[fig\_outflowrate\_rad\_theta\]](figures/outflowRate2D_Gas_rad-theta_mstar_TNG50-1_50_mean_skipzeros-False.pdf){width="3.3in"}
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In Figure \[fig\_outflowrate\_rad\_theta\] we show the dependency of the mass outflow rate (as the background color) on both galactocentric angle and galactocentric distance at $z=1$. Within each panel the two horizontal lines indicate the direction aligned with the minor axis of the galaxy.[^5] The top panel stacks relatively low-mass galaxies with $M_\star \simeq 10^{10}$[M$_{\odot}$]{}($\sim$ 280 systems), a mass regime where outflows are generated by stellar feedback. The bottom panel stacks high-mass galaxies with $M_\star \simeq 10^{11}$[M$_{\odot}$]{}($\sim$ 70 systems), where higher-velocity outflows result almost exclusively from BH feedback.
In both cases the conclusion is the same: the mass outflow rate of winds is not directionally isotropic. Rather, strong $\dot{M}$ outflows are preferentially found aligned with the minor axes of galaxies. This is particularly intriguing in the context of TNG, as it represents an emergent quality of galactic-scale outflows. In the TNG model, wind launching and energy injection from stellar as well as black hole feedback is entirely isotropic, in the time-average, multiple event sense. For stellar feedback driven winds, the directionality of the wind-phase mass is entirely random at launch [see @pillepich18a], while BH energy injection in the low accretion state kinetic mode, which drives the outflows seen here at high-mass, is also entirely random with each injection event [see @weinberger17]. The preferential propagation of outflows along the minor axis of a galaxy therefore represents a natural, hydrodynamical collimation of the flow. White lines demarcate the ‘half mass outflow rate angle’, quantifying a measure of the opening angle; this is $\theta_{\rm 1/2} \simeq 70^{\circ}$ at $r \sim 10$ kpc and $\theta_{\rm 1/2} \simeq 40^{\circ}-50^{\circ}$ at $r \sim 50$ kpc.
Figure \[fig\_outflowrate\_rad\_theta\] quantifies the visual impression provided by Figures \[fig\_timeevo1\] and \[fig\_timeevo2\] for the case of a strong AGN driven outflow. To complement this high mass, high outflow velocity regime, we visualize the velocity structure of a smaller galaxy with $M_\star = 10^{10.3}$[M$_{\odot}$]{}($M_{\rm halo} = 10^{11.8}$[M$_{\odot}$]{}) in Figure \[fig\_vis\_wind\]. Such a system observed at a higher redshift would be similar to a Lyman-break galaxy. Here we show a gas density projection of the galactic and halo gas, alloyed with its instantaneous velocity field using the line-integral convolution technique. Small-scale transport and swirling, vortical motions are revealed in regions without strong bulk outflow or inflow. Overlaid on top, streamlines indicate flow direction and velocity, with outflows reaching up to $\sim$ 350 km/s as they establish a circulatory, galactic fountain like baryon cycle across $r_{\rm vir}/2$ (dotted circle).[^6]
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We speculate that this collimation is largely a ‘path of least resistance’ effect, whereby outflows directed into the plane of a disk, comprised of the densest material, experience enhanced resistance. Consequently, we find that a roughly bipolar or biconical outflow is typical behavior. Sculpting of the flow is evident already by $\sim 10$ kpc in the inner halo, and persists out as far as strong outflows are found ($\gtrsim 50$ kpc at least). The inferred opening angle is large, decreasing for stronger outflow thresholds which become more concentrated along the minor axes. We note that the degree of this collimation appears to evolve with redshift (not shown), becoming stronger at late times and emerging particularly from $z=2$ to $z=1$. It is difficult to identify such a clear signal even at $z=2$. The degree of collimation is likely to depend on galaxy morphology, particularly the amount of rotational support of the system.
Observationally, preferential detection of outflow signatures along the minor axes of galaxies is a common conclusion at low-z [@heckman00] based on SDSS [@chen10; @concas17b; @bae18] and also towards $z \sim 1$ [@martin12; @kornei12; @bordoloi14c; @rubin14]. At the same time, data from even higher redshifts $z \geq 2$ support a lack of collimation at such early times [@law12]. Typical evidence along these lines comes from cold gas-phase tracers such as MgII or NaD, or hydrogen, and the association is typically with stellar feedback driven outflows, although asymmetric outflows from low-luminosity black holes may also be common [@cheung16; @wylezalek17; @roy18]. For hotter phases, [@kacprzak15] find OVI around $z \leq 0.7$ galaxies preferentially along either the major or minor axes (e.g. inflows and outflows, respectively), but not in between; similarly so for colder MgII absorbers at $z \geq 0.3$ [@nielsen15].
In addition to spanning both feedback regimes, we also find no clear temperature dependence of this signal. We therefore expect (i) hotter outflowing gas phases to be similarly collimated, (ii) bipolar geometry outflows to be present across a broad stellar mass range, and (iii) a lack of similar signatures at high redshift ($z>2$; not shown). We speculate that this collimation arises only towards lower redshift due to the rise of ordered rotation and settling of gaseous galactic disks which helps to sculpt the flows.
Theoretically such a natural collimation was anticipated already by [@tomisaka88] whose numerical simulations demonstrated galactic-scale superwind emergence preferentially elongated along the direction of the minor axis of a launching disk, caused by propagation in the direction of the maximum pressure gradient of the confining gaseous halo. Early wind models for global simulations sometimes including such hard-coded directionality by hand [@spr03], including in the original Illustris model [@vog13], but this was removed in TNG mainly as an opportunity to simplify an unnecessarily complex model feature [@pillepich18a].
Relating outflow and host galaxy properties {#sec_results_vsgal}
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With the exception of global stellar mass trends, we have not yet explored the dependence of outflows on additional properties of the galaxies from which they are launched. Here we specifically consider star formation activity (SFR) as well as black hole activity ($L_{\rm bol}$, $\lambda_{\rm edd}$). As both types of feedback activity are a strong function of galaxy mass, we naively expect a strong scaling of most outflow properties such as $v_{\rm out}$ with $M_\star$, and move beyond such zeroth order expectations by predominantly investigating trends at fixed mass.
In Figure \[fig\_vout\_delta\_sfms\] we consider the dependence of outflow velocity on the star formation rate of a galaxy, cast in terms of $\Delta$SFMS, the deviation from the median star-forming main sequence relation. To isolate out the general trend of increasing velocity with $M_\star$, we further normalize $v_{\rm out}$ by its median value in each stellar mass bin. Color then indicates higher (red), lower (blue), or typical (light gray) outflow velocity relative to galaxies at that same mass. As our operating definition of outflow velocity we take $v_{\rm out,50,<5kpc}$, the 50th velocity percentile of outflowing mass near the galaxy. The trends we discuss do not depend strongly on this choice.
First, we find a clear correlation between $v_{\rm out}$ and $\Delta$SFMS for star-forming galaxies, whereby outliers above the main sequence launch faster winds, whereas outliers below the MS drive slower outflows. This statement is true at fixed stellar mass, and holds from $z=1$ up to $z=6$ if not higher.
However, this correlation actually *inverts* at $M_\star \gtrsim 10^{10.5}$[M$_{\odot}$]{}. Galaxies which have grown above this mass scale drive faster outflows if they are below the main sequence, i.e. if they are in the process of quenching towards the red population. In the main panel, this is visible as an excess of red pixels for low $\Delta$SFMS at high $M_\star$. We quantify this anti-correlation in the inset panel, which plots relative $v_{\rm out}$ as a function of $\Delta$SFMS for high-mass galaxies (orange) versus low-mass star forming galaxies (blue). The two slopes have opposite sign, and the strength of this relation is such that $v_{\rm out}$ varies by $30-40$% for outliers offset by $\Delta$SFMS = $\pm 0.5$ dex.
Observationally, [@chisholm15] correlate outflow velocity and SFR in nearby star-forming galaxies, finding that mergers identified via optical morphology drive faster outflows, and that merging systems have the fastest outflows overall. If mergers are preferentially found at $\Delta$SFMS $>$ 0, this may be a similar signal as discussed here. [@cicone16] explicitly compare outflow velocity measurements with $\Delta$SFMS (their Fig. 21) for a ‘normal’ SDSS star-forming population, finding a clear positive trend above the main-sequence, with a weak or flat trend below. Towards higher masses, [@sato09] use a SF-agnostic sample from EGS to uncover a continuation of detectable NaI outflows into the red population, concluding that outflows outlive the star-formation phase and contribute to quenching galaxies en route to the quiescent population. Our analysis here implies that these ‘relic’ outflows may also be mixed with more recent outflows driven by black holes rather than supernovae.
We note that for star-forming galaxies a large number of properties correlate strongly with $\Delta$SFMS. That is, (a) outliers above the main sequence could be outliers because they drive faster outflows, or (b) they could simply have faster winds because they are already part of the $\Delta$SFMS $>$ 0 population for another reason. In this regime, relative to those systems below the median main-sequence at fixed stellar mass, galaxies have: faster outflows (as noted), lower gas-phase metallicities, higher gas fractions, lower central stellar surface densities, lower stellar metallicities, bluer (g-r) colors, larger stellar sizes (half mass radii), lower outflow mass loadings, less rotationally supported gas disks, higher halo masses, and lower mass BHs (all not shown, but see ). Oscillation and quasi-equilibria about the evolving SFMS leads to an interrelation between many of these galaxy properties. Although the root cause of the trend of $v_{\rm out}$ with $\Delta$SFMS is difficult to determine, and to some degree both (a) and (b) must apply, we note that the injection scaling of $v_{\rm out} \propto \sigma_{\rm DM}$ in the TNG model is sensitive to a spatially localized measurement of the dark matter velocity dispersion, which would naturally respond to merger activity (i.e. where $\sigma_{\rm gas}$ would also be enhanced) in contrast to more isolated, dynamically quiet environments. Likewise, the positive correlation of $M_{\rm halo}$ with $\Delta$SFMS implies larger $\sigma_{\rm DM}$ at fixed $M_\star$ [similar to the finding of @matthee17 with the EAGLE model].
In order to reincorporate the dominant trend of increasing $v_{\rm out}$ with stellar mass, Figure \[fig\_fastout\_frac\] presents a view on the fraction of galaxies with ‘fast’ outflows, with respect to their position on the plane. We do this by defining $v_{\rm out,90,10kpc} > 300$ km/s as a somewhat arbitrary though useful threshold. For example, if observational detectability of an outflow depended mainly on the existence of a component traveling above 200 km/s (i.e. above an instrumental limit or intrinsic level), our choice would map to an outflow detection fraction. The color of each pixel shows the fraction of galaxies in that bin which satisfy the velocity cut, from none (dark purple) to 100% (yellow). As a common feature, we find a diagonal boundary across the $\Delta$SFMS$-M_\star$ plane, with fast outflow galaxies residing towards the upper right. This can be understood as the superposition of the $v_{\rm out}-\Delta$SFMS secondary correlation on top of the dominant $v_{\rm out}-M_\star$ relation. A higher threshold velocity moves this boundary to the right, with roughly constant slope.
This trend can be compared in spirit to Fig. 17 of [@rubin14] at $z \sim 0.5$ based on MgII absorption modeling, or Fig. 3 of [@forsterschreiber18b] at $z=0.6 - 2.7$ based on broad emission line decomposition. Both hint towards a tilted trend in $\Delta$SFMS at fixed $M_\star$, in the sense that galaxies above the main sequence have a larger detected outflow fraction, while galaxies approaching $10^{11}$[M$_{\odot}$]{}and higher, even those below the extension of the main-sequence, do likewise. Similarly, Fig. 18 of [@cicone16] based on OIII emission from ionized outflows at $z \lesssim 0.7$ also highlights higher outflow velocities at positive $\Delta$SFMS. We emphasize such similarities are largely speculative and hindered by the current inability to make quantitative comparisons to these observables, as discussed below in Section \[subsec\_discussion\_comparisons\].
![ The fraction of galaxies with ‘fast’ outflows, as a function of their position on the $\Delta$SFMS-M\* plane at $z=1$. Here we take one possible choice and define $v_{\rm out,90,r=10kpc} > 300$ km/s as the threshold for a fast outflow. The color of each pixel then shows the fraction of galaxies in each bin which satisfy this cut, from zero (dark purple) too one hundred percent (yellow). Galaxies with fast outflows exist only above a diagonal threshold, from the upper left to lower right, in this plane. \[fig\_fastout\_frac\]](figures/histofrac2d_x=mstar_30pkpc_log_y=delta_sfms_c=vout_90_10kpc_TNG50-1_50.pdf){width="3.3in"}
Given our current inability to do a full forward-modeling, and the uncertainties inherent with inversion in the opposite direction starting from the observational quantities, our ability to make a robust comparison between the simulations and observations is presently limited. However, we would like to give a sense of the overlap between the outflow properties expected from the TNG model and available observational datasets. We therefore provide a series of suggestive plots, which are intended to be *qualitative only* in nature, and from which *no direct statements can be made* as to the level of (dis)agreement between the simulations and observations. Further modeling efforts from the theory side are needed before we can contrast against, or provide interpretation of, the wealth of high-quality observational data on outflows [see @rupke18].
In Figure \[fig\_outflows\_vs\_obs\] we show a number of correlations between outflow properties and central galaxy or supermassive black hole properties. In these various spaces we add a diverse set of observations, with citations indicated in the caption. These span all redshifts, at least as high as $z \sim 4$, and include many local universe data points from $z \sim 0$. They also span a range of sample selections – in particular, a significant number specifically target some of the most energetic nearby ULIRGs or the most luminous observable starbursts. All detectable gas phases are represented, including molecular, neutral, and ionized tracers. Importantly, we combine datasets where outflows are believed to arise from stellar feedback as well as BH feedback – their separation being often difficult.
In addition, outflow properties including velocity and mass loading factors are measured in a diversity of ways and with a number of definitions across the observational works, and no homogenization has been undertaken. For example, outflow velocities may be derived from line center offsets from systemic, line wings below some percentile of the continuum, or based in some way on line width, with or without inclination corrections. Error bars are excluded in every case for clarity.
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All simulation results are restricted to a consistent minimum stellar mass of $M_\star \geq 10^{9}$[M$_{\odot}$]{}, but otherwise include galaxies of all stellar masses as representative of a mass complete volume-limited sample, and are shown at $z=1$ (colored lines). To highlight the definitional range, we include five velocity percentile ranges ($v_{25}$ to $v_{99}$) for $v_{\rm out}$, or three velocity cut criteria (0 - 350 km/s) for [$\eta_{\rm M}$]{}, as appropriate. Measurements are made at $r = 10$ kpc; varying galactocentric distance would produce an additional fan of lines in each panel. Observational data points are shown as black symbols.
- The upper left panel shows the relationship between $v_{\rm out}$ and star formation rate of the central galaxy. The stellar mass of galaxies in this plane increases monotonically from left to right, while outliers which scatter towards high $v_{\rm out}$ at a given SFR are massive, low-sSFR, and black hole outflow dominated. Although outflow velocity increases with SFR, the slope is fairly shallow: from to the outflow velocity increases by a factor of $\sim 3-5$. Observational datasets which span only a small dynamic range in SFR are unlikely to robustly identify a trend and would conclude a flat $v_{\rm out}$ [as appreciated in e.g. @chen10]. We highlight the results of [@cicone16], where we have taken the LoSVD $v_{\rm 0.1}$ (OIII) values based on a consistent analysis of a large, low-z sample which spans nearly our entire SFR range and shows an encouraging similar scaling as $v_{\rm out,95}$ for instance. We note that due to the finite volume of TNG50, there is only a partial sampling of high $\Delta$SFMS outliers and very few galaxies with star formation rates above 100 $\rm{M}_\odot \,\rm{yr}^{-1}$.
- With the upper right panel we show the dependence on the mass loading factor [$\eta_{\rm M}$]{}with the SFR of the galaxy. As above, the stellar mass of galaxies in this plane increases monotonically from left to right, while outliers which scatter towards high [$\eta_{\rm M}$]{}at a given SFR are quenching (or quenched) and black hole outflow dominated. Note that many observations present [$\eta_{\rm M}$]{}of a specific gas phase which is only a sub-component of the total. Many of these [e.g. @chisholm15] imply to zeroth order a trend of decreasing [$\eta_{\rm M}$]{}with SFR, but as we have shown in Section \[sec\_results\_multiphase\] higher SFR (and so $M_\star$) galaxies progressively shift the majority of their outflowing mass into a difficult to observe hot phase. Without a phase-specific modeling of [$\eta_{\rm M}$]{}we cannot robustly compare to trends in SFR, other than to comment that a large fraction of observed points, which are predominantly at $1.0 < \log(\rm{SFR} / \rm{M}_\odot \,yr^{-1}) < 2.5$ have lower inferred mass outflow rates than the phase-total rates of TNG.
- In the center left panel we show the dependence of outflow velocity on the bolometric luminosity of the black hole. As with extreme starbursts, the volume of TNG50 is also too small to host the brightest luminosity AGN. There are essentially no black holes with $L_{\rm bol} > 10^{45}$ erg/s, while the majority of observations of BH-driven outflows focus on this regime of bright quasars which host $1000$ km/s or greater winds, which are offset from our simulated sample. When observations of ‘normal’ galaxies with black holes exist, inferred values for $v_{\rm out}$ cover the range of velocity percentiles from TNG [@fluetsch18 where $v_{\rm out}$ = FWMH$_{\rm broad}$/2 + $|v_{\rm broad} - v_{\rm narrow}|$]. On this plane, stellar mass increases with $L_{\rm bol}$ from left to right, but also with $v_{\rm out}$ from bottom to top. In particular, the tails visible at low black hole luminosities are the most active BH-driven outflows in the simulation.
- The center right panel shows [$\eta_{\rm M}$]{}as a function of $L_{\rm bol}$ of the central black hole. Depending on velocity threshold, the trend can be slightly decreasing, flat, or increasing with $L_{\rm bol}$, but is weak in any case. In comparing to observed trends, the previous caveats about phase specific mass outflow rate tracers apply. In TNG, the scatter in mass loading increases strongly below $L_{\rm bol} < 10^{42}$ erg/s, where low and high mass galaxies are simultaneously present, with the latter driving high [$\eta_{\rm M}$]{}outflows.
- The lower left panel highlights the relation between [$\eta_{\rm M}$]{}and $\Sigma_{\rm SFR}$, an integrated star formation rate surface density, computed within one times the stellar half mass radius (roughly similar to $R_{\rm e}$). As before, depending on velocity threshold, the trend of mass loading with galaxy-integral $\Sigma_{\rm SFR}$ can be negative, flat, or even possibly positive. At high-z and based on kpc-scale correlations, [@davies18] find an increasing trend, suggestive of a small-scale relationship which warrants an in-depth comparison.
- Finally, the lower right panel shows the relationship between outflow velocity and mass loading factor. The usual scalings of $\eta_{\rm M} \propto v_{\rm out}^{-1}$ (’momentum’ driven) and $\eta_{\rm M} \propto v_{\rm out}^{-2}$ (’energy’ driven) are indicated by the dashed gray lines. The latter is prescribed by the TNG model at injection [@pillepich18a], and we roughly recover this expectation between [$\eta_{\rm M}$]{}of a few and a few tens. In this range, stellar mass increases leftwards, from $M_\star \sim 10^8$[M$_{\odot}$]{}at [$\eta_{\rm M}$]{}of a few tens to $M_\star \sim 10^{10.5}$[M$_{\odot}$]{}at [$\eta_{\rm M}$]{}$\sim$ a few. High mass, quenching or quenched systems populate the top of this plane, distributed across all values of [$\eta_{\rm M}$]{}as high outliers to the median lines. For example, at $v_{\rm out} \sim 1000$ km/s and $\eta_{\rm M} \sim 50$. Many of the observational data points in this regime, for instance of the [@fiore17] compilation, are thought to be black hole driven outflows. The upturn of outflow velocity at $\eta_{\rm M} \gtrsim 10$ occurs when low-mass galaxies disappear and $M_\star > 10^{10.5}$[M$_{\odot}$]{}systems set the median. The monotonic scaling behavior of $v_{\rm out}(\eta_{\rm M})$ breaks down when low-SFR galaxies begin to dominate at a given mass loading.
![ Median radial profiles of star formation rate surface density, in random projections, at $z=1$. Stacked bins of stellar mass, where the half mass radii of the stars ($r_{1/2,\star}$) is given by the short vertical colored lines along the bottom edge of the panel. Similarly, 10% of the virial radius is shown by the short colored lines along the top edge of the panel. \[fig\_radprofiles\_sfr\]](figures/radprofiles_SFR-2Dz-log_pkpc_TNG50-1_50_cen.pdf){width="3.2in"}
Outflow properties are clearly related to galaxy stellar mass, star formation rate, and black hole activity, as well as the distribution of star formation activity. To emphasize the co-existence of stellar driven outflows across the mass range where BH activity also becomes relevant, Figure \[fig\_radprofiles\_sfr\] presents stacked radial profiles of star formation rate surface density $\Sigma_{\rm SFR}(r)$ at $z=1$ for galaxies of different stellar masses, from $10^8$ to $10^{11.5}$[M$_{\odot}$]{}.
Overall, the star formation scales up with increasing $M_\star$ at all radii. Except for the two highest mass bins, $\Sigma_{\rm SFR}$ profiles are always monotonically decreasing away from the galactic center. They have a characteristic break at $\sim 1-2 \, r_{\rm 1/2,\star}$, outside of which profiles steepen noticeably. We draw attention to the behavior within the galaxy body itself at high-mass, where $\Sigma_{\rm SFR}$ flattens at $r < 10$ kpc for $M_\star \gtrsim 10^{11}$[M$_{\odot}$]{}. For the most massive systems, the profile even inverts towards the center. This central inversion is also present in the total gas density profile as well as the gas-phase metallicity profiles (not shown), indicative of dilution of the remaining central gas content by metal-poor inflows.
The drop in star formation (and so gas mass) in the central ISM is another example of the ejective character of the low-accretion mode BH feedback in TNG, which starts to operate at $M_\star \gtrsim 10^{10.5}$[M$_{\odot}$]{}. We previously saw evidence for this in Figure \[fig\_outflowrate\_phase\] as the culprit for the cold phase of the bimodal temperature distribution of outflowing gas. Even though such galaxies are either in the progress of quenching for the first time, or maintaining their quenched state, Figure \[fig\_radprofiles\_sfr\] shows that there can be non-negligible, residual star formation, particularly at characteristic larger radii.
Discussion {#sec_discussion}
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The dichotomy of SN versus BH-driven outflows and the transition to quiescence
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As any star-forming gas is eligible to drive a wind-phase in TNG, galaxies do not by construction require a threshold in $\Sigma_{\rm SFR}$ such as the canonical suggested in [@heckman02] as a requirement to launch a galactic wind [but see @rubin14]. In our case, outflowing mass flux will correlate with the spatial locations of star formation, where the most supernova energy is available. The suppression of central SF at high $M_\star$ implies that stellar feedback driven winds will then become less nuclear and more extended. At the same time, the co-existence of SF-driven and BH-driven outflows also becomes inevitable at $M_\star \gtrsim 10^{10.5}$[M$_{\odot}$]{}[for example, as seen in NGC 3079; @cecil01].
In general, energy inputs from stellar and black hole feedback are essentially always both present, except in the case where the halo is not yet massive enough to host a BH seed. That is, one cannot absolutely separate out the energetic contribution from these two sources to the production of an outflow, neither in simulations nor in observations. If part or all of a measured $\dot{M}_{\rm out}$ is generated due to energy injection from black holes rather then supernovae, then the denominator in our expression for $\eta_{\rm M}^{\rm SN} = \dot{M}_{\rm out} / \dot{M}_\star$ (Eqn. \[eqn\_eta\_M\_SN\]) neglects some if not most of the relevant energetics, as implied by the ‘SN’ superscript.
It would be tempting to write or and so measure the outflow properties relative to the total available energetics. However, SN versus BH feedback operate at different (spatial) scales, and have different effective (galactic or halo-scale) coupling efficiencies. For instance, although $\dot{E}_{\rm BH}$ might dominate $\dot{E}_{\rm SN}$ for some main-sequence galaxies with rapidly accreting black holes, the resulting thermal AGN feedback may have little dynamical impact and so not actually be the cause of any outflows. Conversely, even a quenching or almost entirely quiescent galaxy may have $\dot{E}_{\rm SN} \gg \dot{E}_{\rm BH}$; nonetheless, the relatively small amount of energy available from the low-accretion state BH feedback could in practice be solely responsible for a high mass flux, high velocity outflow, possibly due to the spatial (i.e. nuclear) or temporal (i.e. rapid) coherence of that energy injection, or due to the physical/numerical coupling mechanism.
These differences leave signatures in properties of contemporaneous outflows. In TNG, the process of reddening and migration away from the blue, star-forming population is tied closely to a threshold in black hole and so stellar mass [@nelson18a]. As a result, the galactic-scale winds which arise from galaxies in different locations of the SFR$-M_\star$ plane (and with different BH properties) directly constrain underlying assumptions of the feedback models. Emblematically, Figure \[fig\_vout\_delta\_sfms\] reveals the correlation of $v_{\rm out}$ with $\Delta$SFMS as well as its inversion into an anti-correlation above a particular mass scale. Feedback also leaves signatures in the residual star formation of the galaxy itself – Figure \[fig\_radprofiles\_sfr\] demonstrates that the evolutionary process of quenching proceeds ‘inside-out’ in TNG, with star-formation truncated in the center of the galaxy first. IFU survey data at low redshift will unambiguously measure the spatially resolved progression of SFR through the transition to quiescence [@belfiore17a], providing stringent and complementary constraints to the galactic outflows themselves.
Observationally it has been noted that, at least in the local universe, ‘normal’ AGN-host galaxies do not show a clear difference in outflow prevalence or properties when compared to non-AGN control samples [@sarzi16; @robertsborsani18]. This may come down to the timescale issue, given that the BH need not be ‘on’ and injecting energy at the moment of observation, although a recent (i.e. Myr-scale) outflow could still be present [@forsterschreiber14; @gabor14b]. As shown in Figures \[fig\_timeevo1\] and \[fig\_timeevo2\], Myr-timescale variability at the energy injection scale can produce a coherent, large-scale BH-driven outflow which lasts for at least several hundred million years. In Figure \[fig\_outflows\_vs\_obs\] we noted only weak dependencies of both [$\eta_{\rm M}$]{}and $v_{\rm out}$ on instantaneous black hole luminosity, likely due in part to this issue.
At the same time, it is evident that the highest velocity BH-driven outflows in the TNG50 simulation do not arise from high luminosity quasars (e.g. with space densities ), which are not present in the small volume. Instead, they are generated from low-accretion rate BHs for which the model posits a non-relativistic wind mechanism produced by a radiatively inefficient flow [@blandford99]. In TNG black holes which are slowly accreting and have low bolometric luminosities can drive some of the most powerful outflows.
Observational evidence has emerged in this direction, probing low accretion rate black hole populations possibly underrepresented in previous surveys [@cheung16; @penny18]. Observational constraints on the outflow properties as well as the mass distribution of low luminosity AGN, particularly as a function of location in the SFR$-M_\star$ plane and so in relation to the quenching state of the galaxy, will provide an important assessment of the TNG black hole model in the future.
Towards quantitative comparisons with observations {#subsec_discussion_comparisons}
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Outflows are usually observed through the Doppler shift of specific gas tracers, either through blueshifted features in absorption [e.g. @steidel02], or through broad line components in emission [e.g. @heckman90]. In both cases several complexities arise which make the inference of intrinsic physical values difficult.
First, the outflow signature itself must be separated from the host. Studying kinematics using absorption spectra when the galaxy stellar continuum is the background source is complicated by existence of strong stellar absorption features in the same lines; [@chen10] find for instance that $\sim$ 80% of Na D in stacked SDSS spectra arises from stellar atmospheres – careful subtraction is required and the signal of interest represents a potentially small residual. Handling emission which partially fills in absorption signatures from the ISM or resonantly scattered emission similarly requires careful modeling to back out the intrinsic absorption [e.g. for FeII and MgII; @zhu15]. Outflows derived from emission lines also face difficulties, particularly in the decomposition of narrow and broad-line components, inclination corrections, the treatment of the complex underlying ISM kinematics, and the diverse excitation mechanisms for nebular lines [@newman12]. Significant assumptions are still required to convert observed line properties into e.g. mass outflow rates or mass loading factors; in particular, on the geometry, metal content, and ionization state of the outflowing gas. The uncertainties therein are significant [@murray07; @chisholm16], and ignoring or improperly accounting for galaxy-to-galaxy variation in assumed metallicity, ionization corrections, and outflow radii can introduce a factor of ten uncertainty in $\dot{M}_{\rm out}$, possibly hiding (or producing misleading) trends [@chisholm17]. For a mass conserving outflow with density varying only as a function of distance from the galaxy, the expression for mass outflow rate is
$$\dot{M}_{\rm out} = \Omega \,r^2 \,\mu(r) \,n_{\rm H}(r) \,v_{\rm out}(r)$$
where $n_{\rm H}(r)$ is the hydrogen number density, $v_{\rm out}(r)$ is the dependence of the outflow velocity on distance, $\Omega = 4\pi$ for a unity covering factor, and $\mu(r) / m_{\rm p}$ the mean molecular weight. In terms of a column density $N_{\rm H}$ observed along the line of sight, the mass flow rate across a shell at radius $r$ is
$$\dot{M}_{\rm out}(r) = \Omega \,r \,\mu(r) \,N_{\rm H} \,v_{\rm out}(r).$$
Alternatively, given a total, uniform mass $M_{\rm gas}$ at a constant outflow speed $v_{\rm out}$ within an outer radius of $r_{\rm out}$, dimensional analysis yields an even simpler expression for total outflow rate of
$$\dot{M}_{\rm out} = M_{\rm gas} \,v_{\rm out} \,r_{\rm out}^{-1}.$$
In all cases, observations must assume values for $\Omega$ and $r_{\rm out}$, which are largely or entirely unconstrained [@harrison18]. The functional form of $v_{\rm out}(r)$ is also typically unknown and often assumed to be constant, taking some derivable characteristic velocity. Finally, whatever phase of the gas is observed represents only a fraction of the total outflow [@cicone18], and for metal ion lines this can be a truly minuscule component. Inversion to the total hydrogen mass is therefore crucial, and yet also error prone due to the large correction factors. These complexities, which together accumulate into large uncertainties, motivates a comparison in the other direction. That is, forward modeling of the simulation outcome into the space of direct observables.
From the simulation side, there are several difficulties in modeling the actual gas phases. First, many important observational tracers are very cold, dense phases at $T \lesssim 10^4$ K, including neutral or molecular gas. This is an unresolved regime in many current cosmological models, due primarily to resolution limitations. Second, a standard ionization post-processing treatment, as commonly applied in studies in the low-density CGM for instance, is insufficient. This is because local radiation sources are important so close to the galaxy, from stars as well as possibly a central AGN, which are present in addition to the uniform, meta-galactic background. A treatment here would require an expensive, essentially radiative transfer calculation to derive the total incident radiation on each gas parcel, as well as assumptions on free parameters such as escape fractions from young stars. In addition, the ionization calculation is complicated by non-equilibrium effects due to rapid temporal evolution, as arise in shocks. This is a difficult problem even for specialized photoionization codes [e.g. MAPPINGS; @sutherland18]. It is also unclear how to efficiently capture the needed short time-scale information from a cosmological simulation.
Overcoming these challenges would imply a level of modeling detail which would also enable direct predictions of synthetic nebular emission lines from the ISM itself [@gutkin16; @byler17], as well as extended and extraplanar ionized gas reservoirs [e.g. @jones17], in the context of cosmological galaxy simulations [@hirschmann17]. This is the inevitable direction of future modeling efforts and will be an important step to bridge the gap between hydrodynamical simulations and many of the most accessible and important observational probes of galaxies. For outflows, we will then be able to directly calculate phase-specific predictions – for example, $\eta_{\rm M, MgII}$ or $v_{\rm out,H\alpha}$. This will facilitate direct quantitative comparisons with observables, and allow us to assess the ‘tip of the iceberg’ effect and the importance of gas phases missing in current observations of galactic outflows.
Conclusions {#sec_conclusions}
===========
In this work and together with the companion paper we have presented the new TNG50 simulation, the third and final volume of the IllustrisTNG project. TNG50 has been designed to overcome the resolution limitations inherent in cosmological simulations by sampling a large, statistically representative volume at unprecedented numerical resolution. Leveraging this new resource, we describe a first exploration of the properties of galactic outflows – driven by both supernovae and black hole feedback at $z>1$ – in the cosmological setting and with respect to the galaxies from which they arise. By resolving the internal structure of individual galaxies, TNG50 enables us to study the connection between small-scale (i.e. few hundred pc) feedback and large-scale (i.e. few hundred kpc) outflows. The diversity of galaxies and outflow properties realized in the fully representative galaxy sample of TNG50 highlights the importance of the large cosmological volume. We summarize our principal results:
- The new TNG50 simulation occupies a unique combination of large volume and high resolution, with a 50 Mpc box sampled by $2160^3$ gas cells. This provides a baryon mass resolution of $8 \times 10^4$[M$_{\odot}$]{}and an average spatial resolution of star-forming ISM gas of $\sim 100-200$ parsecs. This resolution approaches or exceeds that of modern ‘zoom’ simulations of individual galaxies, while the volume contains $\sim$ 20,000 resolved galaxies with $M_\star \ga 10^7$[M$_{\odot}$]{}. It offers an unparalleled view into the small-scale structure of galaxies and the co-evolution of dark matter, gas, stars, and supermassive black holes from dwarfs to massive ellipticals. (§\[sec\_sims\] and §\[sec\_results\_tng50\])
- By measuring mass outflow rates around galaxies we quantify the mass loading factor $\eta_{\rm M} = \dot{M}_{\rm out} / \dot{M}_\star$ as a function of stellar mass, redshift, distance, and velocity. Whereas the TNG model input value for [$\eta_{\rm M}$]{}at the injection scale of stellar feedback monotonically decreases with $M_\star$, this is not the case for the emergent outflows at 10 kpc. Instead, the mass loading *inverts* and rises rapidly for $M_\star \gtrsim 10^{10.5}$[M$_{\odot}$]{}. As a result, $\eta_{\rm M}(M_\star)$ has a broken ‘v’ shape: the low-mass behavior is regulated by stellar feedback, while the high-mass behavior is set by the energetics and coupling efficiency of BH feedback, as it begins to drive strong outflows away from quenching, low-SFR systems. Strong outflows reach larger distances towards low redshift, and [$\eta_{\rm M}$]{}decreases monotonically with increasing galactocentric distance for $r > 10$ kpc. (§\[sec\_results\_rates\])
- We extract the velocity of outflows as a function of $M_\star$, redshift, and distance, characterizing heterogeneous $v_{\rm out}$ distributions around individual galaxies through mass flux weighted velocity percentiles such as $v_{\rm out,95}$. In this case, median outflow velocities increase from $\sim$ 200 km/s at $M_\star = 10^{7.5}$[M$_{\odot}$]{}to $\sim$ 1000 km/s at $M_\star = 10^{11}$[M$_{\odot}$]{}. At fixed stellar mass, outflows are faster at higher redshift, by roughly a factor of two from $z=1$ to $z=6$. Despite a ‘minimum’ launch velocity of 350 km/s imposed by the TNG model, low-mass galaxies host much slower outflows due to resolved halo drag. At the Milky Way mass scale, despite high wind velocities of $\gtrsim$ 800 km/s prescribed by the model, the bulk of outflowing material escapes the galaxy at $\sim$ 150 km/s. The provided scalings $v_{\rm out} \propto M_\star^\alpha$ depend on redshift and velocity percentile, and $v_{\rm out}$ always declines with galactocentric distance. At the high-mass end, BH feedback produces high-velocity outflows at speeds exceeding $\sim 3000$ km/s to 20 kpc and beyond into the inner halo. (§\[sec\_results\_vel\])
- Outflows from TNG galaxies are multiphase, and different phases have different kinematics. In the temperature distribution of outflowing gas, the dominant component is a ‘hot’ phase whose peak temperature increases with the virial temperature of the host halo gas. At $M_\star \gtrsim 10^{10.5}$[M$_{\odot}$]{}, however, a second cool outflow component emerges, resulting in a bimodal outflow temperature distribution. This colder gas is also denser, and is a direct consequence of ejective BH-driven feedback launching formerly nuclear ISM material out into the halo. The highest outflow velocities are achieved by the hottest outflow phases. (§\[sec\_results\_multiphase\])
- Despite the directional isotropy of all energy inputs from both SN and BH feedback mechanisms, outflows are found preferentially aligned with the minor axes of galaxies. These bipolar outflows are therefore an emergent feature of our simulations, resulting from hydrodynamical collimation. Strong angular anisotropy is evident already by $\sim$ 10 kpc, and persists out as far as strong outflows are present ($\gtrsim$ 50 kpc at least). It exists at both low and high mass scales, where SN and BH feedback, respectively, dominate the production of galactic-scale outflows. Collimation increases with time, being ubiquitous by $z=1$ but difficult to convincingly demonstrate even at $z=2$. We speculate that this redshift trend is associated with the epoch of disk settling, where the rise of orderly rotating gas structures helps to sculpt outflow propagation. (§\[sec\_results\_angle\])
- We demonstrate a correlation of $v_{\rm out}$ with $\Delta$SFMS, such that star-forming galaxies above the main sequence drive faster winds than otherwise. This relationship inverts for , where quenching galaxies far below the main sequence launch the fastest outflows as a result of BH feedback. (§\[sec\_results\_vsgal\])
- We measure the trends of outflow properties ($v_{\rm out}$ and [$\eta_{\rm M}$]{}) with galactic star formation rate, black hole bolometric luminosity, and the spatial distribution and concentration of star formation activity. Where available we compare against a large, heterogeneous mix of observations spanning all possible galaxy types and selections, redshifts, gas phases, and outflow tracers. While data occupies similar parameter ranges as our intrinsic predictions from TNG, a robust comparison with observations of outflow properties requires additional modeling. In TNG, low luminosity, slowly accreting black holes can drive some of the most powerful outflows. (§\[sec\_results\_vsgal\])
In the future, similar measurements of outflow properties can be made in other hydrodynamical cosmological simulations, facilitating meaningful comparisons of the feedback physics implemented in different models. This will be particularly powerful because the gas dynamics of resolved outflows and halo-scale flows in general are followed faithfully without subgrid prescriptions. It remains to be seen if other models can produce similarly realistic galaxy populations with radically different outflow properties, or vice versa.
In this work we have shown that inputs and parameters of the feedback model at the *injection* scale differ from the emergent properties of galactic-scale outflows. Despite the relative simplicity of the physical assumptions invoked on the smallest scales, the resulting outflow properties are diverse and complex. That is, model parameterizations do not translate directly into observable outflow signatures. The rich phenomenology of outflows encodes the effective functioning of feedback physics and provides a unique way to study the impact of the baryon cycle on galaxy formation.
Acknowledgements {#acknowledgements .unnumbered}
================
DN would like to thank Kate Rubin, Gwen Rudie, and Alice Shapley for insightful discussions and suggestions, as well as the anonymous referee for a constructive report. SG, through the Flatiron Institute, is supported by the Simons Foundation. The primary TNG simulations were realized with compute time granted by the Gauss Centre for Supercomputing (GCS): TNG50 under GCS Large-Scale Project GCS-DWAR (2016; PIs Nelson/Pillepich), and TNG100 and TNG300 under GCS-ILLU (2014; PI Springel) on the GCS share of the supercomputer Hazel Hen at the High Performance Computing Center Stuttgart (HLRS). GCS is the alliance of the three national supercomputing centres HLRS (Universit[ä]{}t Stuttgart), JSC (Forschungszentrum J[ü]{}lich), and LRZ (Bayerische Akademie der Wissenschaften), funded by the German Federal Ministry of Education and Research (BMBF) and the German State Ministries for Research of Baden-W[ü]{}rttemberg (MWK), Bayern (StMWFK) and Nordrhein-Westfalen (MIWF). Additional simulations were carried out on the Draco and Cobra supercomputers at the Max Planck Computing and Data Facility (MPCDF).
Halo-normalized velocities {#sec:appendix}
==========================
![ **(Top)** As in the main panel of Figure \[fig\_massloading1\], the trend of [$\eta_{\rm M}$]{}as a function of $M_\star$ at $z=2$. However, instead of exploring several constant outflow velocity thresholds, we recast these in terms of the increasing halo virial velocity $v_{\rm 200}$. **(Bottom)** As in the main panel of Figure \[fig\_vout\], the trend of $v_{\rm out}$ as a function of $M_\star$ at $z=2$ and at a distance of 20 kpc. \[fig\_appendix\]](figures/massLoading_total_mstar_30pkpc_TNG50-1_33_v4r1_mean_skipzeros-False_v200norm.pdf "fig:"){width="3.4in"} ![ **(Top)** As in the main panel of Figure \[fig\_massloading1\], the trend of [$\eta_{\rm M}$]{}as a function of $M_\star$ at $z=2$. However, instead of exploring several constant outflow velocity thresholds, we recast these in terms of the increasing halo virial velocity $v_{\rm 200}$. **(Bottom)** As in the main panel of Figure \[fig\_vout\], the trend of $v_{\rm out}$ as a function of $M_\star$ at $z=2$ and at a distance of 20 kpc. \[fig\_appendix\]](figures/outflowVelocity_total_mstar_30pkpc_TNG50-1_33_nr1_np3_mean_skipzeros-False_v200norm.pdf "fig:"){width="3.4in"}
In this Appendix we revisit two primary results, namely the trends of mass loading and outflow velocity as a function of stellar mass. For [$\eta_{\rm M}$]{}$(M_\star)$ we change the $v_{\rm rad}$ threshold from a constant value in km/s to a threshold which scales with the halo mass. We likewise present the trends of $v_{\rm out} (M_\star)$ with a halo-based normalization. In both cases, we take a halo virial velocity defined as $v_{\rm 200}$, using the spherical overdensity measurements for each halo. For reference, at $z=2$, $v_{\rm 200}$ increases from $\sim 25$ km/s at $M_\star = 10^8 \,\rm{M}_\odot$ to $\sim 100$ km/s at $M_\star = 10^{11} \,\rm{M}_\odot$, reaching $\sim 150-200$ km/s only for the most massive $M_\star \gtrsim 10^{11.5} \,\rm{M}_\odot$ galaxies.
The top panel of Figure \[fig\_appendix\] shows [$\eta_{\rm M}$]{}as a function of $M_\star$ for four different thresholds: $v_{\rm rad} > \{0,1,5,10\} v_{\rm 200}$. As with a constant threshold, increasingly restrictive choices lead to lower measured mass outflow rates and hence mass loading values. For reasonable choices of velocity threshold, the results of Figure \[fig\_massloading1\] at the high-mass end are qualitatively and quantitatively similar. [$\eta_{\rm M}$]{}still shows a minimum at $M_\star \sim 10^{10.5} \,\rm{M}_\odot$, rapidly increasing above this mass due to strong BH-driven winds.
The main difference arises at the low-mass end, where the use of a halo-normalized velocity threshold allows slow outflows driven by relatively shallow potential halos to still contribute. As a result, no threshold value causes a turnover and subsequent decline of [$\eta_{\rm M}$]{}towards the smallest stellar masses, as was the case for the $v_{\rm rad} > 150$ km/s and $v_{\rm rad} > 250$ km/s choices of Figure \[fig\_massloading1\]. Based on this zeroth order comparison with halo circular velocity, it is clear that low mass galaxies continue to drive outflows which are ‘fast’, relatively speaking.
In the bottom panel of Figure \[fig\_appendix\] we show the trend of outflow velocity as a function of stellar mass, again at $z=2$ and for a fixed distance of 20 kpc from the galaxy. In contrast to Figure \[fig\_vout\], velocities are in each case normalized to $v_{\rm 200}$ of the parent halo. Three different percentiles are contrasted: $v_{\rm out,50}$, $v_{\rm out,75}$, and $v_{\rm out,95}$. As expected, measurements further into the tails of the distribution reveal gas flowing at larger velocities relative to the halo velocity. The trend with $M_\star$ in the first two cases is roughly constant, and the typical outflow velocities are between and , respectively. The high velocity tail starts to reveal different behavior, increasing relative to $v_{\rm 200}$ at both the low mass and high mass ends. The former can be understood in terms of the minimum injection velocity of 350 km/s for the TNG wind model, and the latter in terms of the rise of fast BH-driven outflows. In both cases the fastest moving components of the outflow can exceed at these distances.
[^1]: E-mail: [email protected]
[^2]: <http://www.tng-project.org>
[^3]: For example, $v_{\rm out,95}$ gives the velocity which only 5% of the outflow mass flux exceeds, i.e. the velocity envelope of 95% of the mass flux. In some observational studies, this value might be referred instead to as $v_{\rm out,05}$. As a characterization of the extreme tails of the outflow velocity distribution, it will approach the similar though less well defined $v_{\rm max}$ measure.
[^4]: Some outflowing gas at temperatures even lower than $10^4$ K is also evident. While it would be tempting to interpret this in the context of material which could be neutral or even partially molecular, we remind that the effective cooling floor of TNG is $\simeq$ 10$^4$ K due to the radiative cooling assumptions, which neglect metal fine-structure contributions. As a result, we interpret this gas as representative only of a cool ionized component.
[^5]: We rotate each galaxy to an edge-on orientation, such that the longest axis is aligned with $\hat{x}$ (i.e. the $y=0$ line) when viewed in projection in the $x-y$ plane, and the angle $\theta$ for every gas cell is calculated from its $(x,y)$ coordinates in this projection, such that $\theta = \{0, \pm \pi\}$ corresponds to the $\pm x$ (major) axes, respectively, while $\theta = \pm \pi/2$ corresponds to alignment along the $\pm y$ (i.e. minor) axes, respectively.
[^6]: In addition to this one halo, a comprehensive set of examples, spanning a range of galaxy masses and redshifts, is available on the TNG50 website at [www.tng-project.org/explore/gallery/](www.tng-project.org/explore/gallery/).
|
{
"pile_set_name": "ArXiv"
}
|
Introduction
============
The discovery of quasicrystals in the early 1980s [@SBGC] not only led to a reconsideration of the fundamental concept of a crystal (see [@G15] and references therein), but also highlighted the need for a mathematically robust treatment of the diffraction of systems that exhibit aperiodic order. The foundations for a rigorous approach were laid by Hof [@Hof1]. In particular, the measure-theoretic approach via the autocorrelation and diffraction measures allows for a mathematically rigorous discussion and separation of the different spectral components, the pure point, singular continuous and absolutely continuous part; see [@BG12] for background and examples, and [@TAO Sec. 9] for a systematic exposition. For general background on the theory of aperiodic order, we refer to [@PF; @AS; @Q; @TAO; @KLS; @Morlet18] and references therein.
Within a few years, it was established that regular model sets [@Moody00] (systems obtained by projection from higher-dimensional lattices via cut and project mechanisms with ‘nice’ windows) have pure point diffraction [@Martin; @RS17a]. We refer to the discussion in [@TAO] for details and examples, and to [@BEG] for an instructive application of the cut and project approach to an experimentally observed structure with twelvefold symmetry. The result on the pure point nature of diffraction holds for rather general setups, including cut and project schemes with non-Eucliden internal spaces. It has recently been generalised to weak model sets of extremal densities [@BHS; @RS17b], for which the window may even entirely consist of boundary, that is, has no interior; see also [@Str17; @Str20] for recent work on pure point spectra.
While systems based on a cut and project scheme are generally well understood, this is less so for systems originating from substitution or inflation rules, which constitute another popular method of generating systems with aperiodic order; see [@TAO; @Dirk] and references therein for details. There has been recent progress particularly on substitutions of constant length; see [@Neil; @Bart; @BS19; @BCM; @BGM19; @BS20].
There are familiar examples of inflation-based structures for all spectral types, such as the Fibonacci chain for a pure-point diffractive system, the Thue–Morse chain for a system with purely singular continuous diffraction, and the binary Rudin–Shapiro chain as the paradigm of a system with absolutely continuous diffraction; see [@PF; @AS; @TAO] for details. When one equips the Rudin–Shapiro chain with balanced weights ($\pm 1$), it becomes homometric with the binary Bernoulli chain with random weights $\pm 1$ [@BG09]. It is easy to construct inflation-based systems which combine any of these spectral components in their diffraction; see [@BGG] for examples. As of today, the celebrated Pisot substitution conjecture (which stipulates that an irreducible Pisot substitution has pure point spectrum) remains open; see [@Aki] for a review of the state of affairs.
While diffraction was the first property to be analysed in detail, many other questions from traditional crystallography and lattice theory require an extension to their aperiodic counterparts [@BZ]. In particular, classic counting problems based on lattices, when reformulated for point sets in aperiodic tilings, need both a conceptual reformulation and new tools to tackle them. The key observation is the necessity to employ averaging concepts, and then tools from dynamical systems and ergodic theory [@Q; @Sol97; @BG03]. If one is in the favourable situation of point sets that emerge from either the projection formalism or an inflation procedure, many of these averaged quantities are well defined and can actually be calculated; see [@BG03] and references therein. Despite some progress, many questions in this context remain open.
Let us sketch how this introductory review is organised. Our guiding example in this exposition is the classic, self-similar Fibonacci tiling of the real line. Its descriptions as an inflation set and as a cut and project set are reviewed in Section \[sec:Fibo\]. As a simple example of the role of the window in averaging, we discuss the averaged shelling for this system in Section \[sec:shell\]. This is followed by a brief review of the standard approach to diffraction in Section \[sec:standard\], where we exploit the description of the Fibonacci point set as a cut and project set and the general results for the diffraction of regular model sets.
In Section \[sec:cocycle\], we introduce the recently developed internal cocycle approach. For systems which possess both an inflation and a projection interpretation, such as the Fibonacci tiling, the inflation cocycle can be lifted to internal space. This makes it possible to efficiently compute the diffraction of certain cut and project systems with complicated windows, such as windows with fractal boundaries which are commonly found in inflation structures. We explore this with planar examples, which are based on the Fibonacci substitution, in Section \[sec:planar\].
Finally, in Section \[sec:hyper\], we discuss the use of ‘hyperuniformity’ as a measure of order in Fibonacci systems. This amounts to an investigation of the asymptotic behaviour of the total diffraction intensity near the origin. It turns out that this can dinstinguish between generic and inflation-invariant choices for the window in the cut and project scheme.
The Fibonacci tiling revisited {#sec:Fibo}
==============================
Let us start with a paradigm of aperiodic order in one dimension, the classic Fibonacci tiling. It can be defined via the primitive two-letter inflation rule $$\varrho \colon \quad a\mapsto ab{\hspace{0.5pt}}, \quad b \mapsto a{\hspace{0.5pt}},$$ where $a$ and $b$ represent *tiles* (or intervals) of length $\tau = \frac{1}{2} \bigl( 1 + \mbox{\small $\sqrt{5}$} \, \bigr)$ and $1$, respectively. The corresponding incidence matrix is given by $$\label{eq:fibmat}
M \, = \, \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix},$$ which has Perron–Frobenius eigenvalue $\tau$. Its left and right eigenvectors read $$\label{eq:ev}
\langle u{\hspace{0.5pt}}| \, = \, {\frac{\raisebox{-2pt}{$\tau+2$}}
{\raisebox{0.5pt}{$5$}}} \bigl(\tau,1\bigr)\quad
\text{and} \quad
|{\hspace{0.5pt}}v\rangle \, = \, \bigl(\tau^{-1},\tau^{-2}\bigr)^{T},$$ where we employ Dirac’s intuitive ‘bra-c-ket’ notation, which makes it easy to distinguish row and column vectors. We normalise the right eigenvector $|{\hspace{0.5pt}}v\rangle$ such that $\langle 1 | {\hspace{0.5pt}}v \rangle =1$, which means that its entries are the *relative frequencies* of the tiles. For later convenience, we normalise the left eigenvector $\langle u{\hspace{0.5pt}}|$ by setting $\langle u{\hspace{0.5pt}}|{\hspace{0.5pt}}v\rangle=1$, rather than using the vector of natural tile lengths itself. With this normalisation, we have $$\label{eq:Pdef}
\begin{split}
\lim_{n\to\infty} \tau^{-n} M^n \, &= \, {\frac{\raisebox{-2pt}{$\tau+2$}}
{\raisebox{0.5pt}{$5$}}}
\begin{pmatrix} 1 & \tau^{-1} \\
\tau^{-1} & \tau^{-2}\end{pmatrix}\\[1mm]
&= \, |{\hspace{0.5pt}}v\rangle\langle u{\hspace{0.5pt}}| \, {=\mathrel{\mathop:}}\, P{\hspace{0.5pt}},
\end{split}$$ where $P=P^2$ is a symmetric projector of rank $1$ with spectrum $\{1,0\}$.
Starting from the legal seed $b|a$, where the vertical bar denotes the origin, and iterating the square of the inflation rule $\varrho$, generates a tiling of the real line that is invariant under $\varrho^2$; see [@TAO Ex. 4.6] for details and why it does not matter which of the two fixed points of $\varrho^2$ one chooses. Let us use the left endpoints of each interval as *control points* and denote the set of these points by ${\varLambda}_{a}$ and ${\varLambda}_{b}$, respectively. Clearly, since $0\in{\varLambda}_{a}$ and all tiles have either length $\tau$ or length $1$, all coordinates are integer linear combinations of these two tile lengths, and we have $${\varLambda}_{a,b}\, \subset\, {\mathbb{Z}}[\tau]\, =\, \{m+n\tau : m,n\in{\mathbb{Z}}\}{\hspace{0.5pt}}.$$ While the incidence matrix $M$ only contains information about the number of tiles under inflation, but not about their positions, the latter information can be encoded by introducing the *displacement matrix* $$\label{eq:T}
T \, = \, \begin{pmatrix} \{ 0 \} & \{ 0 \} \\
\{\tau\} & \varnothing \end{pmatrix},$$ where $\varnothing$ denotes the empty set. Note that $T$ is the geometric counterpart of the instruction matrices that are used in the symbolic context [@Q]. The matrix elements of $T$ are sets that specify the relative displacement for all tiles under inflation. For instance, the two entries in the first column correspond to a long tile with relative shift $0$ and a small tile with shift $\tau$ originating from inflating a long tile. Clearly, the inflation matrix $M$ is recovered if one takes the elementwise cardinality of $T$, noting that the empty set has cardinality $0$.
The inflation rule $\varrho$ induces an iteration on pairs of point sets, namely $$\label{eq:vL-rec}
\begin{split}
{\varLambda}^{(n+1)}_{a} \, &= \, \tau{\varLambda}^{(n)}_{a}\cup
\tau{\varLambda}^{(n)}_{b}{\hspace{0.5pt}},\\
{\varLambda}^{(n+1)}_{b}\, &= \, \tau{\varLambda}^{(n)}_{a}{\hspace{-0.5pt}}+\tau{\hspace{0.5pt}},
\end{split}$$ with suitable initial conditions ${\varLambda}^{(0)}_{a,b}$. When one starts with the left endpoints of a legal seed, this iteration precisely reproduces the endpoints of the corresponding, successive inflation steps. In this case, the union on the right-hand side is disjoint. In particular, for the above choice of ${\varLambda}_{a,b}$, one needs ${\varLambda}^{(0)}_{a}=\{0\}$ and ${\varLambda}^{(0)}_{b}=\{-1\}$.
The point sets ${\varLambda}_{a,b}$ also have an interpretation as a cut and project set. Here, we use the natural (Minkowski) embedding of the module ${\mathbb{Z}}[\tau]$ in the plane ${\mathbb{R}{\hspace{0.5pt}}}^{2}$, by associating to each $x=m+n\tau\in{\mathbb{Z}}[\tau]$ its image $x^{\star}=m+n\tau^{\star}=m+n(1-\tau)$ under algebraic conjugation (which maps $\sqrt{5}$ to $-\sqrt{5}\,$). This gives $$\begin{aligned}
{\mathcal{L}}\, &= \, \bigl\{(x,x^{\star}) : x\in{\mathbb{Z}}[\tau]\bigr\}\\
& = \, \bigl\{(m+n\tau,m+n\tau^{\star}) : m,n\in{\mathbb{Z}}\bigr\}\\
& = \, \bigl\{m(1,1) + n(\tau,\tau^{\star}): m,n\in{\mathbb{Z}}\bigr\},\end{aligned}$$ which is a planar lattice with basis vectors $(1,1)$ and $(\tau,\tau^{\star})$; see [@TAO; @BEG] for details and further examples. Here, we refer to the two one-dimensional subspaces of ${\mathbb{R}{\hspace{0.5pt}}}^2={\mathbb{R}{\hspace{0.5pt}}}\times{\mathbb{R}{\hspace{0.5pt}}}$ as the *physical* and the *internal* space, respectively. The physical space hosts our point sets ${\varLambda}_{a,b}$, while the windows are subsets of the internal space, with the providing the relevant link between the two spaces.
![Cut and project description of the Fibonacci chain from the lattice ${\mathcal{L}}$ (blue dots). The windows $W_{a}$ and $W_{b}$ are the cross-sections of the yellow and green strips, repectively.\[fig:fiboproj\]](fiboproj.eps){width="\columnwidth"}
The point sets ${\varLambda}_{a,b}$ are given by the projection of two strips of the lattice ${\mathcal{L}}$; compare Figure \[fig:fiboproj\]. The strips are defined by their cross-sections, usually called *windows*, which are the half-open intervals $W_{a}=[\tau-2,\tau-1)$ and $W_{b}=[-1,\tau-2)$. With $L = {\mathbb{Z}}[\tau]$, the projection of ${\mathcal{L}}$ into physical space, the point sets are thus given by $$\label{eq:vL-ab}
{\varLambda}_{a,b} \, = \, \bigl\{x\in L :
x^{\star}\in W_{a,b}\bigr\}{\hspace{0.5pt}}.$$ The windows $W_{a,b}$, or more precisely their closures, can be obtained as the unique solutions of a contractive iterated function system that arises from the $\star$-image of , $$\label{eq:W-rec}
W_{a} \, = \, \sigma W_{a} \cup \sigma W_{b}{\hspace{0.5pt}},
\qquad W_{b} \, = \, \sigma W_{a} + \sigma{\hspace{0.5pt}},$$ where $\sigma=\tau^{\star}=1-\tau$ satisfies $\lvert\sigma\rvert<1$. One key property, which can be employed to show that these point sets are pure point diffractive, is the fact that the $\star$-images of the point sets ${\varLambda}_{a,b}$ are *uniformly distributed* in the windows $W_{a,b}$, which makes it possible to translate the computation of *average quantities* in physical space to computations in internal space.
Shelling {#sec:shell}
========
Let us discuss a simple example of an averaged quantity, the averaged shelling function for the Fibonacci point set; see [@BG03] for the concept and various applications to aperiodic systems. For a point set, the *shelling* problem asks for the number $n(r)$ of points that lie on shells of radius $r$, taken with respect to a fixed centre. For an aperiodic point set, this generally depends on the choice of the centre. The *averaged shelling* numbers $a(r)$ are obtained by taking the average over all choices of centres, where we limit ourselves to centres that are themselves in the point set. Clearly, since we are dealing with a one-dimensional point set, any shell can have at most two points, so $n(r)\in\{0,1,2\}$ for all $r\in{\mathbb{R}{\hspace{0.5pt}}}$, with $n(r)=0$ if $r\not\in {\mathbb{Z}}[\tau]$, as well as $n(0)=a(0)=1$. Clearly, this also implies that $a(r)\in [0,2]$ for all $r\in{\mathbb{R}{\hspace{0.5pt}}}$, with $a(r)=0$ whenever $r\not\in {\mathbb{Z}}[\tau]$.
Consider a point $x\in{\varLambda}$ and $r=m+n\tau\in{\mathbb{Z}}[\tau]$. To compute $n(r)$, we have to check whether $x\pm r$ are also in the point set ${\varLambda}$. From the model set description, we know that $x^{\star}\in W\!$, and checking whether $x\pm r$ are in ${\varLambda}$ is equivalent to checking whether $x^{\star}\pm r^{\star}\in W\!$. While it is possible to perform this computation for any given value of $x$ and $r$, there is no simple closed formula for these coefficients.
To obtain the averaged shelling number, we have to consider all $x\in{\varLambda}$ as centres, each with the same weight, which means averaging over all $x^{\star}\in W\!$. Define $\nu(r)=\nu(-r)$ as the relative frequency to find a point in ${\varLambda}$ at $x$ as well as at $x+r$, so $a(0)=\nu(0)=1$ and $a(r)=2\nu(r)$ for $r>0$, to account for the points on both sides. Now, for $r\in{\mathbb{Z}}[\tau]$, the frequency $\nu(r)$ of having both $x^{\star}\in
W$ and $x^{\star}+r^{\star}\in W$ can be calculated as the overlap length between the window $W$ and the shifted window $W\! -r^{\star}$, divided by the length of $W\!$, which is $|W|=\tau$. This is correct because the uniform distribution of points in the window [@TAO; @Moody02] implies that the frequency of any configuration is proportional to the length of the corresponding sub-window. Clearly, the length of the overlap between these two intervals is $0$ whenever $\lvert
r^{\star}\rvert>\tau$, and decreases linearly with $\lvert
r^{\star}\rvert$, so we get $$\label{eq:auto-coeff}
\begin{split}
\nu (r) \, & = \, \frac{\bigl| W\cap (W\! -r^{\star})\bigr|}
{\bigl| W\bigr|}\\
& = \, \begin{cases}
1-\frac{|r^{\star}|}{\tau} , &
\text{if $r\in{\mathbb{Z}}[\tau]$ and $\lvert r^{\star}\rvert\leqslant \tau$,}\\
0 , & \text{otherwise.}
\end{cases}
\end{split}$$ Consequently, the averaged shelling numbers for the Fibonacci point set are given by $$a(r)\, =\, \begin{cases} 1, &\text{if $r=0$,}\\
2\bigl(1-\frac{|r^{\star}|}{\tau}\bigr), & \text{if
$r\in{\mathbb{Z}}[\tau]$ with $|r^{\star}|\leqslant \tau$,}\\
0, & \text{otherwise}.
\end{cases}$$ Note that $a(r)$, for $r\in{\mathbb{Z}}[\tau]$, is a simple function of $r^{\star}$, but that it behaves rather erratically if one looks at it as a function of $r$; compare Figure \[fig:fiboshell\]. The reason behind this observation is the total discontinuity of the from physical to internal space.
![Averaged shelling numbers $a(r)$ for the Fibonacci point set as a function of $r$ (left) and $r^{\star}$ (right).\[fig:fiboshell\]](fibophys.eps "fig:"){width="0.49\columnwidth"}![Averaged shelling numbers $a(r)$ for the Fibonacci point set as a function of $r$ (left) and $r^{\star}$ (right).\[fig:fiboshell\]](fibointern.eps "fig:"){width="0.49\columnwidth"}
For the one-dimensional example at hand, the numbers $\nu(r)$ are nothing but the *relative probability* to find two points at a distance $r$, and thus the (relatively normalised) *autocorrelation coefficients* of the point set ${\varLambda}$. As such, they are intimately connected to the diffraction of this point set. Clearly, correlations are much easier handled in internal space, where we can calculate them via volumes of intersections of windows, as we shall see shortly.
Standard approach to diffraction {#sec:standard}
================================
Here, we start with a brief summary of the derivation of the diffraction spectrum for the Fibonacci point set ${\varLambda}={\varLambda}_{a}\cup{\varLambda}_{b}$, considered as a cut and project set ${\varLambda}=\{x\in L : x^{\star}\in W\}$ with $W=W_{a}\cup W_{b}$. Assume that we place point scatterers of unit scattering strength at all points $x\in{\varLambda}$, and consider the corresponding *Dirac comb* $$\omega\, =\, \delta^{}_{\! {\varLambda}} \,{\mathrel{\mathop:}=}\, \sum_{x\in{\varLambda}} \delta^{}_{x}.$$ We associate to $\omega$ the autocorrelation $\gamma = \omega
\circledast \widetilde{\omega}$, where $\widetilde{\omega}$ is the ‘flipped-over’ (reflected) version of $\omega$ and $\circledast$ denotes volume-averaged (or Eberlein) convolution [@TAO Sec. 8.8]. The diffraction measure $\widehat{\gamma}$ is the Fourier transform of the autocorrelation.
From the general diffraction theory for cut and project sets with well-behaved windows, we know that the diffraction measure of this system is a pure point measure, so consists of Bragg peaks only. These Bragg peaks are located on the projection of the entire *dual lattice* $${\mathcal{L}}^{*}\, =\, {\frac{\raisebox{-2pt}{$1$}}
{\raisebox{0.5pt}{$\sqrt{5}$}}}
\bigl\{m (\tau-1,\tau) + n (1,-1) : m,n\in{\mathbb{Z}}\bigr\}$$ to the physical space (the first coordinate), which is $L^{\circledast}=\frac{1}{\sqrt{5}} {\mathbb{Z}}[\tau]$. We call this set the *Fourier module* of the Fibonacci point set; it coincides with the dynamical spectrum (in additive notation) in the mathematical literature. Note that $\frac{1}{\sqrt{5}}=(2\tau-1)/5$, so $L^{\circledast}\subset{\mathbb{Q}}(\tau)$, which means that the $\star$-map is well defined for all $k\in L^{\circledast}$. The Fourier module is a dense subset of ${\mathbb{R}{\hspace{0.5pt}}}$, which means that the diffraction consists of Bragg peaks on a dense set in space, where the intensities are locally summable.
The diffraction measure is thus the countable sum $$\widehat{\gamma} \, = \sum_{k\in L^{\circledast}} \lvert
A(k)\rvert^2 \, \delta^{}_{k}$$ where the diffraction amplitudes, or *Fourier–Bohr* (FB) coefficients, are given by the general formula $$\label{eq:genampli}
A(k) \, = \, \frac{\operatorname{dens}({\varLambda})}{\operatorname{vol}(W)} \,
\widehat{1^{}_{W}}(-k^{\star}) \, = \,
\frac{\operatorname{dens}({\varLambda})}{\operatorname{vol}(W)} \,
\widecheck{1^{}_{W}}(k^{\star})$$ for all $k\in L^{\circledast}$, and vanish otherwise. Here, $1^{}_{W}$ denotes the characteristic function of the window $W\!$, defined by $$1^{}_{W}(x)\, = \, \begin{cases} 1, & \text{if $x\in W\!$,}\\
0, & \text{otherwise,}\end{cases}$$ and $\widehat{g}$ and $\widecheck{g}$ denote the Fourier transform and inverse Fourier transform of an $L^{1}$-function $g$, respectively. With $\operatorname{dens}({\varLambda})=(\tau+2)/5$ and $\operatorname{vol}(W)=\tau$, Eq. evaluates to $$\begin{split}
A(k) \, &= \, \frac{1}{\sqrt{5}}
\int_{-1}^{\tau-1} {{\hspace{0.5pt}}\mathrm{e}}^{2\pi{\mathrm{i}{\hspace{0.5pt}}}k^{\star}y}{\,\mathrm{d}}y\\
&=\,
\frac{\tau}{\sqrt{5}}\,{{\hspace{0.5pt}}\mathrm{e}}^{\pi {\mathrm{i}{\hspace{0.5pt}}}k^{\star}(\tau-2)}\,
\operatorname{sinc}(\pi \tau k^{\star})
\end{split}$$ where $\operatorname{sinc}(x)=\sin(x)/x$. Hence, the diffraction intensities are $$\label{eq:fibointens}
I(k) \, = \, \lvert A(k)\rvert^2 \, = \,
\biggl(\frac{\tau}{\sqrt{5}}\, \operatorname{sinc}(\pi \tau k^{\star})\biggr)^2$$ for all $k\in L^{\circledast}$, and $0$ otherwise. This is illustrated in Figure \[fig:fibodiff\].
![Schematic construction of the diffraction measure of the Fibonacci point set from the dual lattice ${\mathcal{L}}^{*}$ (blue dots). A point $(k,k^{\star})\in{\mathcal{L}}^{*}$ results in a Bragg peak at $k\in
L^{\circledast}$ of intensity given by the value of the function on the right-hand side evaluated at $k^{\star}$.\[fig:fibodiff\]](fibointproj.eps){width="\columnwidth"}
The corresponding autocorrelation measure $\gamma$ can be expressed in terms of the (dimensionless) *pair correlation coefficients* $$\nu(r) \, {\mathrel{\mathop:}=}\,
\frac{\operatorname{dens}\bigl({\varLambda}\cap({\varLambda}-r)\bigr)}
{\operatorname{dens}({\varLambda})} \, = \, \nu(-r) {\hspace{0.5pt}},$$ which are positive for all $r\in{\varLambda}-{\varLambda}\subset {\mathbb{Z}}[\tau]$ and vanish for all other distances $r$. These are precisely the coefficients we defined in Eq. to compute the shelling numbers. The link between the two expressions is provided by the $\star$-map and the uniform distribution of ${\varLambda}^{\star}$ in the window $W\!$. In terms of these pair correlation coefficients, the autocorrelation measure is $$\gamma \, = \, \operatorname{dens}({\varLambda}) \sum_{r\in{\varLambda}-{\varLambda}}\nu(r)\, \delta_{r},$$ which is a pure point measure supported on the difference set ${\varLambda}-{\varLambda}$.
More generally, we may associate two different, in general complex, scattering strengths $u^{}_{a}$ and $u^{}_{b}$ to the points in ${\varLambda}_{a}$ and ${\varLambda}_{b}$, respectively, and consider the weighted Dirac comb $\omega =
u^{}_{a}\delta^{}_{\!{\varLambda}_{a}}+u^{}_{b}\delta^{}_{\!{\varLambda}_{b}}$. In this case, the diffraction intensity for all wave numbers $k\in L^{\circledast}$ is given by the superposition $$\label{eq:Fibo-int}
I(k) \, = \,
\bigl| u^{}_{a} \, A^{}_{a}(k) +
u^{}_{b} \, A^{}_{b}(k)\bigr|^{2}$$ of the corresponding FB amplitudes $$\begin{aligned}
A^{}_{a,b}(k) \, &= \, \frac{\operatorname{dens}({\varLambda}_{a,b})}{\operatorname{vol}(W_{\! a,b})} \,
\widehat{1^{}_{W_{\! a,b}}}(-k^{\star})\\
& = \, \frac{\operatorname{dens}({\varLambda})}{\operatorname{vol}(W)} \,
\widehat{1^{}_{W_{\! a,b}}}(-k^{\star}) \, = \,
\frac{1}{\sqrt{5}} \, \widehat{1^{}_{W_{\! a,b}}}(-k^{\star}).\end{aligned}$$ The corresponding autocorrelation measure can once more be expressed in terms of pair correlation functions, now distinguishing points in ${\varLambda}_{a}$ and ${\varLambda}_{b}$, $$\nu^{}_{\alpha\beta}(r) \, {\mathrel{\mathop:}=}\,
\frac{\operatorname{dens}\bigl({\varLambda}_{\alpha}\cap({\varLambda}_{\beta} -r)\bigr)}
{\operatorname{dens}({\varLambda})} \, = \, \nu^{}_{\beta\alpha}(-r) {\hspace{0.5pt}}.$$ These coefficients are positive for all $r\in{\varLambda}_{\beta}-{\varLambda}_{\alpha}$ and vanish otherwise, and in particular satisfy the relation $\sum_{\alpha,\beta\in\{a,b\}}\nu^{}_{\alpha\beta} (r) = \nu(r)$.
The relation between the FB coefficients and the Fourier transform of the compact windows holds for any regular model set, which is a cut and project set with some ‘niceness’ constraint on the window; see [@TAO Thm. 9.4] for details. While this works well for many of the nice examples with polygonal windows, it becomes practically impossible to compute the FB coefficients in this way if the windows are compact sets with fractal boundaries. Such windows naturally arise for cut and project sets which also possess an inflation symmetry. Indeed, some of the structure models of icosahedral quasicrystals, see [@CdYb] for an example, feature experimentally determined windows whose shapes may indicate first steps of a fractal construction on the boundary.
Let us therefore explain a different approach that will permit an efficient computation of the diffraction also for such, more complicated, situations.
Renormalisation and internal cocycle {#sec:cocycle}
====================================
Let us reconsider our motivating example, the Fibonacci point sets ${\varLambda}^{}_{a,b}$ of Eq. . We will use both their inflation structure and their description as cut and project sets. Here, we make use of the iteration and the corresponding relation for the windows (or, more precisely, the closure of the windows). This inflation structure induces the following relation between the characteristic functions of the windows, $$\label{eq:charfun-rec}
1^{}_{W_{\! a}} \, =\, 1^{}_{\sigma W_{\! a}\cup{\hspace{0.5pt}}\sigma W_{b}}
\quad\text{and}\quad
1^{}_{W_{b}} \, = \, 1^{}_{\sigma W_{\! a} +{\hspace{0.5pt}}\sigma} {\hspace{0.5pt}},$$ where we again set $\sigma=\tau^{\star}=1-\tau$. Since the (closed) windows only share at most boundary points, we observe that $1^{}_{\sigma W_{\! a}\cup{\hspace{0.5pt}}\sigma W_{b}} =1^{}_{\sigma W_{\! a}} +
1^{}_{\sigma W_{b}}$ holds as an equality of $L^{1}$-functions. Now, we can apply the Fourier transform, where it will turn out to be more convenient to work with the *inverse* Fourier transform from the start. Applying the transform yields the relations $$\label{eq:charfun}
\widecheck{1^{}_{W_{\! a}}} \, = \,
\widecheck{1^{}_{\sigma W_{\! a}}} + \widecheck{1^{}_{\sigma W_{b}}}
\quad\text{and}\quad
\widecheck{1^{}_{W_{b}}} \, = \,
\widecheck{1^{}_{\sigma W_{\! a}+{\hspace{0.5pt}}\sigma}}{\hspace{0.5pt}}.$$ Note that, by an elementary change of variable calculation, one has $$\label{eq:affineFT}
\widecheck{1^{}_{\alpha K+\beta}}(y) \, = \,
\lvert\alpha\rvert \, {{\hspace{0.5pt}}\mathrm{e}}^{2\pi{\mathrm{i}{\hspace{0.5pt}}}\beta y} \,
\widecheck{1^{}_{K}}(\alpha{\hspace{0.5pt}}y)$$ for arbitrary $\alpha,\beta\in{\mathbb{R}{\hspace{0.5pt}}}$ with $\alpha\ne 0$ and any compact set $K\subset {\mathbb{R}{\hspace{0.5pt}}}$.
Defining $h^{}_{a,b}{\mathrel{\mathop:}=}\widecheck{1^{}_{W_{\! a,b}}}$ and using Eq. , we can rewrite Eq. as $$\label{eq:self}
\begin{pmatrix} h^{}_{a} \\ h^{}_{b} \end{pmatrix} (y) \, = \,
\lvert\sigma\rvert \, {{\hspace{0.5pt}}\underline{{\hspace{-0.5pt}}B\!}\,}(y)
\begin{pmatrix} h^{}_{a} \\ h^{}_{b} \end{pmatrix} (\sigma y)$$ with the matrix $$\label{eq:sB}
{{\hspace{0.5pt}}\underline{{\hspace{-0.5pt}}B\!}\,}(y) \, {\mathrel{\mathop:}=}\, \begin{pmatrix} 1 & 1 \\
{{\hspace{0.5pt}}\mathrm{e}}^{2\pi{\mathrm{i}{\hspace{0.5pt}}}\sigma y} & 0 \end{pmatrix}.$$ The matrix ${{\hspace{0.5pt}}\underline{{\hspace{-0.5pt}}B\!}\,}$ is obtained by first taking the $\star$-map of the set-valued displacement matrix $T$ of Eq. and then its inverse Fourier transform. For this reason, ${{\hspace{0.5pt}}\underline{{\hspace{-0.5pt}}B\!}\,}$ is called the *internal Fourier matrix* [@BG19c], to distinguish it from the Fourier matrix of the renormalisation approach in physical space [@BG16; @BFGR]; see [@BS18; @BS20] for various extensions with more flexibility in the choice of the interval lengths.
Going back to using Dirac notation, we introduce $|{\hspace{0.5pt}}h\rangle=(h^{}_{a},h^{}_{b})^T$, which satisfies $|{\hspace{0.5pt}}h(0)\rangle=\tau {\hspace{0.5pt}}|{\hspace{0.5pt}}v\rangle$ with the right eigenvector $ |{\hspace{0.5pt}}v\rangle$ of the substitution matrix $M$ from Eq. . Applying the iteration $n$ times then gives $$\label{eq:h-rec}
|{\hspace{0.5pt}}h(y)\rangle \, = \, \lvert\sigma\rvert^{n} {{\hspace{0.5pt}}\underline{{\hspace{-0.5pt}}B\!}\,}^{(n)}(y) \,
|{\hspace{0.5pt}}h(\sigma^{n} y)\rangle$$ where $${{\hspace{0.5pt}}\underline{{\hspace{-0.5pt}}B\!}\,}^{(n)}(y)\, {\mathrel{\mathop:}=}\, {{\hspace{0.5pt}}\underline{{\hspace{-0.5pt}}B\!}\,}(y) {{\hspace{0.5pt}}\underline{{\hspace{-0.5pt}}B\!}\,}(\sigma y) \cdots
{{\hspace{0.5pt}}\underline{{\hspace{-0.5pt}}B\!}\,}(\sigma^{n-1} y){\hspace{0.5pt}}.$$ In particular, these matrices satisfy ${{\hspace{0.5pt}}\underline{{\hspace{-0.5pt}}B\!}\,}^{(1)}={{\hspace{0.5pt}}\underline{{\hspace{-0.5pt}}B\!}\,}$ and ${{\hspace{0.5pt}}\underline{{\hspace{-0.5pt}}B\!}\,}^{(n)}(0)=M^{n}$ for all $n\in{\mathbb{N}}$, where $M$ is the substitution matrix from Eq. , as well as the relations $$\label{eq:Bsplit}
{{\hspace{0.5pt}}\underline{{\hspace{-0.5pt}}B\!}\,}^{(n+m)}(y) \, = \, {{\hspace{0.5pt}}\underline{{\hspace{-0.5pt}}B\!}\,}^{(n)}(y) \,{{\hspace{0.5pt}}\underline{{\hspace{-0.5pt}}B\!}\,}^{(m)}(\sigma^n y)$$ for any $m,n\in{\mathbb{N}}$. Note that ${{\hspace{0.5pt}}\underline{{\hspace{-0.5pt}}B\!}\,}^{(n)}(y)$ defines a matrix cocycle, called the *internal cocycle*, which is related to the usual inflation cocycle (in physical space) by an application of the $\star$-map to the displacement matrices of the powers of the inflation rule; compare [@BGM19; @BG19c] and see [@BS18; @BS20] for a similar approach. Note also that $\lvert\sigma\rvert <1$, which means that $\sigma^n$ approaches $0$ exponentially fast as $n\to\infty$. We can exploit this exponential convergence to efficiently compute the diffraction amplitudes, which are essentially the elements of the vector $|{\hspace{0.5pt}}h\rangle$.
Considering the limit as $n\to\infty$ in Eq. , one can show that $$\label{eq:h}
|{\hspace{0.5pt}}h(y)\rangle=C(y) |{\hspace{0.5pt}}h(0)\rangle$$ with $$\label{eq:C}
C(y) \, {\mathrel{\mathop:}=}\, \lim_{n\to\infty}
\lvert\sigma\rvert^n {{\hspace{0.5pt}}\underline{{\hspace{-0.5pt}}B\!}\,}^{(n)}(y){\hspace{0.5pt}},$$ which exists pointwise for every $y\in{\mathbb{R}{\hspace{0.5pt}}}$. In fact, one has compact convergence, which implies that $C(y)$ is continuous [@BG19c Thm. 4.6 and Cor. 4.7]. Clearly, since ${{\hspace{0.5pt}}\underline{{\hspace{-0.5pt}}B\!}\,}^{(n)}(0)=M^n$, we have $C(0)=P$ with the projector $P=|{\hspace{0.5pt}}v\rangle\langle u{\hspace{0.5pt}}|$ from Eq. .
Using Eq. with $m=1$ and letting $n\to\infty$, one obtains $$\tau \, C(y) \, = \, C(y) M{\hspace{0.5pt}},$$ since $\lvert\sigma\rvert=\tau^{-1}$. This relation implies that each row of $C(y)$ is a multiple of the left eigenvector $\langle u|$ of the substitution matrix $M$ from Eq. , which means that we can define a vector-valued function $|{\hspace{0.5pt}}c(y) \rangle$ such that $$\label{eq:Cu}
C(y)\, =\, |{\hspace{0.5pt}}c(y)\rangle\langle u{\hspace{0.5pt}}|$$ holds with $|{\hspace{0.5pt}}c(y)\rangle = \bigl(c^{}_{a}(y),c^{}_{b}(y)\bigr)^T {\hspace{-0.5pt}}$, where we have $|{\hspace{0.5pt}}c(0)\rangle=|{\hspace{0.5pt}}v\rangle$.
From Eqs. and , we obtain $$|{\hspace{0.5pt}}h(y)\rangle\, =\, |{\hspace{0.5pt}}c(y)\rangle\langle u{\hspace{0.5pt}}|{\hspace{0.5pt}}h(0)\rangle\, =\, \tau {\hspace{0.5pt}}{\hspace{0.5pt}}|{\hspace{0.5pt}}c(y)\rangle$$ and thus the inverse Fourier transforms of the windows, $\widecheck{1^{}_{W_{\! a,b}}}=h^{}_{a,b}$, are encoded in the matrix $C$.
For the Fibonacci case, we can calculate $|{\hspace{0.5pt}}c(y)\rangle$ by taking the Fourier transforms of the known windows $W^{}_{\! a,b}$ to obtain $$\begin{aligned}
c^{}_{a}(y) \, &= \,
{\frac{\raisebox{-2pt}{${{\hspace{0.5pt}}\mathrm{e}}^{2\pi{\mathrm{i}{\hspace{0.5pt}}}(\tau-1)y} - {{\hspace{0.5pt}}\mathrm{e}}^{2\pi{\mathrm{i}{\hspace{0.5pt}}}(\tau-2)y}$}}
{\raisebox{0.5pt}{$2\pi{\mathrm{i}{\hspace{0.5pt}}}y$}}}
\intertext{and}
c^{}_{b}(y) \, &= \,
{\frac{\raisebox{-2pt}{${{\hspace{0.5pt}}\mathrm{e}}^{2\pi{\mathrm{i}{\hspace{0.5pt}}}(\tau-2)y} - {{\hspace{0.5pt}}\mathrm{e}}^{-2\pi{\mathrm{i}{\hspace{0.5pt}}}y}$}}
{\raisebox{0.5pt}{$2\pi{\mathrm{i}{\hspace{0.5pt}}}y$}}} {\hspace{0.5pt}}.\end{aligned}$$ Note that these functions never vanish simultaneously, so $C(y)$ is always a matrix of rank $1$. However, taking the Fourier transform of the windows takes us essentially back to the standard approach.
The main benefit of the internal cocycle approach is that it applies in other situations, where no explicit calculation of the (inverse) Fourier transform of the windows is feasible. This is achieved by *approximating* $C(y)$ by $\lvert \sigma\rvert^n {{\hspace{0.5pt}}\underline{{\hspace{-0.5pt}}B\!}\,}^{(n)}(y)$ for a sufficiently large $n$, such that $\lvert \sigma\rvert^n y$ is small and $C(y)$ is approximated sufficiently well. This works because the (closed) windows are compact sets, so that their (inverse) Fourier transforms are continuous functions. The convergence of this approximation is exponentially fast. We refer to [@BG19c] for further details and an extension of the cocycle approach to more general inflation systems, and [@BG20] for a planar example.
From the general formula for regular model sets, the FB amplitudes are $$\label{eq:FB-h}
A^{}_{\!{\varLambda}_{ a,b}}(k) \, = \,
\frac{h^{}_{a,b}(k^{\star})}{\sqrt{5}} \, = \,
\frac{\tau}{\sqrt{5}}\, c^{}_{a,b}(k^{\star})$$ for $k\in L^{\circledast}$. So, the relevant input is the knowledge of the Fourier module, which determines where the Bragg peaks are located. Then, one can approximate $C$ by evaluating the matrix product in Eq. , for any chosen $k\in L^{\circledast}$, at $y = k^{\star}$ and with a sufficiently large $n$. In what follows, numerical calculations and illustrations are based on this cocycle approach due to its superior speed and accuracy in the presence of complex windows.
Fractally bounded windows {#sec:planar}
=========================
The internal cocycle approach of Section \[sec:cocycle\] was first applied to a ternary inflation tiling with the smallest Pisot–Vijayaraghavan (PV) number (also known as the ‘plastic number’) as its inflation multiplier [@BG20]. In the cut and project description, the internal space of this one-dimensional tiling is two-dimensional, and the windows are *Rauzy fractals* [@PF]. This means that the windows are still topologically regular, so the closure of their interior, but have a fractal boundary of zero Lebesgue measure. Consequently, the general diffraction result for model sets still applies, and the diffraction is given by the Fourier transform of the windows as described above. In turn, this means that the internal cocycle approach applies and can be used to compute the Fourier transforms and the diffraction intensities for such tilings; see [@BG20] for details.
Here, we discuss examples of planar projection tilings with fractally bounded windows, which are based on *direct product variations* (DPVs) of Fibonacci systems, as recently described in [@BFG21]. Clearly, if one considers a direct product structure based on the Fibonacci tiling, one obtains a tiling of the plane, called the *square Fibonacci tiling*. It is built from four prototiles, a large square of edge length $\tau$, a small square of edge length $1$, and two rectangles with a long ($\tau$) and a short ($1$) edge; see Figure \[fig:fibosq\].
![Patch of the square Fibonacci tiling.\[fig:fibosq\]](fibosq.eps){width="0.7\columnwidth"}
![Central part of the diffraction image of the square Fibonacci tiling.\[fig:fibosqdiff\]](fibosqdiff.eps){width="0.7\columnwidth"}
As a direct product of an inflation tiling, this two-dimensional square Fibonacci tiling also possesses an inflation rule, which takes the form $$\label{eq:fibosqrule}
\raisebox{-0.1\columnwidth}{\includegraphics[width=0.6\columnwidth]{fibosqrule.eps}}$$ where we labelled the small and large squares by $0$ and $3$, and the two rectangles by $1$ and $2$, respectively. A DPV is now obtained by modifying these rules while keeping the stone inflation character intact. Clearly, there are two possibilities to rearrange the images of the rectangles by swapping the two tiles, and a close inspection shows that there are altogether $12$ ways of rearranging the image of the large square. This means that there are $48$ distinct inflation rules in total, which all share these prototiles and the same inflation matrix.
Due to the direct product structure, the square Fibonacci tiling clearly possesses a cut and project description. The windows for the four prototiles are obtained as products of the original windows. The product structure thus extends to the diffraction measure, which is supported on the Fourier module $$L^{\circledast}{\hspace{-0.5pt}}\times{\hspace{0.5pt}}L^{\circledast}{\hspace{0.5pt}},$$ where $L^{\circledast}\, =\, \frac{1}{\sqrt{5}}{\mathbb{Z}}[\tau]$ is the Fourier module of the one-dimensional Fibonacci tiling. The diffraction amplitudes are also given by products of those for the one-dimensional system, and thus easy to compute. An illustration of the diffraction pattern is shown in Figure \[fig:fibosqdiff\]. Here, Bragg peaks are represented by disks, centred at the position of the peak, with an area proportional to its intensity.
It turns out that *all* $48$ DPV inflation tilings are regular model sets, and hence are pure point diffractive; see [@BFG21 Thm. 5.2]. They all share the same Fourier module $L^{\circledast}\times L^{\circledast}$. This implies that the Bragg peaks are always located at the same positions. However, their intensities are determined by the Fourier transform of the windows, and it turns out that the windows of these DPVs can differ substantially.
In particular, $20$ of these DPVs possess windows of Rauzy fractal type, of which there are three different types, called ‘castle’, ‘cross’ and ‘island’ in [@BFG21]. They have different fractal dimension of the window boundaries, which are approximately $1.875$, $1.756$ and $1.561$, respectively. As the dimensions are all smaller than two, is it obvious that these boundaries have zero Lebesgue measure.
![Castle-type window for the DPV . The windows for the four types of tiles are distinguished by colour, namely red ($0$), yellow ($1$), green ($2$) and blue ($3$). The outer boxes mark the square $[-\tau, \tau]^2$, with the coordinate axes indicated as well.\[fig:castle\]](castle.eps){width="0.7\columnwidth"}
![Diffraction image of the DPV .\[fig:castlediff\]](castlediff.eps){width="0.7\columnwidth"}
In what follows, we are going to illustrate some properties of these DPVs with three examples, one for each of these fractally bounded window types. The inflation rules for the three examples have the same images for the small square (tile $0$) and both rectangles (tiles $1$ and $2$) as the square Fibonacci rule of Eq. , and thus only differ in the image of the large square (tile $3$). For a discussion of the complete set of $48$ DPVs, we refer to [@BFG21].
![Cross-type window for the DPV .\[fig:cross\]](cross.eps){width="0.7\columnwidth"}
![Diffraction image of the DPV .\[fig:crossdiff\]](crossdiff.eps){width="0.7\columnwidth"}
For the castle-type windows of Figure \[fig:castle\], we use the inflation $$\label{eq:castle}
\raisebox{-0.07\columnwidth}{\includegraphics[width=0.4\columnwidth]{castlerule.eps}}$$ for the large square. Note that this rule dissects the inflated large square such that there is a reflection symmetry along the main diagonal, which will be reflected in a symmetry of the tiling (which maps the squares onto themselves and interchanges the rectangles). This is also apparent for the windows in Figure \[fig:castle\]. The windows for the large and small squares are mapped onto themselves under reflection at the main diagonal, while the windows for the rectangular tiles are interchanged. The diffraction pattern also respects this symmetry; an illustration is shown in Figure \[fig:castlediff\].
![Island-type window for the DPV .\[fig:island\]](island.eps){width="0.7\columnwidth"}
![Diffraction image of the DPV .\[fig:islanddiff\]](islanddiff.eps){width="0.7\columnwidth"}
For the cross-type windows, the inflation of the large square is given by $$\label{eq:cross}
\raisebox{-0.07\columnwidth}{\includegraphics[width=0.4\columnwidth]{crossrule.eps}}$$ which, in contrast to the previous example, has no reflection symmetry. Consequently, neither the windows shown in Figure \[fig:cross\] nor the diffraction image illustrated in Figure \[fig:crossdiff\] have any reflection symmetry.
The same is true for the final example with the island-type window shown in Figure \[fig:island\]. This corresponds to the inflation $$\label{eq:island}
\raisebox{-0.07\columnwidth}{\includegraphics[width=0.4\columnwidth]{islandrule.eps}}$$ of the large square tile. The corresponding diffraction pattern is illustrated in Figure \[fig:islanddiff\].
Comparing the diffraction patterns of Figures \[fig:castlediff\], \[fig:crossdiff\] and \[fig:islanddiff\] with those of the square Fibonacci tiling shown in Figure \[fig:fibosqdiff\], we note that the strongest peaks remain pretty much the same, while the intensities of the weaker peaks show some intriguing behaviour. For the fractally-bounded windows, one generally sees more peaks, which is due to the larger spread of the window in internal space, and the slower asymptotic decay of the Fourier transform of the window. With limited resolution, some of the intensity distributions on these peaks could resemble continuous components, so might potentially be mistaken as such in experiments.
Diffraction and hyperuniformity {#sec:hyper}
===============================
The discovery of quasicrystals highlighted the lack of a clear definition of the concept of *order*. In crystallography, diffraction is the main tool to detect long-range order, and a pure point diffraction is generally associated to an ordered, (quasi)crystalline structure, while absolutely continuous diffraction is typically seen as an indication of random disorder (but see [@Frank03; @BG09; @CG; @CGS] for examples of deterministic structures that show absolutely continuous diffraction). Here, we briefly discuss a related concept that has recently gained popularity.
From the original idea of using the degree of ‘(hyper)uniformity’ in density fluctuations in many-particle systems [@TS] to characterise their order, the *scaling* behaviour of the total diffraction intensity near the origin has emerged as a possible measure to capture long-distance correlations. As far as aperiodic structures are concerned, there are in fact a number of early, partly heuristic, results in the literature [@Luck; @Aubry; @GL]. These have recently been reformulated and extended [@Josh1; @Josh2] and rigorously established [@BG19b], using exact renormalisation relations for primitive inflation rules [@BFGR; @BG16; @Neil; @BGM19; @BGM18; @NeilDiss].
For the investigation of scaling properties, we follow the existing literature and define $$\label{eq:Z-def}
Z (k) \, {\mathrel{\mathop:}=}\, \widehat{\gamma} \bigl( (0,k]\bigr) ,$$ which is a modified version of the *distribution function* of the diffraction measure. Here, $Z(k)$ is the total diffraction intensity in the half-open interval $(0,k]$, and thus ignores the central peak. Due to the reflection symmetry of $\widehat{\gamma}$ with respect to the origin, this quantity can also be expressed as $$Z (k) \, = \, {\frac{\raisebox{-2pt}{$1$}}
{\raisebox{0.5pt}{$2$}}}
\Bigl(\widehat{\gamma} \bigl( [-k,k]\bigr) -
\widehat{\gamma} \bigl( \{0\}\bigr) \Bigr).$$ The interest in the scaling of $Z(k)$ as $k\to 0$ is motivated by the intuition that the small-$k$ behaviour of the diffraction measure probes the long-wavelength fluctuations in the structure. As the latter is related to the variance in the distribution of patches, it can serve as an indicator for the degree of uniformity of the structure [@TS]. It is obvious that any periodic structure leads to $Z(k) = 0$ for all sufficiently small wave number $k$.
Here, we review the result for variants of the one-dimensional Fibonacci model sets considered above, where we now allow for changes of the windows. For a general discussion of this approach and more examples of systems with different types of diffraction, we refer to [@BG19b] and references therein.
Let us look at the diffraction for a cut and project set with the same setup as the Fibonacci tiling considered in Section \[sec:standard\], but with the window $W$ replaced by an arbitrary finite interval of length $s$. Note that these tilings, in general, do *not* possess an inflation symmetry. Nevertheless, the diffraction intensity is still of the form , but now featuring the interval length $s$, and is given by $$I(k) \, = \, I(0) \bigl(\operatorname{sinc}(\pi s k^{\star})\bigr)^2$$ for all $k\in L^{\circledast}$. Now, consider a sequence of positions $\tau^{-\ell} k$ with $k \in L^{\circledast}$ and $\ell\in{\mathbb{N}_{0}^{}}$. Since we have $\operatorname{sinc}(x) = \sin(x)/x = {\mathcal{O}}(x^{-1} )$ as $x \to\infty$, it follows that $I ( \tau^{-\ell}k ) = {\mathcal{O}}\bigl(
\tau^{-2\ell} \bigr)$ as $\ell\to\infty$.
Consequently, the sum of intensities along the series of peaks, $${\varSigma}(k)\, =\, \sum_{\ell=0}^{\infty} I(\tau^{-\ell}k){\hspace{0.5pt}},$$ satisfies the asymptotic behaviour $${\varSigma}(\tau^{-\ell} k) \, \sim\, c(k)\, \tau^{-2\ell}\, {\varSigma}(k)$$ as $\ell\to\infty$, where it can be shown that $c(k) = {\mathcal{O}}(1)$ [@BG19b]. Expressing $Z(k)$ in terms of these sums gives $$Z(k) \; = \! \sum_{\substack{\;\kappa\in L^{\circledast} \\
\frac{k}{\tau} < \kappa \leqslant k }} \!\! {\varSigma}(k){\hspace{0.5pt}},$$ which implies the asymptotic behaviour $$Z(\tau^{-\ell} k) \, = \, \tau^{-2\ell}\, Z(k){\hspace{0.5pt}}.$$ This leads to a scaling behaviour of the form $Z(k) = {\mathcal{O}}(k^2)$ as $k\,\raisebox{2pt}{$\scriptscriptstyle \searrow$}\, 0$.
This generic result remains true if we choose a window which corresponds to a tiling with inflation symmetry, which requires the window to be an interval of length $s\in{\mathbb{Z}}[\tau]$. This obviously holds for our original Fibonacci window $W$ of length $\tau$. However, one gets a stronger result for this case [@BG19b; @Josh1], as we shall now recall.
Choosing $s\in{\mathbb{Z}}[\tau]$ means $s=a+b \tau$ with $a,b\in{\mathbb{Z}}$. For $0\ne
k\in L^{\circledast}$, set $k = \kappa/\mbox{\small $\sqrt{5}$}$ with $\kappa = m + n \tau$ for some $m,n \in {\mathbb{Z}}$, excluding $m=n=0$. Applying the $\star$-map then gives $$I(\tau^{-\ell} k ) \, = \, I(0) \,
\biggl(\operatorname{sinc}\Bigl( {\frac{\raisebox{-2pt}{$\pi \tau^{\ell} s {\hspace{0.5pt}}\kappa^{\star}$}}
{\raisebox{0.5pt}{$\mbox{\small $\sqrt{5}$}$}}}
\Bigr)\biggr)^2 ,$$ with $\ell\in{\mathbb{N}}_0$.
Now, denote by $f_n$ with $n\in{\mathbb{Z}}$ the Fibonacci numbers defined by $f^{}_{0} = 0$, $f^{}_{1} = 1$ and the recursion $f^{}_{n+1} =
f^{}_{n} + f^{}_{n-1}$. They satisfy the well-known formula $$\label{eq:fib-form}
f^{}_n \, = \, {\frac{\raisebox{-2pt}{$1$}}
{\raisebox{0.5pt}{$\mbox{\small $\sqrt{5}$}$}}} \,
\Bigl( \tau^n - \bigl( -1/\tau\bigr)^n \Bigr)$$ for all $n\in{\mathbb{Z}}$. Using this relation, we obtain $$\label{eq:fiboexp}
\begin{split}
\sin \Bigl( {\frac{\raisebox{-2pt}{$\pi \tau^{\ell} s {\hspace{0.5pt}}\kappa^{\star}$}}
{\raisebox{0.5pt}{$\mbox{\small $\sqrt{5}$}$}}}
\Bigr)^2 & = \:
\sin \Bigl( {\frac{\raisebox{-2pt}{$\pi {\hspace{0.5pt}}\lvert s {\hspace{0.5pt}}\kappa^{\star} \rvert$}}
{\raisebox{0.5pt}{$\mbox{\small
$ \sqrt{5}$}$}}} \, \tau^{-\ell} \Bigr)^2 \\
& = \, {\frac{\raisebox{-2pt}{$\pi^2 ( s {\hspace{0.5pt}}\kappa^{\star} )^2$}}
{\raisebox{0.5pt}{$5$}}}
\, \tau^{-2\ell} \, + {\mathcal{O}}\bigl( \tau^{-6 \ell}\bigr)
\end{split}$$ as $\ell\to\infty$. Here, the first step follows by using Eq. to replace $\tau^{\ell}/\mbox{\small $\sqrt{5}$}$ and then reducing the argument via the relation $$\sin (m \pi + x) \, = \, (-1)^m \sin (x){\hspace{0.5pt}},$$ which holds for all $m\in{\mathbb{Z}}$ and $x\in{\mathbb{R}{\hspace{0.5pt}}}$. This is possible because all Fibonacci numbers are integers. The second step then uses the Taylor approximation $\sin (x) = x + {\mathcal{O}}(x^3)$ for small values of $x$.
Now, the same argument as above implies the asymptotic behaviour $$Z( \tau^{-\ell} k )
\, \asymp \tau^{-4\ell}\, Z(k) {\hspace{0.5pt}},$$ and hence $Z(k)={\mathcal{O}}(k^4)$. This results means that, for inflation-invariant projection sets, the distribution function $Z(k)$ of the diffraction intensity vanishes like $k^4$ as $k\,\raisebox{2pt}{$\scriptscriptstyle \searrow$}\, 0$, while in the generic case we find a $k^2$-behaviour. This example illustrates that the behaviour of the diffraction intensity near $0$ can pick up non-trivial aspects of order in this system. This is illustrated for some cases in Figure \[fig:fibopeak\].
![Double logarithmic plot of the intensity ratio $I/I_{0}$ of Bragg peaks located at $k=(m+n\tau)/\sqrt{5}$ with $\max(|m|,|n|)\leqslant 10^4$, where $I_{0}=I(0)$, for windows $W$ of different lengths. The dashed line corresponds to $k^4$ for $\lvert
W\rvert=\tau$ (top) and to $k^2$ for the other two cases.\[fig:fibopeak\]](fibpeaktau4.eps){width="0.9\columnwidth"}
![Double logarithmic plot of the intensity ratio $I/I_{0}$ of Bragg peaks located at $k=(m+n\tau)/\sqrt{5}$ with $\max(|m|,|n|)\leqslant 10^4$, where $I_{0}=I(0)$, for windows $W$ of different lengths. The dashed line corresponds to $k^4$ for $\lvert
W\rvert=\tau$ (top) and to $k^2$ for the other two cases.\[fig:fibopeak\]](fibpeakpi4.eps){width="0.9\columnwidth"}
![Double logarithmic plot of the intensity ratio $I/I_{0}$ of Bragg peaks located at $k=(m+n\tau)/\sqrt{5}$ with $\max(|m|,|n|)\leqslant 10^4$, where $I_{0}=I(0)$, for windows $W$ of different lengths. The dashed line corresponds to $k^4$ for $\lvert
W\rvert=\tau$ (top) and to $k^2$ for the other two cases.\[fig:fibopeak\]](fibpeaksq4.eps){width="0.9\columnwidth"}
Let us briefly comment on the scaling behaviour for other prominent examples of aperiodic order discussed in [@BG19b]. For noble means inflations, we observe the same $k^4$-scaling as for the Fibonacci tiling. The period doubling sequence, which is limit periodic, shows $k^2$-scaling, and a range of scaling exponents is accessible for substitutions of more than two letters. For the Thue–Morse sequence, which is the paradigm of an inflation structure with singular continuous diffraction, we do not obtain a power law, but an exponential scaling behaviour which decays faster than any power; see also [@BGKS] for more on the scaling of the spectrum for this system. Finally, the Rudin–Shapiro sequence, which has absolutely continuous spectrum, shows a linear scaling behaviour, due to the constant density of its diffraction measure.
Acknowledgements {#acknowledgements .unnumbered}
================
It is our pleasure to thank Claudia Alfes-Neumann, Natalie Priebe Frank, Neil Mañibo, Bernd Sing, Nicolae Strungaru and Venta Terauds for valuable discussions. This work was supported by the German Research Foundation (DFG), within the CRC 1283 at Bielefeld University, and by EPSRC, through grant EP/S010335/1.
[111]{}
|
{
"pile_set_name": "ArXiv"
}
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---
abstract: |
A Calderón projector for an elliptic operator $P$ on a manifold with boundary $X$ is a projection from general boundary data to the set of boundary data of solutions $u$ of $Pu=0$. Seeley proved in 1966 that for compact $X$ and for $P$ uniformly elliptic up to the boundary there is a Calderón projector which is a pseudodifferential operator on $\partial X$. We generalize this result to the setting of fibred cusp operators, a class of elliptic operators on certain non-compact manifolds having a special fibred structure at infinity. This applies, for example, to the Laplacian on certain locally symmetric spaces or on particular singular spaces, such as a domain with cusp singularity or the complement of two touching smooth strictly convex domains in Euclidean space. Our main technical tool is the $\phi$-pseudodifferential calculus introduced by Mazzeo and Melrose.
In our presentation we provide a setting that may be useful for doing analogous constructions for other types of singularities.
address:
- 'Institut für Analysis, Leibniz Universität Hannover'
- 'Institut für Mathematik, Carl von Ossietzky Universität Oldenburg'
- 'Institut für Analysis, Leibniz Universität Hannover'
author:
- Karsten Fritzsch
- Daniel Grieser
- Elmar Schrohe
bibliography:
- 'D-N-cusp.bib'
title: The Calderón Projector for Fibred Cusp Operators
---
Introduction
============
A Calderón projector associated with an elliptic partial differential operator $P$ of order $m$ on a compact manifold $X$ with non-empty boundary ${{\partial X}}$ is a projection $C$ in $\Cinf(\partial X)^m$ to the set of boundary data of solutions of the homogeneous equation $$\label{eqn:cauchy data space}
\{(u_{|\partial X},{D_\nu}u_{|\partial X},\dots, {D_\nu}^{m-1} u_{|\partial X}):\, u\in\Cinf(X), Pu = 0\} \,.$$ Here $\nu$ is some choice of transversal vector field in a neighborhood of the boundary and ${D_\nu}= \frac{1}{i}\partial_\nu$.
It was first observed by Calderón [@Cal63] that such a projection exists which is a pseudodifferential operator, with an explicit principal symbol, and that this can be used to study boundary value problems. The first complete proof was given by Seeley in 1966, [@See66; @See69]. The result applies more generally to operators acting between sections of vector bundles.
In the present paper we extend this result to certain settings where the boundary has singularities, that is, we construct Calderón projectors with full control of their behavior near the singularities. We consider certain classes of singularities, often called of fibred cusp type, but our purpose is also to present this construction in a systematic way to lay the foundation for generalizing the analysis to other types of singularities. Typical examples to which our results apply are the Laplace-Beltrami or Dirac type operators on Riemannian manifolds with boundary which locally are of one of the following types:
1. Domains with (incomplete) cusp singularity such as $\{(\xi,\eta)\in\R^m\times\R^k: |\xi|\leq |\eta|^2\}$ near $(\xi,\eta)=0$ where $m,k\in\N$; for $k=1$ this is what is commonly called an incomplete cusp; for $m=1$ this has the same geometry as the complement of two touching strictly convex sets in $\R^{k+1}$ as in Figure \[fig:blow-up example\] left;
2. certain types of domains in locally symmetric spaces of $\Q$-rank one, for example the strip given by $\{|\Re w| \leq \frac14,\ \Im w \geq 1\}$ in the complex plane with the hyperbolic metric (considered locally near infinity),
3. spaces of the form $C\times F$, where $C$ is the far end of a cone (e.g. the outside of a ball in $\R^n$) and $F$ is a compact Riemannian manifold with non-empty boundary, or more generally fibre-bundles of this sort.
In the second and third example the singularity of the boundary is at infinity, so here we give a uniform description of the behavior of the Calderón projector at infinity. We also allow spaces to have several such singularities of different types, as well as similar types of singularities away from the boundary, for example the exterior of a smooth bounded domain in $\R^n$. More details on these examples are provided in Section \[ssec:ex\], and the precise class of spaces and operators is described below.
The main motivation for the study of Calderón projectors is their use in the analysis of regularity and Fredholm properties of boundary value problems. In particular, if $P$ is the Laplace-Beltrami operator $\Delta$ associated to some Riemannian metric on a compact manifold $X$ with boundary, then Seeley’s result implies, by standard pseudodifferential calculus techniques, that the Dirichlet-Neumann operator, defined by $$\label{eqn:def DN op}
{\mathcal{N}}: \Cinf(\partial X)\to\Cinf(\partial X),\ f \mapsto \partial_\nu u\ \text{ where } \ \Delta u = 0, u_{|\partial X} = f \,,$$ with the outward unit normal derivative $\partial_\nu$, is a pseudodifferential operator of order 1, with principal symbol $|\xi|$.
We will apply our results on Calderón projectors to boundary value problems and the Dirichlet-Neumann operator in the fibred cusp setting in a separate paper. The Dirichlet-Neumann operator occurs in a number of contexts, such as the Calderón inverse problem or the theory of water waves. It also appears in the context of the plasmonic eigenvalue problem, [@Gri14], and here the fibred cusp geometry is of particular importance. This two-sided boundary value problem describes the coupling of electromagnetic fields to the electron gas of a conducting body. Since the geometry of the body can be used to specifically tailor properties of the resulting surface waves, its solutions on more singular spaces have seen great interest in recent years. See for instance [@GR09; @GUB+09; @Sav12; @CCN18; @Sch18] and the references therein. If this body consists of two (nano-meter sized) balls that touch each other then we are led precisely to the geometry studied in this paper. This is also considered in [@Sch18] and the first author’s thesis [@Fri14]. In the latter, the method of layer potentials is studied in the light of manifolds with corners and conormal distributions. See also [@Fri19] for the geometrically simpler case of the half-space.
Main theorems {#subsec:main theorems}
-------------
The differential operators we consider have, in suitable local coordinates near the singularities, the form $$\label{eqn:phi-op local coords}
P = x^{-cm} \sum_{k+|\alpha|+|\beta|\leq m} a_{k\alpha\beta}(x,y,z) (x^2 D_x)^k (x D_y)^\alpha D_z^\beta \,.$$ Here $x>0$, $y=(y_1,\dots,y_b)\in U\subset\R^b$, $z=(z_1,\dots,z_f)\in V$ where $V\subset \R^f$ or $V\subset[0,\infty)\times\R^{f-1}$ with $b\in\N_0$, $f\in\N$ and open subsets $U$, $V$. As usual, we let $D_x=\frac1i\partial_x$ etc. The singularity corresponds to $x=0$, and the case $V\subset[0,\infty)\times\R^{f-1}$ corresponds to a neighborhood of the boundary, which then is $z_1=0$. Also, $m\in\N$ is the order of $P$ and $c\in\Z$ is a parameter. The coefficients $a_{k\alpha\beta}$ are assumed to be smooth up to $x=0$, and $P$ is assumed to be elliptic, in a uniform sense as $x\to0$ to be described later. The singular structure of the operator is reflected in the $x$-factors in $x^2D_x$ and $xD_y$, and to a lesser extent in the prefactor $x^{-cm}$.
Globally it is useful to describe $P$ as acting on a compact manifold with corners $X$ which has two types of boundary hypersurfaces: the ‘singular boundary’ ${{\partial_{{\mathrm{s}}}X}}$, given by $x=0$, and the ‘boundary at which boundary conditions could be imposed’, the ‘[[<span style="font-variant:small-caps;">bc</span>]{}]{}-boundary’ denoted by ${{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}$ and given locally by $z_1=0$. These intersect in their common boundary ${{\partial_{{\mathrm{s}},{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}} X}}$. Note that $P$ acts on functions defined on $X\setminus{{\partial_{{\mathrm{s}}}X}}$ only, but adding the boundary ${{\partial_{{\mathrm{s}}}X}}$ to the space allows us to express the singular behavior of $P$ and other objects efficiently. The difference in the roles of the $y$ and $z$ variables in is geometrically and globally described by an additional piece of data: a fibration of the singular boundary $$F - {{\partial_{{\mathrm{s}}}X}}\overset{\phi}\to B$$ with compact base $B$ and compact fibre $F$. Locally, $\phi$ is just the map $\phi:(y,z)\mapsto y$, so $y$ are base coordinates and $z$ are fibre coordinates. The fibre $F$ has boundary given locally by $z_1=0$. The restriction of $\phi$ to $\partial({{\partial_{{\mathrm{s}}}X}}) = {{\partial_{{\mathrm{s}},{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}} X}}$ is again denoted by $\phi$ and defines a fibration $\partial F - {{\partial_{{\mathrm{s}},{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}} X}}\overset{\,\phi\,}\to B$. We call such a space $X$ a $\phi$-[[<span style="font-variant:small-caps;">bc</span>]{}]{}-**manifold** or $\phi$-**manifold with** [[<span style="font-variant:small-caps;">bc</span>]{}]{}-**boundary** and an operator as in , with $c=0$, a **$\phi$-differential operator** or short **$\phi$-operator**. The definition extends in a straightforward way to operators acting between sections of vector bundles $E,E'$ over $X$. The class of these operators of order $m$ is denoted $$\Diff_\phi^m(X;E,E') \,.$$ See Section \[sec:setting\] and Appendix \[sec:app mwc\] for basics on manifolds with corners and more details, including how the examples above fit into this framework.
In the case that $X$ has empty [[<span style="font-variant:small-caps;">bc</span>]{}]{}-boundary, i.e. if the fibres of $\phi$ do not have a boundary, the class of $\phi$-operators was introduced and studied by Mazzeo and Melrose [@MM98] (see also [@GH09; @GH14]). They showed that interior regularity extends to conormal boundary regularity if the operator is **$\phi$-elliptic**, i.e. its $\phi$-principal symbol is invertible, and that $P$ is Fredholm in naturally associated $L^2$ spaces if and only if it satisfies the stronger condition of being **fully elliptic**, which in addition requires invertibility of a family of differential operators, called **normal family** of $P$, on the fibres of $\phi$. These results are proved via construction of a pseudodifferential operator ($\Psi$DO) calculus adapted to the singular structure. The space of such $\phi$-$\Psi$DOs of order $m\in\R$ is denoted $\Psi^m_\phi(X)$. Its definition and properties are recalled in Appendix \[sec:app phi ops\].
In the setting of $\phi$-[[<span style="font-variant:small-caps;">bc</span>]{}]{}-manifolds with non-empty [[<span style="font-variant:small-caps;">bc</span>]{}]{}-boundary, ${{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}$ with the restricted fibration is a $\phi$-manifold (without [[<span style="font-variant:small-caps;">bc</span>]{}]{}-boundary), so $\Psi^*_\phi({{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}})$, the space of $\phi$-$\Psi$DOs with respect to the fibration $\phi : \partial({{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}) = {{\partial_{{\mathrm{s}},{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}} X}}\rightarrow B$, is defined.
When considering Calderón projectors on a singular or non-compact space we need to specify the growth behavior at the singular set which we allow for the functions $u$ and for the vector field $\nu$ in . The geometrically natural condition on $\nu$ is that it is a $\phi$-vector field, see , transversal to ${{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}$. For the functions $u$ there are various ways to restrict the growth behavior at ${{\partial_{{\mathrm{s}}}X}}$. We use the letter $\calF$ to denote any choice of function space encoding such behavior. Thus, we have spaces $\calF(X)$, $\calF({{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}})$ of functions on $X$ and ${{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}$, and similarly spaces of sections of vector bundles. We call $\calF$ **admissible** if it behaves well under restriction to ${{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}$ and under the action of $\phi$-$\Psi$DOs, see Definition \[def:admissible function space\]. Also, we require all functions to be smooth in $X\setminus{{\partial_{{\mathrm{s}}}X}}$. A simple example is $\calF(X) = x^\alpha \Cinf(X)$, where $\alpha\in\R$. This would allow $x^{\alpha}$ decay (if $\alpha>0$) or growth (if $\alpha<0$) at the singular set. Other choices include spaces characterized by $L^2$- or $L^\infty$-based bounds (conormality) or full asymptotic expansions (polyhomogeneity) at ${{\partial_{{\mathrm{s}}}X}}$, introduced in Appendix \[sec:app mwc\]. The smallest choice of $\calF$ is ${{\dot C^\infty_{{\mathrm{s}}}}}$, the space of functions vanishing to infinite order at ${{\partial_{{\mathrm{s}}}X}}$ (in the geometric setting (iii) this corresponds to functions rapidly decreasing at infinity with all derivatives), and the largest choice is $\calA_{\mathrm{s}}$, the space of functions conormal at ${{\partial_{{\mathrm{s}}}X}}$.
We fix a $\phi$-vector field $\nu$ on $X$ transversal to ${{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}$ throughout the paper. Then for any $m \in \N$ we have the boundary data map for admissible $\calF$ $$\label{eqn:first.gamma}
\gamma : \calF(X) \longrightarrow \calF({{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}})^m \,,\quad
u \longmapsto \gamma u = \big(u_{|{{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}},{D_\nu}u_{|{{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}}, \dotsc, {D_\nu}^{m-1}u_{|{{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}}\big) \,.$$ In order to define $\gamma$ on sections of a bundle $E$, we also need to chose a connection on $E$ if $m>1$. We denote the **$\calF$-boundary data space** of $P \in \Diff_\phi^m(X;E,E')$ by $$\label{eqn:first.bdy.data}
\calB_{P,\calF} :=\{\gamma u :\, u\in\calF(X;E),\ Pu=0\} \subset \calF({{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}};E)^m$$ and we define an **$\calF$-Calderón projector** for $P$ to be a projection in $\calF({{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}};E)^m$ to $\calB_{P,\calF}$. Note that there are many such projections.
A particular issue in dealing with the Calderón projector is that $P$ may have **shadow solutions**, i.e. sections $u\not\equiv0$ satisfying $Pu=0$ whose boundary data at ${{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}$ vanish. This is a type of failure of unique continuation for $P$. While this is not a problem for $P$ itself, we need to exclude it for the normal families of $P$ and its adjoint $P^\star$. A precise formulation is given in Assumption \[UCNF\]. Our main result is:
\[thm:Calderon\] Let $X$ be a $\phi$-manifold with non-empty [[<span style="font-variant:small-caps;">bc</span>]{}]{}-boundary and $E,E'$ be complex vector bundles over $X$. Let $c\in\Z$ and $$P =x^{-cm}\tilde P,\ \tilde P\in\Diff_\phi^m(X;E,E') \,,$$ where $\tilde P$ is $\phi$-elliptic and satisfies Assumption \[UCNF\]. Then there is an operator $C\in\Psi^*_\phi({{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}};E^m)$ which for any choice of admissible function space $\calF$ is an $\calF$-Calderón projector for $P$.
Considering $C$ as acting between $m$-tuples of sections of $E$, the operator $C$ is an $m\times m$ matrix $(C_{kl})_{k,l=1\dots m}$ where $C_{kl} \in \Psi_\phi^{k-l}({{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}};E)$.
Theorem \[thm:Calderon\] is proven in Subsection \[ssec:Calderon\]. We also give an explicit description of the $\phi$-symbol and the normal family of $C$, see Propositions \[prop:Calderon.symbol\] and \[prop:Calderon.normal\]. All pseudodifferential operators in this paper are classical, i.e. their symbols have complete expansions in positively homogeneous terms.
The different choices of $\calF$, i.e. of growth behavior at the singular boundary, yield different boundary data spaces $\calB_{P,\calF}$. However, from an $L^2$ perspective these spaces are not very different: Recall the definition of $L^2_\phi$ and of the $\phi$-Sobolev space $H^k_\phi$ for a $\phi$-manifold (without [[<span style="font-variant:small-caps;">bc</span>]{}]{}-boundary), see Appendix \[sec:app mwc\]. The space $$\calH= \bigoplus_{k=0}^{m-1}H^k_\phi({{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}},E)$$ is a natural space for boundary data in the $L^2$-setting, which for $m=1$ reduces to $L^2_\phi({{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}},E)$. Any operator $C$ as constructed in Theorem \[thm:Calderon\] is a bounded projection $\calH\to\calH$. We call its range the **$L^2$-boundary data space** of $P$ and denote it by $\calB_{P,L^2}$. The following corollary shows in particular that this space does not depend on the choice of $C$.
\[cor:L2 closure of BC spaces\] Let $X$ and $P$ be as in Theorem \[thm:Calderon\]. If $\calF({{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}) \subset L^2_\phi({{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}})$ then $\calB_{P,\calF}\subset\calH$, and its closure is equal to $\calB_{P,L^2}$.
In particular, the closure of $\calB_{P,\calF}$ in $\calH$ is the same for all such choices of $\calF$. The smallest $\calF$ satisfying $\calF({{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}) \subset L^2_\phi({{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}})$ is ${{\dot C^\infty_{{\mathrm{s}}}}}$, other choices include $\calF=x^\alpha\Cinf$ with $\alpha>\frac {\dim B}2+1$. There is also a more general statement with $L^2_\phi$ and $\calH$ replaced by the weighted spaces $x^\beta L^2_\phi$, $x^\beta\calH$ for any $\beta\in\R$.
We now discuss questions of uniqueness and canonical choices of a Calderón projector. This is relevant for instance when considering parameter-dependent problems or problems invariant under a group action. In general, there is no canonical choice of Calderón projector $C$. This is reflected in the fact that the construction of $C$ in the proof of Theorem \[thm:Calderon\] involves several choices. However, we have a uniqueness result:
\[prop:Calderon.unique\] The construction used in the proof of Theorem \[thm:Calderon\] determines $C$ uniquely modulo $\Psi^{-\infty}_\phi({{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}},E^m)$.
The full $\phi$-symbol of $C$, i.e. the element $[C]$ of $\Psi^*_\phi({{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}},E^m)/\Psi^{-\infty}_\phi({{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}},E^m)$ fixed in this way, is determined constructively by the infinite order jet of $P$ at ${{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}$, i.e. by its equivalence class modulo ${{\dot C^\infty_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}}}(X)\Diff^m_\phi(X,E)$. Here ${{\dot C^\infty_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}}}(X)$ is the space of smooth functions on $X$ vanishing to infinite order at ${{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}$.
The normal family of $C$ can not be expected to be uniquely determined since it is a Calderón projector for the normal family of $P$, and again there is no canonical choice of such a projector. So the space $\Psi^{-\infty}_\phi({{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}},E^m)$ can not be replaced by $x\Psi^{-\infty}_\phi({{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}},E^m)$. Proposition \[prop:Calderon.unique\] is proven in Subsection \[ssec:Calderon.properties\]. Interestingly, the proof shows that it is much easier to construct the equivalence class $[C]$ than an operator $C$ which is an actual projection to the boundary data space.
With additional data there are also ways to distinguish special choices of $C$. For simplicity we consider the $L^2$-setting. Since only the range of $C$ is determined by the given data $X$ and $P$ (and $\nu$), the possible projections $C$ are parametrized by their kernels, which must be subspaces of $\calH$ complementary to the $L^2$-boundary data space of $P$. There are (at least) two approaches in the literature how to choose additional data to fix a kernel and hence a projection, in the setting of a compact manifold with boundary:
1. If an extension of $X$ to a closed manifold $\Xhat$, as well as extensions of the bundles $E,E'$ to $\Xhat$ and of $P$ to an elliptic, *invertible* operator $\Phat$ on $\Xhat$ are given, then the boundary data space for $\Phat$ from the other side, $\Xhat\setminus X$, is complementary to the boundary data space of $P$ on $X$, and the corresponding projection is a Calderón projector.
2. If one chooses additional geometric data at ${{\partial X}}$ which defines an $L^2$ scalar product then one may look at the *orthogonal* projector.[^1]
The projectors in both cases are pseudodifferential. The one in (i) is in the class described in Proposition \[prop:Calderon.unique\], but the one in (ii) will in general even have a different principal symbol. We start by extending (i) to the singular setting. We first remark on the notion of invertibility. By the general theory of $\phi$-pseudodifferential operators, the following conditions on a fully elliptic operator $\Phat\in\Psi^m_\phi(\Xhat)$ on a $\phi$-manifold $\Xhat$ are equivalent:[^2]
- $\Phat$ is invertible in $\Psi^*_\phi(\Xhat)$, i.e. there is $Q\in\Psi^{-m}_\phi(\Xhat)$ so that $\Phat Q=Q\Phat = I$.
- $\Phat:H^m_\phi(\Xhat)\to L^2_\phi(\Xhat)$ is invertible.
- $\Phat: \calF(\Xhat)\to \calF(\Xhat)$ is invertible for any choice of admissible function space $\calF$.
If this is satisfied, we simply say that $\Phat$ is invertible. For the following theorem recall the definition of $\calH$ and $\calB_{P,L^2}$ before Corollary \[cor:L2 closure of BC spaces\].
\[thm:calderon canonical inv extension\] Assume that $P$ has an extension to an invertible, fully elliptic $\phi$-$\Psi$DO $\Phat$ on a $\phi$-manifold (without [[<span style="font-variant:small-caps;">bc</span>]{}]{}-boundary) $\Xhat$ extending $X$ across ${{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}$, acting between extended bundles $E,E'$. Denote by $\calB_{P,L^2}^+$, $\calB_{P,L^2}^-$ the $L^2$-boundary data spaces of $P$ and of $\Phat_{|\Xhat\setminus X}$, respectively. Then
- $\calB_{P,L^2}^+ \oplus\calB_{P,L^2}^- = \calH$.
- The projection $C_{\Phat}$ in $\calH$ with range $\calB_{P,L^2}^+$ and kernel $\calB_{P,L^2}^-$ is a $\phi$-$\Psi$DO as in Theorem \[thm:Calderon\], and its normal family is the Calderón projector analogously defined using the $\pm$-boundary data spaces of the normal family of $\Phat$.
- If $\calF$ is any admissible function space then $\calB_{P,\calF}^+ \oplus\calB_{P,\calF}^- = \calF({{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}},E)^m$ and $C_{\Phat}$ acts as a projection in $\calF({{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}},E)^m$ with range $\calB_{P,\calF}^+$ and kernel $\calB_{P,\calF}^-$.
Note that $\Phat$ is allowed to be a *pseudo*differential operator. See Section \[ssec:extensions\] for a precise definition of ‘extension’, which implies in particular that the restriction of $\Phat$ to $\Xhat\setminus X$ makes sense also in this case. Theorem \[thm:calderon canonical inv extension\] is proven in Subsection \[ssec:Calderon\]. For example, it applies to Laplace-Beltrami operators for metrics $x^{2c}g$ where $g$ is a $\phi$-metric, as in the examples (i)-(iii) above, see Subsection \[ssec:laplacian.extension\]. See the paragraph around for the definition of $\phi$-metrics.
We now consider (ii), i.e. orthogonal Calderón projectors. For simplicity we only consider $m=1$, which includes the important class of Dirac type operators.
\[thm:orthogonal\] In the setting of Theorem \[thm:Calderon\] assume a $\phi$-metric is given on $X$, as well as a hermitian metric on $E$. Assume $m=1$. Then the orthogonal projection in $L^2_\phi({{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}};E)$ to $\calB_{P,L^2}$ is in $\Psi^0_\phi({{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}},E)$.
See Proposition \[thm:orth.Calderon\] for a more precise statement and the proof, including the calculation of the principal symbol and normal family. Note that the principal symbol is, in general, different from that of the projector in Theorem \[thm:Calderon\].
We remark that both the singular boundary ${{\partial_{{\mathrm{s}}}X}}$ and the [[<span style="font-variant:small-caps;">bc</span>]{}]{}-boundary ${{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}$ are allowed to have several connected components, see Definition \[def:bv-mfd\]. For ${{\partial_{{\mathrm{s}}}X}}$ this would correspond to several singularities. In fact, the dimensions of fibres and bases could vary from component to component, and some components may have empty intersection with ${{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}$, so that their fibres have empty boundary. These would correspond to interior singularities (in contrast to boundary singularities). Assumption \[UCNF\] localizes to each component of ${{\partial_{{\mathrm{s}}}X}}$, and at components not intersecting ${{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}$ it is equivalent to the operator $P$ being fully elliptic there, and is also a necessary condition for our theorem to be provable within the $\phi$-calculus.
On the other hand, it is not clear whether our theorems also hold without Assumption \[UCNF\]. At least our methods do not carry over to that case, see Remark \[rem:need UCNF\].
A side result of our analysis is the following fact. Recall that $u\not\equiv0$ is called a shadow solution of $Pu=0$ if it is smooth and has zero boundary data at ${{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}$.
\[prop:shadow solutions\] Let $X$ and $P$ be as in Theorem \[thm:Calderon\]. Then any shadow solution of $Pu=0$ is rapidly decreasing at the singular boundary ${{\partial_{{\mathrm{s}}}X}}$; more precisely, it lies in the space ${{\dot C^\infty_{{\mathrm{s}},{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}}}}(X;E)$ defined in Subsection \[ssec:function.spaces\].
Finally, we remark that, as in the classical constructions by Seeley and by Hörmander, the operator $P$ only needs to be differential *near* ${{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}$, i.e. $P$ is an elliptic $\phi$-$\Psi$DO and there is a neighborhood $U$ of ${{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}$ on which $P$ is a $\phi$-differential operator, and $\supp Pu \subset X\setminus U$ whenever $\supp u\subset X\setminus U$.
Outline of the paper {#ssec:outline}
--------------------
In Section \[sec:setting\] we introduce the setting in detail and explain how the examples mentioned above fit into this framework. The proof of Theorem \[thm:Calderon\] proceeds along the lines of Seeley’s proof in the non-singular case, [@See66] and [@See69], combined with the construction by H[ö]{}rmander, [@Hor85 Sec. 20]. In Section \[sec:outline\] we give an outline of this construction. In Section \[sec:aug-ext-mod\] we introduce the notions of augmenting, extending and modifying, leading to generalized extensions of spaces, bundles and operators. These abstract notions help to clarify Seeley’s construction, and we hope that they will be useful for extending our results to different geometries. In Section \[sec:inv.ext\] we construct an invertible generalized extension $\Phat$ of $P$. Its properties imply Proposition \[prop:shadow solutions\] as shown in Subsection \[ssec:pf prop shadow\]. Then we use its inverse to construct a Calderón projector $C$ with the stated properties in Section \[sec:Calderon\]. The construction of the Calderón projector from the inverse of $\Phat$ requires considering the transmission property for $\phi$-$\Psi$DOs. This is straightforward once we rephrase the transmission property in the language of conormal distributions (as opposed to $\Psi$DOs, as is usually done). This is carried out in Section \[ssec:transmission\] and may be of independent interest. Theorems \[thm:Calderon\] and \[thm:calderon canonical inv extension\] and Corollary \[cor:L2 closure of BC spaces\] are proved in Section \[ssec:Calderon\]. Theorem \[thm:orthogonal\] follows from Theorem \[thm:Calderon\] via standard properties of the $\phi$-$\Psi$DO calculus and by applying a formula for orthogonal projections as for instance in [@BLZ09].
We deal with some parts of the construction in the greater generality of manifolds with [[<span style="font-variant:small-caps;">bc</span>]{}]{}-boundary in order to facilitate generalization to other types of singularities.
Setting and Examples {#sec:setting}
====================
We now introduce the setting in detail. We use basic notions of manifolds with corners, which are collected in Appendix \[sec:app mwc\]. The setting is characterized by a class of spaces together with additional structure, which defines a class of partial differential operators on them, and a notion of ellipticity for these operators. All of these will be introduced in Subsection \[ssec:phi-geom\]. Then, in Subsection \[ssec:function.spaces\], we introduce the type of function spaces we consider as well as the associated spaces of boundary data and shadow solutions.
-[BC]{}-geometry {#ssec:phi-geom}
----------------
The geometric structure is described by the following definitions. It may be helpful for the reader to look at Figure \[fig:blow-up example\] for illustration.
\[def:bv-mfd\] A **manifold with** [[<span style="font-variant:small-caps;">bc</span>]{}]{}-**boundary**, or [[<span style="font-variant:small-caps;">bc</span>]{}]{}-**manifold**, is a smooth compact manifold with corners $X$, with a choice of a disjoint union of boundary hypersurfaces of $X$ designated as its [[<span style="font-variant:small-caps;">bc</span>]{}]{}-**boundary** and denoted by ${{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}$. We then denote ${{\partial_{{\mathrm{s}}}X}}= \overline{\partial X \setminus {{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}}$, the union of the remaining boundary hypersurfaces, and call this the **singular boundary**.
A **$\phi$-manifold with** [[<span style="font-variant:small-caps;">bc</span>]{}]{}-**boundary**, or short $\phi$-[[<span style="font-variant:small-caps;">bc</span>]{}]{}-**manifold**, is a [[<span style="font-variant:small-caps;">bc</span>]{}]{}-manifold $X$ so that ${{\partial_{{\mathrm{s}}}X}}=\bigcup_i {{\partial_{{\mathrm{s}},i}X}}$ is also a disjoint union of boundary hypersurfaces of $X$, together with the following data:
1. For each $i$ a fibration $ F_i - {{\partial_{{\mathrm{s}},i}X}}\overset{\phi_i}\to B_i$ where the base $B_i$ is a smooth closed manifold and the fibre $F_i$ is a smooth compact manifold with boundary, whose boundary corresponds to the [[<span style="font-variant:small-caps;">bc</span>]{}]{}-boundary ${{\partial_{{\mathrm{s}},{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}},i} X}}:= {{\partial_{{\mathrm{s}},i}X}}\cap {{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}$ of ${{\partial_{{\mathrm{s}},i}X}}$: $$\label{eqn:fibrations}
\begin{tikzcd}[ampersand replacement=\&]
F_i \arrow[r,dash] \& {{\partial_{{\mathrm{s}},i}X}}\ \arrow[dr,"\phi_i"] \\[-4mm]
\&\& B_i \\[-4mm]
\partial F_i \arrow[uu,hook] \arrow[r,dash] \& {{\partial_{{\mathrm{s}},{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}},i} X}}\arrow[uu,hook] \arrow[ur,swap,"{{\phi_{{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}},i}}}" near start]
\end{tikzcd}$$ where ${{\phi_{{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}},i}}}$ is the restriction of $\phi_i$ to ${{\partial_{{\mathrm{s}},{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}},i} X}}$.
2. A boundary defining function $x$ for ${{\partial_{{\mathrm{s}}}X}}$, i.e. $x:X\to\R_+=[0,\infty)$ is smooth, ${{\partial_{{\mathrm{s}}}X}}=\{x=0\}$ and $dx\neq0$ at ${{\partial_{{\mathrm{s}}}X}}$.
To simplify the notation we will assume for the most part that ${{\partial_{{\mathrm{s}}}X}}$ is connected and then leave out the index $i$, and will point out adjustments for the disconnected case only when they are not obvious. We will in general simply write $\phi$ instead of ${{\phi_{{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}}}}$ to keep the notation short. The fibration $\phi$ of ${{\partial_{{\mathrm{s}}}X}}$ can be extended to a product neighborhood of ${{\partial_{{\mathrm{s}}}X}}$, and we will fix such an extension throughout.
Note that for a [[<span style="font-variant:small-caps;">bc</span>]{}]{}-manifold $X$, $\partial X = {{\partial_{{\mathrm{s}}}X}}\cup {{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}$ and any boundary hypersurface is contained in precisely one of ${{\partial_{{\mathrm{s}}}X}}$ or ${{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}$. The classical non-singular case (compact manifold with boundary) corresponds to ${{\partial_{{\mathrm{s}}}X}}= \emptyset$. In the definition we allow the [[<span style="font-variant:small-caps;">bc</span>]{}]{}-boundary to be empty, in which case we obtain manifolds with corners resp. **$\phi$-manifolds** (in that case $\partial F=\emptyset$ also). The class of $\phi$-manifolds was introduced by Mazzeo and Melrose in [@MM98], and they are called manifolds with fibred boundary there.
For a $\phi$-[[<span style="font-variant:small-caps;">bc</span>]{}]{}-manifold $X$, only ${{\partial_{{\mathrm{s}},{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}} X}}$ can be a codimension two corner and there are no higher codimension corners. ${{\partial_{{\mathrm{s}},{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}} X}}$ is a closed manifold. If non-empty, the [[<span style="font-variant:small-caps;">bc</span>]{}]{}-boundary ${{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}$ is a $\phi$-manifold whose boundary fibrations are given by the bottom line in .
The [[<span style="font-variant:small-caps;">bc</span>]{}]{}-boundary of a [[<span style="font-variant:small-caps;">bc</span>]{}]{}-manifold $X$ should be thought of as a boundary as it appears in classical boundary value problems, for example, while ${{\partial_{{\mathrm{s}}}X}}$ serves as a smooth model for the singularities. Different components ${{\partial_{{\mathrm{s}},i}X}}$ of ${{\partial_{{\mathrm{s}}}X}}$ may model different kinds of singularities, e.g. the fibres $F_i$ may have varying dimensions. In some cases, e.g. in the example in Figure \[fig:blow-up example\], the analysis on $X$ models that on the singular space $\tilde X$ obtained by collapsing the fibres of $\phi$ to points, and conversely $X$ is obtained from a singular space $\tilde X$ by a suitable blow-up of its singular set.
Generally we consider objects on $X$, e.g. functions or metrics, which are defined on $X\setminus {{\partial_{{\mathrm{s}}}X}}$ and smooth up to ${{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}\setminus{{\partial_{{\mathrm{s}},{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}} X}}$, while they may be undefined at ${{\partial_{{\mathrm{s}}}X}}$. One way to express smoothness up to the [[<span style="font-variant:small-caps;">bc</span>]{}]{}-boundary is via extensions across that boundary, so we define:
![An example of a $\phi$-[[<span style="font-variant:small-caps;">bc</span>]{}]{}-manifold $X$ (exterior of the central picture) and how it arises via blow-up from a singular space (exterior of left picture, cf. Example \[ex:example touching domains\](i)). Here the boundary face ${{\partial_{{\mathrm{s}}}X}}$ of $X$ created by the blow-up is fibred with base $B=S^1$ (bottom) and fibre $F=[-1,1]$ (right).[]{data-label="fig:blow-up example"}](spheres-blow-up2.pdf)
\[def:bv-ext\] If $X$ is a manifold with non-empty [[<span style="font-variant:small-caps;">bc</span>]{}]{}-boundary, then a [[<span style="font-variant:small-caps;">bc</span>]{}]{}-**extension** of $X$ is a smooth compact manifold with corners $\Xhat$ containing $X$ and of the same dimension as $X$ and so that ${{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}$ is an interior $p$-submanifold of $\Xhat$ and ${{\partial_{{\mathrm{s}}}X}}\subset \partial \Xhat$.
If $X$ is a $\phi$-manifold with non-empty [[<span style="font-variant:small-caps;">bc</span>]{}]{}-boundary and with fibration $\phi: {{\partial_{{\mathrm{s}}}X}}\rightarrow B$, then a $\phi$-[[<span style="font-variant:small-caps;">bc</span>]{}]{}-**extension** of $X$ is a [[<span style="font-variant:small-caps;">bc</span>]{}]{}-extension $\Xhat$ of $X$ so that $\Xhat$ is a $\phi$-manifold (without [[<span style="font-variant:small-caps;">bc</span>]{}]{}-boundary) whose fibration ${{\hat{\phi}}}: \partial \Xhat \rightarrow B$ has the same base as $\phi$ and extends $\phi$, i.e., $\phi$ is the restriction of ${{\hat{\phi}}}$ to ${{\partial_{{\mathrm{s}}}X}}$.
Note that the fibres $F$ of $\phi$ are [[<span style="font-variant:small-caps;">bc</span>]{}]{}-manifolds (with $\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}F = F \cap {{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}= \partial F$ and $\partial_{\mathrm{s}}F = \emptyset$) and that the fibres $\Fhat$ of ${{\hat{\phi}}}$ are [[<span style="font-variant:small-caps;">bc</span>]{}]{}-extensions of the fibres $F$ of $\phi$. [[<span style="font-variant:small-caps;">bc</span>]{}]{}– and $\phi$-[[<span style="font-variant:small-caps;">bc</span>]{}]{}-extensions may be obtained by doubling across the [[<span style="font-variant:small-caps;">bc</span>]{}]{}-boundary, see Lemma \[lem:mfd.extension\].
Whenever we express things in local coordinates, then near any point $q$ of ${{\partial_{{\mathrm{s}}}X}}$ we will use **adapted coordinates** $x\geq0$, $y\in\R^b$, $z\in\R^f$ (or $z\in\R_+\times\R^{f-1}$ if $q \in {{\partial_{{\mathrm{s}},{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}} X}}$) centered at $q$, where $b=\dim B$, $f=\dim F$. Here $x$ is the given boundary defining function for ${{\partial_{{\mathrm{s}}}X}}$, $y$ is pulled back from a local coordinate system for $B$, and $z$ are remaining coordinates, locally parametrizing the points in each fibre.
We now turn to differential operators on a $\phi$-[[<span style="font-variant:small-caps;">bc</span>]{}]{}-manifold $X$. This is an immediate extension of the definitions for $\phi$-manifolds from [@MM98] – we simply take everything smooth up to the [[<span style="font-variant:small-caps;">bc</span>]{}]{}-boundary. The geometric data define the Lie algebra of $\phi$-**vector fields**: $$\label{eqn:def phi-vector field}
\begin{aligned}
\calV_\phi(X) = \{V\in \Cinf(X,TX):\, &Vx=O(x^2), \\ &V\text{ is tangential to the fibres of }\phi \text{ at }{{\partial_{{\mathrm{s}}}X}}\}
\end{aligned}$$ They are unconstrained away from ${{\partial_{{\mathrm{s}}}X}}$ but in adapted coordinates near a point of ${{\partial_{{\mathrm{s}}}X}}$ are $\Cinf(X)$-linear combinations of the following vector fields: $$\label{eqn:basis phi-tangent}
x^2\partial_x,\ x\partial_{y_i}, \partial_{z_j},\quad i=1,\dotsc,b,\ j=1,\dotsc,f$$ Composing these vector fields and adding functions we obtain $\phi$-**differential operators**: these have the form, near ${{\partial_{{\mathrm{s}}}X}}$, $$\label{eqn:phi operator}
P = \sum_{k+|\alpha|+|\beta|\leq m} a_{k,\alpha,\beta}(x,y,z) (x^2D_x)^k (xD_y)^\alpha D_z^\beta,\quad a_{k,\alpha,\beta} \text{ smooth, }$$ where $D_x=\frac1i\partial_x$ etc., $\alpha,\beta$ are multi-indices and $m\in\N_0$ is the order of $P$, cf. . The set of these operators is denoted $\Diff_\phi^m(X)$. The class of operators acting between sections of vector bundles $E,E^\prime$ on $X$ is defined similarly and denoted $\Diff^m_\phi(X;E,E^\prime)$. The $\phi$-**principal symbol** of $P\in\Diff^m_\phi(X)$ is the standard principal symbol in the interior, and near ${{\partial_{{\mathrm{s}}}X}}$ where holds it is the function $${{}^\phi\sigma}_m(P) = \sum_{k+|\alpha|+|\beta| = m} a_{k,\alpha,\beta}(x,y,z) \tau^k \eta^\alpha \zeta^\beta$$ for $\tau\in\R$, $\eta\in\R^b$, $\zeta\in\R^f$. Invariantly and globally, ${{}^\phi\sigma}_m(P)$ can be made sense of as a function (or section of a suitable vector bundle of homomorphisms) on a rescaled cotangent bundle ${{}^\phi T}^*X$ whose local basis near ${{\partial_{{\mathrm{s}}}X}}$ is below. $P$ is called $\phi$-**elliptic** if its $\phi$-principal symbol is invertible outside the zero section $(\tau,\eta,\zeta)=0$.
The **normal family** of $P\in \Diff^m_\phi(X)$ captures its behavior at the singular boundary. It is a family of differential operators on the fibres $F_y$, parametrized by $y\in B$, $\tau\in\R$ and $\eta\in T_y^*B$, and having coefficients smooth up to $\partial F_y$. In adapted local coordinates where holds it is given by $$\label{eqn:def normal op diff}
N(P)(\tau;y,\eta) = \sum_{k+|\alpha|+|\beta|\leq m} a_{k,\alpha,\beta}(0,y,z) \tau^k \eta^\alpha D_z^\beta$$ where now $\eta\in\R^b$.[^3] If ${{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}=\emptyset$ (e.g. for a [[<span style="font-variant:small-caps;">bc</span>]{}]{}-extension of our given $\phi$-[[<span style="font-variant:small-caps;">bc</span>]{}]{}-manifold) then $P$ is called **fully elliptic** if it is $\phi$-elliptic and $N(P)(\tau;y,\eta)$ is invertible for all $\tau,y,\eta$. Full ellipticity is equivalent to $P$ having a parametrix in the $\phi$-calculus, modulo errors that are smoothing and small near ${{\partial_{{\mathrm{s}}}X}}$ in a suitable sense, and also to being Fredholm between $\phi$-Sobolev spaces. See Appendix \[sec:app phi ops\] for details on the $\phi$-calculus and for an extension of the $\phi$-symbol and normal family to $\phi$-pseudodifferential operators.
The main motivation for considering $\phi$-differential operators is that geometric operators, e.g. the Laplacian and Dirac operators, for $\phi$-metrics are in this class (see [@MM98] and [@Mel90]). A $\phi$-**metric** is a Riemannian metric on $X\setminus{{\partial_{{\mathrm{s}}}X}}$ that, near ${{\partial_{{\mathrm{s}}}X}}$, can be written as positive definite quadratic form in terms of $$\label{eqn:basis phi-cotangent}
\frac{dx}{x^2},\ \frac{dy_i}{x},\ dz_j,\quad i=1,\dotsc,b,\ j=1,\dotsc,f$$ with smooth coefficients (where smoothness and positive definiteness hold up to $x=0$). More generally, the Laplacian for a metric of the form $x^{2c}g$, where $c\in\Z$ and $g$ is a $\phi$-metric, is of the form $x^{-2c}P$ where $P\in\Diff^2_\phi(X)$ is $\phi$-elliptic.
We will fix a $\phi$-[[<span style="font-variant:small-caps;">bc</span>]{}]{}-manifold $X$ throughout the main construction. To simplify the exposition we will also choose a background $\phi$-metric $g$ on $X$, smooth up to ${{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}$. This is useful in two ways:
1. The volume form $\dvol_g$ allows us to interpret Schwartz kernels of operators as distributions and avoid densities.
2. Having the $L^2$-space will allow us to talk about adjoints of operators, which will be useful in the construction.
We also fix a $\phi$-vector field $\nu$ transversal to ${{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}$ in order to define the boundary data map $\gamma$, see . Given a $\phi$-[[<span style="font-variant:small-caps;">bc</span>]{}]{}-extension $\Xhat$ of $X$ we extend $\nu$ to a $\phi$-vector field on $\Xhat$. The flow of $\nu$ defines a trivialization $$\label{eqn:trivialize bd}
U \cong (-1,1)_\rho\times {{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}$$ (after scaling $\nu$ if needed) of a neighborhood $U$ of ${{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}$ in $\Xhat$ which is compatible with $\phi$ (that is, the image of a $\hat\phi$-fibre is a $\phi_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}$-fibre times $(-1,1)$) and with the chosen boundary defining function $x$ and with $\rho\geq0$ in $X$, and $D_\nu=D_\rho$ then. In adapted coordinates we may and will take $z_1=\rho$.
Function spaces and boundary data {#ssec:function.spaces}
---------------------------------
Throughout, spaces of smooth functions vanishing at (parts of) the boundary of a [[<span style="font-variant:small-caps;">bc</span>]{}]{}-manifold $X$ will play an important role. We denote $$\label{eqn:Cdotinf.BC}
{{\dot C^\infty_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}}}(X) = \big\{\, u \in \Cinf(X) \,:\, \text{$u$ vanishes to infinite order at ${{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}$}\,\big\}$$ and similarly denote by ${{\dot C^\infty_{{\mathrm{s}}}}}(X)$, ${{\dot C^\infty_{{\mathrm{s}},{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}}}}(X)$ the spaces of smooth functions on $X$ vanishing to infinite order at ${{\partial_{{\mathrm{s}}}X}}$ and at ${{\partial_{{\mathrm{s}}}X}}\cup {{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}$, respectively. If $x$ is a boundary defining function for ${{\partial_{{\mathrm{s}}}X}}$, ${{\dot C^\infty_{{\mathrm{s}}}}}(X) = x^\infty \Cinf(X)$ and ${{\dot C^\infty_{{\mathrm{s}},{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}}}}(X) = x^\infty {{\dot C^\infty_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}}}(X) = {{\dot C^\infty}}(X)$, where ${{\dot C^\infty}}(X)$ is the space of smooth functions vanishing to infinite order at $\partial X$. As all of these spaces are local $\Cinf(X)$-modules, we can define for instance ${{\dot C^\infty_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}}}(X;E)$ for a vector bundle $E \rightarrow X$.
As mentioned in the introduction, we also need to specify growth conditions on functions on a [[<span style="font-variant:small-caps;">bc</span>]{}]{}-manifold $M$ near the singular boundary. These are encoded in function spaces which are admissible in the sense of the following definition. Recall from Appendix \[sec:app mwc\] the definition of $\calA_s(M)$, the space of functions conormal at ${{\partial_{{\mathrm{s}}}M}}$.
\[def:admissible function space\] An assignment $\calF$ of a function space $\calF(M)$ to any [[<span style="font-variant:small-caps;">bc</span>]{}]{}-manifold $M$ is called **admissible** if it satisfies the following conditions, for every $M$:
1. ${{\dot C^\infty_{{\mathrm{s}}}}}(M) \subset \calF(M) \subset \calA_{\mathrm{s}}(M)$
2. If ${{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}M}}\neq\emptyset$ and $\Mhat$ is a [[<span style="font-variant:small-caps;">bc</span>]{}]{}-extension of $M$ then the restriction maps $$\calF(\Mhat) \to \calF(M), \qquad \calF(M) \to \calF({{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}M}})$$ are defined and surjective.
3. $\calF(M)$ is a local $\Cinf(M)$-module.
In the context of $\phi$-[[<span style="font-variant:small-caps;">bc</span>]{}]{}-manifolds we require (ii) to hold for $\phi$-[[<span style="font-variant:small-caps;">bc</span>]{}]{}-extensions and in addition:
1. If $M$ is a $\phi$-manifold (i.e. ${{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}M}}=\emptyset$) then $\calF(M)$ is stable under $\Psi^*_\phi(M)$, i.e. $u\in\calF(M)$, $P\in\Psi^*_\phi(M)\Rightarrow Pu\in\calF(M)$.
Given such an admissible choice $\calF$, we define in analogy to ${{\dot C^\infty_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}}}(X)$: $$\label{eqn:dfn.FdotBC}
{\dot{\calF}_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}}(M) := \{u\in\calF(M):\,u \text{ vanishes to infinite order at }{{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}M}}\}$$
Note that (i) implies that functions $u\in\calF(M)$ are smooth up to the [[<span style="font-variant:small-caps;">bc</span>]{}]{}-boundary, but unrestricted otherwise, away from ${{\partial_{{\mathrm{s}}}M}}$. Also, functions in $\calF(M)$ can be paired with functions in ${{\dot C^\infty_{{\mathrm{s}}}}}(M)$, with respect to a $\phi$-volume form. Condition (ii) is used in the construction of Calderón projectors. Condition (iii) ensures that spaces of sections $\calF(M,E)$ of vector bundles $E\to M$ are defined, and they have analogous properties. Condition (iv) will be used for $M$ being either the [[<span style="font-variant:small-caps;">bc</span>]{}]{}-boundary or a $\phi$-[[<span style="font-variant:small-caps;">bc</span>]{}]{}-extension of a $\phi$-manifold with non-empty [[<span style="font-variant:small-caps;">bc</span>]{}]{}-boundary.
Admissible choices for $\calF$ include spaces of conormal functions $\calA^a$, polyhomogeneous functions $\calA^\calE$ or functions in $x^k \Cinf$, $k \in \Z$. Their definitions are recalled in Appendix \[sec:app mwc\]. Here the conormality order $a$ and index family $\calE$ refer only to the singular boundary. Note that the order and index family do not change under restriction to the [[<span style="font-variant:small-caps;">bc</span>]{}]{}-boundary. This is different from the Sobolev order which would decrease by $\frac12$.
Let us check that the conditions on $\calF$ imply that $\gamma$ defined in maps as stated: Let $u\in\calF(X)$. Since $\nu$ is a $\phi$-vector field, $D_\nu$ is a $\phi$-differential operator, so $D_\nu^k u\in\calF(X)$ by condition (iv) in Definition \[def:admissible function space\]. Then by condition (ii) $D_\nu^ku_{|{{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}}\in\calF({{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}})$, so $\gamma u \in \calF({{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}})^m$. Note that this clearly implies $\calB_{P,\calF} \subset \calF({{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}})^m$ as in .
Examples {#ssec:ex}
--------
In this section we give some examples of settings to which our results apply. Examples (i)-(iii) elaborate on the examples mentioned in the introduction. In all examples metrics of the form $x^{2c}g$ arise, with $c\in\{0,1,2\}$ and $g$ a $\phi$-metric, so the Laplacian is $x^{-2c}$ times an elliptic $\phi$-operator.
### Example (i): Incomplete fibred cusps {#ex:example touching domains .unnumbered}
Consider the set $X_0=\{(\xi,\eta)\in\R^m\times\R^k: |\xi|\leq |\eta|^2, |\eta|<1\}$ where $m,k\in\N$, whose boundary $ |\xi|= |\eta|^2$ has a singularity at $\xi=0,\eta=0$. We introduce quasihomogeneous polar coordinates by writing $$X = [0,1)\times {\mathbb{S}}^{k-1}\times {\mathbb{B}}^m\,,\ \
\beta: X\to X_0,\ (x,\omega,z) \mapsto (\xi=x^2z,\eta = x\omega)\,,$$ where ${\mathbb{S}}^{k-1}=\{\omega\in\R^k:\,|\omega|=1\}$ and ${\mathbb{B}}^m=\{z\in\R^m:\, |z|\leq 1\}$.[^4] Except for the non-compactness at $x=1$, the space $X$ is a $\phi$-[[<span style="font-variant:small-caps;">bc</span>]{}]{}-manifold, with ${{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}=[0,1)\times {\mathbb{S}}^{k-1}\times {\mathbb{S}}^{m-1}$ corresponding to $|\xi|=|\eta|^2$ and ${{\partial_{{\mathrm{s}}}X}}=\{0\}\times {\mathbb{S}}^{k-1}\times {\mathbb{B}}^m$, and $\phi:{{\partial_{{\mathrm{s}}}X}}\to B={\mathbb{S}}^{k-1}$ the projection, so the fibre is ${\mathbb{B}}^m$. If $y$ are local coordinates on the sphere then the Euclidean metric $|d\xi|^2+|d\eta|^2$ pulls back to a positive definite quadratic form in $dx$, $x dy$, $x^2 dz$, with coefficients smooth in $x\geq0$, hence is $x^4 g$ for a $\phi$-metric $g$ on $X$.
Any bounded domain in $\R^{m+1}$ whose boundary is smooth except for isolated outward-pointing cuspidal singularities is locally of this type with $k=1$.
If $m=1$ then $X_0$ is the complement of the two solid paraboloids $\{\pm\xi>|\eta|^2\}$. The same resolution and type of metric arises on $X_0=\R^{k+1}\setminus(\Omega_1\cup\Omega_2)$ where $\Omega_1,\Omega_2$ are domains with smooth boundary whose closures intersect in a single point $p$ and which are simply tangent there, see Figure \[fig:blow-up example\] for an example with $k=2$. Note that the geometry at infinity of $\R^{k+1}$ is also of $\phi$-type (see Example (iii)). However, the Laplacian is not fully elliptic at infinity, so if both $\Omega_i$ are bounded then $X$ does not satisfy Assumption \[UCNF\] there. We expect our result to hold anyway. Our results do apply to the part of $X_0$ lying in a large ball, which has an additional [[<span style="font-variant:small-caps;">bc</span>]{}]{}-boundary.
We remark that there is an obvious extension of $X_0$ without [[<span style="font-variant:small-caps;">bc</span>]{}]{}-boundary, namely $\R^{m+k}$, and one may construct the Calderón projector using an inverse of the Laplacian on $\R^{m+k}$ in the construction as outlined in Section \[sec:outline\]. However, it is not clear that $C$ is a $\phi$-$\Psi$DO then. (In [@Fri14], it is shown that the boundary layer potentials which can be used to construct $C$ are elements of the full $\phi$-calculus only, but it is the small calculus we need here. [@Fri19] studies the same problem in the setting of half-spaces, thus in a different geometry.) The point is that the complement $\R^{m+k}\setminus X_0$ has a very different geometric structure near the singularity. Our double of $X_0$ is really a ’thin’ double, i.e. it still has a cusp (though without [[<span style="font-variant:small-caps;">bc</span>]{}]{}-boundary).
### Example (ii): Domains in locally symmetric spaces {#example-ii-domains-in-locally-symmetric-spaces .unnumbered}
Consider $X_0=\{w\in{\mathbb{C}}:\,|\Re w| \leq c,\ \Im w \geq 1\}$ for some $c>0$ with the hyperbolic metric $g_0=\frac{dw\,d\wbar}{(\Im w)^2}$. Introduce coordinates $x=\frac1{\Im w}$, $z=\Re w$ and compactify $X_0$ by adding a singular boundary ${{\partial_{{\mathrm{s}}}X}}=\{x=0\}$ to obtain $X=[0,1]_x\times[-c,c]_z$. The metric is $x^2\left(d(\frac1x)^2 + dz^2\right) = x^2g$ where $g=\frac{dx^2}{x^4} + dz^2$. This has the form of a $\phi$-metric without $y$-coordinates, so for the fibration whose base is a point. This makes $X$ into a $\phi$-manifold with ${{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}=\{|z|=c\}\cup \{x=1\}$, apart from the corners at $\{1\}\times\{\pm c\}$ which could be rounded without affecting the geometry near $x=0$. A similar geometry arises from domains in other locally symmetric spaces of $\Q$-rank one, if the boundary of the domain extends into the cusps.
### Example (iii): Fibre bundles over cones {#example-iii-fibre-bundles-over-cones .unnumbered}
Let $B\subset{\mathbb{S}}^{n-1}$ be a compact submanifold and $C=\{rw:\,w\in B, r>1\}\subset\R^n$. In polar coordinates the metric on $C$ induced by the Euclidean metric is $g_C=dr^2+r^2g_0$, where $g_0$ is the induced metric on $B$. To describe its behavior near infinity we introduce the coordinate $x=\frac1r$. Then the metric is $\frac{dx^2}{x^4} + \frac{g_0}{x^2}$, which is a $\phi$-metric for the space $\Cbar$ defined as $C$ with a copy of $B$ added at $x=0$ as the singular boundary; thus, $\Cbar$ is the radial compactification of $C$. Since there are no $z$-coordinates we take $\phi$ to be the fibration with point fibres, so the base is $B$ and $\phi$ is the identity.
The product $X=\Cbar\times F$ with a compact Riemannian manifold $F$ with boundary is a $\phi$-[[<span style="font-variant:small-caps;">bc</span>]{}]{}-manifold, where $\phi:B\times F\to B$ is the projection and ${{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}=\Cbar\times\partial F$, and the product metric is a $\phi$-metric. A simple example is the slab $\R^n\times[0,1]$. A closely related embedded example is obtained by thickening the conical set $C$ in the orthogonal direction, i.e. choosing $C_\eps=\{p+z:\,p\in C, z\perp T_pC, |z|\leq\eps\}$ for $\eps>0$. If $\eps$ is sufficiently small then this is an $n$-dimensional submanifold with boundary of $\R^n$. It can be parametrized as $(1,\infty)\times N^\eps B\to C_\eps$, $(r,(w,z)) \mapsto rw + z$ where $N^\eps B =\{(w,z)\in NB:\, |z|\leq\eps\}$ and $NB$ is the normal bundle of $B$ in ${\mathbb{S}}^{n-1}$. Adding a copy of $N^\eps B$ at $x=0$ (i.e. $r=\infty$) as singular boundary again we obtain a $\phi$-manifold with $\phi:N^\eps B\to B$ the bundle projection with fibre a ball, and it follows from standard calculations (see e.g. [@Gra:T]) that the Euclidean metric on $C_\eps$ is a $\phi$-metric. An example of a compact space $X$ to which our results apply directly is obtained by capping off $\Cbar$ smoothly near the origin of $\R^n$.
Note that the closed half space, radially compactified, is not an example of a $\phi$-[[<span style="font-variant:small-caps;">bc</span>]{}]{}-manifold. Its geometry at infinity is that of a $\phi$-manifold where the base is a half sphere, thus has boundary (and not the fibres, which are points).
### Example (iv): Exterior problem for a smooth domain {#example-iv-exterior-problem-for-a-smooth-domain .unnumbered}
If $\Omega\subset\R^n$ is a bounded domain with smooth boundary then our results apply to the operator $-\Delta + 1$ on $\R^n\setminus\Omega$. Here the [[<span style="font-variant:small-caps;">bc</span>]{}]{}-boundary $\partial\Omega$ and the singular boundary (at infinity) are disjoint. The geometry at infinity is that of $C$ in Example (iii) with $B={\mathbb{S}}^{n-1}$. Note that since the [[<span style="font-variant:small-caps;">bc</span>]{}]{}-boundary does not extend to infinity, Assumption \[UCNF\] requires full ellipticity for the operator, which is satisfied by $-\Delta+1$, but not by $\Delta$.
Outline of the Construction {#sec:outline}
===========================
For the convenience of the reader we give an outline of the classical construction of the Calderón projector, ignoring all technical details, in the case of a compact manifold with boundary (no singularities, no bundles), and then point to the various technical issues that arise both in the classical and in the singular setting. We carry out the construction of a Calderón projector as in [@Hor85 20.1] but should also mention [@Gru96 Ex. 1.9], which is closer to the original construction in [@See66].
Let $P$ be an elliptic differential operator of order $m$ on the compact manifold $X$ with boundary ${{\partial X}}$. By doubling $X$ across its boundary we obtain a compact manifold without boundary, $\Xhat \supset X$.
Constructing a Calderón projector from an invertible extension {#ssec:outline Calderon constr}
--------------------------------------------------------------
Assume first that $P$ can be extended to an invertible elliptic differential operator $\Phat$ on $\Xhat$. Similarly to [@Hor85], but using the exact inverse of $\Phat$ rather than a parametrix, one can then explicitly construct a pseudodifferential Calderón projector for $P$ as follows. Identify a neighborhood of ${{\partial X}}$ in $\Xhat$ with $(-1,1)_\rho\times{{\partial X}}$, where $\rho=0$ corresponds to ${{\partial X}}\subset\Xhat$. Consider the boundary data map (cf. ) $$\label{eqn:cauchy.data}\begin{array}{rlrcl}
\gamma: & \Cinf(X) \to \Cinf({{\partial X}})^m \,,
&u &\mapsto &(u_{|\partial},D_\rho u_{|\partial},\dots, D_\rho^{m-1} u_{|\partial})
\end{array}$$ where $m$ is still the order of $P$ and denote ‘$\delta$ extension from the boundary’ by (ignoring density factors) $$\label{eqn:delta.data}\begin{array}{rlrcl}
\gamma^\star:& \Cinf({{\partial X}})^m \to \calD'(\Xhat) \,,
&U=(U_0,\dots,U_{m-1}) &\mapsto &\sum_{l=0}^{m-1} D^l_\rho\delta(\rho) \otimes U_l
\end{array}$$ where $\delta(\rho)$ is the Dirac measure of ${{\partial X}}$. For $u\in\Cinf(X)$, denote by $u^0$ its extension by $0$ to $\Xhat$. The construction now proceeds as follows:
1. The main point is to observe that $\Phat (u^0) = (Pu)^0 + \gamma^\star{\hspace{-2pt}\mathscr{J}}_P \gamma u$ for a differential operator ${\hspace{-2pt}\mathscr{J}}_P: \Cinf({{\partial X}})^m\to \Cinf({{\partial X}})^m$ of order $m-1$ determined by $P$, since extending $u$ by zero introduces a jump singularity (hence the letter ${\hspace{-2pt}\mathscr{J}}$) which yields (normal derivatives of) delta distributions when differentiated. If $Pu=0$ then the first term on the right vanishes.
2. If $Pu = 0$, applying $\Phat^{-1}$ and taking boundary data yields $\gamma u = \gamma\Phat^{-1}\gamma^\star{\hspace{-2pt}\mathscr{J}}_P \gamma u$. Here $\gamma$ is understood as the limit when approaching ${{\partial X}}$ from the interior of $X$. Denoting by $\calB$ the boundary data space defined in we see that the operator $$\label{eqn:Calderon formula intro}
C := \gamma\Phat^{-1}\gamma^\star{\hspace{-2pt}\mathscr{J}}_P : \ \calB\to\calB$$ is the identity. Now the idea is to show that the formula defining $C$ defines an operator $\Cinf({{\partial X}})^m\to \Cinf({{\partial X}})^m$. Note that this would be given as the composition $$\label{eqn:Calderon formula intro 2}
\begin{array}{rccccccccl}
\qquad C: &\Cinf({{\partial X}})^m& \stackrel{\,{\hspace{-2pt}\mathscr{J}}_P}\to & \Cinf({{\partial X}})^m & \stackrel{\gamma^\star}\to &\calD'(\Xhat) & \stackrel{\Phat^{-1}}\to & \calD'(\Xhat) & \stackrel{\gamma}\to & \Cinf({{\partial X}})^m \\[1mm]
& U &\mapsto& {\hspace{-2pt}\mathscr{J}}_PU &\mapsto& f & \mapsto & v &\mapsto & \gamma v=CU
\end{array}$$ where all maps are well-defined except the last, since restriction of general distributions does not make sense. However, $\gamma$ only needs to be applied to $v=\Phat^{-1}f$ where $f=\gamma^\star{\hspace{-2pt}\mathscr{J}}_P U$ and $U\in\Cinf({{\partial X}})^m$. Now by standard $\Psi$DO theory, the operator $\Phat^{-1}$ is pseudodifferential, hence pseudolocal. Since $f$ is smooth (in fact, zero) in $\interior{X}$, so is $v$, and although $v$ is singular at ${{\partial X}}$, a further argument involving the transmission property of $\Phat^{-1}$ shows that $v_{|\interior{X}}$ extends to a smooth function on $X$, so in this sense $\gamma v$ is defined, and that the operator $C:U\mapsto \gamma v$ is pseudodifferential. Finally, it remains to check that $CU\in\calB$ for all $U\in\Cinf({{\partial X}})^m$. This is clear since $\Phat v=f$ and $f=0$ in $\interior{X}$, so $Pv=0$ in $\interior{X}$. Since $v$ extends smoothly to ${{\partial X}}$ it follows that $\gamma v\in\calB$.
3. The orthogonal Calderón projector $C_o$ defined by additional geometric data is given by the formula $C_o = C (\Id + C - C^\star)^{-1}$, where the adjoint $C^\star$ also depends on the geometric data; see [@BLZ09 Lem. 3.5] for instance. It follows from standard $\Psi$DO theory again that $C_o$ is pseudodifferential as well.
Constructing an invertible extension {#subsec:inv ext}
------------------------------------
Above we assumed the existence of an invertible extension $\Phat$. However, such an invertible extension might not exist, at least not as a differential operator.
One obstruction is that the equation $Pu=0$ might have shadow solutions as in Proposition \[prop:shadow solutions\]. Then by ellipticity $u$ is in $\dot C^\infty(X)$, the space of smooth functions vanishing to infinite order at ${{\partial X}}$, and any differential operator $\Phat$ extending $P$ would have $u^0$ in its kernel. Another obstruction to the existence of $\Phat$ is topological: An elliptic differential operator $P$ need not have an [elliptic]{} extension to $\Xhat$ as differential operator. [^5]
Both of these obstacles can be overcome by constructing $\Phat$ as an elliptic *pseudo*differential operator and by generalizing the notion of extension to also allow enlarging the bundles and adding certain smoothing operators. Here is an outline of such a construction close to that of Seeley ([@See66] and the appendix of [@See69]) and used in this paper:
1. Setting $\Pbar = P \oplus P^\star$ one obtains a formally self-adjoint elliptic differential operator on $X$. We call this an augmentation of $P$; here one needs to choose an auxiliary background metric on $X$. Selfadjointness allows to connect the symbol of $\Pbar$ at ${{\partial X}}$ via a homotopy through elliptic symbols to $|\xi|^m$ times the identity, and this can be used to define an elliptic *pseudo*differential operator $\Phat_0$ on $\Xhat$ that extends $\Pbar$. Augmenting again one obtains a self-adjoint elliptic operator $\Ptilde$ on $\Xhat$.
2. If $\Ptilde$ is invertible then the construction above yields a Calderón projector for the restriction of $\Ptilde$ to $X$, and this easily yields one for $P$.
3. If $\Ptilde$ is not invertible the strategy is to perturb $\Ptilde$ to make it invertible, without changing the boundary data space $\calB_\Ptilde$. One idea to make $\Ptilde$ invertible is to add an orthogonal projection to its kernel; however, this may change $\calB_\Ptilde$. Instead one can add an orthogonal projection ${\Pi_{\mathrm{comp}}}$ to any subspace ${V_{\mathrm{comp}}}$ complementary to the range of $\Ptilde$. If $\Ptilde$ has no shadow solutions then ${V_{\mathrm{comp}}}$ can be chosen to consist of functions supported in $X^-=\overline{\Xhat\setminus X}$, and then $\calB_{\Ptilde+{\Pi_{\mathrm{comp}}}}=\calB_{\Ptilde}$, so we are done.
4. In general, if ${V_{\mathrm{sh}}}\subset\ker \Ptilde$ is the space of shadow solutions and ${\Pi_{\mathrm{sh}}}$ is the orthogonal projection to ${V_{\mathrm{sh}}}$ then $\Ptilde+{\Pi_{\mathrm{sh}}}$ has no shadow solutions and $\calB_{\Ptilde+{\Pi_{\mathrm{sh}}}}=\calB_{\Ptilde}$. So choosing ${V_{\mathrm{comp}}}$ as in (ii’), but for $\Ptilde+{\Pi_{\mathrm{sh}}}$ instead of $\Ptilde$, we get that $\Phat':=\Ptilde+{\Pi_{\mathrm{sh}}}+{\Pi_{\mathrm{comp}}}$ is invertible and has the same boundary data space as $\Ptilde$.
Note also that the boundary data space is only defined for operators $\Xhat$ that may be restricted to $X$. This restriction is defined for ${\Pi_{\mathrm{sh}}}$ and ${\Pi_{\mathrm{comp}}}$ since ${V_{\mathrm{sh}}}$, ${V_{\mathrm{comp}}}$ consist of functions supported on one side of ${{\partial X}}$, but not for the projection to the kernel.
The details about this generalized extension procedure are given in Section \[sec:aug-ext-mod\].
Additional issues in the singular setting
-----------------------------------------
In our singular setting everything needs to be made to work within the $\phi$-$\Psi$DO calculus. First, doubling $X$ (and the fibration $\phi$) across its [[<span style="font-variant:small-caps;">bc</span>]{}]{}-boundary we obtain a $\phi$-manifold $\Xhat$ without [[<span style="font-variant:small-caps;">bc</span>]{}]{}-boundary. Once we have constructed an invertible generalized extension $\Phat'$ on $\Xhat$ which is a fully elliptic $\phi$-$\Psi$DO, we can apply the standard $\phi$-calculus of Mazzeo and Melrose [@MM98] to prove that its inverse is also a $\phi$-$\Psi$DO. After generalizing the transmission condition to the $\phi$-setting we can carry out steps (i)-(iii) essentially as before.
The main new ingredient in our singular setting is the following: In order to ensure that the inverse of $\Phat'$ is a $\phi$-$\Psi$DO, we need to construct $\Phat'$ in such a way that it is **fully** elliptic, i.e. that its normal family is invertible. For this one would like to do steps (iii’), (iv’) above for each fibre and each operator in the normal family separately. However, the kernels of the normal family will in general not define a vector bundle. This problem can be overcome for step (iii’) since there is some flexibility here, but not for step (iv’), and it is at this point where we need to make the additional Assumption \[UCNF\] about the unique continuation property of the normal family.
Augmenting, Extending and Modifying Operators {#sec:aug-ext-mod}
=============================================
As explained in Section \[sec:outline\] a central step in the construction of a Calderón projector for $P\in\Diff^m_\phi(X;E,E')$ is extending $P$ to an operator on a [[<span style="font-variant:small-caps;">bc</span>]{}]{}-extension of $X$, e.g. its [[<span style="font-variant:small-caps;">bc</span>]{}]{}-double. However, such an extension might not exist (cf. Subsection \[subsec:inv ext\]), so we need to generalize the notion of extension. This generalization has three ingredients:
- **Augmenting** the bundles $E,E'$ to larger bundles, and correspondingly the operator.
- Adding a projection onto a finite-dimensional subspace of ${{\dot C^\infty_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}}}(X;E)$ (respectively ${\dot{\calF}_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}}(X;E)$) to an operator. We call this a **modification** of the operator.
- **Extending** the space $X$ to a $\phi$-manifold without [[<span style="font-variant:small-caps;">bc</span>]{}]{}-boundary, and correspondingly bundles and operators.
Augmentation is used to create formally self-adjoint operators, which then can be extended. Modification is needed to deal with operators having shadow solutions, since these cannot have an invertible extension. See Section \[sec:inv.ext\] for more details.
Combinations of these also occur; a combination of all three is called a **generalized extension**.
In this section we explain these notions in detail and show how to obtain a Calderón projector for $P$ from a Calderón projector for a modified augmentation of $P$.
Most of the arguments in this section are of a general functional analytic nature, about linear operators on some function space. With the application to $m$-th order differential operators in mind we fix $m\in\N$, a boundary data map $\gamma$ and an admissible choice $\calF$ of function space, see Subsection \[ssec:function.spaces\]. Thus, we also have the $\calF$-boundary data space $\calB_{T,\calF}$ of order $m$ of a linear operator $T : \calF(X;E) \to \calF(X;E')$ and the notion of an $\calF$-Calderón projector for $T$.
We adopt the following conventions for notation: a bar is used for an augmented object, e.g. $\Ebar$, and a hat is used for an extended object, e.g. $\Ehat$ (and also for an augmented extended object).
Augmentations {#ssec:augmentations}
-------------
In this subsection $X$ denotes a manifold with [[<span style="font-variant:small-caps;">bc</span>]{}]{}-boundary, see Definition \[def:bv-mfd\]. No $\phi$-structure is needed here.
If $E,F\to X$ are vector bundles then we can form the direct (or Whitney) sum $\Ebar=E\oplus F\to X$. It will be useful to phrase the relation of $E$ and $\Ebar$ without mentioning $F$, using the inclusion $E\hookrightarrow \Ebar$ and projection $\Ebar\twoheadrightarrow E$ with kernel $F$ instead:
\[def:augmentation\]
1. Let $E\to X$ be a vector bundle. An **augmentation** of $E$ is a vector bundle $\Ebar\to X$ together with vector bundle maps $$\iota: E \hookrightarrow \Ebar,\quad \pi: \Ebar \twoheadrightarrow E,\quad
\pi \circ \iota = \id_E$$ where $\iota$ is injective and $\pi$ is surjective.
2. Let $E,E'\to X$ be vector bundles and $T:\calF(X;E) \to \calF(X;E')$ be a linear operator. An **augmentation** of $T$ is a linear operator $\Tbar : \calF(X;\Ebar) \to \calF(X;\Ebar')$ between augmentations $\Ebar, \Ebar'$ of $E, E'$ for which $$\label{eqn:def augment op}
\pi' \Tbar = T\pi\,,\quad \Tbar \iota = \iota' T \,.$$ Here $\iota$ also denotes the induced map $\calF(X;E) \to \calF(X;\Ebar)$, and similarly for $\pi, \iota', \pi'$.
If we identify $E$ with $\iota E$ then $\Ebar=E\oplus F$ where $F=\ker\pi$, and similarly $\Ebar'=E'\oplus F'$. Then an augmentation of $T$ is a direct sum (or block diagonal) operator $$\label{eqn:augment op direct sum}
\Tbar = T \oplus U: E \oplus F \to E' \oplus F'$$ for some operator $ U:F\to F'$. We will use this characterization in this section. Later it will be useful to have both descriptions available.
We now consider the effect of augmentation on boundary data spaces and Calderón projectors.
\[prop:augment B C\] Let $\Tbar : \calF(X;\Ebar) \to \calF(X;\Ebar')$ be an augmentation of the linear operator $T : \calF(X;E) \to \calF(X;E')$. Then the maps $\iota: E \hookrightarrow \Ebar$, $\pi: \Ebar \twoheadrightarrow E$ induce an injection and surjection $$\iota: \calB_{T,\calF} \hookrightarrow \calB_{\Tbar,\calF},\quad \pi: \calB_{\Tbar,\calF} \twoheadrightarrow \calB_{T,\calF} \,.$$ Also, if $\Cbar: \calF({{\partial X}};\Ebar)^m\to \calB_{\Tbar,\calF}$ is a Calderón projector for $\Tbar$ then $$C = \pi \Cbar \iota: \calF({{\partial X}};E)^m\to \calB_{T,\calF}$$ is a Calderón projector for $T$.
If we represent $\Ebar=E\oplus F$ and $\Tbar=T\oplus U$ as in then clearly $\calB_{\Tbar,\calF} = \calB_{T,\calF} \oplus \calB_{U,\calF}$, which implies the first claim. Also $\calF({{\partial X}};\Ebar) = \calF({{\partial X}};E)\oplus\calF({{\partial X}},F)$, so the $\calF({{\partial X}};E)^m\to \calB_{T,\calF}$ part of $\Cbar: \calF({{\partial X}};E)^m\oplus\calF({{\partial X}},F)^m\to \calB_{T,\calF}\oplus\calB_{U,\calF}$, which is $\pi \Cbar \iota$, is a projection to $\calB_{T,\calF}$.
Modifications {#ssec:modifications}
-------------
As in Subsection \[ssec:augmentations\], in this subsection $X$ denotes a manifold with [[<span style="font-variant:small-caps;">bc</span>]{}]{}-boundary. Let $E\to X$ be a vector bundle. Operators in this subsection map sections of $E$ to sections of $E$ (so $E'=E$ in the previous notation). As before we fix $m\in\N$, a boundary data map $\gamma$ of order $m$ and an admissible $\calF$.
\[def:modification\] Let $T : \calF(X;E) \to \calF(X;E)$ be linear. A **modification** of $T$ is an operator $T+\Pi$ where $\Pi$ is a projection in $\calF(X;E)$ with $$\label{eqn:modification}
\rg \Pi \subset \ker T \cap \ker\gamma \,.$$
\[prop:modification B\] Let $T$ be as in Definition \[def:modification\] and $T+\Pi$ be a modification. Assume moreover that $\rg\Pi \cap \rg T = \{0\}$. Then $$\calB_{T+\Pi,\calF} = \calB_{T,\calF}\,.$$ If, in addition, equality holds in then $$\label{eqn:modification no kernel}
\ker (T+\Pi) \cap \ker\gamma = \{0\} \,.$$ The condition $\rg\Pi \cap \rg T = \{0\}$ is satisfied, for example, if $T$ restricts to an operator ${{\dot C^\infty_{{\mathrm{s}}}}}(X;E) \to {{\dot C^\infty_{{\mathrm{s}}}}}(X;E)$ and satisfies $\ker T \cap \ker \gamma \subset {{\dot C^\infty_{{\mathrm{s}}}}}(X;E)$ and $$\label{eqn:form selfadjoint}
\langle Tu,v\rangle = \langle u,Tv \rangle\quad \text{ for } u\in {{\dot C^\infty_{{\mathrm{s}}}}}(X;E), v\in\calF(X;E), \gamma u=0 \,.$$
\[rem:shadow solutions\] If $P$ is an elliptic differential operator of order $m$ then $\ker P \cap \ker \gamma = \ker P \cap {\dot{\calF}_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}}(X;E)$ by ellipticity. The proposition says that these shadow solutions can be removed by adding to $P$ a projection $\Pi_{sh}$ onto $\ker P \cap \ker \gamma$ and that this does not alter the boundary data space, assuming the additional condition $\rg \Pi_{sh}\cap\rg P=\{0\}$, which follows from formal self-adjointness.
The idea is that a ‘shadow solution’ $u \in \rg \Pi$ is brought to light in $T+\Pi$ since $(T+\Pi)u=\Pi u=u$. The additional condition is needed to ensure that no new shadow solutions (or other solutions of $(T+\Pi)u=0$ altering $\calB$) are created.
Proof of $\calB_{T,\calF}\subset\calB_{T+\Pi}$: If $U\in\calB_{T,\calF}$ then $U=\gamma u$ with $Tu=0$. Let $v=u-\Pi u$. We claim that $(T+\Pi)v=0$ and $\gamma v=\gamma u$, which implies $U\in\calB_{T+\Pi}$. First, $\Pi v=0$ implies $(T+\Pi)v=Tv=Tu-T\Pi u=0$ since $Tu=0$ and $T_{|\rg\Pi}=0$. Second, $\gamma_{|\rg\Pi}=0$ implies $\gamma v=\gamma u$.
Proof of $\calB_{T+\Pi}\subset\calB_{T,\calF}$: If $U\in\calB_{T+\Pi}$ then $U=\gamma u$ with $(T+\Pi)u=0$. Then $Tu=-\Pi u$, and $\rg \Pi\cap\rg T=\{0\}$ implies $Tu=0$, so $U\in\calB_{T,\calF}$.
Proof of $\ker (T+\Pi) \cap \ker\gamma = \{0\}$: If $(T+\Pi)u=0$ and $\gamma u=0$ then $Tu=0$, $\Pi u=0$ as before, so $u\in \ker T\cap\ker\gamma=\rg\Pi$. Since also $u\in\ker\Pi$ and $\Pi$ is a projection it follows that $u=0$.
Proof of $\rg \Pi \cap \rg T = \{0\}$: Suppose that $T$ restricts to an operator ${{\dot C^\infty_{{\mathrm{s}}}}}(X;E) \to {{\dot C^\infty_{{\mathrm{s}}}}}(X;E)$ and let $u \in {{\dot C^\infty_{{\mathrm{s}}}}}(X;E)$, $v \in \calF(X;E)$. Then also $Tu \in {{\dot C^\infty_{{\mathrm{s}}}}}(X;E)$ and $Tv \in \calF(X;E)$ and because the $L^2$-scalar product extends to a pairing of ${{\dot C^\infty_{{\mathrm{s}}}}}(X;E)$ and $\calF(X;E)$, the scalar products $\langle Tu, v \rangle$ and $\langle u, Tv \rangle$ are well-defined. Now additionally assume $\ker T \cap \ker \gamma \subset {{\dot C^\infty_{{\mathrm{s}}}}}(X;E)$ and . If $u\in\rg\Pi\cap\rg T$ then $u\in \rg\Pi\subset\ker T\cap \ker\gamma\subset {{\dot C^\infty_{{\mathrm{s}}}}}(X;E)$ and $u=Tv$ for some $v\in\calF(X;E)$, so $Tu=0$ gives $0=\langle Tu,v\rangle = \langle u,Tv \rangle = \|u\|^2$, so $u=0$.
Extensions {#ssec:extensions}
----------
### Extensions of spaces and bundles {#sssec:extspaces}
Given a [[<span style="font-variant:small-caps;">bc</span>]{}]{}-extension $\Xhat$ of $X$ as in Definition \[def:bv-ext\] we will sometimes denote $$X^+ = X\,,\quad X^- = \overline{\Xhat\setminus X}\,.$$ We need to show that ($\phi$-[[<span style="font-variant:small-caps;">bc</span>]{}]{}-)extensions exist. One way to prove this is by doubling across the [[<span style="font-variant:small-caps;">bc</span>]{}]{}-boundary. For a compact manifold with non-empty boundary, $M$, this is standard: join two copies $M^\pm$ of $M$ along their identical boundaries: $\hat{M} = M^- \sqcup_{\partial M} M^+$ and use a collar neighborhood of $\partial M$ to define the smooth structure of $\Mhat$ near $\partial M$, yielding an extension of $M$. This extends to [[<span style="font-variant:small-caps;">bc</span>]{}]{}-manifolds, and also to $\phi$-[[<span style="font-variant:small-caps;">bc</span>]{}]{}-manifolds:
\[lem:mfd.extension\] Let $X$ be a $\phi$-manifold with non-empty [[<span style="font-variant:small-caps;">bc</span>]{}]{}-boundary and let $\Xhat = X^- \sqcup_{{{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}} X^+$ be its double across ${{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}$. Then the $\phi$-structure of $X$ extends canonically to a $\phi$-structure on $\Xhat$ by doubling the fibres, making $\Xhat$ a $\phi$-[[<span style="font-variant:small-caps;">bc</span>]{}]{}-extension of $X$ in the sense of Definition \[def:bv-ext\].
We first choose a collar neighborhood of the [[<span style="font-variant:small-caps;">bc</span>]{}]{}-boundary compatible with $\phi$, that is, a diffeomorphism from a neighborhood $U\subset X$ of ${{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}$ $$\label{eqn:collar nbhd1}
S:U\to[0,1)\times{{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}},\quad S_{|{{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}} = 0\times\id \,,$$ which at ${{\partial_{{\mathrm{s}}}X}}$ respects the fibres of $\phi$ (this makes sense since a diffeomorphism must map $U\cap{{\partial_{{\mathrm{s}}}X}}$ to ${{\partial_{{\mathrm{s}},{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}} X}}\times[0,1)$), as in but only for $\rho\geq0$.
The smooth structure on $\Xhat$ is inherited from that of $X$ away from ${{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}$, and near ${{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}$ is defined by choosing a collar neighborhood $S$ as above and demanding the map $$\widehat S: \widehat U = U^- \sqcup_{{{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}} U^+ \longrightarrow (-1,1)\times{{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}$$ to be a diffeomorphism, where $\widehat S$ is defined on $U^+$ as $S$ and on $U^-$ as $S$ followed by sign reversal in the first coordinate. Then, defining $${{\hat{\phi}}}: {{\partial_{{\mathrm{s}}}X}}^- \sqcup_{{{\partial_{{\mathrm{s}},{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}} X}}} {{\partial_{{\mathrm{s}}}X}}^+ = \partial \Xhat \longrightarrow B$$ to be $\phi$ on both ${{\partial_{{\mathrm{s}}}X}}^\pm$, we see that on $\widehat U$ it maps $\widehat S^{-1}(\rho,p)\mapsto \phi(p)$ for all $(\rho,p)\in(-1,1)\times{{\partial_{{\mathrm{s}},{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}} X}}$, hence is still a fibration. By a similar reasoning the boundary defining function $x$ for ${{\partial_{{\mathrm{s}}}X}}\subset X$ on each copy $X^\pm$ yields a boundary defining function for $\partial\Xhat\subset \Xhat$.
We emphasize that at ${{\partial_{{\mathrm{s}}}X}}$ the doubling ‘happens in the fibres’ only. The base $B$ is a manifold without boundary.
Let $E$ be a vector bundle over a $\phi$-manifold with non-empty [[<span style="font-variant:small-caps;">bc</span>]{}]{}-boundary, $X$, and let $\Xhat$ be a [[<span style="font-variant:small-caps;">bc</span>]{}]{}-extension of $X$. An **extension** of $E$ is a vector bundle $\Ehat$ over $\Xhat$ so that $\Ehat_{|X} = E$.
Note that if we take for $\Xhat$ the double of $X$ across ${{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}$ then any vector bundle $E$ has an extension to $\Xhat$: simply use $E$ on $X^-$ and glue using a trivialization of $E$ on a collar neighborhood of ${{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}$. Nevertheless, we will generally use any [[<span style="font-variant:small-caps;">bc</span>]{}]{}-extension $\Xhat$ and then assume that the bundles under consideration extend as well.
### Extensions of operators
If $\Xhat$, $\Ehat$ are ([[<span style="font-variant:small-caps;">bc</span>]{}]{}-)extensions of $X$, $E$, then any section $\uhat\in\calF(\Xhat;\Ehat)$ can be restricted to a section $\uhat_{|X}\in\calF(X;E)$, by the assumptions on $\calF$. If $u=\uhat_{|X}$ then we also say that $\uhat$ extends $u$. For operators on $\Xhat$ to be restrictable to $X$ is an extra condition.
\[def:extension operator\] Let $\Xhat$ be a [[<span style="font-variant:small-caps;">bc</span>]{}]{}-extension of $X$ and $\Ehat,\Ehat^\prime\to\Xhat$ be extensions of vector bundles $E,E^\prime\to X$. We say that a linear operator $\hat{T} : \calF(\Xhat;\Ehat) \to \calF(\Xhat;\Ehat^\prime)$ **restricts** to $X$ if for all $\uhat\in\calF(\Xhat;\Ehat)$ $$\begin{gathered}
\label{eqn:extension of ops 1}
(\That\uhat)_{|X} \text{ only depends on } \uhat_{|X}\\
\label{eqn:extension of ops 2}
\supp \uhat \subset X \Rightarrow \supp \That \uhat \subset X\,.\end{gathered}$$ In this case we denote the restriction by $$\That_X: \calF(X;E) \to \calF(X;E^\prime)\,.$$ Given $T: \calF(X;E) \to \calF(X;E^\prime)$ we say that $\That$ **extends** $T$ if $\That_X=T$.
Note that $\That_X$ is unique if it is defined since restriction to $X$ is surjective $\calF(\Xhat;E)\to\calF(X;E)$. Condition says that for $u\in\calF(X;E)$, $\That_X u :=(\That \uhat)_{|X}$ is independent of the choice of extension $\uhat$ of $u$, so $\That_X$ is well-defined. This is clearly equivalent to condition with $X$ replaced by $X^-=\overline{\Xhat\setminus X}$. Therefore, $T$ restricts to $X$ if and only if it restricts to $X^-$.[^6] Also, note that is equivalent to $$\That_X : {\dot{\calF}_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}}(X;E) \to {\dot{\calF}_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}}(X;E^{\prime}) \,,$$ with ${\dot{\calF}_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}}(X;E)$ as in , since we can identify, via extension by zero, $$\label{eqn:CdotBCinf identific}
{\dot{\calF}_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}}(X;E)
= \{\uhat\in \calF(\Xhat;\Ehat):\, \supp \uhat\subset X \} \,.$$ We close with additional remarks regarding Definition \[def:extension operator\]:
\[rmk:extension operator\]
1. If $\That$ is given in terms of a Schwartz kernel $K$, a distribution on $\Xhat\times\Xhat$ (as for $\Psi$DOs), then $\That$ restricts to $X$ if and only if $\supp K \subset X^2 \cup (X^-)^2$, i.e if and only if $K$ has block-diagonal structure with respect to the $\pm$ decomposition of $\Xhat$. So in this case (which is all we care about) Definition \[def:extension operator\] is independent of the choice of function space $\calF$.
2. If $\That$ restricts to $X$ then so does its adjoint (with respect to any smooth measure on $\Xhat$ and hermitian metrics on the bundles).
3. A projection $\Pi$ to a finite dimensional subspace $K\subset\calF(\Xhat;\Ehat)$ restricts to $X$ if
- either all $u\in K$ have support in $X$, or all $u\in K$ have support in $X^-$, and
- $\Pi$ is an orthogonal projection with respect to some metrics on $\Xhat$ and $\Ehat$.
One way to see this is that the Schwartz kernel of $\Pi$ is $\sum_j u_j\otimes \overline{u_j}$, for an orthonormal basis $(u_j)_j$ of $K$.
Constructing an Invertible Generalized Extension {#sec:inv.ext}
================================================
In this section we prove the following theorem.
\[thm:construction inv ext\] Let $X$ be a $\phi$-[[<span style="font-variant:small-caps;">bc</span>]{}]{}-manifold with non-empty [[<span style="font-variant:small-caps;">bc</span>]{}]{}-boundary and suppose $P\in\Diff_\phi^m(X;E,E')$ is a $\phi$-elliptic differential operator on $X$ satisfying Assumption \[UCNF\]. Let $\Xhat$ be a $\phi$-[[<span style="font-variant:small-caps;">bc</span>]{}]{}-extension of $X$ and assume that the bundles $E$ and $E^{\prime}$ extend to $\Xhat$. Then there is an augmented extension $\Ehat\to\Xhat$ of the bundles $E,E'\to X$ and an operator $\Phat\in\Psi^m_\phi(\Xhat;\Ehat)$ as well as a $\phi$-measure on $\Xhat$ and a hermitian metric on $\Ehat$ so that
1. $\Phat$ restricts to $X$ in the sense of Definition \[def:extension operator\] and $\Phat_X$ is a $\phi$-differential operator on $X$ augmenting $P$,
2. $\ker \Phat_X \cap \ker \gamma$ is a finite dimensional subspace of ${{\dot C^\infty_{{\mathrm{s}},{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}}}}(X;\Ehat)$,
3. $\Phat+\Pi_{sh}$ is fully elliptic, self-adjoint and invertible in $L^2_\phi(\Xhat;\Ehat)$, where $\Pi_{sh}$ is the orthogonal projection from $L^2_\phi(\Xhat;\Ehat)$ to the space of shadow solutions in (b).
In (b) the kernel of $\Phat_X$ is taken in $\calA_{\mathrm{s}}(X)$. In (c) we consider ${{\dot C^\infty_{{\mathrm{s}},{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}}}}(X;\Ehat)$ as a subspace of $L^2_\phi(\Xhat;\Ehat)$ as in . For $\Xhat$ one can always take the double of $X$, but other extensions are possible.
In the case that $P=-\Delta_g$ is the Laplacian of a $\phi$-metric $g$ there is a quite straight-forward proof of Theorem \[thm:construction inv ext\], which we present in Subsection \[ssec:laplacian.extension\]. It motivates the last steps (achieving full ellipticity, Subsection \[ssec:ext.full\]) of the general construction, which is much more involved.
We now give an outline of the general construction of $\Phat$. It proceeds in several steps, summarized as follows:
$$\label{eqn:inv ext outline}
\newcommand{\move}[1]{\hspace{-#1pt}} \newcommand{\nxt}[1][3]{\move{#1}&\move{#1}}
\begin{array}{ccccccccccccccc}
P \nxt[5] \stackrel{\text{aug}}{{\rightsquigarrow}} & \Pbar & \stackrel{\text{ext}}{\rightsquigarrow}\nxt \Phat_0
\nxt \stackrel{\text{aug}}{\rightsquigarrow}\nxt {{\bar\Phat}}_0 \nxt \stackrel{\text{add}}{\rightsquigarrow}\nxt[5] \Phat_1 \nxt[5] \stackrel{\text{aug}}{\rightsquigarrow}\nxt {{\bar\Phat}}_1
\nxt \stackrel{\text{add}}{\rightsquigarrow}\nxt[5] \Phat \nxt \stackrel{\text{mod}}{\rightsquigarrow}\nxt[1] \Phat + \Pi_{sh} \\[2pt]
\nxt[5] \nxt\text{\small s.a.} \nxt \nxt \text{\small $\phi$-ell} \nxt \nxt[5] \text{\small + s.a.}
\nxt[5] \nxt[5] \text{\small fully ell} \nxt[5] \nxt \text{\small + s.a.} \move{5} & \nxt \text{\small + supp${}^+$} \nxt \nxt \text{\small + inv} \\[2pt]
E,E' \nxt[5] \nxt \Ebar \nxt \nxt \Ehat_0 \nxt \nxt[5] \Ehat_1 \nxt[5] \nxt[5] \Ehat_1 \nxt[5] \nxt \Ehat
\nxt[5] \nxt \Ehat \nxt \nxt \Ehat
\end{array}$$
Here ‘aug’ means augmentation (enlarging the bundle), ‘ext’ means extension (from $X$ to $\Xhat$), ‘add’ means that a smoothing perturbation term is added that is supported in $X^-$ (i.e. restricts to $X$ as the zero operator) while ‘mod’ means modification. The term ‘s.a.’ means formally self-adjoint, ‘ell’ means elliptic, ‘supp${}^+$’ means that all functions in the kernel are supported in $X$ and ‘inv’ means invertible.
There are three main steps here: constructing a $\phi$-elliptic extension $\Phat_0$, achieving full ellipticity with $\Phat_1$ and then achieving invertibility of $\Phat+\Pi_{sh}$. In each of these constructions it is useful to start with a formally self-adjoint operator, so each step is preceded by an augmentation whose only purpose is to make the previous operator formally self-adjoint; this is achieved by considering the augmentation $\Pbar=
\big(\begin{smallmatrix}
0 & P^\star \\ P & 0
\end{smallmatrix}\big)
$. For this purpose auxiliary metrics will be chosen. These augmentations and the vector bundles (third line in ) are introduced in Subsection \[ssec:inv ext bundles aug\].
The construction of $\Phat_0$ from $P$ via $\Pbar$ is the same as done by Seeley, adapted to the $\phi$-setting, cf. (i’) in Section \[subsec:inv ext\], and also the step from full ellipticity to invertibility ($\Phat_1$ to $\Phat+\Pi_{sh}$) is analogous to Seeley’s construction, except for a slight simplification afforded by constructing ${{\bar\Phat}}_1$ first. We do this in Subsections \[ssec:extension phi-elliptic\] and \[ssec:ext.inv\].
The main new contribution in this paper is the construction of $\Phat_1$, carried out in Subsection \[ssec:ext.full\]. We now give an outline of this construction. Recall that the normal family of ${{\bar\Phat}}_0$ consists of $\Psi$DOs $N({{\bar\Phat}}_0)(\mu)$ on the fibres $F_y$, where $\mu=(\tau;y,\eta)\in\R\times T^*B$. So the task is to perturb this normal family to make it invertible. The first idea would be to proceed as in step (iii’) in Section \[subsec:inv ext\], i.e.to add a projection to a subspace $V_\mu$ which is complementary to the range of $N({{\bar\Phat}}_0)(\mu)$, for each $\mu$. However, the dimension of the kernels of $N({{\bar\Phat}}_0)(\mu)$ may vary with $\mu$, so there is no continuous family of such projections. We circumvent this problem by adding $i=\sqrt{-1}$ times an orthogonal projection, which yields invertibility under the weaker condition that $V_\mu+\rg N({{\bar\Phat}}_0)(\mu)$ is the full space (the sum not necessarily being direct), see Lemma \[lem:proj.inversion\]. Construction of a smooth finite rank bundle $(V_\mu)_{\mu\in\R\times T^*B}$ satisfying this condition is possible since $N({{\bar\Phat}}_0)(\mu)$ is invertible for $\mu$ outside a compact set. Then we obtain $\Phat_1$ from ${{\bar\Phat}}_0$ by adding a smoothing operator whose normal family at $\mu$ is the orthogonal projection to $V_\mu$. In order to ensure that $\Phat_1$ still extends the original operator $P$, we need to construct $V_\mu$ so that its elements are supported in $X^-$. This is where Assumption \[UCNF\] is needed, see Lemma \[lem:complement in M-\]. See also Remark \[rem:need UCNF\].
We remark that (a) in Theorem \[thm:construction inv ext\] could be strengthened to $\Phat$ being a $\phi$-differential operator *near* $X$, in the sense that its Schwartz kernel has support in $\diag_\Xhat \cup \interior{X^-}^2$. This follows from the fact that the space $W$ constructed in Lemma \[lem:complement in M-\](ii) is actually contained in ${{\dot C^\infty_{{\mathrm{s}},{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}}}}(X^-;\Ehat)$. However, we don’t need this extra information, and our statement seems cleaner.
Throughout this section we fix a background $\phi$-metric $\ghat$ on $\Xhat$, with respect to the extended fibration ${{\hat{\phi}}}$ of $\partial \Xhat$, and also hermitian metrics on the vector bundles $E,E'$.
The case of the Laplacian {#ssec:laplacian.extension}
-------------------------
If $P=-\Delta_g$ is the scalar Laplacian of a $\phi$-metric $g$ then there is a much simpler proof of Theorem \[thm:construction inv ext\] than in the general case. We present this first as it motivates the last steps, construction of $\Phat_1$ and of $\Phat$, of the general construction. The other steps are trivial in this case. The bundles are all trivial line bundles, but the same construction also works for the Hodge Laplacian on forms.
First, $P$ has a natural extension to $\Xhat$ as $-\Delta_{\ghat}$ for a $\phi$-metric $\ghat$ extending $g$. This is a $\phi$-elliptic, self-adjoint differential operator on $\Xhat$. However, it is not fully elliptic since its normal family is $N(-\Delta_{\ghat}) (\tau;y,\eta) = - \Delta_{\Fhat_y} + |(\tau,\eta)|^2$, with the induced metric $|\cdot|$ on $(\tau,\eta)$ space, and for $(\tau,\eta)=0$ this operator vanishes on the constants. This can be remedied easily with the help of the positivity of the Laplacian: Choose a function $a \in \Cinf(\Xhat, \R)$ satisfying
1. \[lapl.ext.item1\] $a=0$ on $X$,
2. \[lapl.ext.item2\] $a\geq 0$ everywhere on $\Xhat$ and
3. \[lapl.ext.item3\] $a_{|\Fhat_y}$ does not vanish identically for each $y \in B$,
and set $\Phat = -\Delta_{\ghat} + a$. This is a $\phi$-differential operator whose normal family is $$N(\Phat) (\tau;y,\eta) = - \Delta_{\Fhat_y} + |(\tau,\eta)|^2 + a_{|\Fhat_y} \in \Diff^2(\Fhat_y)$$ which is non-negative by and even positive by , for each $ (\tau;y,\eta)\in \R \times T^*B$, so $\Phat$ is fully elliptic. Also, $\Phat$ is positive for the same reason, so it is invertible. Finally, $\Phat$ extends $P$ by , so it satisfies the claims of Theorem \[thm:construction inv ext\], where $\ker \Phat_X=\{0\}$ and hence $\Pi_{sh}=0$ since the Laplacian has the unique continuation property.
The augmentations and vector bundles {#ssec:inv ext bundles aug}
------------------------------------
We first discuss the three augmentation steps in and define the vector bundles on which the operators act. Each operator in except $P$ acts from sections of the bundle noted underneath it to sections of the same bundle. Recall that we consider $P$ as an operator from $\calF(X;E)$ to $\calF(X;E^\prime)$.
First, we let $$\label{eqn:def.Pbar}
\Pbar =
\begin{pmatrix}
0 & P^\star\\ P & 0
\end{pmatrix} : \calF(X;\Ebar)\to\calF(X;\Ebar),\quad \Ebar=E\oplus E'$$ where $P^\star$ is the formal adjoint of $P$ with respect to the chosen $\phi$-metric on $X$ and hermitian metrics on $E,E'$. Thus, $\Pbar$ is an augmentation of $P$ with respect to the bundle maps $$\label{eqn:def i_0 pi_0}
E \stackrel{\iota_0}\longhookrightarrow \Ebar \stackrel{\pi_0}\longtwoheadrightarrow E,
\quad
E' \stackrel{\iota_0'}\longhookrightarrow \Ebar
\stackrel{\pi_0'}\longtwoheadrightarrow E'$$ which are injection as and projection to the first and second factor. Next, we extend $\Ebar\to X$ to $\Ehat_0\to\Xhat$ using the extensions of $E,E'$ to $\Xhat$. For $j=0,1$ and assuming that $\Phat_j$ is already constructed, we now let $$\label{eqn:def.Phatbar}
{{\bar\Phat}}_j =
\begin{pmatrix}
0 & \Phat_j^\star\\ \Phat_j & 0
\end{pmatrix} : \calF(\Xhat;\Ehat_{j+1})\to\calF(\Xhat;\Ehat_{j+1}),\quad \Ehat_{j+1}=\Ehat_j\oplus \Ehat_j$$ where $$\label{eqn:def i_j pi_j}
\Ehat_j \stackrel{\iota_{j+1}}\longhookrightarrow \Ehat_{j+1} \stackrel{\pi_{j+1}}\longtwoheadrightarrow \Ehat_j,
\quad
\Ehat_j' \stackrel{\iota_{j+1}'}\longhookrightarrow \Ehat_{j+1} \stackrel{\pi_{j+1}'}\longtwoheadrightarrow \Ehat_j'$$ are injection as and projection to the first and second factor. We also write $\Ehat=\Ehat_2$.
Recall that our notion of restriction, Definition \[def:extension operator\], is preserved under taking adjoints. Therefore, if $\Phat_j$ restricts to $X$ then so does ${{\bar\Phat}}_j$, and if $(\Phat_j)_X$ is a $\phi$-differential operator then so is $({{\bar\Phat}}_j)_X$.
On $\Ebar$ we use the hermitian product metric, starting with the given metrics on $E$ and $E'$. The metric on $\Ehat_0$ is any extension of this metric. On $\Ehat_1$, $\Ehat=\Ehat_2$ we use the hermitian product metrics. Then $\Pbar$, ${{\bar\Phat}}_0$, ${{\bar\Phat}}_1$ are formally self-adjoint $\phi$-elliptic operators.
Over $X$, the bundles $E,E'$ are related to $\Ehat$ via the compositions $$\label{eqn:relation E Ehat}
E \stackrel{\iota}\longhookrightarrow \Ehat_X \stackrel{\pi}\longtwoheadrightarrow E\,,\qquad \iota = \iota_2\iota_1\iota_0\,,\quad \pi = \pi_0\pi_1\pi_2$$ and similarly $E' \stackrel{\iota'}\longhookrightarrow \Ehat_X
\stackrel{\pi'}\longtwoheadrightarrow E'$ where $\iota' = \iota_2'\iota_1'\iota_0'$, $\pi' = \pi_0'\pi_1'\pi_2'$. The restrictions $({{\bar\Phat}}_1)_X$, $({{\bar\Phat}}_2)_X$ could be represented as $4\times 4$ and $8\times8$ matrices whose only non-zero entries are an alternating sequence of $P$ and $P^\star$ on the antidiagonal. The ’original’ $P$ is included as lower left corner, or formally: $$\label{eqn:inv ext P in P2}
P = \pi' \,({{\bar\Phat}}_2)_X \,\iota\,.$$
Two functional analytic lemmata
-------------------------------
The following elementary facts are used at several places in the construction.
\[lem:proj.inversion\] Let $H$ be a complex Hilbert space, $T : \mathrm{dom}\, T \subset H \to H$ a densely defined self-adjoint Fredholm operator and $\Pi$ a finite rank orthogonal projection in $H$. Let $\alpha>0$. Then:
1. $T+\alpha\Pi$ is invertible if $\rg T\oplus\rg \Pi=H$.
2. $T + i\alpha\Pi$ is invertible if and only if $\rg T + \rg \Pi = H$.
Statement (a) has a weak converse: $T+\alpha\Pi$ invertible implies $\rg T + \rg \Pi = H$ (same proof as for (b)), but the sum need not be direct as the example $T=\Pi=\id$ on a finite dimensional space shows.
The operators in (a) and (b) are Fredholm with index zero since $T$ has this property and $\Pi$ has finite rank. So for invertibility it suffices to check injectivity in each case. Also, observe that $\rg T+\rg \Pi=H$ implies $\ker T\cap\ker \Pi=\{0\}$ since $(\ker T \cap \ker \Pi)^\perp = (\ker T)^\perp + (\ker \Pi)^\perp = \rg T + \rg\Pi$.
\(a) If $(T+\alpha\Pi)u=0$ then $Tu=-\alpha\Pi u$, so $Tu=\Pi u=0$ since $\rg T\cap\rg \Pi=\{0\}$. Then $\ker T\cap\ker \Pi=\{0\}$ implies $u=0$.
\(b) If $(T + i\alpha\Pi)u=0$ then $\langle Tu,u\rangle + i\alpha\langle \Pi u,u\rangle=0$, and because both $T$ and $\Pi$ are self-adjoint this implies $\langle \Pi u, u\rangle = 0$, hence $\Pi u = 0$ since $\Pi$ is an orthogonal projection. Inserting this into $(T+\alpha i\Pi)u=0$ we get $T u = 0$, and again $\ker T\cap\ker \Pi=\{0\}$ implies $u=0$.
For the converse suppose that $T + i\alpha\Pi$ is invertible. Given $g \in H$, there is $f \in H$ so that $(T + i\alpha\Pi) f = g$. Then $g = Tf+\Pi(i\alpha f)\in \rg T+\rg\Pi$.
The following lemma allows us to find projections as in Lemma \[lem:proj.inversion\].
\[lem:complement in M-\] Let $M$ be a manifold with [[<span style="font-variant:small-caps;">bc</span>]{}]{}-boundary and $\Mhat$ be a [[<span style="font-variant:small-caps;">bc</span>]{}]{}-extension of $M$.
Then for any finite-dimensional subspace $K\subset {{\dot C^\infty_{{\mathrm{s}}}}}(\Mhat)$ the following are equivalent:
1. $ u\in K,\ \supp u \subset M \Rightarrow u = 0$
2. There is a subspace $W \subset {{\dot C^\infty_{{\mathrm{s}},{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}}}}(M^-)$ satisfying $$W \oplus K^\perp = L^2(\Mhat)\,.$$
The analogous statements hold for sections of a hermitian vector bundle over $\Mhat$.
Here we use the $L^2$ scalar product defined by any conormal density on $\Mhat$. The lemma says that if all non-trivial elements of $K$ are already non-trivial on $M^-$ then the complement $K$ of $K^\perp$ may be replaced by a space of sections supported in $M^-$. Note that condition (i) may be expressed as ’$u\in K$ is determined by $u_{|M^-}$’ and therefore is a unique continuation condition.
In the applications of the lemma $K$ is the kernel of a self-adjoint ($\phi$-)elliptic $\Psi$DO $\That$ on $\Mhat$, and then $W$ is a complement to its range. We need this lemma in two settings: In the first setting $\That$ is an operator in the normal family of ${{\bar\Phat}}_0$ and $M$ is a fibre $F$ (proof of Lemma \[lem:bundle.ext\]). Here ${{\partial_{{\mathrm{s}}}M}}=\emptyset$. In the second setting $\That$ is a modification of ${{\bar\Phat}}_1$ and $M=X$ (see the proof of Theorem \[thm:construction inv ext\] in Subsection \[ssec:ext.inv\]). We will not need the analogous statement for general function spaces $\calF$ though.
The lemma is a smooth version of a simple fact about Hilbert spaces: Let $H=L^2(\Mhat)$, $H^\pm=L^2(M^\pm)$. Then $H = H^+ \oplus H^-$ (orthogonal direct sum), and for a closed subspace $K \subset H$ we have from $H^-=(H^+)^\perp$ that $$K\cap H^+ = \{0\} \iff K^\perp + H^- = H \iff \exists W\subset H^-:\ K^\perp \oplus W = H\,.$$
(ii)$\Rightarrow$(i): If $W\subset {{\dot C^\infty_{{\mathrm{s}},{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}}}}(M^-)$ satisfies (ii) and $u\in K$ is supported in $M$ then $u\perp W$ and $u\perp K^\perp$, so $u\perp W\oplus K^\perp=L^2(\Mhat)$, hence $u=0$.
(i)$\Rightarrow$(ii): Let $\rho:\Mhat\to\R$ be a defining function for $\partial_{{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}}M$ so that $M=\{\rho\geq0\}$. We first show that we can enlarge $M$ slightly in condition (i), that is, there is $\eps>0$ so that $$\label{eqn:cond(i)enlarged}
u\in K, \supp u \subset \{\rho\geq-\eps\} \Rightarrow u=0\,.$$ To show this, assume it was wrong. Then we could find a sequence $u_m\in K$ with $\supp u_m\subset\{\rho\geq-\frac1m\}$ and $\|u_m\|=1$. Since $\dim K<\infty$, there would be a convergent subsequence $u_{m'}\to u$ with $u\in K$, $\|u\|=1$ by compactness, with convergence in $C(\Mhat)$, so $\supp u\subset\{\rho\geq0\}=M$. This would contradict assumption (i).
Now choose $\chi\in\Cinf(\Mhat,\R)$ supported in $M^-$ and equal to 1 on $\{\rho\leq-\eps\}$. Then implies that the maps $u\mapsto \chi u$, $u\mapsto \chi^2 u$ are injective on $K$. Let $W=\chi^2 K$. We claim that $W$ satisfies $W \oplus K^\perp = L^2(\Mhat)$. To prove this, it suffices to check $W \cap K^\perp = \{0\}$ because $\dim W=\dim K$. Now if $w\in W$ then $w=\chi^2 u$ with $u\in K$, so if $w\in K^\perp$ also then $0=\langle w,u\rangle=\langle \chi^2 u,u\rangle = \langle \chi u, \chi u\rangle$, so $\chi u=0$, hence $u=0$.
Constructing a -elliptic augmented extension {#ssec:extension phi-elliptic}
--------------------------------------------
Here we construct the operator $\Phat_0$ in . Recall from that we already constructed $\Pbar$.
\[prop:ext1\] Let $\Pbar\in\Diff^m_\phi(X;\Ebar)$ be formally self-adjoint and $\phi$-elliptic. Let $\Xhat$ be a $\phi$-[[<span style="font-variant:small-caps;">bc</span>]{}]{}-extension of $X$ and $\Ehat_0\to\Xhat$ an extension of $\Ebar$. Then there is a $\phi$-elliptic extension $\Phat_0\in\Psi^m_\phi(\Xhat;\Ehat_0)$ of $\Pbar$.
Extend the hermitian metric on $\Ebar$ to $\Ehat_0$. Since $\Pbar$ is assumed to have coefficients smooth up to ${{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}$, we may extend it to a neighborhood $\tilde X$ of $X$ in $\Xhat$ as a formally self-adjoint $\phi$-elliptic differential operator $\Ptilde$.
Since the $\phi$-principal symbol ${\mathfrak{p}}= {{}^\phi\sigma}_m(\Ptilde)$ is self-adjoint over $\tilde{X}$, we can extend it to a $\phi$-elliptic symbol ${\mathfrak{p}}_1$ of order $m$ over $\Xhat$ by setting ${\mathfrak{p}}_1=|\xi|^m\Id_{\Ehat_0}$ outside a neighborhood of the closure of $\tilde{X}$ and connecting endomorphisms of fibres of $\Ehat_0$ to the identity by moving their spectrum through the upper complex half plane, cf. [@See69 p. 298].
We choose an operator $$\text{$\Ptilde' \in \Psi_\phi^m(\Xhat;\Ehat_0)$ so that ${{}^\phi\sigma}_m(\Ptilde') = {\mathfrak{p}}_1$.}$$ This operator need not extend $\Pbar$ although its principal symbol extends the principal symbol of $\Pbar$. Therefore, we ’glue’ $\Ptilde'$ away from $X$ with $\Ptilde$ on $\Xtilde$, by choosing cut-off functions $\psi_+$, $\psi_- \in \Cinf(\Xhat,\R_+)$ so that $\psi_+ + \psi_- = 1$ on $\Xhat$, $\psi_- = 0$ in a neighborhood of $X$ and $\psi_- = 1$ outside of $\tilde{X}$, and setting $$\Phat_0 = \psi_+ \Ptilde \psi_+ + \psi_- \Ptilde' \psi_- \,.$$ Then $\Phat_0 \in \Psi^m_\phi(\Xhat;\Ehat_0)$ is $\phi$-elliptic and an extension of $\Pbar$ in the sense of Definition \[def:extension operator\].
The extension $\Phat_0$ cannot in general be chosen to be a differential operator. Also, the construction yields a non-selfadjoint operator in general since the extended symbol ${\mathfrak{p}}_1$ cannot in general be chosen to be self-adjoint; the reason for this is that the set of self-adjoint invertible $N\times N$ matrices is not connected for any $N$.
Achieving full ellipticity {#ssec:ext.full}
--------------------------
We now construct the fully elliptic operator $\Phat_1$ from ${{\bar\Phat}}_0$ in , where ${{\bar\Phat}}_0$ is an augmentation as in of the operator $\Phat_0$ just constructed. For this we need to perturb the normal family, and it is in this step that we need to make an assumption on our original operator $P$. Let us say that a differential operator $T$ on a compact manifold $F$ with ([[<span style="font-variant:small-caps;">bc</span>]{}]{}-)boundary has the **unique continuation property (UCP) with respect to the boundary** if $$\label{eqn:UCP def}
\ker T \cap {{\dot C^\infty}}(F,E) = \{0\} \,.$$ That is, if any solution $u$ of $Tu=0$ vanishing to infinite order at $\partial F$ must vanish identically. If $T$ is elliptic of order $m$ then this is equivalent to $Tu=0, \gamma u =0\Rightarrow u=0$. We also express this by saying that **$T$ has no shadow solutions.**
Recall that the normal family of $P$ is a family of differential operators on $F$, $N(P)(\mu):\Cinf(F_y,E_y) \to \Cinf(F_y,E_y')$, and that $N(P^\star)(\mu)=N(P)(\mu)^\star$ (see Appendix \[sec:app phi ops\]).
\[UCNF\] We assume that the normal families of $P$ and $P^\star$ have the unique continuation property at the boundary of $F$.
See Remark \[rem:need UCNF\] for some considerations on this assumption. Note that to define the adjoint we need to choose a $\phi$-metric on $X$ and bundle metrics on $E,E'$. However, the assumption on $N(P)^\star$ is independent of the choice of metrics on $X$ and $E, E'$. This can be proved as follows. First, by considering $T=N(P)(\mu)$ and $T^\star$ as operators $V \to V'$ resp. $V'\to V$ where $V={{\dot C^\infty_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}}}(F_y,E_y)$, $V'={{\dot C^\infty_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}}}(F_y,E_y')$ and using $\ker T^\star = (\rg T)^\perp$ we see that the failure of the UCP for $T^\star$ is equivalent to the existence of $v\in V'\setminus\{0\}$ such that $v\perp \rg T$. Here we still use the $L^2$ scalar product in $V'$. Next, changing the metrics on $X$ and $E'$ amounts to replacing the $L^2$-scalar product $\langle\ ,\ \rangle$ on $V'$ by $\langle u,v\rangle_1 = \langle u,Av\rangle$ where $A$ is a bundle automorphism of $E'$. This implies that $v\perp_1 \rg T \iff Av\perp \rg T$. Now $v\mapsto Av$ is an invertible linear map $V'\to V'$, and the claim follows.
Assumption \[UCNF\] implies that the normal family of $({{\bar\Phat}}_0)_X$ also has the unique continuation property from the boundary since this operator is a direct sum of two copies of $P$ and $P^\star$ each. Therefore, we need to prove the following proposition.
\[prop:construction fully elliptic\] Let ${{\bar\Phat}}_0\in\Psi_\phi^m(\Xhat;\Ehat_1)$ be $\phi$-elliptic and selfadjoint. Also, assume that it restricts to $X$ (Definition \[def:extension operator\]) and that the normal family of $({{\bar\Phat}}_0)_X$ has the unique continuation property at the boundary (as defined before Assumption \[UCNF\]).
Then there is a *fully* elliptic $\Phat_1\in\Psi_\phi^m(\Xhat;\Ehat_1)$ which restricts to $X$ and so that $$\begin{gathered}
\label{eqn:Phat1 and Phatbar0}
(\Phat_1)_X = ({{\bar\Phat}}_0)_X\,,\quad \Phat_1 - {{\bar\Phat}}_0 \in \Psi^{-\infty}_\phi (\Xhat;\Ehat_1)\end{gathered}$$
Note that Definition \[def:extension operator\] involves an implicit choice of function space $\calF$ but following Remark \[rmk:extension operator\], this choice does not matter for $\phi$-$\Psi$DOs, and that is equivalent to the Schwartz kernel of $\Phat_1-{{\bar\Phat}}_0$ being a smooth section on $\Xhat^2_\phi$ supported in $(X^-)^2_\phi$.
For the proof of the proposition we need two lemmata.
\[lem:ind.sect\] Let $\calV \to Z$ be a smooth vector bundle of infinite rank over a manifold $Z$ and let $w_1, \dotsc, w_k$ be smooth sections of $\calV$. Then there are smooth sections $s_1, \dotsc, s_k$ of $\calV$ so that, for all $\eps \neq 0$, $$w_1 + \eps s_1, \dotsc, w_k + \eps s_k$$ are linearly independent sections of $\calV$.
For $z \in Z$, let $W(z) = \{ w_1(z), \dotsc, w_k(z)\}$. As both the base $Z$ and $\operatorname{span}W(z)$ are finite dimensional, we can choose a section $s_1$ of $\calV$ so that $s_1(z) \not\in \operatorname{span}W(z)$ for all $z \in Z$. Step by step, choose sections $s_1, \dotsc, s_k$ of $\calV$ so that $$\label{eqn:ind.sect.1}
s_j(z) \not\in \operatorname{span}\big( W(z) \cup S_{j-1}(z) \big) \quad\text{for all $z \in Z$,}$$ where $S_l(z) = \{s_1(z), \dotsc, s_l(z) \}$. Fix $z$. Then inductively $\operatorname{span}W(z) \cap \operatorname{span}S_l(z)=\{0\}$ for each $l$. Then if $\eps \neq 0$ and $\sum_i \lambda_i \big(w_i(z) + \eps s_i(z)\big) = 0$ it follows that $\sum_i \lambda_i s_i(z) = -\frac1\eps \sum_i \lambda_i w_i(z)$, so both sides must be zero, hence $\lambda_i=0$ for all $i$ since the $s_i(z)$ are linearly independent. So we even get that $w_1(z) + \eps s_1(z), \dotsc, w_k(z) + \eps s_k(z)$ are linearly independent for each $z$.
\[lem:bundle.ext\] In the setting of Proposition \[prop:construction fully elliptic\] there is a smooth finite rank vector bundle $V \to \R \times T^*B$ so that for each $\mu= (\tau;y,\eta) \in \R \times T^*B$: $$\begin{gathered}
\label{eqn:V in F-}
V_\mu \subset {{\dot C^\infty_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}}}(F^-_y,\Ehat_1) \\
\label{eqn:bundle.ext.2}
V_{\mu} + \rg N({{\bar\Phat}}_0) (\mu)
= L^2(\Fhat_y,\Ehat_{1y})\,.
\end{gathered}$$
In this proof write $T={{\bar\Phat}}_0$ and leave out bundles from the notation. Since $T$ is $\phi$-elliptic there is a compact set $\calK \subset \R \times T^*B$ so that $N(T)(\mu)$ is invertible for $\mu\not\in\calK$, see [@Mel95a Prop. 2].
Fix $\mu=(\tau;y,\eta)\in\calK$. We apply Lemma \[lem:complement in M-\] with $\Mhat=\Fhat_y$ and $K=\ker N(T)(\mu)$. Condition (i) in the lemma is simply the unique continuation property at the [[<span style="font-variant:small-caps;">bc</span>]{}]{}-boundary, so it is satisfied by assumption. Now $N(T)(\mu)$ is an elliptic $\Psi$DO on the closed manifold $\Fhat_y$, hence Fredholm in $L^2(\Fhat_y)$, so $\dim K<\infty$ and $K^\perp = \rg N(T)(\mu)$ by self-adjointness, so (ii) of the lemma gives a subspace $$V_\mu' \subset{{\dot C^\infty_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}}}(F^-_y) \text{ so that } V_\mu' \oplus \rg N(T)(\mu) = L^2(\Fhat_y)\,.$$ Then by continuity there is an open neighborhood $U_\mu$ of $\mu$ in $\R\times T^*B$ so that $$\label{eqn:V'compl local}
V_\mu' + \rg N(T)(\tilde\mu) = L^2(\Fhat_\ytilde)$$ (even $\oplus$ here) for all $\tilde\mu=(\tautilde;\ytilde,\etatilde)\in U_\mu$, where we identify nearby fibres $\Fhat_\ytilde$.
We now combine the $V_\mu'$ using a compactness argument and Lemma \[lem:ind.sect\] to obtain $V$: As $\calK$ is compact, finitely many of the $U_\mu$ suffice to cover this set, say $U_1, \dotsc, U_N$ where $U_j = U_{\mu_j}$. Let $\psi_1, \dotsc, \psi_N\in\Cinf(\R\times T^*B)$ be a partition of unity subordinate to the cover $U_1, \dotsc, U_N$ of $\calK$. Choose a basis $(\omega_{ij}')_i$ for each $V_{\mu_j}'$ and apply Lemma \[lem:ind.sect\] to the set of $\omega_{ij} = \psi_j \omega_{ij}'$ over all $i,j$, where $Z=\R\times T^*B$ and the bundle $\calV\to\R\times T^*B$ is given by $\calV_\mu = {{\dot C^\infty_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}}}(F^-_y)$ for $\mu=(\tau;y,\eta)$. The lemma gives sections $s_{ij}$ of $\calV$ so that for each $\eps>0$ the sections $(\omega_{ij}+\eps s_{ij})_{i,j}$ are linearly independent, so span a subbundle $V^\eps$ of $\calV$. Moreover, for each $j$ condition , with $\mu=\mu_j$ and $\mutilde$ in the compact set $\supp\psi_j$, is stable under small perturbations of and under enlarging $V_\mu'$, hence is satisfied with $V_\mu'$ replaced by $V^\eps$ for $\eps$ sufficiently small. Therefore we may choose $\eps$ sufficiently small so that , with $V_\mu'$ replaced by $V^\eps_\mu$ and $\mutilde=\mu$, holds for all $\mu \in \calK$. Since $\rg N(T)(\mu) = L^2(\Fhat_y)$ for $\mu \not\in \calK$ the bundle $V=V^\eps$ satisfies the claim of the proposition.
Construct a bundle $V$ as in Lemma \[lem:bundle.ext\]. For each $\mu=(\tau;y,\eta)\in \R\times T^*B$ let $\widetilde{\Pi}_\mu$ be the orthogonal projection $$\widetilde{\Pi}_\mu : L^2(\Fhat_y,\Ehat_{1y}) \longrightarrow V_{\mu} \,.$$
We now assemble the Schwartz kernels of these projections to construct the desired perturbation of ${{\bar\Phat}}_0$.
Choose a compact set $\calK\subset\R\times T^*B$ so that $N({{\bar\Phat}}_0)(\mu)$ is invertible for $\mu\not\in\calK$ (see the beginning of the proof of Lemma \[lem:bundle.ext\]) and a smooth, compactly supported function $e : \R \times T^*B \to \R_+$ so that $e >0$ on $\calK$. Let $\kappatilde(\mu;z,z')$, $z,z'\in \Fhat_y$, be the Schwartz kernel of $\widetilde{\Pi}_\mu$. By it is supported in $(F^-)^2$. Taking the $(\tau,\eta)\mapsto (T,Y)$ inverse Fourier transform of $e\,\kappatilde$ we obtain a smooth function $\kappa(T,y,Y;z,z')$ vanishing rapidly as $|(T,Y)|\to\infty$. Recall that $(T,y,Y;z,z')$ are coordinates on the interior of the $\phi$-front face $\phi f$ of $\Xhat^2_\phi$, and the boundary of $\phi f$ is the bundle of spheres at infinity $|(T,Y)|=\infty$. Therefore $\kappa$ is smooth on $\phi f$ and vanishes to infinite order at its boundary faces, and is supported in $(z,z')\in (F^-)^2$ (see Appendix \[sec:app phi ops\]). Hence $\kappa$ can be extended to a smooth section, which we still denote by $\kappa$, on all of $\Xhat_\phi^2$ which vanishes to infinite order at all boundary hypersurfaces except $\phi f$ and which is supported in $(X^-)^2_\phi$. Then the operator $\widetilde{\Pi}$ with Schwartz kernel $\kappa$ is in $\Psi^{-\infty}_\phi(\Xhat;\Ehat_1)$ and has normal family $$N(\widetilde{\Pi})(\mu) = e(\mu)\widetilde{\Pi}_\mu$$ by construction.
Define $\Phat_1={{\bar\Phat}}_0+i\widetilde{\Pi}$. As ${\widetilde\Pi}$ is of order $-\infty$, $\Phat_1$ is still $\phi$-elliptic. Moreover, $N(\Phat_1)(\mu)=N({{\bar\Phat}}_0)(\mu) + i e(\mu)\widetilde{\Pi}_\mu$ is invertible for all $\mu$ by Lemma \[lem:proj.inversion\](b) because of and since $e(\mu)>0$ for all $\mu$ where $N({{\bar\Phat}}_0)(\mu)$ is not invertible. By construction, $\widetilde{\Pi}$ is supported in $X^-$ so we obtain .
Achieving invertibility; proof of Theorem \[thm:construction inv ext\] {#ssec:ext.inv}
----------------------------------------------------------------------
We now prove Theorem \[thm:construction inv ext\]. We follow the steps outlined in and around equation : We choose a $\phi$-metric $g$ on $X$ and hermitian metrics on $E$, $E'$. Then we augment $P$ to $\Pbar=
\big(\begin{smallmatrix}
0&P^\star\\ P & 0
\end{smallmatrix}\big)
$ on $\Ebar\to X$, where $P^\star$ is the formal adjoint and $\Ebar=E\oplus E'$ (with the product metric) as explained in Subsection \[ssec:inv ext bundles aug\]. We extend $X$ to $\Xhat$, $\Ebar$ to $\Ehat_0$, and also the metrics. Then we use Proposition \[prop:ext1\] to extend $\Pbar$ to a $\phi$-elliptic operator $\Phat_0\in\Psi^m_\phi(\Xhat;\Ehat_0)$. We augment $\Phat_0$, $\Ehat_0$ (as explained in Subsection \[ssec:inv ext bundles aug\]) to obtain the self-adjoint, $\phi$-elliptic operator ${{\bar\Phat}}_0\in\Psi^m_\phi(\Xhat;\Ehat_1)$. Then ${{\bar\Phat}}_0$ restricts to $X$ as the $\phi$-differential operator $\Pbar$ which augments $P$. Then we use Proposition \[prop:construction fully elliptic\] to find a fully elliptic $\Phat_1\in\Psi^m_\phi(\Xhat;\Ehat_1)$ which agrees with ${{\bar\Phat}}_0$ on $X$. In this step we use the unique continuation property at the boundary for $N(P)$ and $N(P)^\star$, Assumption \[UCNF\]. Now we augment $\Phat_1$, $\Ehat_1$ again to obtain ${{\bar\Phat}}_1\in \Psi^m_\phi(\Xhat;\Ehat)$ where $\Ehat = \Ehat_1\oplus\Ehat_1$. Then ${{\bar\Phat}}_1$ is self-adjoint, fully elliptic and restricts to $X$ as a $\phi$-differential operator which augments $P$.
It remains to make the operator invertible, i.e. to construct $\Phat + \Pi_{sh}$ as described in Theorem \[thm:construction inv ext\] from ${{\bar\Phat}}_1$. First we recall from Appendix \[sec:app phi ops\] that $${{\bar\Phat}}_1 \text{ fully elliptic} \,\Rightarrow\, \ker {{\bar\Phat}}_1 \subset {{\dot C^\infty_{{\mathrm{s}}}}}(\Xhat;\Ehat)\,,$$ the space of smooth sections vanishing to infinite order at the (singular) boundary $\partial\Xhat$, and that this kernel has finite dimension. Let $V_+=\{u\in\ker{{\bar\Phat}}_1:\, \supp u \subset X\}$. Since $({{\bar\Phat}}_1)_X$ is a $\phi$-elliptic differential operator we have $$\label{eqn: ker Phatbar1}
V_+ = \ker ({{\bar\Phat}}_1)_X \cap \ker\gamma \,.$$ Let $\Pi_{sh}$ be the orthogonal projection to $V_+$ in $L^2_\phi(\Xhat;\Ehat)$. Then $({{\bar\Phat}}_1)_X+\big(\Pi_{sh}\big)_X$ is a modification of $({{\bar\Phat}}_1)_X$ in the sense of Definition \[def:modification\], and the conditions around in Proposition \[prop:modification B\] are satisfied, so by we have $$\label{eqn:Phatbar1 eqn1}
\ker(({{\bar\Phat}}_1)_X+\big(\Pi_{sh}\big)_X) \cap \ker\gamma = \{0\}\,.$$ This implies that $K := \ker({{\bar\Phat}}_1+\Pi_{sh})$ satisfies (i) of Lemma \[lem:complement in M-\] (where $M=X$). Also, $V_+\subset {{\dot C^\infty_{{\mathrm{s}}}}}(\Xhat;\Ehat)$ implies that $\Pi_{sh} \in x^\infty\Psi^{-\infty}_\phi(\Xhat;\Ehat)$, so ${{\bar\Phat}}_1+\Pi_{sh}$ is fully elliptic, hence Fredholm, so $K$ is finite-dimensional and $ K\subset {{\dot C^\infty_{{\mathrm{s}}}}}(\Xhat;\Ehat).$ Therefore, by (ii) of Lemma \[lem:complement in M-\] we can choose a subspace $W\subset {{\dot C^\infty_{{\mathrm{s}},{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}}}}(X^-,\Ehat)$ complementing $K^\perp = \rg({{\bar\Phat}}_1+\Pi_{sh})$ in $L^2_\phi(\Xhat;\Ehat)$. Let $\Pi_{comp}$ be the orthogonal projection to $W$ in $L^2_\phi(\Xhat;\Ehat)$ and define $$\Phat = {{\bar\Phat}}_1 + \Pi_{comp}\,.$$ Then $\Phat + \Pi_{sh} = ({{\bar\Phat}}_1 + \Pi_{sh} ) + \Pi_{comp}$ is invertible by Lemma \[lem:proj.inversion\] (applied with $T={{\bar\Phat}}_1+\Pi_{sh}$ and with $\Pi_{sh}$ replaced by $\Pi_{comp}$), it is fully elliptic since $W\subset {{\dot C^\infty_{{\mathrm{s}},{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}}}}(X^-,\Ehat)$ and so $\Pi_{comp}\in x^\infty\Psi^{-\infty}_\phi(\Xhat;\Ehat)$, and it is self-adjoint since ${{\bar\Phat}}_1$ is self-adjoint and $\Pi_{sh}$, $\Pi_{comp}$ are orthogonal projections. Since $\Pi_{comp}$ restricts to $0$ on $X$ it is clear that the spaces of shadow solutions for $\Phat$ and ${{\bar\Phat}}_1$ coincide, so (c) in Theorem \[thm:construction inv ext\] holds. Furthermore, (a) holds since it holds for ${{\bar\Phat}}_1$ and again because $\Pi_{comp}$ restricts to zero on $X$. Finally, $({{\bar\Phat}}_1)_X=\Phat_X$ implies that $V_+=\ker\Phat_X\cap\ker\gamma$, so (b) follows. $\square$
\[rem:need UCNF\] We do not know if Assumption \[UCNF\] is necessary for our theorems to hold. In our proof we need it in an essential way: Our strategy for constructing $C$ is to make the extension $\Phat$ fully elliptic and invertible. The first obstruction to invertibility of $\Phat$ is the existence of shadow solutions for $P$. If $P$ has shadow solutions, we can deal with them by adding the projection $\Pi_{sh}$ to the space of shadow solutions and using Proposition \[prop:modification B\].
The second obstruction to invertibility of $\Phat$ (in the $\phi$-calculus) is that $N(\Phat)(\mu)$ must be invertible for all $\mu$. In particular, $N(P)(\mu)$ must not have shadow solutions for all $\mu$.
Now if some $N(P)(\mu)$ had shadow solutions then we could try adding an analogous projection to it.
However, there are two problems with this: First, while adding this projection does not alter the boundary data space of $N(P)(\mu)$ (by the same argument as for $P$), it is unclear how it affects the boundary data space of $P$ itself. Second, the space of shadow solutions is unstable under perturbations, so generically its dimension will vary with $\mu$. This means that the associated family of projections is not continuous in $\mu$, so does not define a $\Psi$DO. Note that here it is not possible to use ’too big’ projections as in the proof of Proposition \[prop:construction fully elliptic\] where we modified $N(\Phat)(\mu)$ on the minus side of $\Fhat$, since by (the proof of) Proposition \[prop:modification B\] a projection which is too big will change the boundary data space. Also, it is not possible to use a smaller projection since then it would not remove all shadow solutions.
We also mention that an assumption analogous to Assumption \[UCNF\] was used in [@BosFur:MIFADSFF] (see also [@BLZ09]) to ensure that the boundary data spaces of a family of operators vary continuously with the parameter.
Also, if $\partial F=\emptyset$, i.e. in the case of an interior singularity, Assumption \[UCNF\] is equivalent to $\ker N(P)(\mu)=\ker N(P)(\mu)^\star=0$, i.e. invertibility of $N(P)(\mu)$ for each $\mu$. This is a necessary condition for the invertibility on $\phi$-Sobolev spaces, and therefore very natural in the context of the $\phi$-calculus. The same remark applies to any connected component of ${{\partial_{{\mathrm{s}}}X}}$ which does not intersect ${{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}$.
Proof of Proposition \[prop:shadow solutions\] {#ssec:pf prop shadow}
----------------------------------------------
By Theorem \[thm:construction inv ext\](b) the space of shadow solutions of $\Phat_X$ is contained in ${{\dot C^\infty_{{\mathrm{s}},{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}}}}(X;\Ehat)$. This implies the corresponding statement for $P$ since $\Phat_X$ augments $P$, so $\ker P\subset \ker \Phat_X$ with respect to the inclusion $E\hookrightarrow\Ehat$. $\square$
Calderón Projectors {#sec:Calderon}
===================
In this section we prove Theorems \[thm:Calderon\], \[thm:calderon canonical inv extension\] and \[thm:orthogonal\] and Corollary \[cor:L2 closure of BC spaces\], following the outline given in Section \[sec:outline\]. The arguments in the non-singular case carry over without essential changes because of two facts: On the one hand, the Schwartz kernels of operators in the $\phi$-calculus behave in a uniform way near the $\phi$-face of $X^2_\phi$, which corresponds to the singularities, and on the other hand, the [[<span style="font-variant:small-caps;">bc</span>]{}]{}-face ${{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}$ and the singular face ${{\partial_{{\mathrm{s}}}X}}$ are transversal by assumption. Then, for instance, the transmission condition is still the correct condition to ensure that limits of restrictions (as in Proposition \[prop:transm.cond\] below) still define $\phi$-$\Psi$DOs.
The transmission property {#ssec:transmission}
-------------------------
We will need to generalize the transmission property of $\Psi$DOs from the classical to the singular setting. For this purpose it is useful to state it in terms of conormal distributions, whose definition is recalled in Appendix \[sec:app mwc\].
\[def:transm.prop\] Let $Z$ be a manifold with corners and $Y\subset Z$ an interior p-submanifold. We say that a conormal distribution $u \in I^t_{{\mathrm{cl}}}(Z,Y)$ satisfies the **strong transmission condition** at $Y$ if $\tau := t + \frac14\dim Z - \frac12{\operatorname{codim}}Y$ is an integer and in its local representations the complete symbol satisfies for all $\lambda\in\R\setminus\{0\}$.
That is, $\lambda$ is also allowed to be negative in . The complete symbol depends on the coordinate system of course, but this condition is easily seen to be independent of coordinates. The condition $\tau\in\Z$ is imposed to allow a consistent choice of powers $\lambda^{\tau-j}$ for negative $\lambda$: taking $\lambda=-1$ and using for $\eta''$ and $-\eta''$ implies that we should have $(-1)^{2\tau}=1$, so $\tau\in\Z$.
In the case where $Z=\Xhat^2$ for a compact manifold $\Xhat$ and $Y=\diag_\Xhat$ the diagonal, distributions in $I^t_{{\mathrm{cl}}}(Z,Y)$ are Schwartz kernels of classical pseudodifferential operators $P\in \Psi^t(\Xhat)$. In this case it is well-known (see [@Hor85 18.2], [@Bou:COPVBIPT], [@GruHor:TP]) that if $t \in \Z$ (note that $\tau = t$ here) then the strong transmission condition implies the transmission property for $P$ with respect to any hypersurface $W\subset\Xhat$ that divides $\Xhat$ into manifolds with boundary $X^\pm$; this says that for any $f\in\Cinf(\Xhat)$ the functions $\left[P(\chi_{X^\pm}f)\right]_{|\interior{X^\pm}}$, where $\chi_{X^\pm}$ is the characteristic function of $X^\pm$, extend smoothly to $X^\pm$. In fact, the transmission property for $P$ for a given $W$ is equivalent to a similar condition on the full symbol of $P$, called the transmission condition, only at the conormal bundle of $W$. Our condition does not refer to an a priori choice of $W$. It could be refined to the transmission condition as in [@Hor85], but we don’t need this here.
\[prop:transm.cond\] Let $Z$ be a manifold with corners and $Y\subset Z$ an interior p-submanifold. Furthermore, let $H,H'\subset Z$ be interior p-hypersurfaces such that $H,H',Y$ intersect pairwise normally transversally[^7]. Also, assume the conormal bundle of $H$ is orientable, allowing the choice of a ’positive side’ of $H$.
Define $\tilde Z = H\cap H'$, $\tilde Y = \tilde Z \cap Y$, and assume that $$\label{eqn:diag intersection}
\text{for each $p\in \tilde Y$}, \quad N_p^*\tilde Z\cap N_p^*Y \text{ has dimension }1$$ Then $\tilde Y\subset\tilde Z$ is a p-submanifold. If $u\in I^t_{{\mathrm{cl}}}(Z,Y)$ then the restriction $u_{|H'}$ is well-defined, and if $u$ satisfies the strong transmission condition then the limit (in the sense of distributions) of $u_{|H'}$ when approaching $H$ from the positive side is well-defined and defines an element of $I^{t+1}_{{\mathrm{cl}}}(\tilde Z,\tilde Y)$.
![The p-submanifolds $Y$, $H$ and $H^{\prime}$ of $Z$ of Prop. \[prop:transm.cond\] in local coordinates . Here, $\widetilde Y$ is given by the black dot that is the origin of the chosen coordinate system and the arrows next to $u_{|H^{\prime}}$ indicate the direction $y_1 \to 0+$.[]{data-label="fig:transm.cond"}](transm-cond-sketch.pdf)
For the last statement it clearly suffices that $u$ satisfy the strong transmission condition at $Y\cap U$ where $U$ is a neighborhood of $\tilde Z$.
In the case where $Z=\hat X^2$, $Y=\diag_{\hat X}$, $H=W\times \hat X$, $H'=\hat X\times W$ for a separating hypersurface $W\subset X$ we get $\tilde Z = W^2$, $\tilde Y = \diag_{W}$, and we recover a standard result on $\Psi$DOs closely related to the transmission property mentioned above, see [@Hor85 Thm. 18.2.17], generalized below in Corollary \[cor:transmission property\].
Clearly the restriction and limit are well-defined outside $\tilde Y$ and the limit is smooth there, so we consider a neighborhood of a point $p\in \tilde Y$. It is easy to see that normal transversality and condition imply that there are adapted local coordinates $x,y$ for $Z$ near $p$ in terms of which locally $$\label{eqn:HH'Y coords}
H = \{y_1=0\},\ H' = \{y_2=0\},\ Y = \{y_1=y_2, y''=0\}$$ where $y=(y_1,y_2,y',y'')$. Then locally $\tilde Z= \{y_1=y_2=0\}$, $\tilde Y= \{y_1=y_2=0, y''=0\}$, in particular $\tilde Y$ is a p-submanifold of $\tilde Z$. The assumption $u\in I_{{\mathrm{cl}}}^t(Z,Y)$ means that, near $p$, $$\label{eqn:TC proof}
u(x,y_1,y_2,y',y'') = \int e^{i(y_1-y_2)\eta + iy''\eta''} a (x,y_2,y'; \eta,\eta'')\, d\eta d\eta''$$ where $a$ is a classical symbol of order $\tau = t + \tfrac{1}{4}\dim Z - \tfrac{1}{2}{\operatorname{codim}}Y$.
Restriction to $H'$ means setting $y_2=0$, which clearly yields a well-defined conormal distribution $u_{|y_2=0}$ on $H'$ with respect to $Y\cap H'$. Now assume that $u$ satisfies the strong transmission condition. We need to show that the limit of $u_{|y_2=0}$ as $y_1\to0+$ exists and defines a distribution $\tilde u$ conormal with respect to $y'' = 0$. This follows from the considerations before Theorem 18.2.17 in [@Hor85]: the $\eta$-integral in (with $y_2=0$) exists for $y_1>0$ in the sense of Lemma 18.2.16 (loc.cit.) as $b(x,y_1,y';\eta''):=\int^+ e^{iy_1\eta} a(x,0,y';\eta,\eta'')\,d\eta$, this is a classical symbol of order $\tau+1$ uniformly in $y_1\geq0$, and the limit $\tilde u$ of $u_{|y_2=0}$ as $y_1\to0+$ is the conormal distribution with symbol $b(x,0,y';\eta'')$. The order of $\tilde u$ is $t + 1$ since the symbol $b$ has order $\tau+1$ and $\dim \tilde Z = \dim Z - 2$, $\dim \tilde Y = \dim Y-1$, so $\frac14\dim \tilde Z - \frac12{\operatorname{codim}}\tilde Y=\frac14\dim Z - \frac12{\operatorname{codim}}Y$.
\[cor:transmission property\] Let $X$ be a $\phi$-[[<span style="font-variant:small-caps;">bc</span>]{}]{}-manifold with non-empty [[<span style="font-variant:small-caps;">bc</span>]{}]{}-boundary and $\Xhat$ be an extension of $X$ across the [[<span style="font-variant:small-caps;">bc</span>]{}]{}-boundary. Let $Q\in\Psi^t_\phi(\Xhat)$, $m\in\Z$, and assume that the Schwartz kernel $K_Q\in I^t_{{\mathrm{cl}}}(\Xhat^2_\phi,\diag_\phi)$ satisfies the strong transmission condition in a neighborhood of $\diag_{\phi,\Xhat}\cap ({{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}})^2_\phi$.
Fix a trivialization $(-1,1)_\rho\times{{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}$ as in . If $v\in{{\dot C^\infty_{{\mathrm{s}}}}}({{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}})$ then $Q(\delta(\rho)\otimes v)$, which is smooth in $X\setminus{{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}$, has a limit at ${{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}$ from $\rho>0$, and the operator $$v \mapsto Q(\delta(\rho)\otimes v)_{|{{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}}$$ thus defined is in $\Psi^{m+1}_\phi({{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}})$.
Note that the strong transmission condition is satisfied if $Q$ is a parametrix for a differential operator.
Apply Proposition \[prop:transm.cond\] with $Z=\Xhat^2_\phi$, $Y=\diag_{\phi,\Xhat}$, $u=K_Q$ and $H,H'$ the lifts to $\Xhat_\phi^2$ of ${{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}\times\Xhat$ and $\Xhat\times{{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}$, respectively, and the positive side of $H$ defined by $X\times\Xhat$. Transversality in the interior of $Z$ is obvious, and at the boundary of $Z$ the pairwise intersections are subsets of the interior of $\phif$, where we can use coordinates $T,x,Y,y',z,z'$ as in , with $z_1$ defining ${{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}$. In these coordinates $H=\{z_1=0\}$, $H'=\{z_1'=0\}$ and $Y=\{T=0,Y=0, z=z'\}$, so the assumptions are satisfied, with $T,Y,z_2-z_2',\dots,z_f-z_f'$ the coordinates $y''$ in . Also, $\tilde Z = ({{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}})^2_\phi$ and $\tilde Y$ is its $\phi$-diagonal.
We may choose the coordinates such that $z_1=\rho$. Then the Schwartz kernel of the operator $v\mapsto Q(\delta(\rho)\otimes v)$ is $u_{|H'}$. By the proposition the limit $\tilde u$ of $u_{|H'}$ at $H$ from the positive side exists, so the limit of $Q(\delta(\rho)\otimes v)$ at ${{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}$ from $X\setminus{{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}$ exists, and is given (as operator applied to $v$) by the Schwartz kernel $\tilde u$. By the proposition again, we have $\tilde u\in I^{t+1}_{{\mathrm{cl}}}(({{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}})^2_\phi,\diag_{\phi,{{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}})$. Also, since $u=K_Q$ vanishes to infinite order at all boundary hypersurfaces of $\Xhat^2_\phi$ except $\phif$, the analogous statement holds for $\tilde u$.
Construction of a Calder[ó]{}n projector {#ssec:Calderon}
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We now prove Theorems \[thm:Calderon\] and \[thm:calderon canonical inv extension\] and Corollary \[cor:L2 closure of BC spaces\]. First, note that as the boundary data spaces $\calB_{P,\calF}$ are defined in terms of the homogeneous equation $Pu=0$, which does not see the $x^{-cm}$ factor, we may assume $c=0$.
Let $\Xhat$ be the [[<span style="font-variant:small-caps;">bc</span>]{}]{}-double of $X$ with extended $\phi$-structure and extensions of the bundles $E,E'$ as discussed in Subsection \[sssec:extspaces\]. Let $\Ehat$, $\Phat$ and $\Pi$ be as in Theorem \[thm:construction inv ext\]. Since $\Xhat$ has no [[<span style="font-variant:small-caps;">bc</span>]{}]{}-boundary, standard $\phi$-$\Psi$DO theory (see Appendix \[sec:app phi ops\]) implies that $(\Phat+\Pi)^{-1}\in \Psi^{-m}_\phi(\Xhat;\Ehat)$. We apply the construction explained in Subsection \[ssec:outline Calderon constr\] to $\Phat+\Pi$ and then show that it yields a Calderón projector for $\Phat$ and then for $P$.
Choose a trivialization $(-1,1)_\rho\times{{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}$ of a neighborhood of ${{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}$ in $\Xhat$ as in and define $\gamma$, $\gamma^\star$ by the formulas in , . Define the jump operator ${\hspace{-2pt}\mathscr{J}}_\Phat$ for $\Phat$, a differential operator of order $m-1$ from boundary data to boundary data, by the equation $$\label{eqn:dfn.jump.op}
\Phat(u^0)=(\Phat u)^0 + \gamma^\star{\hspace{-2pt}\mathscr{J}}_\Phat\gamma u$$ where $u$ is a function on $X$ and $u^0$ its extension to $\Xhat$ by zero. The same equation then holds with $\Phat$ replaced by $\Phat+\Pi$ everywhere and with ${\hspace{-2pt}\mathscr{J}}_{\Phat+\Pi}={\hspace{-2pt}\mathscr{J}}_\Phat$. This can be seen as follows. is equivalent to $[\Phat,\chi]=\gamma^\star{\hspace{-2pt}\mathscr{J}}_\Phat\gamma$ where $\chi:\Xhat\to\R$ is the characteristic function of $X$. Since the Schwartz kernel of $\Pi$ is smooth and supported in $X\times X$, we have $[\Pi,\chi]=0$, so we get $[\Phat+\Pi,\chi]=\gamma^\star{\hspace{-2pt}\mathscr{J}}_\Phat\gamma$ also, which was to be shown.
Let $$\label{eqn:dfn.Chat}
\Chat = \gamma (\Phat+\Pi)^{-1} \gamma^\star {\hspace{-2pt}\mathscr{J}}_{\Phat+\Pi}\,.$$ Here $\gamma (\Phat+\Pi)^{-1} \gamma^\star$ is an $m\times m$ matrix whose $k,p$ entry when applied to $v\in{{\dot C^\infty_{{\mathrm{s}}}}}({{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}},\Ehat)$ is defined as the limit, at ${{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}$ from the interior of $X$, of $D_\rho^{k-1} (\Phat+\Pi)^{-1} D_\rho^{p-1}(\delta(\rho) \times v)$. The strong transmission condition is satisfied near ${{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}$ since $\Phat$ is a differential operator in a neighborhood of ${{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}$ and $\Pi$ is smoothing. So applying Corollary \[cor:transmission property\] to $Q=D_\rho^{k-1} (\Phat+\Pi)^{-1} D_\rho^{p-1}$ we conclude that this limit exists and defines an element of $\Psi_\phi^{-m+k+p-1}({{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}},\Ehat)$. Since the $p,l$ entry of ${\hspace{-2pt}\mathscr{J}}_{\Phat+\Pi}$ is a differential operator of order $m+1-p-l$ if this is non-negative and equals zero otherwise it follows that $\Chat\in\Psi^*_\phi({{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}};\Ehat^m)$, where the order of the $k,l$ component is $k-l$.
Now fix an admissible function space $\calF$. We apply the arguments in Section \[ssec:outline Calderon constr\] to $\Phat+\Pi$, using additionally that $\gamma$ and ${\hspace{-2pt}\mathscr{J}}_{\Phat+\Pi}={\hspace{-2pt}\mathscr{J}}_\Phat$ respect $\calF$, to conclude that $\Chat$ is a projection in $\calF({{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}};\Ehat)^m$ to $\calB_{\Phat_X+\Pi_X,\calF}$. Next, Proposition \[prop:modification B\] implies $$\label{eqn:B equality}
\calB_{\Phat_X+\Pi_X,\calF} = \calB_{\Phat_X,\calF} \,,$$ so we conclude that $\Chat$ is an $\calF$-Calderón projector for $\Phat$. The assumption of the proposition is satisfied since $\Pi$ projects to $\ker\Phat_X\cap\ker\gamma$, so the operator $\Phat_X+\Pi_X$ is a modification of $\Phat_X$, and since $\Phat$ is self-adjoint.
Finally, in view of the fact that $\Phat_X$ augments $P$, where the vector bundles $\Ehat$ and $E$ are related as in , Proposition \[prop:augment B C\] implies that $$\calB_{P,\calF} = \pi\calB_{\Phat_X,\calF} \text{ and $C = \pi\Chat\iota$}$$ is an $\calF$-Calderón projector for $P$. Now $\Chat\in\Psi^*_\phi({{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}};\Ehat^m)$ implies $C\in\Psi^*_\phi({{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}};E^m)$, with the same orders of the components. This completes the proof of Theorem \[thm:Calderon\].
First, $\calF({{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}) \subset L^2_\phi({{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}})$ implies $\calB_{P,\calF}\subset\calH$ since $u\in\calF(X)$ implies $\gamma u\in \calF({{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}})^m$ (see , which is proved after Definition \[def:admissible function space\]) and $L^2_\phi({{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}})\subset H^k_\phi({{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}})$ for all $k\geq0$.
Now choose a Calderón projector as in Theorem \[thm:Calderon\]. Recall that $C=(C_{kl})_{k,l=1\dots m}$ where $C_{kl} \in \Psi_\phi^{k-l}({{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}};E)$. In particular, $C_{kl}$ is bounded as an operator $H^{l-1}_\phi\to H^{k-1}_\phi$, so $C$ is bounded $\calH\to\calH$. Now $\calF^m\subset\calH$ is dense for any admissible $\calF$ since $({{\dot C^\infty_{{\mathrm{s}}}}})^m\subset\calH$ is dense. It is a simple exercise to show that for a bounded projection on a Hilbert space the image of a dense subspace is a dense subspace of the image. Therefore, $\calB_{P,\calF}$ is dense in the range of $C$, considered as an operator on $\calH$.
The proof of Theorem \[thm:calderon canonical inv extension\] is completely analogous to the proof in the non-singular case, see e.g. [@See66 Lem. 5]. For completeness we recall the argument in our setting.
Let $P$ and $\Phat$ be as in Theorem \[thm:calderon canonical inv extension\] and $C_\pm$ be the Calderón projectors for $\Phat_{X^\pm}$ as constructed in the proof of Theorem \[thm:Calderon\], but without the augmentations and modifications. Thus, given $U \in {{\dot C^\infty_{{\mathrm{s}}}}}({{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}};E)^m$, let $u \in \calD^{\prime}(\Xhat;E)$ be the solution of $\Phat u = \gamma^\star {\hspace{1pt}\hat{\hspace{-3pt}{\hspace{-2pt}\mathscr{J}}}}U$, with ${\hspace{3pt}\hat{\hspace{-3pt}{\hspace{-2pt}\mathscr{J}}}}:={\hspace{-2pt}\mathscr{J}}_\Phat$ defined in . The restriction of $\Phat u$ to $\interior{\Xhat_\pm}$ is $0$, and because $\Phat^{-1}$ has the transmission property, the restriction of $u$ to $\interior{\Xhat_\pm}$ extends to a smooth function $u_\pm$ on $\Xhat_\pm$. Then $C_\pm U = \gamma_\pm u$. Moreover, $u = \Phat^{-1} \gamma^\star {\hspace{3pt}\hat{\hspace{-3pt}{\hspace{-2pt}\mathscr{J}}}}U$ and since $\gamma^\star {\hspace{3pt}\hat{\hspace{-3pt}{\hspace{-2pt}\mathscr{J}}}}U \in H_\phi^{1/2-m-\eps}(\Xhat;E)$ for any $\eps > 0$, this gives $u \in H_\phi^{1/2-\eps}(\Xhat;E)$. In particular, $u \in L^2_\phi(\Xhat;E)$ and so $u = u_+ + u_-$ where $u_\pm$ are extended by zero to $\Xhat$.
Now and its $\Xhat^-$ counterpart read $$\label{eqn:jump.op.ibp.2}
\Phat u_\pm = \big(\Phat u_\pm\big)^0 + \gamma^\star {\hspace{1pt}\hat{\hspace{-3pt}{\hspace{-2pt}\mathscr{J}}}}\gamma_\pm u_\pm \,.$$ The first term on the right vanishes since $\Phat u = \gamma^\star {\hspace{3pt}\hat{\hspace{-3pt}{\hspace{-2pt}\mathscr{J}}}}U$ vanishes outside ${{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}$ and hence so does $\Phat u_\pm$. Therefore, using $u = u_+ + u_-$ and $\gamma_\pm u_\pm = \gamma_\pm u$, $$\label{eqn:jump.op.ibp.3}
\gamma^\star {\hspace{3pt}\hat{\hspace{-3pt}{\hspace{-2pt}\mathscr{J}}}}U = \Phat u = \sum_{\pm} \Phat u_\pm = \sum_{\pm} \gamma^\star {\hspace{1pt}\hat{\hspace{-3pt}{\hspace{-2pt}\mathscr{J}}}}\gamma_\pm u_\pm = \sum_{\pm} \gamma^\star {\hspace{1pt}\hat{\hspace{-3pt}{\hspace{-2pt}\mathscr{J}}}}\gamma_\pm u \,.$$ Since $\gamma^\star$ and ${\hspace{3pt}\hat{\hspace{-3pt}{\hspace{-2pt}\mathscr{J}}}}$ are easily seen to be injective, this gives $U = \gamma_+ u + \gamma_- u$. As $C_\pm U = \gamma_\pm u$ we obtain $C_+ + C_- = \Id$ on ${{\dot C^\infty_{{\mathrm{s}}}}}({{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}};E)^m$, hence as elements of $\Psi^*_\phi({{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}};E)$. It also follows that $C_\Phat=C_+$. As $C_\pm$ are bounded projections in $\calH$ with range $\calB^\pm_{P,L^2}$ we obtain the first claim and the first part of the second claim. Theorem \[thm:Calderon\] also gives the third claim.
The normal families satisfy $N(C_+)+N(C_-)= N(\Id)=\Id$, and since $N(C_\pm)(\mu)$ is a Calderón projector for $N(\Phat_{X^\pm})(\mu)$ for each $\mu$ (see Proposition \[prop:Calderon.normal\]) the second part of the second claim also follows.
The symbol and the normal family {#ssec:Calderon.properties}
--------------------------------
In this subsection we study the $\phi$-principal symbol and the normal family of the Calderón projector constructed above, and prove that the full $\phi$-symbol is independent of the choices made in the construction.
Given a $\phi$-elliptic operator $P \in \Diff_\phi^m(X;E,E^\prime)$ and a Calderón projector $C$ as in Theorem \[thm:Calderon\], we can follow Hörmander’s arguments (see [@Hor85 Thm. 20.1.3]) to shed light on the $\phi$-principal symbol ${{}^\phi\sigma}_*(C)$ of $C$.
Recall that ${{}^\phi\sigma}_*(C)$ is a function on (or rather a section of a homomorphism bundle over) ${{}^\phi T}^*{{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}$. To describe it we use the identification of a neighborhood of ${{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}$ in $\Xhat$ with $(-1,1)\times{{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}$ from , which induces an isomorphism $${{}^\phi T}^*_{p} X \cong \R\times{{}^\phi T}^*_{p}{{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}},\quad \xi \mapsto (\omega,\xi')$$ for $p \in {{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}$. Recall that ${{}^\phi\sigma}_m(P)$ is defined on ${{}^\phi T}^* X$. For each $(p,\xi')\in {{}^\phi T}^*{{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}$ consider the ordinary differential equation $$\label{eqn:symbol.ode}
{{}^\phi\sigma}_m(P)(p;D_t,\xi^{\prime}) v(t) = 0 \,,\quad t\in\R\,.$$ Let ${{\dot C^\infty}}_\pm(\R;E_{p})$ be the space of smooth functions $v: \R \rightarrow E_{p}$ for which $v(t)$ vanishes rapidly with all its derivatives as $t \to \pm\infty$. As in [@Hor85 Thm. 20.1.3] the $\phi$-principal symbol of $C$ is expressed in terms of solutions of :
\[prop:Calderon.symbol\] Let $(p,\xi^{\prime}) \in {{}^\phi T}^*{{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}$ with $\xi^{\prime}\neq 0$. The map ${{}^\phi\sigma}_*(C)(p,\xi^{\prime})\in \End(E_p^m)$ is a Calderón projector for the differential operator ${{}^\phi\sigma}_m(P)(p;D_t,\xi^{\prime})$ on ${{\dot C^\infty}}_+(\R;E_{p})$.
More precisely, let $\calB_{{{}^\phi\sigma}(P)}^\pm(p,\xi^{\prime})$ denote the space of boundary data at $t=0$ of solutions to with $v \in {{\dot C^\infty}}_\pm(\R;E_{p})$. Then ${{}^\phi\sigma}_*(C)(p,\xi^{\prime})$ is the projection in $(E_{p})^m$ with range $\calB_{{{}^\phi\sigma}(P)}^+(p,\xi^{\prime})$ and kernel $\calB_{{{}^\phi\sigma}(P)}^-(p,\xi^{\prime})$.
The proof in [@Hor85], i.e. in the non-singular case, is a local calculation near $\diag_{{{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}}$, the core of which is quantifying the statement of Corollary \[cor:transmission property\] on the level of symbols. Just as this corollary carries over to the $\phi$-case, as stated, because the local geometry near the $\phi$-diagonal $\diag_{\phi,{{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}}$ (cf. Figure \[fig:transm.cond\]) stays the same uniformly for $x\to 0$, this calculation also carries over to the singular case, so we do not repeat it here. The $\phi$-principal symbol arises since it is the symbol in the representation, as a conormal distribution, of the Schwartz kernel at the $\phi$-diagonal, and this symbol is defined on the dual of the normal bundle of this diagonal, which in turn is identified with ${{}^\phi T}^*{{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}$ under the projection to ${{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}$.
A genuinely new feature in our singular context is the boundary symbol or normal family $N(C)$ of $C$. It turns out that the normal family of $C$ is a Calderón projector as well.
\[prop:Calderon.normal\] For each $ \mu \in \R \times T^*B$, $N(C) (\mu)$ is a Calderón projector for $N(P)(\mu)$ on $\Cinf(F_y;E)$. More precisely, it is the Calderón projector resulting from the construction of Section \[ssec:outline Calderon constr\] when starting with $N(P)(\mu)$ instead of $P$.
Taking the normal family defines an algebra homomorphism and since both of $\gamma (\Phat + \Pi_{sh})^{-1} \gamma^\star$ and ${\hspace{3pt}\hat{\hspace{-3pt}{\hspace{-2pt}\mathscr{J}}}}:={\hspace{-2pt}\mathscr{J}}_\Phat$ are (matrices of) $\phi$-$\Psi$DOs on ${{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}$ with respect to the fibration $\phi : {{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}\to B$, we have, for each $\mu=(\tau;y,\eta)$, $$\label{eqn:normal.calderon.decomp}
N(\hat{C}) (\mu) = N(\gamma (\Phat + \Pi_{sh})^{-1} \gamma^\star) (\mu) \, N({\hspace{3pt}\hat{\hspace{-3pt}{\hspace{-2pt}\mathscr{J}}}}) (\mu) \,.$$ Note that all three operators in operate on sections over $\partial F_y$, which is locally given by $z_1=0$. Recall from Appendix \[sec:app phi ops\] the definition of the normal family for $\phi$-pseudodifferential operators, see Equation . We use this with $X$ replaced by ${{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}$ and $F_y$ replaced by $\partial F_y$ to examine $N(\gamma(\Phat + \Pi_{sh})^{-1}\gamma^\star)(\mu)$ more closely. Note that the function $g$ in is constant along the fibre $\partial F_y$ and that $\gamma$, $\gamma^\star$ act in the fibre direction $z_1$ only. Therefore, $\gamma$ and $\gamma^\star$ commute with multiplication by $e^{\pm i g}$. Now define $\gamma_y$ and $\gamma_y^\star$ analogously to $\gamma$ and $\gamma^\star$, but fibre-wise for $\partial F_y \subset F_y$. Then if $V \in C^\infty({{\partial_{{\mathrm{s}},{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}} X}};\Ehat)^m$ and $\tilde V$ denotes a smooth extension of $V$ to ${{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}$, we see that $\widetilde{\gamma_y^\star V}:=\gamma^\star\tilde V$ restricts to $\gamma_y^\star V$ at ${{\partial_{{\mathrm{s}},{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}} X}}$. Therefore, $$\begin{aligned}
N(\gamma (\Phat &+ \Pi_{sh})^{-1} \gamma^\star) (\mu) V
= \big[ e^{-ig} \gamma (\Phat + \Pi_{sh})^{-1} \gamma^\star (e^{ig}\tilde V)\big]_{|\partial F_y} \\
&= \big[\gamma e^{-ig} (\Phat + \Pi_{sh})^{-1} (e^{ig}\gamma^\star \tilde V)\big]_{|\partial F_y}
= \gamma_y \big[ e^{-ig} (\Phat + \Pi_{sh})^{-1} (e^{ig} \widetilde{\gamma_y^\star V})\big]_{|\partial F_y} \\
&= \gamma_y N((\Phat + \Pi_{sh})^{-1})(\mu)\gamma_y^\star \, V \,.
\end{aligned}$$ As, locally near ${{\partial_{{\mathrm{s}},{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}} X}}$, the construction of the jump terms ${\hspace{3pt}\hat{\hspace{-3pt}{\hspace{-2pt}\mathscr{J}}}}= {\hspace{-2pt}\mathscr{J}}_{\Phat} = {\hspace{-2pt}\mathscr{J}}_{\Phat + \Pi_{sh}}$ involved the fibre direction $z_1$ only, by a similar argument we see that $N({\hspace{3pt}\hat{\hspace{-3pt}{\hspace{-2pt}\mathscr{J}}}}) (\mu)$ are the jump terms for $N(\Phat + \Pi_{sh})$, that is ${\hspace{-2pt}\mathscr{J}}_{N(\Phat+\Pi_{sh})}$, compare . Since $N((\Phat + \Pi_{sh})^{-1}) = (N(\Phat+\Pi_{sh}))^{-1}$ and $N(\Pi_{sh})=0$ (as the Schwartz kernel of $\Pi_{sh}$ vanishes rapidly at $({{\partial_{{\mathrm{s}}}X}})^2$), this shows that $$N(\hat{C}) = \gamma_y N((\Phat+\Pi_{sh})^{-1}) \gamma_y^\star {\hspace{-2pt}\mathscr{J}}_{N(\Phat+\Pi_{sh})}
= \gamma_y (N(\Phat))^{-1} \gamma_y^\star {\hspace{-2pt}\mathscr{J}}_{N(\Phat)}$$ is, by the same arguments that lead to Theorem \[thm:Calderon\], a Calderón projector for $N(\Phat)$ or more precisely for $N(\Phat)_{|X} = N(\Phat_{|X})$.
Now, using again, we see that taking the normal family commutes with the bundle maps $\pi$ and $\iota$, i.e., $\pi N(\Chat) \iota = N(\pi \Chat \iota) = N(C)$ by Proposition \[prop:augment B C\] and $\pi N(\Phat_{|X}) \iota = N(\pi \Phat_{|X} \iota) = N(P)$ by Theorem \[thm:construction inv ext\]. Also, taking inverses and products and taking ${\hspace{-2pt}\mathscr{J}}$ is compatible with the $\pi$-$\iota$-restriction, as is most easily seen using the direct summand characterization of augmentation, Equation . By Proposition \[prop:augment B C\] again, we obtain the claim and the proof is complete.
\[Proof of Proposition \[prop:Calderon.unique\]\] As in the proof of Proposition \[prop:Calderon.normal\] we use that $C=\pi \Chat\iota$ where $\Chat = \gamma (\Phat + \Pi_{sh})^{-1} \gamma^\star{\hspace{3pt}\hat{\hspace{-3pt}{\hspace{-2pt}\mathscr{J}}}}$. For $p\in\Xhat$ denote by ${{}^\phi\sigma_{\mathrm{full}}}(Q)(p)$ the full $\phi$-symbol of a $\phi$-$\Psi$DO $Q$ at $p$. Using a choice of coordinates near $p$ this is defined as a function on ${}^\phi T^*_p\Xhat$ modulo Schwartz functions. By the standard parametrix construction, for any $k$ the $k$-jet of ${{}^\phi\sigma_{\mathrm{full}}}((\Phat+\Pi_{sh})^{-1})$ at $p$ is determined by the infinity-jet of ${{}^\phi\sigma_{\mathrm{full}}}(\Phat+\Pi_{sh})={{}^\phi\sigma_{\mathrm{full}}}(\Phat)$ at $p$. Since the same holds for ${{}^\phi\sigma_{\mathrm{full}}}({\hspace{3pt}\hat{\hspace{-3pt}{\hspace{-2pt}\mathscr{J}}}})$, it follows for $p\in{{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}$ that ${{}^\phi\sigma_{\mathrm{full}}}(\Chat)(p)$ is determined by the infinity-jet of ${{}^\phi\sigma_{\mathrm{full}}}(\Phat)$, hence of ${{}^\phi\sigma_{\mathrm{full}}}(\Phat_X)$, at $p$. Applying $C=\pi \Chat\iota$ and $\pi\Phat_X\iota=P$ we obtain as at the end of the proof of Proposition \[prop:Calderon.normal\] that ${{}^\phi\sigma_{\mathrm{full}}}(C)(p)$ is determined by the infinity-jet of ${{}^\phi\sigma_{\mathrm{full}}}(P)$ at $p$. This completes the proof. Note that instead of the inverse of $\Phat+\Pi_{sh}$ a parametrix modulo $\Psi_\phi^{-\infty}$ in a neighborhood of ${{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}$ suffices to fix $C$ modulo $\Psi_\phi^{-\infty}({{\partial X}})$.
The orthogonal Calderón projector {#ssec:orth proj}
---------------------------------
We now prove Theorem \[thm:orthogonal\]. It follows from the following proposition.
\[thm:orth.Calderon\] Let $C$ be a Calderón projector for $P \in \Diff_\phi^1(X;E,E^{\prime})$ as constructed in Theorem \[thm:Calderon\]. Denote the $L^2_\phi$-adjoint of $C$ by $C^\star$.
Then $\Id + C - C^\star$ is a fully elliptic and invertible $\phi$-$\Psi$DO and $$\label{eqn:orth.calderon}
C_o = C ( \Id + C - C^\star)^{-1} : L^2_\phi({{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}};E) \longrightarrow \calB_{P,L^2}$$ is the orthogonal projection (with respect to the $L^2_\phi$-scalar product).
We have $C_o \in \Psi_\phi^0({{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}};E)$, with the $\phi$-principal symbol and the normal family of $C_o$ being the orthogonalizations (obtained as in ) of the $\phi$-principal symbol respectively the normal family of $C$.
The main tool in proving Proposition \[thm:orth.Calderon\] is part b) of Lemma 3.5 of [@BLZ09] dealing with orthogonalizations of bounded projections: it states that whenever $T : H \to H$ is a bounded projection in a Hilbert space $H$ and $T^\star$ is its Hilbert space adjoint, then $\Id + T - T^\star$ is invertible and $$\label{eqn:orth.formula}
T_o = T (\Id + T - T^\star)^{-1} : H \longrightarrow H$$ is the orthogonal projection with range $\rg T$.
By construction $C$ is a projection, and since $C \in \Psi_\phi^0({{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}};E)$, it is bounded on $L^2_\phi({{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}};E)$. The operators ${{}^\phi\sigma}_0(C)(p^\prime,\xi^\prime)$ and $N(C)(\mu)$ are bounded projections as well, by Propositions \[prop:Calderon.symbol\] and \[prop:Calderon.normal\].
We now apply part b) of Lemma 3.5 of [@BLZ09] separately to $C$, its $\phi$-principal symbol and its normal family. In the following, we omit mentioning the variables $(p^\prime,\xi^\prime)$ and $(\tau;y,\eta)$ for the $\phi$-principal symbol and normal family and denote by a star $\star$ the appropriate Hilbert space adjoint. For $C$ and $N(C)$, this will be the $L^2_\phi$- respectively $L^2$-adjoint and for ${{}^\phi\sigma}_0(C)$, this will be the adjoint of an endomorphism of the hermitian bundle $E$.
As the $\phi$-principal symbol- and normal family-maps are $\star$-algebra homomorphisms (see [@MM98]), we have $$\begin{aligned}
{{}^\phi\sigma}_0(\Id+C-C^\star)
&= \Id + {{}^\phi\sigma}_0(C) - {{}^\phi\sigma}_0(C)^\star \,, \\
N(\Id + C - C^\star)
&= \Id + N(C) - N(C)^\star \,,
\end{aligned}$$so by Lemma 3.5 (loc.cit.) these operators, as well as $\Id + C - C^\star$, are invertible. Thus, $\Id + C - C^\star$ is fully elliptic and invertible on $L^2_\phi({{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}};E)$, whence its inverse is in $\Psi_\phi^0({{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}};E)$ again. But then $$\label{eqn:orth.Calderon.red}
C_o = C \big( \Id + C - C^\star\big)^{-1} \in \Psi_\phi^0({{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}};E)$$ with $\phi$-principal symbol $$\label{eqn:orth.Calderon.red.symb}
{{}^\phi\sigma}_0(C_o)
= {{}^\phi\sigma}_0(C) \big( \Id + {{}^\phi\sigma}_0(C) - {{}^\phi\sigma}_0(C)^\star\big)^{-1}$$ and normal family $$\label{eqn:orth.Calderon.red.normal}
N(C_o)
= N(C) \big( \Id + N(C) - N(C)^\star\big)^{-1} \,,$$ and by the lemma again, , and are the orthogonalizations of $C$, ${{}^\phi\sigma}_0(C)$ and $N(C)$, respectively. Clearly, $C_o$ has the same range as $C$, i.e. $\calB_{P,L^2}$.
\[rem:orthogonal\]
1. We chose to formulate the theorem and proposition on orthogonal projections only for $m=1$ since for $m>1$ one needs to introduce scalar products in the Sobolev spaces $\calH$ resp. $H^k_\phi({{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}};E)$, which seems (even) less natural. However, it is easy to generalize the statements and proofs to this case, using order reduction operators in the $\phi$-calculus.
2. In general, the $\phi$-principal symbols of $C$ and $C_o$ will be different. Heuristically this is clear since the range and kernel spaces $\calB_{{{}^\phi\sigma}(P)}^\pm(p^{\prime},\xi^{\prime}) \subset (E_{p'})^m$ of ${{}^\phi\sigma}(C)(p',\xi')$ (see Proposition \[prop:Calderon.symbol\]) cannot be expected to be orthogonal to each other for each value of $\xi'$, as they would have to be if ${{}^\phi\sigma}(C)={{}^\phi\sigma}(C_o)$. For example, if $P$ is the Laplacian of a $\phi$-metric then a simple calculation shows that this is indeed not the case.
Basics on Manifolds with Corners and Blow-Ups {#sec:app mwc}
=============================================
We give a quick summary of basic notions on manifolds with corners. Details can be found in [@Mel:APSIT], [@Mel:DAMWC] or in the introductory text [@Gri:BBC].
A **manifold with corners** of dimension $n$, denoted $X$ in the sequel, is defined like a manifold except that local charts are defined on open subsets of model spaces $\R^n_k:=\R^{n-k}\times\R_+^k$ for various $k\in\{0,\dots,n\}$ where $\R_+=[0,\infty)$, and an additional global condition is satisfied, see below. To define smoothness of transition maps or of maps between manifolds with corners we say that a map $$\text{$U_1\to U_2$, where $U_i\subset\R^{n_i}_{k_i}$,}$$ is **smooth** if it extends to a smooth map $$\text{$\Utilde_1\to \Utilde_2$, where $\Utilde_i\subset\R^{n_i}$ are open and $U_i=\tilde U_i\cap \R^{n_i}_{k_i}$.}$$ If $p\in X$ then there is a unique $k$, called the **codimension** of $p$, so that there is a coordinate system (inverse of a chart) mapping $p$ to $0\in\R^n_k$. The coordinates are then sometimes called **adapted** to $X$, and are often denoted $x=(x_1,\dots,x_k)$, $y=(y_1,\dots,y_{n-k})$ where $x_i\geq0$ and $y_j\in\R$ for all $i,j$.
A **face** of $X$ of codimension $k$ is the closure of a connected component of the set of points of codimension $k$. A **boundary hypersurface** is a face of codimension one. The global condition on a manifold with corners is that boundary hypersurfaces be embedded (rather than immersed) submanifolds. Equivalently, for each boundary hypersurface $H$ there is **boundary defining function** $\rho$, i.e. a smooth function $\rho:X\to\R_+$ satisfying $\rho^{-1}(0)=H$ and $d\rho_{|p}\neq0$ for all $p\in H$.
A **p-submanifold** (where p is for product) of $X$ is a subset $Y$ so that for each $p\in Y$ there is an adapted coordinate system on $X$ in which $Y$ is locally a coordinate subspace. That is, adapted coordinates $z=(x,y)$ can be chosen and regrouped as $(z',z'')$ so that $Y=\{z''=0\}$. Also, $Y$ is called a **boundary p-submanifold** if $Y\subset{{\partial X}}$, otherwise it is an **interior p-submanifold**. In the latter case only $y$ variables occur among the $z''$ variables. For example, faces of $X$ are boundary p-submanifolds.
If $Y\subset X$ is a p-submanifold then the **blow-up** of $X$ in $Y$ is a new manifold with corners, denoted $[X,Y]$, together with a smooth map $\beta:[X,Y]\to X$, called **blow-down map**, which restricts to a diffeomorphism $[X,Y]\setminus\ff \to X\setminus Y$, where $\ff:=\beta^{-1}(Y)$ is called the **front face**, and so that near any $p\in Y$ with coordinates $(z',z'')$ as above the map $\beta$ is locally near $\beta^{-1}(p)$ modelled by the polar coordinates map in the $z''$-coordinates, i.e. if $z'\in\R^{n'}_{k'}$, $z''\in\R^{n''}_{k''}$ then locally $\beta: \R^{n'}_{k'}\times \R_+\times {\mathbb{S}}^{n''-1}_{k''}\to\R^{n'}_{k'}\times\R^{n''}_{k''}, (z',r,\omega)\mapsto (z',r\omega)$ where ${\mathbb{S}}^{n''-1}_{k''}\subset\R^{n''}_{k''}$ is the unit sphere. Locally the front face is $\R^{n'}_{k'}\times{\mathbb{S}}^{n''-1}_{k''}$ and has local boundary defining function $r$. In practice, it is better to use **projective coordinates**. Examples are given in Appendix \[sec:app phi ops\] for the double spaces $X^2_b$ and $X^2_\phi$. Blow-ups can also be iterated, i.e. if $Z$ is a p-submanifold of $[X,Y]$ then one can form $[[X,Y],Z]$ etc.
If $Z$ is a connected subset of $X$ then the **lift** of $Z$ under the blow-up of a p-submanifold $Y\subset X$, denoted $\beta^*(Z)$, is defined as $\beta^{-1}(Z)$ if $Z\subset Y$ and as the closure of $\beta^{-1}(Z\setminus Y)$ otherwise. If $Z$ is a p-submanifold meeting $Y$ **cleanly** (i.e. so that for each $p\in Y\cap Z$ there is an adapted coordinate system in which both $Y$ and $Z$ are coordinate subspaces) then $\beta^*(Z)$ is a p-submanifold of $[X,Y]$. However, also subsets $Z$ which are not p-submanifolds can become such after (possibly iterated) blow-up, and then we say that $Z$ is **resolved** by the (iterated) blow-up. An important example is the diagonal $\{x=x'\}$ in $\R^2_+$, which is not a p-submanifold but is resolved by blowing up the origin. A p-submanifold $Y$ meets any face of $X$ cleanly, and the boundary hypersurfaces of $[X,Y]$ are $\ff$ and the lifts of the boundary hypersurfaces of $X$.
The space of all smooth vector fields on $X$ which at each $p\in X$ are tangent to all boundary hypersurfaces containing $p$ is denoted $\calV_b(X)$. Interpreting vector fields as first order differential operators and taking finite sums of smooth functions and compositions $V_1\circ\dots\circ V_l$ with all $V_i\in\calV_b(X)$, for $l\leq m$, we obtain the space of **b-differential operators** of order at most $m$, denoted $\Diff_b^m(X)$. Also $\Diff_b^*(X) := \bigcup_m \Diff_b^m(X)$. In adapted coordinates these are combinations of expressions $x_i\partial_{x_i}$, $\partial_{y_j}$ with smooth coefficients.
We now define various function spaces on a manifold with boundary $X$.[^8] First, $\Cinf(X)$ denotes the space of functions $X\to{\mathbb{C}}$ which are smooth up to the boundary. Let $x$ be a boundary defining function. The following spaces consist of functions only defined and smooth on the interior of $X$ but having a certain prescribed behavior near the boundary:
- The space of **functions conormal to the boundary** of order $a\in\R$: $$\calA^a(X)= \{u\in\Cinf(\interior{X}): \Diff_b^*(X)u \subset x^a L^\infty(X)\}\,.$$ There is also the $L^2$-variant of $\calA^a(X)$, where $L^\infty$ is replaced by $L^2$ in the definition. Also, the space of all conormal functions is denoted $$\calA(X) = \bigcup_{a\in\R}\calA^a(X)\,.$$
- Spaces of functions **polyhomogeneous** at the boundary: $$\calA^\calE(X) = \{u\in\calA(X):\, u\sim \sum_{(z,k)\in \calE} a_{z,k} \,x^z\log^k\!x\}$$ where $\calE$ is an **index set**, i.e. a discrete subset of ${\mathbb{C}}\times\N_0$ satisfying certain additional conditions, and $a_{z,k}\in\Cinf(X)$. The asymptotics means that the difference of $u$ and the finite sum over $\Re z<N$ is in $\calA^N(X)$, for each $N$. If $\calE=\N_0\times\{0\}$ then $\calA^\calE(X)=\Cinf(X)$. If the boundary is disconnected then each component can have its own index set (resp. order $a$ for $\calA^a(X)$).
If $X$ is a manifold with [[<span style="font-variant:small-caps;">bc</span>]{}]{}-boundary (see Definition \[def:bv-mfd\]) whose singular boundary ${{\partial_{{\mathrm{s}}}X}}$ is a disjoint union of boundary hypersurfaces then we denote by $$\calA^a_{\mathrm{s}}(X),\ \calA_{\mathrm{s}}(X),\ \calA^\calE_{\mathrm{s}}(X)$$ the corresponding spaces where we assume the respective behavior only to occur at ${{\partial_{{\mathrm{s}}}X}}$, smoothly up to ${{\partial_{\ensuremath{\text{\resizebox{!}{4pt}{\tiny\upshape BC}}}}X}}$. Alternatively, elements of $\calA^a_{\mathrm{s}}(X)$ are restrictions to $X$ of elements of $\calA^a(\Xhat)$ where $\Xhat$ is the [[<span style="font-variant:small-caps;">bc</span>]{}]{}-double of $X$, cf. Section \[ssec:extensions\], and similarly in the other cases.
We denote by $L^2_\phi(X)$ the $L^2$-space with respect to the volume form of a $\phi$-metric, and define $\phi$-Sobolev spaces $H^k_\phi(X):=\{u:\,\Diff^k_\phi(X)u\subset L^2_\phi(X)\}$ for $k\in\N_0$.
For the definition of pseudodifferential operators we need conormal distributions. A distribution $u$ on a manifold with corners $Z$ is **classical conormal** of order $m\in\R$ with respect to an interior p-submanifold $Y$ if it is smooth on $Z\setminus Y$ and near any point of $Y$ and in any adapted coordinate system $x,y=(y',y'')$ for $Z$ in terms of which $Y=\{y''=0\}$ locally, $$\label{eqn:def conormal}
u(x,y',y'') = \int e^{iy''\eta''}a(x,y';\eta'')\,d\eta''$$ for a classical symbol $a$ of order $\mu=m+\frac14\dim Z - \frac12{\operatorname{codim}}Y$. The space of these distributions is denoted $I^m_{{\mathrm{cl}}}(Z,Y)$. We only need the case $\dim Z=2\dim Y$, then $\mu=m$. Here classical means that $a$ has a symbol expansion $a\sim \sum_{j=0}^\infty a_j$ where for each $j$ $$\label{eqn:pos homogeneous}
a_j (x,y';\lambda\eta'') = \lambda^{\mu-j} a_j(x,y';\eta'')$$ for all $\lambda>0$ and all $x,y'$ and $\eta''\neq0$. The $a_j$ are uniquely determined by $u$.
Background on Fibred Cusp Operators {#sec:app phi ops}
===================================
In Section \[ssec:phi-geom\] we defined $\phi$-manifolds, $\phi$-metrics, $\phi$-vector fields, $\phi$-differential operators and their $\phi$-symbol and normal family. Here we describe the $\phi$-pseudodifferential calculus introduced in [@MM98]. We assume that $M$ is a $\phi$-manifold without [[<span style="font-variant:small-caps;">bc</span>]{}]{}-bondary, i.e. simply a manifold with boundary (here we write ${{\partial M}}$ rather than ${{\partial_{{\mathrm{s}}}M}}$), which at first we assume to be connected, equipped with a fibration $\phi:{{\partial M}}\to B$ and a boundary defining function $x$.
We want to define a pseudodifferential calculus, i.e. a set of operators closed under composition, which contains $\Diff_\phi^*(M)$, as defined in Section \[ssec:phi-geom\], as well as parametrices of fully elliptic elements of this space. By the general philosophy on singular pseudodifferential calculi introduced by R. Melrose, such a calculus is defined by a set of Schwartz kernels, which are distributions on the (interior of the) double space $M^2=M\times M$, and whose boundary behavior is restricted by requiring that their pull-backs to a suitable blow-up, $M^2_\phi$, of $M^2$ satisfy certain smoothness and vanishing conditions at the boundary hypersurfaces of $M^2_\phi$ and have a conormal singularity at the diagonal, uniformly up to the boundary.
The $\phi$-double space, $M^2_\phi$, is defined as follows: First, blow up $({{\partial M}})^2\subset M^2$. This yields the b-double space with blow-down map $$\beta_b: M^2_b := [M^2,({{\partial M}})^2] \to M^2 \,.$$ Its front face, denoted by ${\mathrm{bf}}$, is naturally diffeomorphic to $(0,\infty)\times({{\partial M}})^2$ where the first coordinate is $t=\frac{x'}x$, with $x,x'$ the pull-backs of the boundary defining function $x$ on $M$ to the first and second factor in $M^2$. Let $\diag_M\subset M^2$ be the diagonal and $\diag_{b}$ be its lift to $M^2_b$. It meets the boundary of $M^2_b$ in the interior of ${\mathrm{bf}}$, in the set $\{1\}\times\diag_{{\partial M}}$. The larger interior submanifold $D_\phi:=\{1\}\times\{(p,p')\in({{\partial M}})^2:\phi(p)=\phi(p')\}$ of ${\mathrm{bf}}$ is called the fibre diagonal. We blow this up and define the $\phi$-double space $$\beta_{D_\phi}: M^2_\phi := [M^2_b,D_\phi]\to M^2_b,\quad \beta_\phi:= \beta_b\circ\beta_{D_\phi}:M^2_\phi\to M^2\,.$$ The front face created by this blow-up is denoted $\phif$. The diagonal $\diag_{b}$ lifts to a p-submanifold $\diag_\phi$ of $M^2_\phi$ which meets the boundary in the interior of $\phif$.
We describe these spaces locally, using adapted local coordinates $x,y,z$ (see Section \[ssec:phi-geom\]; recall that $\phi(x,y,z)=(x,y)$). Starting from coordinates $(x,y,z; x',y',z')$ on $M^2$ we have $({{\partial M}})^2= \{x=x'=0\}$, so coordinates near interior points of ${\mathrm{bf}}$ are $$t=\frac {x'}{x}, x, y,z,y',z'\,,$$ with $x$ defining ${\mathrm{bf}}$ there. The diagonal in $M^2$ is $\{x=x',y=y',z=z'\}$ and lifts to $\{t=1,y=y',z=z'\}$ The fibre diagonal $D_\phi$ is locally $\{t=1, x=0, y=y'\}$. So projective coordinates near the interior of $\phif$ are $$\label{eqn:phi double coords}
T := \frac{t-1}{x}, x, Y:=\frac{y-y'}{x}, y, z,z'$$ with $x$ defining $\phif$ there. In these coordinates the diagonal $\diag_\phi$ is given by $\{T=0, Y=0, z=z'\}$, and $|(T,Y)|\to\infty$ corresponds to the boundary of $\phif$, which is its intersection with the lift of ${\mathrm{bf}}$. The fundamental reason for considering the space $M^2_\phi$ is that $\phi$-vector fields on $M$, when considered as vector fields on $M^2$ in the $x,y,z$ variables, i.e. pulled back from the left factor, lift under $\beta_\phi$ to smooth vector fields on $M^2_\phi$ that span a rank $\dim M$-bundle and at any $\gamma\in \diag_\phi$ span a subspace of $T_\gamma M^2_\phi$ transversal to $T_\gamma\diag_\phi$.
We consider operators $P$ acting on functions on $\interior{M}=M\setminus{{\partial M}}$ which are given by Schwartz kernels $K_P$, which are distributions on $(\interior M)^2=\interior{M^2}$, in the sense that $$\label{eqn:schwartz kernel}
(Pu)(p) = \int_{\interior M} K_P(p,p') u(p')\, \nu(p')$$ where $\nu$ is some fixed density on $\interior M$. We choose (and fix once and for all) for $\nu$ a smooth positive $\phi$-density, i.e. locally $\nu = a \frac{dx}{x^2}\frac{dy}{x^b}dz$ with $a>0$ smooth up to the boundary $x=0$ and $b = \dim B$. For example, $\nu$ could be the volume density of a $\phi$-metric. The reason for this choice is that for $P=\Id$ (and, say, $a=1$), we have $K_P=(x')^{b+2}\delta(x-x')\delta(y-y')\delta(z-z')$, which in coordinates is $\delta(T)\delta(Y)\delta(z-z')$, and since this has no $x$-factor, it extends from the interior of $M^2_\phi$ to a distribution on $M^2_\phi$ as a smooth non-vanishing delta-function for the submanifold $\diag_\phi$. The lifting property of $\phi$-vector fields mentioned above then implies that the Schwartz kernels for $P\in\Diff^m_\phi(M)$ lift to $\interior{M^2_\phi}$ and extend to $M^2_\phi$ to be delta-functions of order at most $m$ for $\diag_\phi$. Explicitly, if $P$ is given in coordinates by and $\nu=\frac{dx}{x^2}\frac{dy}{x^b}dz$ then $$\label{eqn:kernel of phidiff op}
K_{P} = \sum_{k+|\alpha|+|\beta|\leq m} a_{k,\alpha,\beta}(0,y,z) D_T^k\delta(T) D_Y^\alpha \delta(Y) D_z^\beta\delta(z-z') + O(x)$$ We define $\phi$-pseudodifferential operators by replacing delta-functions by the larger space of classical conormal distributions, defined in Appendix \[sec:app mwc\]:
Let $M$ be a $\phi$-manifold and $m\in\R$. The space $\Psi_\phi^m(M)$ is defined as the set of operators whose Schwartz kernels $K_P$ lift to $M^2_\phi$ to elements of $$I^m_{{\mathrm{cl}}}(M^2_\phi, \diag_\phi)$$ that vanish to infinite order at all boundary hypersurfaces of $M^2_\phi$ except $\phif$.
There is also a more general definition without ${{\mathrm{cl}}}$, but all operators occurring in this paper are classical. By the remarks before the definition we have $\Diff_\phi^*(M) \subset \Psi_\phi^*(M)$. The definition of the $\phi$-principal symbol extends in a straight-forward way using the local representation . The definition of the normal family of $P\in\Diff_\phi^m(M)$ given in does not extend directly to $\Psi_\phi^*(M)$. However, reinterpreting this formula in terms of oscillatory testing allows to extend it as follows: For $\tau\in\R$, $\eta\in\R^b$ and $y_0\in B$ let $g(x,y)=-\frac\tau x + \frac\eta x (y-y_0)$ in coordinates on $B$ near $y_0$. Then $x^2D_x e^{ig} = \left(\tau+\eta(y-y_0)\right)e^{ig}$ and $xD_y e^{ig} = \eta e^{ig}$, and this implies that for $P$ as in and $u\in\Cinf(M)$ we have $$\label{eqn:normal op osc testing}
\left[e^{-ig}P(e^{ig}u)\right]_{|F_{y_0}} = N(P)(\tau;y_0,\eta)(u_{|F_{y_0}})$$ as functions on the fibre $F_{y_0}$, since $F_{y_0}$ is given by $x=0, y=y_0$. It can be shown that the left hand side is well-defined and smooth for $P\in \Psi^*_\phi(M)$ and only depends on $u_{|F_{y_0}}$, and that $N(P)$ so defined is a (standard) pseudodifferential operator with parameter $(\tau,
\eta)$ on $F_{y_0}$, and varies smoothly in $y_0$. Also, the definition makes sense invariantly when considering $\eta\in T_{y_0}B$. By the normal family for $P\in\Diff^m_\phi(M)$ vanishes if and only if $P\in x\Diff^m_\phi(M)$, and an analogous statement holds for $P\in\Psi^* _\phi(M)$. A short calculation shows that the Schwartz kernel $K_{N(P)}(\tau;y,\eta;z,z')$ of $N(P)(\tau;y,\eta)$ is the $(T,Y)\to(\tau,\eta)$ Fourier transform of the restriction of $K_P$ to $\phif$, when writing $K_P$ in coordinates .
As for $\phi$-differential operators, a $\phi$-pseudodifferential operator is called $\phi$-elliptic if its $\phi$-principal symbol is invertible outside the zero section and fully elliptic if in addition its normal family is invertible for all $\tau,y,\eta$.
The main facts about the $\phi$-pseudodifferential calculus are:
1. $\Psi^*_\phi(M):= \bigoplus_{m\in\R}\Psi^m_\phi(M)$ is an $\R$-graded $\star$-algebra, i.e. a vector space and closed under adjoints and composition, with orders adding under composition.
2. The $\phi$-principal symbol $P\mapsto {{}^\phi\sigma}(P)$ and the normal family $P\mapsto N(P)$ are $\star$-algebra homomorphisms, i.e. they are linear and respect composition and the involution $\star$.
3. Operators in $\Psi^m_\phi(M)$ are bounded $H^s_\phi(M)\to H^{s-m}_\phi(M)$ for all $s$ and map each of the spaces $\calA^a(M)$, $\calA^{\calE}(M)$ into itself, for any $a\in\R$ and index set $\calE$.
4. An element of $x^a\Psi^m_\phi(M)$ is a compact operator in $L^2_\phi(M)$ if and only if $m<0$ and $a>0$.
5. An operator $P\in \Psi^m_\phi(M)$ is $\phi$-elliptic if and only if it has a parametrix with remainders in $\Psi^{-\infty}_\phi(M)$.
6. An operator $P\in \Psi^m_\phi(M)$ is fully elliptic if and only if it has a parametrix with remainders in $x^\infty\Psi^{-\infty}_\phi(M)$, if and only if it is Fredholm as a map $H^s_\phi(M)\to H^{s-m}_\phi(M)$ for any $s$. In particular, $\ker P\subset {{\dot C^\infty_{{\mathrm{s}}}}}(M)$ in this case.
7. If $P\in\Psi^m_\phi(M)$ is invertible as an operator $H^{s}_\phi(M)\to H^{s-m}_\phi(M)$ for some $s$ then its inverse is in $\Psi^{-m}_\phi(M)$.
We make some additional remarks. Above we assumed that the boundary ${{\partial M}}$ is connected. More generally, if ${{\partial M}}$ has components $H_1,\dots,H_r$ (each one with its own base and fibre) then $\Psi^m_\phi(M)$ is defined in the same way, except that in the definition only submanifolds meeting the diagonal are blown up: $M^2_b$ is defined by blowing up $H_i^2$ for $i=1,\dots,r$ (but not $H_i\times H_j$ with $i\neq j$), and similarly for $M^2_\phi$. This yields a disjoint collection of $\phi$-faces $\phif_i$, and a normal family for each boundary component $H_i$.
The $\phi$-double space $M^2_\phi$ is defined also when $M$ is a $\phi$-manifold with [[<span style="font-variant:small-caps;">bc</span>]{}]{}-boundary. Note that the lifted diagonal $\diag_b$ is not a p-submanifold in $M^2_b$. However, the fibre diagonal $D_\phi$ is a p-submanifold (essentially since the [[<span style="font-variant:small-caps;">bc</span>]{}]{}-boundary arises from fibres having boundary, not the base), so the second blow-up used to define $M^2_\phi$ is still defined. Of course $\phi$-pseudodifferential operators on $M$ are not defined unless the [[<span style="font-variant:small-caps;">bc</span>]{}]{}-boundary is empty.
[^1]: Such a construction seems quite unnatural except in the case of Dirac operators which involve choices of geometric data anyway.
[^2]: For the proof use that a fully elliptic $\Phat$ has a parametrix $Q\in\Psi^{-m}_\phi(\Xhat)$ so that the remainder terms $Q\Phat-I$, $\Phat Q-I$ are in $x^\infty\Psi^{-\infty}_\phi(\Xhat)$ and are projections to the kernel and cokernel of $\Phat$.
[^3]: In the literature the notation $\hat N(P)$ is sometimes used for the normal family, with $N(P)$ denoting the normal *operator*, where $\tau,\eta$ are replaced by differentiations $D_T$, $D_Y$ in auxiliary variables $T\in\R$, $Y\in\R^b$. We do not use the normal operator.
[^4]: Alternatively, one may resolve $X_0$ by two standard blow-ups: first blow up the point $(\xi,\eta)=0$, then the intersection of the lift of the $\xi=0$ plane with the front face.
[^5]: For example, consider the operator $P=\frac12(\partial_x+i\partial_y)$ on the unit disk $X\subset\R^2$. The double of $X$ is the 2-sphere ${\mathbb{S}}^2$, and there is no scalar elliptic first order scalar differential operator on ${\mathbb{S}}^2$ since its principal symbol would furnish a trivialization $T^*{\mathbb{S}}^2\to{\mathbb{C}}\setminus\{0\}$ of the cotangent bundle of $S^2$, which does not exist by the ‘hairy ball theorem’. Of course this obstruction can be overcome by extending the trivial bundle on $X$ as the antiholomorphic 1-form bundle on ${\mathbb{S}}^2={\mathbb{C}}P^1$, then the $\overline\partial$ operator taking values in sections of this bundle is an elliptic extension.
[^6]: In order to define $\That_X$, only condition would be needed. However, we add condition for the sake of symmetry, since it will be useful when considering adjoints.
[^7]: That is, near any point $p$ of the intersection of two of them there are adapted coordinates in which both are coordinate subspaces, and the two tangent spaces at $p$ together span $T_pZ$.
[^8]: The definitions can be extended to manifolds with corners, but we don’t need this.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Nonlinear interaction between normal modes dramatically affects energy equipartition, heat conduction and other fundamental processes in extended systems. In their celebrated experiment Fermi, Pasta and Ulam (FPU, 1955) observed that in simple one-dimensional nonlinear atomic chains the energy must not always be equally shared among the modes. Recently, it was shown that exact and stable time-periodic orbits, coined $q$-breathers (QBs), localize the mode energy in normal mode space in an exponential way, and account for many aspects of the FPU problem. Here we take the problem into more physically important cases of two- and three-dimensional acoustic lattices to find existence and principally different features of QBs. By use of perturbation theory and numerical calculations we obtain that the localization and stability of QBs is enhanced with increasing system size in higher lattice dimensions opposite to their one-dimensional analogues.'
author:
- 'M. V. Ivanchenko$^1$, O. I. Kanakov$^1$, K. G. Mishagin$^1$ and S. Flach$^2$'
title: '$q$-breathers in finite two- and three-dimensional nonlinear acoustic lattices'
---
Nonlinearity induced interaction between normal modes of extended systems is crucial for many fundamental dynamical and statistical phenomena like thermalization, thermal expansion of solids or turbulence in liquids. It is also important in many artificial systems where one aims at controlling the energy flow among the normal modes, preventing or efficiently channeling energy pumping due to resonances with external perturbations etc. Among the many accumulated results in this area a seminal one is due to Fermi, Pasta and Ulam (FPU) who reported on the absence of thermalization of chains of atoms connected by weakly nonlinear springs [@fpu]. They observed that the energy of an initially excited normal mode with frequency $\omega_q$ and wave number $q$ did not spread over all other normal modes, staying almost completely locked within a few neighbouring modes in the normal mode space [@Ford; @chaosfpu]. Longer waiting times yielded another puzzle of energy reccurrence to the originally excited mode. Many efforts to explain the FPU paradox resulted in an extraordinary progress in this field: the observation of solitons [@Zabusky], size-dependent stochasticity thresholds [@Izrailev_Chirikov], nonlinear resonances [@Chirikov], KAM tori, Arnold diffusion, and many other issues [@deLuca; @Shepel; @italian; @kantz; @lcmcsmpegdc97]. The efforts to carry all these concepts into two-dimensional lattices have been also reported [@2dim]. However, according to Ford [@Ford], despite the richness of new topics, the original FPU problem was still waiting to be understood.
Recently it was shown that the major ingredients of the FPU problem can be addressed within the promising concept of $q$-breathers (QBs), which are exact time-periodic solutions in the nonlinear FPU chain [@we]. These solutions are exponentially localized in the $q$-space of normal modes and preserve stability for small enough nonlinearity. They continue from their trivial counterparts for zero nonlinearity at finite energy. In that limit they correspond to one mode with the seed wave number $q_0$ being excited, and all the other modes being at rest. The stability threshold of QB solutions coincides with the weak chaos threshold in [@deLuca]. Persistence of exact stable QB modes surrounded by almost quasiperiodic trajectories explains the origin of FPU recurrences and the absence of energy equipartition. The scaling of the localization exponents with the relevant control parameters relates to results on higher order nonlinear resonance overlaps [@Shepel]. But perhaps the most important result was, that it needs only one ingredient to obtain QBs in FPU chains: a discrete nonequidistant frequency spectrum of normal modes, as induced by a finite system. Thus the door is opened to apply the concept of QBs to a variety of nonlinear finite systems - truly common objects in nature and applications. One of such challenges is an extension of the notion of QBs into more physically realistic two- and three-dimensional FPU-type systems.
In this Letter we report on the existence and remarkable properties of $q$-breathers in finite two-and three-dimensional nonlinear acoustic lattices. For fixed energy, nonlinearity coefficient and seed wave number the QB localization length stays finite in the 2d case and tends to zero in the 3d case with increasing size of the system. For both 2d and 3d cases the nonlinearity coefficient at the QB stability threshold increases in the same limit. This comprises a crucial difference from the 1d case [@we] where QBs delocalize and the nonlinearity threshold of the instability tends to zero in the limit of large chains.
We consider quadratic and cubic lattices of $N^d$ ($d=2$ and $3$ respectively) equal masses coupled by nonlinear springs with the Hamiltonian $$\label{eq1}
\begin{aligned}
& H=\frac{1}{2}\sum\limits_{\boldsymbol{n}}(p_{\boldsymbol{n}}^2
+\sum\limits_{\boldsymbol{m}\in D(\boldsymbol{n})}[
\frac{1}{2}(x_{\boldsymbol{m}}-x_{\boldsymbol{n}})^2+\frac{\beta}{4}(x_{\boldsymbol{m}}-x_{\boldsymbol{n}})^4])
\end{aligned}$$ where $x_{\boldsymbol{n}}(t)$ is the displacement of the $\boldsymbol{n}=(n_1,\ldots,n_d)$-th particle from its original position, $p_{\boldsymbol{n}}(t)$ its momentum, $D(\boldsymbol{n})$ is the set of its nearest neighbors, and fixed (zero) boundary conditions are taken: $x_{\boldsymbol{n}}=0$ if $n_l=0$ or $n_l=N+1$ for any of the components of $\boldsymbol{n}$.
A canonical transformation $$\label{eq1.5}
x_{\boldsymbol{n}}(t)=\left(\frac{2}{N+1}\right)^{d/2}\sum\limits_{q_{1},\ldots,
q_{d}=1}^N Q_{\boldsymbol{q}}(t)\prod\limits_{i=1}^d
\sin{\left(\frac{\pi q_i n_i}{N+1}\right)}$$ takes into the reciprocal wave number space with $N^d$ normal mode coordinates $Q_{\boldsymbol{q}}(t)\equiv Q_{q_1,\ldots,q_d}(t)$. The normal mode space is spanned by $\boldsymbol{q}$ and represents a d-dimensional lattice similar to the situation in real space. The equations of motion then read $$\label{eq2}
\Ddot{Q}_{\boldsymbol{q}}+\Omega_{\boldsymbol{q}}^2
Q_{\boldsymbol{q}}=-\frac{16\beta}{(2N+2)^d}\sum\limits_{\boldsymbol{p},\boldsymbol{r},\boldsymbol{s}}
C_{\boldsymbol{q},\boldsymbol{p},\boldsymbol{r},\boldsymbol{s}}Q_{\boldsymbol{p}}Q_{\boldsymbol{r}}
Q_{\boldsymbol{s}}\;.$$ Here $\Omega_{\boldsymbol{q}}^2=4\sum\limits_{i=1}^d\omega_{q_i}^2$ are the squared normal mode frequencies with $\omega_{q_{i}}=\sin{(\pi
q_{i}/2(N+1))}$. Note, that all linear modes but the diagonal ones $Q_{q_1=\ldots=q_d}$ are at least $d$-fold degenerate with respect to their frequencies. The coupling coefficients $C_{\boldsymbol{q},\boldsymbol{p},\boldsymbol{r},\boldsymbol{s}}$ induce a selective interaction between distant modes in the normal mode space similar to the 1d case.
For small amplitude excitations the nonlinear terms in the equations of motion can be neglected, and according to (\[eq2\]) the $q$-oscillators get decoupled, each conserving its harmonic energy $E_{\boldsymbol{q}}=\frac{1}{2}\left(\dot{Q}_{\boldsymbol{q}}^2+\Omega_{\boldsymbol{q}}^2
Q_{\boldsymbol{q}}^2\right)$ in time. Especially, we may consider the excitation of only one of the $q$-oscillators, i.e. $E_{\boldsymbol{q}} \neq 0$ for $\boldsymbol{q}\equiv
\boldsymbol{q}_0$ only. Such excitations are trivial time-periodic and $q$-localized solutions (QBs) for $\beta=0$.
For the 1d chain, such periodic orbits can be continued into the nonlinear case at fixed total energy [@we] because the nonresonance condition $n \Omega_{\boldsymbol{q}_0} \neq
\Omega_{\boldsymbol{q} \neq \boldsymbol{q}_0}$ (here $n$ is an integer) holds for any finite size [@Tiziano]. This argument can be used straightforwardly for $d=2,3$ and large seed wave numbers $\boldsymbol{q}$ on the main diagonal such that $2\Omega_{\boldsymbol{q}} > 4$. For all other seed wave numbers on the main diagonal we checked that the nonresonance condition holds as well. For seed wave numbers off the main diagonal the above mentioned $d$-fold degeneracies could be lifted by considering anisotropic lattices. In fact these degeneracies are lifted by the nonintegrability of the nonlinear lattice (\[eq2\]), and a discrete set of periodic orbits will remain. QBs are successfully observed in numerical experiments in the presence of the degeneracy and we did not find substantial difficulties in computing them.
In the following we compute QBs as well as their Floquet spectrum numerically using the same algorithms as for the 1d atomic chain [@we], and compare the results with analytical predictions, derived by means of asymptotic expansions.
First, we turn to the case $d=2$. We obtain various symmetric (with $(q_0)_1=(q_0)_2$, Fig.\[fig1\],\[fig2\]) and asymmetric (with $(q_0)_1\neq(q_0)_2$) QBs in the lower frequency range, which are exponentially localized in $q$-space. Note, that due to the parity symmetry of the model (Eq.(\[eq1\]) is invariant under $x_{\boldsymbol{n}} \rightarrow
-x_{\boldsymbol{n}}$ for all $\boldsymbol{n}$) only the modes with odd components $(q_1,q_2)$ are excited by the $\boldsymbol{q}_0=(3,3)$ mode. In contrast to $d=1$, the decay of the energy distribution (especially along the diagonal $\boldsymbol{q}(n)=(2n-1)\boldsymbol{q}_0$) remains almost constant with increase of the lattice size (Fig.\[fig2\]).
The abovementioned degeneracy of the frequency spectrum supports the existence of multi-mode QBs, namely those, which have two (or more) excited seed modes. Indeed, we continued multi-mode periodic solutions of the linear lattice $E_{\boldsymbol{q}} \neq 0$ for $\boldsymbol{q}\in S(\boldsymbol{q}_0)=\{\boldsymbol{q}:
\Omega_{\boldsymbol{q}}=\Omega_{\boldsymbol{q}_0}\}$ into the nonlinear regime. For example, the set $\boldsymbol{q}_0=(2,3),\boldsymbol{q}_0^*=(3,2)$ allows for two (asymmetric single-mode) QB solutions with the energy mainly concentrated in one of the two seed modes and two symmetric multi-mode QB solutions with the same energy in each of the seed modes, and oscillations being in- or out-of-phase.
By an asymptotic expansion of the solution to (\[eq2\]) in powers of the small parameter $\sigma=\beta/(N+1)^2$ (in analogy to [@we]) we estimate the shape of a QB solution $\hat{Q}_{\boldsymbol{q}}(t)$ with a low-frequency seed mode number $\boldsymbol{q}_0$. The energies of the modes on the diagonal of the QB $\boldsymbol{q}_0$, $3\boldsymbol{q}_0$,…,$(2n+1)\boldsymbol{q}_0$,…$\ll(N,N)$ read $$\label{eq4}
E_{(2n+1)\boldsymbol{q}_0}=\lambda_{d}^{2n}E_{\boldsymbol{q}_0}\;,\;
\lambda_{d}=\frac{3\beta
E_{\boldsymbol{q}_0}N^{2-d}}{2^{2+d}\pi^2|\boldsymbol{q}_0|^2}\;\;.$$ Dashed lines in Fig.\[fig2\](a) are obtained using (\[eq4\]) and show very good agreement with the numerical results. The energy distribution between other modes, involved in the QB is more complicated, but the decay along the diagonal (\[eq4\]) gives a good estimate for it (Fig.\[fig2\](b)). Note, that the localization along the diagonal is the weakest, at least for large $N$. The shape of the QB in the $q$-space then possesses the following properties: (i) the localization remains constant when the lattice size tends to infinity with all other parameters fixed, in contrast to the 1d case where QBs delocalize as $\lambda\propto(N+1)$, (ii) in the limit of constant [*energy density*]{} $\varepsilon=E_{\boldsymbol{q}_0}/(N+1)^2$ and [*wave vector*]{} of the QB $\boldsymbol{\kappa}_0=\boldsymbol{q}_0/(N+1)$ the localization remains constant for $N\rightarrow\infty$ (similar to the 1d case), (iii) for smaller $\beta, E_{\boldsymbol{q}_0}$ and larger $\boldsymbol{q}_0$ QBs compactify. These results are a consequence of a more fundamental scaling property of finite to infinite systems.
We analyze the stability of the obtained periodic orbits with standard methods of linearizing the phase space flow around the solutions and computing the eigenvalues and eigenvectors of the corresponding symplectic Floquet matrix [@we]. The orbits are stable if all complex eigenvalues lie on the unit circle. The absolute values of the Floquet eigenvalues of QBs for symmetric $\boldsymbol{q}_0=(3,3)$ and asymmetric $\boldsymbol{q}_0=(2,3)$ are plotted versus $\beta$ for different system sizes $N$ in Fig.\[fig4\]. Similar to the 1d case, QBs are stable for sufficiently weak nonlinearities. When $\beta$ exceeds a certain threshold $\beta^*$, some eigenvalues get absolute values larger than unity (and some of them less than unity) and the QB becomes unstable. In remarkable contrast to the 1d case, $\beta^*$ rapidly increases with the size of the system Fig.\[fig4\](a). For a series of computationally accessible large lattices $N=20, 30, 40$ (not plotted in Fig.\[fig4\](a)) we found the QB with $\boldsymbol{q}_0=(3,3)$ to be stable at least up to $\beta=10.0$. For insufficiently large $N$ the dependence $\beta^*(N)$ may become non-monotonous (Fig.\[fig4\](b)). It may be quite sensitive to the chosen seed wave number $\boldsymbol{q}_0$, compare Figs.\[fig4\](a) and (b).
The observed instabilities can be traced analytically, similar to the 1d case. Using standard secular perturbation techniques we approximate the frequency of the QB solution as $ \hat{\Omega} =
\Omega_{\boldsymbol{q}_0}(1+9h\rho)+O(h^2)$, where $h=3\beta
E_{\boldsymbol{q}_0}/(N+1)^2$ is a small parameter and $\rho=(w_{(q_0)_1}^2+w_{(q_0)_2}^2)/\Omega_{\boldsymbol{q}_0}^4$. Linearizing the equations of motion (\[eq2\]) around a QB solution $Q_{\boldsymbol{q}}=\hat{Q}_{\boldsymbol{q}}(t)+\xi_{\boldsymbol{q}}$ we obtain $$\label{eq4a}
\Ddot{\xi}_{\boldsymbol{q}}+\Omega_{\boldsymbol{q}}^2
\xi_{\boldsymbol{q}}=-4h(1+\cos{2\hat{\Omega}t})\sum\limits_{\boldsymbol{p}}
C_{\boldsymbol{q},\boldsymbol{q}_0,\boldsymbol{q}_0,\boldsymbol{p}}/
\Omega_{\boldsymbol{q}_0}^2\xi_{\boldsymbol{p}}+O(h^2)$$ The strongest instability, caused by primary parametric resonance in , comes from pairs of resonant modes $\boldsymbol{q}+\boldsymbol{p}=2\boldsymbol{q}_0$ with a nonzero vector $\boldsymbol{k}=\boldsymbol{q}-\boldsymbol{q}_0=
\boldsymbol{q_0}-\boldsymbol{p}$. The bifurcation point and the absolute values of the Floquet multipliers involved in the resonance in its vicinity are represented by a complex expression, which demonstrates good agreement with the numerical results (Fig.\[fig4\]). We assume $|k_{1,2}|<<(q_0)_{1,2}<<N$ and approximate $$\label{eq5}
\begin{aligned}
&3\beta^*
E_{\boldsymbol{q}_0}/(N+1)^2=8(\Omega_{\boldsymbol{q}}+\Omega_{\boldsymbol{p}}-
2\Omega_{\boldsymbol{q}_0})/\Omega_{\boldsymbol{q}_0}\\
& \approx 8\left(\frac{(q_0)_2k_1-(q_0)_1k_2}{\boldsymbol{q}_0^2}\right)^2
\end{aligned}$$ This estimate explains the following basic features of the observed instability. The instability of the type $\boldsymbol{k}$ that minimizes $|(q_0)_2k_1-(q_0)_1k_2|$ is the first to occur. It results in a non-monotonous and discontinuous dependence of $\beta^*(\boldsymbol{q}_0,N)$ for small lattices. It monotonously increases with $N$ while $\boldsymbol{k}$ is constant (Fig.\[fig4\](a)); when $\boldsymbol{k}$ changes $\beta^*$ changes as well (Fig.\[fig4\](b)). Besides, the instability threshold scales as $\beta^*\propto\boldsymbol{q}_0^{-2}$ and as $\beta^*\propto N^2$.
The 2d lattice supports also the existence of QBs with $\boldsymbol{q}_0$ located in the intermediate and high-frequency parts of the normal mode spectrum.
Let us turn to the case $d=3$. We again compute QB solutions as in the 2d case. A similar analysis shows, that for the low-frequency seed mode $\boldsymbol{q}_0$ the decay of the normal mode energies along the leading direction $3\boldsymbol{q}_0$,…,$(2n+1)\boldsymbol{q}_0$,…$\ll(N,N,N)$ is given by (\[eq4\]) with $d=3$, and fits well the shape of the numerically computed QBs with $\boldsymbol{q}_0=(3,3,3)$ (Fig.\[fig6\]). In contrast to the 2d case, the localization length even decreases with increasing lattice size $\lambda_{3d}\propto(N+1)^{-1}$. For constant [*energy density*]{} $\varepsilon=E/(N+1)^3$ and [*wave vector*]{} of the QB $\boldsymbol{\kappa}_0$, the degree of localization is independent of the lattice size, as it is for the 1d and 2d cases. Despite having difficulties when calculating the stability of QBs in sufficiently large 3d lattices due to limited machine performance, the analytical treatment has been done similar to lower lattice dimensions. We find that $\beta^*\propto(N+1)^3$ for constant energy of the lattice and does not increase for fixed $\varepsilon$ and $\boldsymbol{q}_0$. We also predict its sensitive dependence upon the choice of the seed mode number of the QB and the size of the lattice when being small.
The main reason for the crucial improvement of localization and stability of QBs in higher dimensions relies on the fact that the effective strength of nonlinear intermode coupling decreases exponentially due to the factor $(N+1)^{-d}$.
In conclusion, we report on the existence and remarkable properties of $q$-breathers as exact time-periodic solutions in nonlinear two- and three-dimensional acoustic systems, thus extending the concept of one-dimensional QBs to more physically relevant objects. They are exponentially localized in the $q$-space of the normal modes and preserve stability for small enough nonlinearity. In the limit of infinite lattice size the localization length stays constant for $d=2$ and tends to zero for $d=3$ when fixing $\boldsymbol{q}_0$ and the total energy. At the same time the localization length does not depend on the lattice size $N$ when fixing the energy density and the wave vector $\boldsymbol{\kappa}_0=\boldsymbol{q}_0/(N+1)$. In contrast to the one-dimensional case the observed instability threshold in the nonlinear coupling strength increases with increasing lattice size, and stays constant if the energy density is fixed.
M.I., O.K., and K.M. appreciate the warm hospitality of the Max Planck Institute for the Physics of Complex Systems. M.I. and O.K. acknowledges support of the “Dynasty” foundation.
[99]{}
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Degeneracy effects for bosons are more important for smaller particle mass, smaller temperature and higher number density. Bose condensation requires that particles be in the same lowest energy quantum state. We propose a cosmic background Bose condensation, present everywhere, whith its particles having the lowest quantum energy state, $\hbar c / \lambda$, with $\lambda$ about the size of the visible universe, and therefore unlocalized. This we identify with the quantum of the self gravitational potential energy of any particle, and with the bit of information of minimum energy. The entropy of the universe ($\sim 10^{122} \ bits$) has the highest number density ($ \sim 10^{36} \ bits / cm^3$) of particles inside the visible universe, the smallest mass, $\sim 10^{-66} g$, and the smallest temperature, $\sim 10^{-29} K$. Therefore it is the best candidate for a Cosmic Background Bose Condensation (CBBC), a completely calmed fluid, with no viscosity, in a superfluidity state, and possibly responsible for the expansion of the universe.'
author:
- 'Antonio Alfonso-Faus'
- Màrius Josep Fullana i Alfonso
title: 'Cosmic Background Bose Condensation (CBBC) '
---
Introduction
============
[@Wei] advanced a clue to suggest that large numbers are determined by both, microphysics and the influence of the whole universe. He constructed a mass using the physical constants $G, \hbar, c$ and the Hubble parameter $H$. This mass was not too different from the mass of a typical elementary particle (like a pion) and is given by
$$m \approx (\hbar^2 H /Gc)^{1/3}
\label{e1}$$
In our work here we consider a general elementary particle of mass $m$. This particle may include not only baryons but the possible quantum masses of dark matter and dark energy in the universe. Since the mass $m$ will disappear from the resultant relation, the conclusion is totally independent on the kind of elementary particle that we may consider. The self gravitational potential energy $E_g$ of this quantum of mass $m$ (and size its Compton wavelength $ \hbar /mc$) is given by
$$E_g = G m^2 / (\hbar / mc) = G m^3 c / \hbar
\label{e2}$$
This relation has been previously used in another context (@Siv [-@Siv]). Combining (\[e1\]) and (\[e2\]) we eliminate the mass $m$ to obtain
$$E_g \approx H \hbar
\label{e3}$$
Here $\hbar$ is Planck’s constant, usually interpreted as the smallest quantum of action (angular momentum). Since $H$ is of the order of $1/t$, $t$ the age of the universe ($t$ being a maximum time today), (\[e3\]) is the lowest quantum energy state that it may exist. It is equivalent to $\hbar c / \lambda$ with $\lambda$ of the order of the size of the visible universe (it is the lowest quantum energy state with $\lambda \approx ct$). We identify it with the quantum of the self gravitational potential energy of any quantum particle (@AAFa [-@AAFa]). We also identify it with the bit, the unit of information with minimum energy (@AAF2 [-@AAF2]). [@LLo], about 10 years ago, stated that [*Merely by existing, all physical systems register information*]{} and about 25 years ago [@Lan], as cited by [@LLo], stated [*Information is physical”. And today we say here: All physical systems of mass M (energy $Mc^2$) are equivalent to an amount of information in number of bits of the order of* ]{} $$Number \ of \ bits \ \approx M c^2 / E_g \approx Mc^2 / (H \hbar)
\label{e4}$$
The equivalence between information and energy, as implied by the above relation, can be interpreted as the result of a recent experiment (@Fun [-@Fun]) where it is shown that entanglement can produce a gain in thermodynamic work, the gain being determined by a change of information content. Also a link between information theory and thermodynamics has been experimentally verified (@Ber [-@Ber]). Previously, in 2010, an experimental demonstration of information-to-energy conversion was also published (@Toy [-@Toy]). Relation (\[e4\]) has general, universal validity. In this sense the unit of energy, that should naturally be taken as the minimum quantum of energy $H \hbar$, implies that the relativistic energy of any mass $M$ has $N$ times this minimum quantum of energy, $N H \hbar$, being $N$ its number of information bits. Therefore, $N H \hbar$ corresponds to the energy of all the information $N$ that carries the physical system. And as far as entropy $S$ is concerned, this number is also the same as $S/ (k_B \log 2)$, $k_B$ being Boltzmann constant, as we can talk about a generalized relation for entropy, in accordance with the ideas introduced by [@Lan2]:
$$S = N k_B \log 2 = \frac{Mc^2}{H \hbar} \ k_B \log 2
\label{e5}$$
We see that the extensive property of entropy is preserved because, in accordance with our proposal, it comes to be proportional to the relativistic energy of the system. In this work we propose a Cosmic Background Bose condensation (CBBC), present everywhere, where the particle with minimum energy $E_g$ and mass $m_g = E_g /c^2$ is defined in (\[e3\]). It is composed of very low energy and temperature components, with very high number density, and of course all in the same state.
[@Bek] found an upper bound for the ratio of the entropy $S_B$ to the energy $E = Mc^2$ of any bounded system with effective size $R$: $$S_B / E < 2 \pi k_B R / \hbar c
\label{eq.1}$$
About ten years later (@Hoo [-@Hoo]; @Sus [-@Sus]), a holographic principle was proposed giving a bound for the entropy $S_h$ of a bounded system of effective size $R$ as $$S_h \leq \pi k_B c^3 R^2 / \hbar G
\label{eq.2}$$
The Bekenstein bound (\[eq.1\]) is proportional to the product $MR$, while the holographic principle bound (\[eq.2\]) is proportional to the area $R^2$. If the two bounds are identical (hence $M$ is proportional to $R$) here we prove that the system obeys the Schwarzschild condition for a black hole. We analyze this conclusion for the case of a universe with finite mass $M$ and a Hubble size $R \approx ct$, $t$ being the age of the universe. $M$ and $R$ are obviously the maximum values they can have in our universe, the visible universe.
Also, for the case of a black hole we find that the Hawking and Unruh temperatures are the same. Then, for our universe we obtain the mass of the gravity quanta, of the order of $10^{-66} g$.
Consequences of the identification of the two bounds
====================================================
Identifying the bounds (\[eq.1\]) and (\[eq.2\]) we get $$2 M = c^2 R / G
\label{eq.3}$$ which is the condition for the system $(M, R)$ to be a black hole. Then, its entropy is given by the Hawking relation (@Haw [-@Haw]) $$S_H = \frac{4 \pi k_B}{\hbar c} \ G M^2
\label{eq.4}$$ that coincides with the two bounds (\[eq.1\]) and (\[eq.2\]). The conclusions here are exclusively related to the Schwarzschild radius within the context of Einstein’s general relativity.
The mass of the universe $M_u$ is a maximum. And so is its size $R$. A bounded system implies a finite value for both. Using present values for $M_u \approx 10^{56} g$ and $R \approx 10^{28} cm$ they fulfill the Schwarzschild condition (\[eq.3\]). This is an evidence for the Universe to be a black hole (@AAF1 [-@AAF1]). And its entropy today is about $10^{122} k_B \log 2$.
The case for the Unruh and Hawking temperatures
===============================================
The fact that a black hole has a temperature, and therefore an entropy (@Haw [-@Haw]) implies that an observer at its surface, or event horizon, sees a perfect blackbody radiation, a thermal radiation with temperature $T_H$ given by $$T_H = \hbar c^3 / (8 \pi GMk)
\label{eq.5}$$ where $\hbar$ is the Planck’s constant, $c$ the speed of light, $G$ the gravitational constant, $k$ Boltzmann constant and $M$ the mass of the black hole. This observer feels a surface gravitational acceleration $R''$. According to the [@Unr] effect an accelerated observer also sees a thermal radiation at a temperature $T_U$, proportional to the acceleration $R''$ and given by $$T_U = \hbar R'' / (2 \pi ck)
\label{eq.6}$$
Based upon the similarity between the mechanical and thermo dynamical properties of both effects, (\[eq.5\]) and (\[eq.6\]), we identify both temperatures and find the relation: $$R'' = c^4 / (4GM)
\label{eq.7}$$
Identifying the Unruh acceleration to the surface gravitational acceleration: $$R'' = GM/R^2
\label{eq.8}$$ and substituting in (\[eq.7\]) we finally get $$2GM/c^2 = R
\label{eq.9}$$
This is the condition for a black hole. Since the Hawking temperature refers to a black hole this result confirms the validity of the identification of the two temperatures, (\[eq.5\]) and (\[eq.6\]), as well as the interpretation of the Unruh acceleration in (\[eq.8\]).
Application of the cosmological principle
=========================================
The cosmological principle may be stated with the two special conditions of the universe: it is homogeneous and isotropic. This means that, on the average, all places in the universe are equivalent (at the same [*time*]{}) and that observing the universe at one location it looks the same in any direction. The cosmic microwave background radiation is a good example, a blackbody radiation at about $2.7 K$. This implies that there is no center of the universe, or equivalently, that any local place is a center.
We have seen that the universe may be taken as a black hole, and therefore that it makes sense to think that there must be an event horizon, a two dimensional bounding surface around each observer. If all places in the universe are equivalent then all places can say this, and the natural event horizon is the Hubble sphere, with radius $R$ about $c / H \approx 10^{28} cm$ today. Then at any point in the universe it can be interpreted as a two dimensional spherical surface, may be in a [*virtual*]{} sense. Following the holographic principle, all the information of the three dimensional world, as we see it, is contained in this spherical surface. And any observer, following the cosmological principle, can be seen as being at the center of the three dimensional [*sphere*]{}. To combine both principles, the cosmological and the holographic, we can think of an isotropic, spherically symmetric, acceleration present at each point in the universe given by (\[eq.8\]). This is a change of view from a 3 dimensional one to a 2 dimensional world. Also we have an isotropic temperature given by $$T = \hbar c^3 / (8 \pi GMk) = (1/4 \pi k) 1 / R \approx 10^{-29} K
\label{eq.10}$$
This is the temperature of the gravitational quanta (@AAF2 [-@AAF2]) at the present time. The equivalent mass of one quantum of gravitational potential energy is then from (\[eq.10\]) found to be about $10^{-66} g$. This may be interpreted as the ultimate quantum of mass. Its wavelength (in the Compton sense) is of the order of the size of the universe, $ct \approx 10^{28} cm$ and therefore it is a gravity quanta unlocalized in the universe, as the gravitational field (@Mis [-@Mis]). It is a boson and not a photon, the photon being the quantum of the electromagnetic field that should be localized in the universe.
The scale factor between the Planck scale and our universe today is about $10^{61}$. Multiplying the temperature found in (\[eq.10\]) by this numerical scale factor we get the Planck’s temperature $T_p \approx 10^{32} K$ at the Planck’s time $10^{-44} s$, when the universe had the Planck’s size $l_p \approx 10^{-33} cm$.
Cosmic Background Bose Condensation (CBBC)
==========================================
We now present the conditions favorable to have a Bose condensation as derived from the energy (@Eis [-@Eis]) per particle relation $E/N$:
$$E/N = \frac{3}{2} \ kT \left \{ 1 - \frac{1}{2^{5/2}} \ \frac{N h^3}{V (2 \pi m k_B T)^{3/2}} \right \}
\label{e5}$$
The term beyond 1 in the above bracket is the deviation of the Bose gas from the classical gas, the degeneracy effect. As we can see in the formulation (\[e5\]) this effect for bosons is more important the smaller the particle mass is ( $\sim 10^{-66} g$ in our case), the smaller the temperature is ( $\sim 10^{-29} K$ in our case) and the higher the number density is ($\sim 10^{36} \ particles/cm^3$ ). Then, the particle we are presenting here appears to be a good candidate for a universal Bose condensation background.
The CBBC and the expansion of the universe
==========================================
As we have seen in (\[e3\]) the gravity quanta proposed here does not depend on the related origin: baryons, dark matter or dark energy. Usually the expansion of the universe is considered to be related to the cosmological constant $\Lambda$. This constant was introduced by Einstein to avoid the collapse of the universe due to attractive gravitation by means of an outward pressure given by $\Lambda$. Within this context we know today that the percentages of baryonic content in the universe, dark matter and dark energy, are respectively about 4%, 27% and 69%. This is given in terms of the usual dimensionless $\Omega$: $$\Omega_b \approx 0.04, \ \ \ \ \ \Omega_{DM} \approx 0.27, \ \ \ \ \ \Omega_{\Lambda} \approx 0.69
\label{e6}$$ all of them adding up to 1. Given that our approach here does not discriminate between baryons, dark matter and dark energy, we may attribute the CBBC expansion effect as due to its pressure as related to the baryon component and possibly dark matter quanta or even dark energy quanta. We do not know today if there are quanta in the dark components. The maximum effect would correspond to consider that the total, critical density, is responsible for the gravity quanta presented here as the ground component of the CBBC, and then its effect would correspond to $\Omega_{CBBC} = 1$.
Conclusions
===========
The universe can be seen as a black hole. From each observer in it we can interpret that he/she is at the center of a sphere, with the Hubble radius. From this we interpret the spherical surface as the event horizon, a two dimensional surface that follows the physics of the holographic principle.
The isotropic acceleration present at each point in the universe, and given by (\[eq.8\]), implies that there is no distortion for the spherically distributed acceleration, as imposed by the cosmological principle. However, the presence of a nearby important mass, like the sun, will distort this spherically symmetric picture. With respect to the probes Pioneer 10/11, that detected an anomalous extra acceleration towards the sun of value ($8.74 \pm 1.33) \times 10^{-8} cm/s^2$ (@And [-@And]), we can see that this value is only a bit higher than the one predicted by (\[eq.8\]), which is about $7.7 \times 10^{-8}cm/s^2$. This difference is an effect that can be explained by the presence of the sun converting the isotropic acceleration to an anisotropic one.
Similarly there may be a factor, due to the influence of nearby masses (i.e. a massive black hole at the center of the galaxy), in the cases of the observed rotation curves in spiral galaxies. They imply that the speed of stars, instead of decreasing with the distance $r$ from the galactic center, is constant or even increases slowly when far from the central luminous object (@Dre [-@Dre]). For the case of globular clusters (@Sca [-@Sca]), where no dark matter is expected to be present, we have stronger evidence in support of the existence of the acceleration field. Also the escape velocity at the sun location, with respect to our galaxy, is higher than expected. The earth-moon distance increases with time and there is a residual part not explained by tidal effects. And the same occurs for the planets in the solar system. We present this evidence in support of the universal field of acceleration, $R''$ (@AAF3 [-@AAF3]).
Finally, for the case of a black hole we find that the Hawking and Unruh temperatures are the same. For our universe we obtain the mass of the gravity quanta, of the order of $10^{-66} g$, with a wavelength that corresponds to its size. It may be identified with the information bit, with entropy $k$. A Cosmic Background Bose Condensation fits well with the properties found here for the quantum of gravitational potential energy.
Alfonso-Faus, A., 2010a, “Universality of the self gravitational potential energy of any fundamental particle”, Astrophysics and Space Science, 337, 363
Alfonso-Faus, A., 2010b, “The case for the Universe to be a quantum black hole”, Astrophysics and Space Science, 325, 113
Alfonso-Faus, A., 2010c, “Galaxies: kinematics as a proof of the existence of a universal field of minimum acceleration”, preprint, arXiv:0708.0308
Alfonso-Faus, A., 2011, “Quantum gravity and information theories linked by the physical properties of the bit”, preprint, arXiv: 1105.3143
Anderson, J. D., et al., 1998, “Indication, from Pioneer 10/11, Galileo, and Ulysses Data, of an Apparent Anomalous, Weak, Long-Range Acceleration”, Physical Review Letters, 81, 2858
Bekenstein, J. D., 1981, Physical Review D, 23 (2), 287
Bérut, A. et al., 2012, “Experimental verification of Landauer’s principle linking information and thermodynamics”, Nature, 483, 187
Drees, M., & Chung-Lin, S., 2007, “Theoretical Interpretation of Experimental Data from Direct Dark Matter Detection”, Journal of Cosmology and Astroparticle Physics, 0706, 011
Eisberg, R. and Resnick, R., 1985, “Quantum Physics of atoms, molecules, solids, nuclei and particles” 1985, Second Edition, John Wiley & Sons Inc., New York
Funo, K., Watanabe, Y. & Ueda, M., 2012, “Thermodynamic Work Gain from Entanglement”, preprint, quant-ph/1207.6872
Hawking, S. W., 1974, “Black hole explosions?” , Nature, 248, 30
Landauer, R., 1961, “Irreversibility and Heat Generation in the Computing Process”, IBM Journal of Research and Development, 5, 183
Landauer, R., 1988, “Dissipation and noise immunity in computation and communication”, Nature, 335, 779
Lloyd, S., 2002, “Computational capacity of the universe”, Physical Review Letters 88, 237901
Misner, C.W., Thorne, K.S. & Wheeler, J.A., 1973, “Gravitation”, W.H. Freeman and Company, Reading (England), p.466 (“Why the energy of the gravitational field cannot be localized”)
Scarpa, R. & Falomo, R., 2010, “Testing Newtonian gravity in the low acceleration regime with globular clusters: the case of omega Centauri revisited”, Astronomy and Astrophysics, 523, A43
Sivaram, C., 1982, “Cosmological and quantum constraint on particle masses”, American Journal of Physics, 50, 279
Susskind, L., 1995, “The World as a hologram”, Journal of Mathematical Physics, 36, 6377
’t Hooft, G., 1993, “Dimensional Reduction in Quantum Gravity”, preprint, arXiv:gr-qc/9310026
Toyabe, S. et al., 2010, “Experimental demonstration of information-to-energy conversion and validation of the generalized Jarzynski equality”, Nature Physics, 6, 988
Unruh, W.G., 1976, “Notes on black-hole evaporation”, Physical Review D, 14 (4), 870
Weinberg, S., 1972, “Gravitation and Cosmology: Principles and applications of the General Theory of Relativity”, John Wiley & Sons Inc., New York, p.619
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
The pseudospectral method is a powerful tool for finding highly precise solutions of Schrödinger’s equation for few-electron problems. Previously we developed the method to calculate fully correlated S-state wave functions for two-electron atoms [@GrabowskiChernoff2010]. Here we extend the method’s scope to wave functions with non-zero angular momentum and test it on several challenging problems. One group of tests involves the determination of the nonrelativistic electric dipole oscillator strength for the helium $1^1$S $\to 2^1$P transition. The result achieved, $0.27616499(27)$, is comparable to the best in the literature. The formally equivalent length, velocity, and acceleration expressions for the oscillator strength all yield roughly the same accuracy because the numerical method constrains the wave function errors in a local fashion.
Another group of test applications is comprised of well-studied leading order finite nuclear mass and relativistic corrections for the helium ground state. A straightforward computation reaches near state-of-the-art accuracy without requiring the implementation of any special-purpose numerics.
All the relevant quantities tested in this paper – energy eigenvalues, S-state expectation values and bound-bound dipole transitions for S and P states – converge exponentially with increasing resolution and do so at roughly the same rate. Each individual calculation samples and weights the configuration space wave function uniquely but all behave in a qualitatively similar manner. Quantum mechanical matrix elements are directly and reliably calculable with pseudospectral methods.
The technical discussion includes a prescription for choosing coordinates and subdomains to achieve exponential convergence when two-particle Coulomb singularities are present. The prescription does not account for the wave function’s non-analytic behavior near the three-particle coalescence which should eventually hinder the rate of the convergence. Nonetheless the effect is small in the sense that ignoring the higher-order coalescence does not appear to affect adversely the accuracy of any of the quantities reported nor the rate at which errors diminish.
author:
- 'Paul E. Grabowski'
- 'David F. Chernoff'
bibliography:
- 'OscStrength.bib'
title: 'Pseudospectral Calculation of Helium Wave Functions, Expectation Values, and Oscillator Strength'
---
Introduction
============
The aim of this work is to test and validate the pseudospectral method as a high-precision few-electron problem solver, capable of calculating state-of-the-art precision matrix elements. The helium atom has been studied extensively since the birth of quantum mechanics and so makes a great testbed problem. High-precision work continues to this day to infer fundamental constants such as the fine structure constant (see Ref. [@PachuckiYerokhin2011]) and the electron-proton mass ratio (see Ref. [@KorobovZhong2009]) by comparing theoretical and experimental measurements. Any theoretical method which may be applied to a variety of problems ([*e.g.*]{} high-precision relativistic corrections, different interaction potentials, excitation levels, symmetries, etc.) without tinkering with or modifying the basis and which has direct, rigorous control of local errors serves as a complementary approach to the variational method.
Methods based on the variational principle, in which the expectation value of the Hamiltonian is minimized with respect to the parameters of a trial wave function, are the most widely used techniques for finding an approximate representation of the ground state. The calculated energy is an upper bound to the exact energy.[^1] If one regards the best approximate wave function as first order accurate then the variationally determined energy eigenvalue is second order accurate. Small errors in the energy eigenvalue of a given state imply that the square of the wave function is accurate in the energy-weighted norm but it does not follow that local wave function errors are also small. In practical terms, while the variational approach excels at determining energy eigenvalues it does not generally achieve comparable accuracy in quantum mechanical matrix elements formed from the wave function.
To achieve ever-more accurate energies and/or wave functions in the variational approach one must select a sequence of trial functions capable of representing the exact solution ever-more closely. The choice of a good sequence entails more than a little art and intuition, especially for a nonstandard problem where one may have only a vague idea what the ultimate limit looks like. A sequence of increasing basis size $n$ may be said to converge exponentially if the errors are proportional to $e^{-a n}$ for some positive constant $a$. This most favorable outcome is achieved only if the basis can reproduce the analytic properties of the exact wave function. Otherwise, convergence is expected to be algebraic, i.e. $\propto n^{-2}$, or worse.
Recently, we applied pseudospectral methods to solve the nonrelativistic Schrödinger equation for helium and the negatively charged hydrogen ion with zero total angular momentum [@GrabowskiChernoff2010]. We found exponentially fast convergence of most quantities of interest including the energy eigenvalues, local energy errors ([*e.g.*]{} $(\hat{H} \Psi)/\Psi - E$ as a function of position) and Cauchy wave function differences. Only the error in the logarithmic derivative near the triple coalescence point had discernibly slower convergence, presumably due to the logarithmic contributions located there [@Bartlett1937; @Fock1954; @Fock1958]. The key virtues of the pseudospectral approach were: no explicit assumptions had to be made about the asymptotic behavior of the wave function near cusps or at large distances, the Schrödinger equation was satisfied at all grid points, local errors decreased exponentially fast with increasing resolution, and no fine tuning was required.
In this article, we extend our previous work to higher angular momentum calculations and utilize the results to evaluate matrix elements for combinations of states. To be systematic, we consider two sorts of matrix elements: the dipole absorption oscillator strength (between S and P states) and first-order mass polarization and $\alpha^2$ relativistic corrections to the nonrelativistic finite-nuclear-mass Hamiltonian (for the S ground state). All have been the subject of extensive investigation. Our main focus is on testing the pseudospectral method’s capabilities by recalculating these quantities and comparing to effectively “exact” published results.
The plan of the paper is as follows. The first four sections are largely background: §\[RevPSMethods\] provides an overview of the pseudospectral method; §\[NR2electronatom\] describes the two-electron atom, the Bhatia-Temkin coordinate system, the expansion of the wave function in terms of eigenstates and the form of the Hamiltonian; §\[DipoleReview\] defines length, velocity and acceleration forms for the oscillator strength and related sum rules. The next two sections detail our pseudospectral method of calculation and those readers primarily interested in seeing the results may skip to §\[EandOscResults\]. §\[VarsAndDomains\] gives a prescription for how to choose coordinates and subdomains for second order partial differential equations and outlines the special coordinate choices needed to deal with the Coulomb singularities. §\[BoundaryConditions\] schematically describes how overlapping and touching grids are coupled together and how symmetry is imposed on the wave function. §\[EandOscResults\] presents the first group of test results on energies and oscillator strengths. The convergence rate of all quantities is studied in detail. §\[CorrectionsHamiltonian\] and §\[MassPolRelCorrCalcs\] review lowest-order corrections to the Hamiltonian due to finite nuclear mass and finite $\alpha$. §\[ExpectationValues\] presents the second group of test results for individual corrections to the ground state of He. §\[Conclusions\] summarizes the capabilities and promise of the pseudospectral method.
The appendix is divided into four parts. Appendix \[BTAppendix\] gives the explicit form of the Hamiltonian operator used in this article. Appendix \[MatrixMethods\] describes how the Hamiltonian matrix problem is solved, gives details of the eigenvalue solver method, and how quantum mechanical matrix elements are calculated once the wave function is determined. Appendix \[BTOscStrength\] gives the particular equations for calculating the oscillator strengths and expectation values. Appendix \[OscStrengthTable\] discusses and tabulates past work done to calculate oscillator strengths.
Review of pseudospectral methods {#RevPSMethods}
================================
Pseudospectral methods have proven success in solving systems of partial differential equations germane to the physics in a wide variety of fields including fluid dynamics [@CanutoEtAl1988], general relativity [@KidderFinn2000; @PfeifferEtAl2003], and quantum chemistry [@Friesner1985; @Friesner1986; @Friesner1987; @RingnaldaEtAl1990; @GreeleyEtAl1994; @MurphyEtAl1995; @MurphyEtAl2000; @KoEtAl2008; @HeylThirumalai2009]. Some problems in one-electron quantum mechanics [@Borisov2001; @BoydEtAl2003] have been treated but only recently has the method been applied to the case of fully correlated, multi-electron atoms [@GrabowskiChernoff2010]. Pseudospectral methods are discussed in some generality in Refs. [@Boyd2000; @Fornberg1996; @Orszag1980; @PfeifferEtAl2003; @NumericalRecipes; @GrabowskiChernoff2010].
The pseudospectral method is a grid-based finite difference method in which the order of the finite differencing is equal to the resolution of the grid in each direction. As the grid size increases it becomes more accurate than any fixed-order finite difference method. If a solution is smooth over an entire domain (or smooth in each subdomain) the pseudospectral method converges exponentially fast to the solution. A spectral basis expansion and a pseudospectral expansion of the same order are nearly equivalent having differences that are exponentially small.
The grid points in the pseudospectral method are located at the roots of Jacobi polynomials or their antinodes plus endpoints. They are clustered more closely near the boundary of a domain than in its center. Such an arrangement is essential for the method to limit numerical oscillations sourced by singularities beyond the numerical domain [@Fornberg1996sec34]. These singularities typically occur in the analytic continuation of solutions to non-physical regimes and/or from the extension of coordinates beyond the patches on which they are defined to be smooth and differentiable. The grid point arrangement facilitates a convergent representation of a function and its derivative across the domain of interest. The interpolated function is more uniformly accurate than is possible using an equal number of equidistant points, as is typical for finite difference methods.
Consider the problem of the pseudospectral representation of an operator like the Hamiltonian. The full domain is multi-dimensional but focus for the moment on a single dimension of the domain. Let $\{{\mathbf{{{\rm X}}}}^k\}_{k=1,2,\ldots N}$ be the roots of an $N$th order Jacobi polynomial enumerated by $k$. Let ${\mathbf{{{\rm X}}}}$ stand for an arbitrary coordinate value in the dimension of interest. Define the one dimensional cardinal functions $$C_j[{{\rm X}}]=\prod_{\substack{k=1\\k\ne j}}^N \frac{{{\rm X}}-{{\rm X}}^k}{{{\rm X}}^j-{{\rm X}}^k}$$ and note the relation $$C_j[{{\rm X}}^k]=\delta_j^k$$ follows. Now let the $n_d$-dimensional grid be the tensor product of the individual, one dimensional coordinate grids labeled by $X_{(i)}$ for $i=1$ to $n_d$. The corresponding cardinal functions are $$\label{EffectiveBasis}
{\cal C}_{J}[{\mathbf{{{\rm X}}}}]=\prod_{i=1}^{n_d}C_{j_{(i)}}[{{\rm X}}_{(i)}],$$ where subscript $J=\{j_{(1)}, j_{(2)},\ldots,j_{(n_d)}\}$ and unadorned ${\mathbf{{{\rm X}}}}=\{{{\rm X}}_{(1)},{{\rm X}}_{(2)},\ldots,{{\rm X}}_{(n_d)}\}$. These multi-dimensional Cardinal functions have the property $${\cal C}_J[{\mathbf{{{\rm X}}}}^K]=\delta_J^K,$$ where the grid point ${\mathbf{{{\rm X}}}}^K = \{{{\rm X}}_{(1)}^{k_1},{{\rm X}}_{(2)}^{k_2},\ldots,{{\rm X}}_{(n_d)}^{k_{n_d}}\}$. They form a basis in the sense that a general function $f$ can be written $$f[{\mathbf{{{\rm X}}}}]=\sum_Jf[{\mathbf{{{\rm X}}}}^J]{\cal C}_J[{\mathbf{{{\rm X}}}}],$$ where $f[{{{\rm X}}}^J]$ is a pseudospectral coefficient (“pseudo” because it is more easily identified as the function value at the grid point).
Let the position ${\mathbf{{{\rm X}}}}^K$ and cardinal ${\cal C}_J$ eigenstates be denoted $|{\mathbf{{{\rm X}}}}^K\rangle$ and $|{\mathbf{\cal C}}_J\rangle$, respectively. The pseudospectral approximation to the Hamiltonian is $$\label{PSMatrix}
\hat{H}_{PS}=\sum_{JK}|{\mathbf{{{\rm X}}}}^K\rangle\langle{\mathbf{{{\rm X}}}}^K|\hat{H}|{\cal C}_J\rangle\langle{\cal C}_J|,$$ where $\hat{H}$ is the full Hamiltonian operator. In practice, the matrix $\langle{\mathbf{{{\rm X}}}}^K|\hat{H}_{PS}|{\cal C}_J\rangle$ is truncated and then diagonalized to find the energy eigenvalues. When the wave function is represented by a pseudospectral expansion the eigenvectors are simply the function values at the grid points. In a spectral representation, by contrast, the eigenvectors are sums of basis functions. It is often more convenient and efficient to work with the local wave function values directly. On the other hand, the truncated operator $\hat{H}_{PS}$ need not be Hermitian at finite resolution, a property that may introduce non-physical effects, e.g. $\langle{\mathbf{{{\rm X}}}}^K|\hat{H}_{PS}|{\cal C}_J\rangle$ may possess complex eigenvalues. Generally, unphysical artifacts quickly reveal themselves as resolution increases. An examination of the eigenvalue spectrum shows that the complex eigenvalues do not converge, permitting separation of physical and unphysical values.
The nonrelativistic two-electron atom {#NR2electronatom}
=====================================
Two-electron atoms are three-particle systems requiring nine spatial coordinates for a full description. In the absence of external forces, three coordinates are eliminated by taking out the center-of-mass motion. In the infinite-nuclear-mass and nonrelativistic approximations the Hamiltonian is $$\label{Hamiltonian0}
\hat{H}_0=-\frac{1}{2}(p_1^2+p_2^2)+\hat{V},$$ where ${\mathbf{p}}_{1,2}$ are the momenta of the two electrons and the potential is $$\hat{V}=-\frac{Z}{r_1}-\frac{Z}{r_2}+\frac{1}{r_{12}},$$ where $Z$ is the nuclear charge, and $r_1$, $r_2$, and $r_{12}$ are the magnitudes of the vectors pointing from the nucleus to each electron and of the vector pointing from one electron to the other, respectively. Here and throughout this article, atomic units are used. For the infinite-nuclear-mass approximation, the electron mass is set to unity; for a finite nuclear mass, the reduced mass of the electron and nucleus is set to one. The fully correlated wave functions are six-dimensional at this stage.
A further reduction is straightforward for S states. Hylleraas [@Hylleraas1929] proposed the ansatz that the wave function be written in terms of three internal coordinates. Typical choices for these coordinates are $r_1$, $r_2$, and $r_{12}$. Alternatively, $r_{12}$ may be replaced by $\theta_{12}$, the angle between the two electrons. The S state is independent of the remaining three coordinates that describe the orientation of the triangle with vertices at the two electrons and nucleus.
The situation for states of general angular momentum is more complicated. Bhatia and Temkin [@BhatiaTemkin1964] introduced a particular set of Euler angles $\{\Theta,\Phi,\Psi\}$ to describe the triangle’s orientation. They defined[^2] a set of generalized spherical harmonics $D_{\kappa lm}^\nu$ which are eigenstates of operators for the total angular momentum, its $z$ component, total parity ($\{{\mathbf{r}}_1,{\mathbf{r}}_2\}\to\{-{\mathbf{r}}_1,-{\mathbf{r}}_2\}$), and exchange (${\mathbf{r}}_1\leftrightarrow {\mathbf{r}}_2$): $$\begin{aligned}
\label{DEigenStates}
\hat{L}^2D_{\kappa lm}^\nu &=& l(l+1)D_{\kappa lm}^\nu\\
\hat{L}_zD_{\kappa lm}^\nu &=& mD_{\kappa lm}^\nu\\
\hat{\Pi}D_{\kappa lm}^\nu &=&(-1)^\kappa D_{\kappa lm}^\nu\\
\hat{{\cal E}}_{12}D_{\kappa lm}^\nu&=&(-1)^{l+\kappa+\nu}D_{\kappa lm}^\nu .\end{aligned}$$ The superscript $\nu$ takes on values $\nu=0$ and $1$ while the integer subscript $\kappa$ obeys $0 \le \kappa \le l$. The quantum number $\kappa$ is the absolute value of an angular momentum-like quantum number about the body-fixed axis of rotation. Even/odd $\kappa$ determines the parity eigenvalue while the combination $l+\kappa+\nu$ determines the exchange eigenvalue. This basis is especially useful since each of the four operators above commutes with the atomic Hamiltonian, $\hat{H}_0$. The spatial eigenfunction $\psi_{klms}[{\mathbf{r}}_1,{\mathbf{r}}_2]$ for total spin $s$, total angular momentum $l$, $z$-component of angular momentum $m$, and parity $k=\pm 1$ satisfies $$\begin{aligned}
\hat{L}^2\psi_{klms}[{\mathbf{r}}_1,{\mathbf{r}}_2] &=& l(l+1)\psi_{klms}[{\mathbf{r}}_1,{\mathbf{r}}_2]\\
\hat{L}_z\psi_{klms}[{\mathbf{r}}_1,{\mathbf{r}}_2] &=& m\psi_{klms}[{\mathbf{r}}_1,{\mathbf{r}}_2]\\
\hat{\Pi}\psi_{klms}[{\mathbf{r}}_1,{\mathbf{r}}_2] &=& k\psi_{klms}[{\mathbf{r}}_1,{\mathbf{r}}_2]\\
\label{PsiEigenStates}
\hat{{\cal E}}_{12}\psi_{klms}[{\mathbf{r}}_1,{\mathbf{r}}_2] &=& (-1)^s\psi_{klms}[{\mathbf{r}}_1,{\mathbf{r}}_2].\end{aligned}$$ Equations \[DEigenStates\]-\[PsiEigenStates\] imply $$\label{WaveFunction}
\psi_{klms}[{\mathbf{r_1}},{\mathbf{r_2}}]=\sum_{\nu=0}^1
\sideset{}{'} \sum_{\kappa=\nu}^l
g_{\kappa ls}^\nu[r_1,r_2,\theta_{12}]D_{\kappa lm}^\nu[\Theta,\Phi,\Psi],$$ where the prime on the sum means that $\kappa$ is restricted to even ($k=1$) or odd ($k=-1$) numbers if parity is even or odd, respectively, and $g_{\kappa ls}^\nu$ is a real function of the internal coordinates. The convenience of the Bhatia and Temkin [@BhatiaTemkin1964] coordinate choice is most evident in how one imposes total antisymmetry of the wave function. The spin singlet (triplet) must have a symmetric (antisymmetric) spatial wave function. The properties of the $D_{\kappa lm}^\nu$ functions reduce this requirement to $$\label{SpatialSymmetry}
\hat{{\cal E}}_{12}g_{\kappa ls}^\nu=(-1)^{\nu+\kappa+l+s}g_{\kappa ls}^\nu.$$ The total antisymmetry of a wave function with given $k$, $l$, $m$ and $s$ follows by imposing the above requirement under $r_1
\leftrightarrow r_2$ on each radial function for each $\nu$ and $\kappa$. Note that $(-1)^{\kappa + l + s}$ is fixed directly by the wave function’s $k$, $l$ and $s$. The same requirement applies to both singlet and triplet states up to the difference in the value of $s$.
The full six-dimensional Schrödinger equation for given $l$, $s$, even/odd parity, and any $m$ yields $l$ or $l+1$ (depending on these quantum numbers) coupled three-dimensional equations for $g_{\kappa ls}^\gamma$. The indices for $g$ satisfy $\gamma$ = $0$ or $1$ and $0 \le \kappa \le l$ with even or odd $\kappa$ for even or odd parity. The equations are $$0 = (\hat{H}_S-E)g_{\kappa ls}^\gamma+\sum_{\nu=0}^1\sum_{n=-1}^1\hat{H}_{\nu \kappa n}^\gamma g_{\kappa+2n,l,s}^\nu ,$$ where $\hat{H}_S$ is the part of the Hamiltonian operator that survives for S states. The summation enumerates couplings with $\gamma \ne \nu$ and/or different $\kappa$ as well as terms that are intrinsic to non-S-states.
Appendix \[BTAppendix\] gives the explicit forms of the operators $\hat{H}_S$ and $\hat{H}_{\nu \kappa
n}^\gamma$.
Review of the oscillator strength and dipole radiative transitions {#DipoleReview}
==================================================================
The oscillator strength quantifies the coupling between two eigenstates of $\hat{H}_0$ on account of interactions with a perturbing electromagnetic field. It is fundamental for interpreting spectra, including the strength and width of atomic transitions and the lifetimes of atomic states. Sites generating spectra of interest are ubiquitous. They include earth-based laboratories, photospheres of the Sun and distant stars, and the near vacuum between the stars where traces of interstellar matter radiate. The specific applications of the oscillator strength are correspondingly diverse. For example, in laboratories the technique of laser spectroscopy is used to measure energy splittings and frequency-dependent photoabsorption cross sections of highly excited states. Knowledge of the transition probability matrices is needed to interpret which states have been directly and indirectly generated. The transitions are driven by collisional and radiative processes, the latter given in terms of oscillator strengths. In an astrophysical context, on the other hand, observations of stellar emission require oscillator strengths for inferring chemical abundances from absorption or emission of radiation [@Smith1973; @BiemontGrevesse1977]. Oscillator strengths have widespread utility.
The practical difficulty in calculating the oscillator strength value is the accurate representation of the initial and final wave functions. Almost from the very beginning of the development of quantum mechanics helium, having but two electrons, has served as a testing ground for new theoretical approaches. Appendix \[OscStrengthTable\] presents a brief, schematic description of the rich history of such improvements in the service of oscillator strength calculations.
Following Baym [@Baym1969] and Bethe and Salpeter [@BetheSalpeter1957], the nonrelativistic Hamiltonian of a two-electron atom in the presence of an electromagnetic field (infinite-nuclear-mass approximation) is $$\label{eq:hamiltonian}
\hat{H}_{EM}=\hat{H}_0+\hat{H}_{\textrm{int}},$$ where $\hat{H}_0$ is the Hamiltonian for the isolated atom (Eq. \[Hamiltonian0\]) and $\hat{H}_{\textrm{int}}$ describes the interaction of the atom with radiation, $$\hat{H}_{\textrm{int}} = \sum_{i} \left(
-\frac{{\mathbf{p}}_i \cdot {\mathbf{A}}_i
+ {\mathbf{A}}_i \cdot {\mathbf{p}}_i}{2c} - \frac{A_i^2}{2c^2} + \varphi_i
\right),
\label{eq:hint}$$ where ${\mathbf{A}}_i$ and $\varphi_i$ are the vector and scalar potential, respectively, at the location of the $i$th electron (excluding the atomic Coulomb interactions included in $V$), and $c$ is the speed of light. If the photon number density is small then the second term, corresponding to two-photon processes, is much smaller than the first and if one adopts the transverse gauge then the third term is zero. With these assumptions the non-zero terms are the ones linear in the vector potential.
Only electric dipole-mediated transitions and the associated $f$’s are considered in this article. The length, velocity and acceleration forms for the oscillator strength [@Hibbert1975] are $$\begin{aligned}
\label{OscStrengthEq}
f_{ij}^l&=&\frac{2}{3}(E_j-E_i)|\langle j|{\mathbf{R}}|i\rangle|^2\\
f_{ij}^v&=&\frac{2}{3}\frac{1}{E_j-E_i}|\langle j|{\mathbf{P}}|i\rangle|^2\\
f_{ij}^a&=&\frac{2}{3}\frac{1}{(E_j-E_i)^3}\left|\left\langle j\left|{\mathbf{A}}
\right|i\right\rangle\right|^2 .\end{aligned}$$ Here $E_i$ and $E_j$ are the energies of the initial and final states. The two-particle operators are $$\begin{aligned}
{\mathbf{R}}& =& {\mathbf{r_1}}+{\mathbf{r_2}} \\
{\mathbf{P}}& =& {\mathbf{p_1}}+{\mathbf{p_2}} \\
{\mathbf{A}}& =& -\frac{Z{\mathbf{r_1}}}{r_1^3}-\frac{Z{\mathbf{r_2}}}{r_2^3} ,\end{aligned}$$ i.e. the position, momentum and acceleration electron operators. Appendix \[BTOscStrength\] presents explicit expressions for $f$ used in the calculations.
If the wave functions, energies, and operators were exact, all three forms would give identical results. However, in a numerical calculation the agreement may be destroyed whenever the operator commutator rule $${\mathbf{P}}=i[\hat{H}_0,{\mathbf{R}}]$$ is violated. Approximations to the operators ($\hat{H}_0$, ${\mathbf{P}}$, or ${\mathbf{R}}$) and to the initial and final eigenstates are possible sources of error. Good agreement between the three forms at a fixed resolution has sometimes been taken to be an indication of an accurate answer. Such agreement is ultimately necessary as resolution improves but the closeness of the agreement is insufficient to infer the accuracy at a fixed resolution [@SchiffEtAl1971; @Hibbert1975]. A more stringent approach involves two steps: first, for each form check that the matrix element converges with resolution or basis size and, second, that the converged answers for different forms agree.
The oscillator strengths $f_{0n}$ for transitions, $1^1$S $\to n^1$P of helium obey a family of sum rules. For integer $k$ define $$\label{SDefinition}
S(k) \equiv \sum_n|\Delta E_{0n}|^k f_{0n}.$$ where the summation is over all P states, including the continuum. Here, $\Delta E_{0n}$ is the energy difference with respect to the ground state. The rules [@DalgarnoLynn1957; @Drake1996] include $$\begin{aligned}
\label{SumRulesA}
S(-1)&=&\frac{2}{3}\langle({\mathbf{r}}_1+{\mathbf{r}}_2)^2\rangle,\\
S(0)&=&2,\\
\label{SumRulek1}S(1)&=&-\frac{4}{3}\langle \hat{H}_0-{\mathbf{p}}_1\cdot{\mathbf{p}}_2\rangle,\\
\label{SumRulesB}
S(2)&=&\frac{2\pi Z}{3}\langle \delta({\mathbf{r}}_1)+\delta({\mathbf{r}}_2)\rangle,\end{aligned}$$ where the expectation values on the right hand side refer to the ground state.
In principle, these sum rules provide consistency checks on theoretically calculated oscillator strengths. However, the explicit evaluation of $S(k)$ (Eq. \[SDefinition\]) is difficult. Multiple methods are needed to handle all the final states, which include a finite number of low energy highly correlated states, a countably infinite number of highly excited states, and an uncountably infinite number of continuum states. Ref. [@Berkowitz1997] inferred that the two sides of Eqs. \[SumRulesA\]-\[SumRulesB\] agree to about one percent based on a combination of the most reliable theoretical and/or experimental values for $f_{0n}$.
This article exemplifies the capabilities of the pseudospectral approach by evaluating the $1^1$S $\to
2^1$P oscillator strength, a physical regime in which strong electron correlations are paramount, and a set of expectation values for operator forms, some of which appear on the right hand side of the sum rules.
\[VarNDom\]Variables and Domains {#VarsAndDomains}
================================
This section details an important element of the application of the pseudospectral method: the choice of coordinates and computational domains.
To achieve exponentially fast convergence with a pseudospectral method, it is imperative that the solution be smooth. The presence of a singular point may require a special coordinate choice in the vicinity of the singularity or a different choice of effective basis. Handling multiple singularities typically requires several individual subdomains, each accommodating an individual singularity. It is useful to have a guide for choosing appropriate coordinates.
The ordinary differential equation $$\label{OneDimRegDifEq}
\left(
{\frac{d^2{}}{d{{{\rm X}}}^2}}+\frac{p_a[{{\rm X}}]}{{{\rm X}}-a}{\frac{d{}}{d{{{\rm X}}}}}+\frac{q_a[{{\rm X}}]}{({{\rm X}}-a)^2}\right)f=0$$ with $p_a[{{\rm X}}]$ and $q_a[{{\rm X}}]$ analytic at ${{\rm X}}=a$ has a regular singular point at ${{\rm X}}=a$. The basic theory of ordinary differential equations (ODE’s) [@Coddington1961] states that $f$ has at least one Frobenius-type solution about ${{\rm X}}=a$ of the form $$f[{{\rm X}}]=({{\rm X}}-a)^{t_a}\sum_{n=0}^\infty c_n ({{\rm X}}-a)^n,$$ where the coefficients $c_n$ can be derived by directly plugging into Eq. \[OneDimRegDifEq\] and $t_a$ is the larger of the two solutions to the indicial equation $$t_a(t_a-1)+p_a[a]t_a+q_a[a]=0.$$ Exponential convergence of the pseudospectral method for a differential equation of the form of Eq. \[OneDimRegDifEq\] requires $t_a$ be a non-negative integer. This must hold at each singularity $a$ in the domain (as well as all other points).[^3]
A simple example is the Schrödinger equation for a hydrogenic atom expressed in spherical coordinates $\{{{\rm X}}_1,{{\rm X}}_2,{{\rm X}}_3\}=\{r,\theta,\phi\}$. The radial part of the full wave function $R_{nl}[r]$ satisfies $$\left({\frac{d^2{}}{d{r}^2}}+\frac{2}{r}{\frac{d{}}{d{r}}}-\frac{l(l+1)-2Zr-2Er^2}{r^2}\right)R_{nl}=0.$$ A comparison with Eq. \[OneDimRegDifEq\] yields $p_0[0]=2$ and $q_0[0]=-l(l+1)$, which gives $t_0=l$, the well known result for hydrogenic wave functions. The reduction of the partial differential equation (PDE) into an ODE having non-negative integer $t_0$ tells us that spherical coordinates are a good choice for solving hydrogenic wave functions using pseudospectral methods. A bad choice would be Cartesian coordinates $\{{{\rm X}}_1,{{\rm X}}_2,{{\rm X}}_3\}=\{x,y,z\}$. The ground state has the form $$\psi \propto e^{-Z\sqrt{x^2+y^2+z^2}}.$$ This solution has a discontinuity in its first derivatives at $x=y=z=0$: $$\lim_{x,y,z\to 0^+}\frac{\partial\psi}{\partial x,y,z}\ne\lim_{x,y,z\to 0^-}\frac{\partial\psi}{\partial x,y,z}.$$ Other solutions have a discontinuity of first or higher derivatives at the same point. The pseudospectral method would not handle these well and convergence would be limited to being algebraic.
An arbitrary second order PDE may have singularities that occur on complicated hypersurfaces of different dimensionality. Deriving the analytic properties of a solution near such a surface is a daunting task. The general idea is to seek a coordinate system such that the limiting form of the PDE near the singularity looks like an ODE of the sort that pseudospectral methods are known to handle well.
For example, in a three-dimensional space, assume the singularity lies on a two-dimensional surface. First, seek a coordinate system such that the surface occurs at ${{\rm X}}_1=a$.[^4] Second, focusing on ${{\rm X}}_1$, seek coordinates so that is possible to rewrite the PDE in the form $$\label{NDimRegDifEq}
\left({\frac{\partial^2}{\partial{{{\rm X}}_1}^2}}+\frac{\hat{P}_{a}[{\mathbf{{{\rm X}}}}]}{{{\rm X}}_1-a}{\frac{\partial}{\partial{{{\rm X}}_1}}}+\frac{\hat{Q}_{a}[{\mathbf{{{\rm X}}}}]}{({{\rm X}}_1-a)^2}\right)f=0$$ where $\hat{P}_{a}$ and $\hat{Q}_{a}$ are linear second order differential operators that do not include derivatives with respect to ${{\rm X}}_1$. Finally, seek coordinates such that $\hat{P}_{a}$ and $\hat{Q}_{a}$ are analytic with respect to ${{\rm X}}_1$ at $a$.
Unfortunately, even if one succeeds in finding such a coordinate system, the theorem of ODEs does not generalize to PDEs, i.e. there is no guarantee that $f$ is analytic near $a$. A celebrated example is exactly the problem of concern here, i.e. the Schrödinger equation for two-electron atoms. Three coordinates are needed to describe the S state. In hyperspherical coordinates ($\{X_1, \dots\}=\{\rho, \dots\}$ where $\rho = \sqrt{r_1^2 + r_2^2}$), Schrödinger’s equation matches the form of Eq. \[NDimRegDifEq\] for ${{\rm X}}_1=\rho$ and $a = 0$. This is the triple coalescence point, a point singularity in the three-dimensional subspace spanned by the coordinates $r_1$, $r_2$, and $r_{12}$. The electron-nucleus and electron-electron singularities (two-body coalescence points) are one-dimensional lines in this subspace that meet at $\rho=0$. Bartlett [@Bartlett1937] proved that no wave function of the form $$\psi = \sum_{n=0}^\infty A_n \rho^n,$$ where $A_n$ is an analytic function of the remaining variables will satisfy the PDE. Fock’s form for the solution [@Fock1954; @Fock1958] is $$\psi = \sum_{n=0}^\infty\sum_{m=0}^{\lfloor n/2 \rfloor}B_{nm}\rho^n(\log\rho)^m,$$ where $B_{nm}$ is an analytic function of the remaining variables. The presence of the $\log \rho$ terms in the wave function is an important qualitative distinction between a solution having two- and three-body coalescence points.
Some properties of the solution near $\rho=0$ have been reviewed in our previous article [@GrabowskiChernoff2010]. For example, Myers [*et al.*]{} [@MyersEtAl1991] showed that the logarithmic terms allow the local energy $(\hat{H}\psi)/\psi$ near $\rho=0$ to be continuous. Despite this property, they have only a slight effect on the convergence of variational energies [@Schwartz2006]. By many measures of error the triple coalescence point does not affect pseudospectral calculations until very high resolutions [@GrabowskiChernoff2010].
As a point of principle, however, no simple coordinate choice can hide the problems that occur at the triple coalescence point, and no special method for handling this singularity is given here. Elsewhere ($\rho \ne 0$) our rule of thumb is the following: coordinates are selected so that the singularity may be described by $X_i=a$ with $\hat{P}_{a}$ and $\hat{Q}_{a}$ satisfying $$\begin{aligned}
\label{PCondition}
\hat{P}_{a}&=&\sum_{n=0}^\infty ({{\rm X}}_i-a)^n \hat{p}_{an}\\
\label{QCondition}
\hat{Q}_{a}&=&\sum_{n=0}^\infty ({{\rm X}}_i-a)^n \hat{q}_{an},\end{aligned}$$ in a neighborhood about ${{\rm X}}_i = a$. Here, $\hat{p}_{an}$ and $\hat{q}_{an}$ are linear differential operators not containing ${{\rm X}}_i$ or its derivatives.
The singularities of the Hamiltonian, given in detail in Appendix \[BTAppendix\], are of two types. The physical singularities at $r_1$, $r_2$, and $r_{12}=0$ were explored in Ref. [@GrabowskiChernoff2010]. One of the essential virtues of hyperspherical coordinates is that $\rho \ne 0$ implies these coalescences have separate neighborhoods. Therefore, the prescription is to seek separate coordinates satisfying eqs. \[PCondition\] and \[QCondition\] in the vicinity of each singularity.
There are also coordinate singularities at $\theta_{12}=0$ and $\pi$ which correspond to collinear arrangements of the two electrons and nucleus. These singularities were completely absent in our previous treatment of S states [@GrabowskiChernoff2010] where $C=-\cos\theta_{12}$ and $B=-\cos\beta_{12}$ ($\beta_{12}$ is defined below) were the third coordinates in different subdomains. Now, to accommodate the singularities’ presence in the Hamiltonian for general angular momentum make the slight change to use $\theta_{12}$ and $\beta_{12}$ instead.
Starting with the internal coordinates $r_1$, $r_2$ and $\theta_{12}$ one defines $\rho$, $\phi$, $\zeta$, and $x$ by $$\begin{aligned}
r_1 & = & \rho \cos{\phi}\\
r_2 & = & \rho \sin{\phi}\\
r_{12}&=& \rho \sqrt{2}\sin\zeta\\
\sqrt{2}\sin{\zeta} & = & \sqrt{1-\cos\theta_{12}\sin{2\phi}}\\
\cos\beta_{12} & = &-\frac{\cos{2\phi}}{\sqrt{1-\cos^2\theta_{12}\sin^2{2\phi}}}\\
x & = &\frac{1-\rho}{1+\rho}.\end{aligned}$$ The full ranges of these variables are $$\label{varRanges}
\begin{array}{c}
0\le r_1,r_2,\rho < \infty\\
|r_1-r_2|\le r_{12} \le r_1+r_2\\
0\le \theta_{12},\beta_{12} \le \pi\\
0\le \phi,\zeta\le \pi/2 \\
-1\le x\le 1.
\end{array}$$ The purpose of coordinate $x$ is to map the semi-infinite range of $\rho$ to a finite interval.
Eqs. \[PCondition\] and \[QCondition\] are satisfied by selecting $\{X_1,X_2,X_3\}=\{x,\phi,\theta_{12}\}$ or $\{x,\zeta,\beta_{12}\}$ in three separate domains $$\begin{array}{lccc}
D_1:&-1\le x\le 1, & 0\le\phi\le\frac{1}{2}, & -1\le \cos\theta_{12}\le 1\\
D_2:&-1\le x\le 1, & \frac{1}{2}\le\phi\le\frac{\pi}{4}, & -1\le\cos\theta_{12}\le\frac{2}{3}\\
D_3:&-1\le x\le 1, & 0\le\zeta\le\frac{1}{2}, & -1\le \cos\beta_{12}\le 0,
\end{array}$$ spanning only half the space defined by the inequalities (\[varRanges\]) due to the symmetry in the Hamiltonian about $r_1=r_2$. Fig. \[GridPoints\] illustrates the layout of the three domains at fixed $\rho$. The coordinate systems in domains $D_1$ and $D_3$ were developed to handle the electron-proton and electron-electron singularities, respectively. The choice of coordinates in domain $D_2$ was more arbitrary, and for simplicity was chosen to be the same as in domain $D_1$. This particular choice allows for no overlap between domains $D_1$ and $D_2$ and makes the symmetry condition (Eq. \[SpatialSymmetry\]) at $r_1=r_2$, $\phi=\pi/4$, or $\beta_{12}=\pi/2$ easy to apply. The remaining electron-nucleus singularity, $r_1=0$, is implicitly accommodated by the spatial symmetry of the wave function. The three domains must jointly describe the full rectangle but the specific choice for edges at $\phi=\zeta=1/2$ is arbitrary.
![\[GridPoints\] (Color online). This is the arrangement of grid points of the three domains at a constant value of $\rho$ in $\phi$ and $\theta_{12}$ coordinates for $n=20$. Note that the point density becomes larger at the boundary of each subdomain and that no grid points sit on the Coulomb singularities. The blue circles, red crosses, and green pluses belong to domains $D_1$, $D_2$, and $D_3$, respectively. $D_1$ and $D_2$ are rectangular domains, while $D_3$ has the curved boundary in $\phi$, $\theta_{12}$ coordinates but is rectangular in $\zeta$, $\beta_{12}$ coordinates. The electron-proton singularity occurs on the left side (solid line at $\phi=0$). The entire line corresponds to one physical point. The electron-electron singularity occurs at the lower right hand corner (solid disk at $\phi=\pi/4, \theta_{12}=0$). A line of symmetry falls on the right side (dashed line at $\phi=\pi/4$ where $r_1=r_2$).](D3Grid.eps){width="\linewidth"}
Boundary conditions {#BoundaryConditions}
===================
Internal boundary conditions
----------------------------
It is necessary to ensure continuity of the wave function and its normal derivative at internal boundaries. There are two ways in which the subdomains can touch: they can overlap or they can barely touch. For clarity, consider a one-dimensional problem with two domains. Let the first domain be domain $1$ and the second be domain $2$ with extrema ${{\rm X}}_{1,\textrm{min}}<{{\rm X}}_{2,\textrm{min}}\le {{\rm X}}_{1,\textrm{max}}<{{\rm X}}_{2,\textrm{max}}$, where the $1$ and $2$ refer to domain number. The first case corresponds to ${{\rm X}}_{2,\textrm{min}}<{{\rm X}}_{1,\textrm{max}}$ and the second to ${{\rm X}}_{2,\textrm{min}}={{\rm X}}_{1,\textrm{max}}\equiv {{\rm X}}_*$. For both cases, exactly two conditions are needed to make the wave function and its derivative continuous. The simplest choice for the first case is $$\begin{aligned}
\psi_1[{{\rm X}}_{1,\textrm{max}}]&=&\psi_2[{{\rm X}}_{1,\textrm{max}}]\\
\psi_1[{{\rm X}}_{2,\textrm{min}}]&=&\psi_2[{{\rm X}}_{2,\textrm{min}}],\end{aligned}$$ and for the second case is $$\begin{aligned}
\psi_1[{{\rm X}}_{*}]&=&\psi_2[{{\rm X}}_{*}]\\
\frac{d}{d{{\rm X}}}\psi_1[{{\rm X}}_{*}]&=&\frac{d}{d{{\rm X}}}\psi_2[{{\rm X}}_{*}].\end{aligned}$$
For multi-dimensional grids, the situation is analogous. The conditions are applied on surfaces of overlap. In this case the derivatives are surface normal derivatives or any derivative not parallel to the boundary surface. On a discrete grid, a finite number of conditions are given which, in the limit of an infinitely fine mesh, would cover the entire surface. Additional discussion and illustrations of the technique are in Ref. [@GrabowskiChernoff2010].
The symmetry condition
----------------------
The Hamiltonian (see appendix \[BTAppendix\]) is symmetric with respect to particle exchange ($r_1\leftrightarrow r_2$). Therefore, there are two types of eigenstates: those with symmetric spatial wave functions (singlets) and those with antisymmetric spatial wave functions (triplets). The radial wave functions $g_{\kappa ls}^\nu$ satisfying the appropriate symmetry must obey Eq. \[SpatialSymmetry\]. More explicitly $$0=\left\{\begin{array}{cc}
\left.{\frac{\partial{g_{\kappa l s}^\nu}}{\partial{\phi}}}\right|_{\phi=\pi/4}=\left.{\frac{\partial{g_{\kappa l s}^\nu}}{\partial{\beta_{12}}}}\right|_{\beta_{12}=\pi/2}
& \textrm{ if } \xi \textrm{ is even}\\
\left.g_{\kappa l s}^\nu\right|_{\phi=\pi/4}=\left.g_{\kappa l s}^\nu\right|_{\beta_{12}=\pi/2}
& \textrm{ if } \xi \textrm{ is odd}
\end{array}\right.,$$ where $\xi=\nu+\kappa+l+s$.
Energy and oscillator strength results {#EandOscResults}
======================================
This article generalizes the pseudospectral methods previously developed for S states to the general angular momentum case, calculates oscillator strengths for transitions, and tests how different measures of wave function errors vary with resolution.
![\[Energy\] (Color online). The logarithm base 10 of the energy error ($\Delta Q$) of both the lowest energy S state and P state of helium. The dark blue circles are for the $1^1$S state and the light red crosses for the $2^1$P state with dashed blue and dotted red fits, respectively (see Tab. \[ConvTabEOS\]).](Energy.eps){width="\linewidth"}
The most widely quoted number to ascertain convergence is the energy which gives a global measure of accuracy. Figure \[Energy\] shows the energy errors for the 1$^1$S and 2$^1$P states of helium. Here and throughout the results sections the high precision values of Drake [@Drake1996] are taken to be exact. The energy error for both states decreases exponentially with resolution. Convergence for the S state is similar to that reported in Ref. [@GrabowskiChernoff2010] with slight differences related to a different choice of coordinates. The current calculation extends to basis size $n=23$ for S states and $n=20$ for P states instead of $n=14$ for only S states in Ref. [@GrabowskiChernoff2010].
A common feature of the energy convergence and all other convergence plots in this article is non-monotonic convergence. This method is not variational, so there is no reason to expect monotonic convergence. Calculated quantities can fall above or below their actual value, with error quasi-randomly determined by the exact grid point locations. The jumps decrease in magnitude as the resolution is increased.
![\[OscStrength\] (Color online). The logarithm base 10 of the error ($\Delta Q$) in the oscillator strength of the $1^1$S $\to 2^1$P transition of helium. The dark blue circles are for the length form, the light red crosses are for the velocity form, and the green pluses for the acceleration form with dashed blue, dotted red, and dot-dashed green fits, respectively (see Tab. \[ConvTabEOS\]).](OscStrength.eps){width="\linewidth"}
As described in Sec. \[DipoleReview\], there are three commonly used forms for the oscillator strength. The length, velocity, and acceleration forms depend most strongly on the value of the wave function at positions in configuration space corresponding to large, medium, and small separations. Sometimes the relative errors are used to infer where the wave function is more or less accurate. It has been observed that for most variational calculations, the acceleration form tends to be much less accurate than the other two forms, suggesting errors in the wave function at small separation that have little effect on the variational energy. The length and velocity forms give results of roughly comparable accuracy.
The oscillator strength of the $1^1$S $\to 2^1$P transition was calculated using all three forms and Fig. \[OscStrength\] displays the errors. Here, all three forms give roughly the same results. At most resolutions the points lie nearly on top of one another and their fits are indistinguishable, indicating the wave function errors for small, medium, and large separations have roughly equal contributions to the numerically calculated oscillator strength. This may be due to the pseudospectral method’s equal treatment of all parts of configuration space.
It should be noted that the value used as the exact value [@Drake1996] is given to seven decimal places. Consequently, the errors inferred for the highest resolution calculations in Fig. \[OscStrength\] are not too precise. There is little practical need for additional digits since a host of other effects including finite nuclear mass, relativistic, and quadrupole corrections would confound any hypothetical, experimental measurement of the oscillator strength to such high precision even if a perfect measurement could be made. Actual experiments struggle to obtain two percent precision [@ZitnikEtAl2003], an error larger than these effects.
![\[OSStat\] (Color online). The logarithm base 10 of the error ($\Delta Q$) in the average of the length, velocity, and acceleration forms of the oscillator strength (dark blue circles) and their standard deviation (light red crosses) for the $1^1$S $\to 2^1$P transition of helium, with dashed blue and dotted red fits, respectively (see Tab. \[ConvTabEOS\]).](OSStat.eps){width="\linewidth"}
As pointed out by Schiff [*et al.*]{} [@SchiffEtAl1971] and reviewed by Hibbert [@Hibbert1975], the assumption that using the differences between the oscillator strength values from the different forms as a measure of the accuracy is not valid. Agreement is necessary but not sufficient. They suggest comparing calculated and extrapolated values. This latter procedure is not straightforward for a pseudospectral method with non-monotonic convergence. We present a similar suitable check. Fig. \[OSStat\] shows the average and standard deviation of the error for the three forms as a function of resolution. The standard deviation is about an order of magnitude (with a large scatter about that factor of ten) less than the average error at low and moderate resolutions but the trend lines suggest that the standard deviation may be approaching the average at the higher resolutions. A possible explanation is that the calculation at the highest resolutions is starting to become sensitive to the wave function truncation (see appendix \[MatEigSol\]). This destroys the expected equality between the forms and each form converges to its own incorrect asymptotic value. The individual errors and the standard deviation become comparable. So at $n=20$, we assume the standard deviation and total error are equal and get a value for the oscillator strength of $0.27616499(27)$ which compares favorably to Drake’s $0.2761647$ [@Drake1996].
$Q$ Figure $A$ $\beta$
---------------- ----------------- --------------------- ---------
$E(1^1$S$)$ \[Energy\] $2.5\times 10^{-9}$ 0.40
$E(2^1$P$)$ \[Energy\] $5.2\times 10^{-9}$ 0.42
$f_{12}^l$ \[OscStrength\] $8.4\times 10^{-8}$ 0.40
$f_{12}^v$ \[OscStrength\] $9.2\times 10^{-8}$ 0.39
$f_{12}^a$ \[OscStrength\] $8.6\times 10^{-8}$ 0.40
$f_{12}^{avg}$ \[OSStat\] $8.7\times 10^{-8}$ 0.40
$f_{12}^{SD}$ \[OSStat\] $2.2\times 10^{-8}$ 0.34
: \[ConvTabEOS\] The fit parameters to all the convergence plots of quantities $Q$ in this section.
All convergence data were fit to functions of the form $\Delta Q = A\times 10^{-\beta(n-20)}$ using the same procedure as in Ref. [@GrabowskiChernoff2010]. Because of uncertainty in the errors for the largest resolutions ($n=19$ and $n=20$) these points were not used in the fits of $f_{12}^l$, $f_{12}^v$, $f_{12}^a$, and $f_{12}^{avg}$. The $\beta$ parameter, which corresponds to the slope of the fits in the convergence graphs is roughly the same for all fits, with the exception of the standard deviation of the oscillator strength forms. This behavior is consistent with our discussion of errors in the previous paragraph.
Corrections to the Hamiltonian {#CorrectionsHamiltonian}
==============================
Two small parameters appear in the full physical Hamiltonian: the ratio of the reduced mass of the electron-nucleus pair to the nuclear mass, $\mu/M=1.37074563559(58)\times 10^{-4}$ [@MohrEtAl2008a; @MohrEtAl2008b] (for ${}^4$He) and the fine structure constant $\alpha=7.2973525376(50)\times 10^{-3}$ [@MohrEtAl2008a; @MohrEtAl2008b]. Here, the lowest order corrections in $\mu/M$ and $\alpha$ are considered. For very high-precision work, one needs the perturbative corrections in powers of each small quantity.
Finite nuclear mass correction {#FiniteNuclearMass}
------------------------------
The nonrelativistic ($\alpha^0$) Hamiltonian for two-electron atoms is $$\hat{H}_{\textrm{nr}}=\hat{H}_0+\hat{H}_{\textrm{cm}}+\hat{H}_{\textrm{mp}},$$ where $\hat{H}_0$ is the fixed-nucleus approximation to the Hamiltonian with the electron mass set to $\mu$, $\hat{H}_{\textrm{cm}}$ is the kinetic energy of the center of mass, and $\hat{H}_{\textrm{mp}}$ is the mass polarization term: $$\begin{aligned}
\label{MassHamiltonian}
\hat{H}_0&=&\frac{1}{2}(p_1^2+p_2^2)+\hat{V}\\
\hat{H}_{\textrm{cm}}&=&\frac{1}{2(M+2m_e)}{p}_{\textrm{cm}}^2\\
\label{HMassPolarization}\hat{H}_{\textrm{mp}}&=&\frac{1}{M}{\mathbf{p}}_1\cdot{\mathbf{p}}_2,\end{aligned}$$ where $\hat{V}$ is the potential energy operator, $m_e$ is the electron mass, ${\mathbf{p}}_{\textrm{cm}}$ is the momentum operator of the center of mass, and reduced mass atomic units ($\mu=1$) are being used. The second term is removed in center-of-mass coordinates and the last term provides the dominant nontrivial correction for finite nuclear mass (the trivial one being the scaling of the energy by $m_e/\mu$).
Relativistic corrections {#RelativisticCorrections}
------------------------
The Schrödinger equation is a nonrelativistic approximation to the true equation of motion. The lowest order relativistic corrections enter at order $(\alpha^2)$, as summarized in Ref. [@Fischer1996] and repeated here. Note, all references in this article to orders in $\alpha$ are in Rydbergs. The Breit-Pauli Hamiltonian encapsulates the correction $$\hat{H}_{\textrm{BP}}=\hat{H}_{\textrm{nr}}+\hat{H}_{\textrm{rel}},$$ where $\hat{H}_{\textrm{nr}}$ is the usual nonrelativistic Hamiltonian used in Schrödinger’s equation and $\hat{H}_{\textrm{rel}}$ is the lowest order relativistic correction. The latter can be further divided into non-fine-structure (NFS) and fine-structure (FS) contributions: $$\begin{aligned}
\hat{H}_{\textrm{NFS}}&=&\hat{H}_{\textrm{mass}}+\hat{H}_{\textrm{D}}+\hat{H}_{\textrm{SSC}}+\hat{H}_{\textrm{OO}}\\
\hat{H}_{\textrm{FS}}&=&\hat{H}_{\textrm{SO}}+\hat{H}_{\textrm{SOO}}+\hat{H}_{\textrm{SS}}.\end{aligned}$$ The separate contributions to the Hamiltonian are the mass-velocity (mass), two-body Darwin (D), spin-spin contact (SSC), orbit-orbit (OO), spin-orbit (SO), spin-other-orbit (SOO), and the spin-spin (SS) terms. These are explicitly given by $$\begin{aligned}
\label{HMassVelocity}\hat{H}_{\textrm{mass}}&=&-\frac{\alpha^2}{8}\sum_ip_i^4\\
\label{HDarwin}\hat{H}_{\textrm{D}}&=&-\frac{\alpha^2Z}{8}\sum_i\nabla_i^2r_i^{-1}+\frac{\alpha^2}{4}\sum_{i<j}\nabla_i^2r_{ij}^{-1}\\
\label{HSpinSpinContact}\hat{H}_{\textrm{SSC}}&=&-\frac{8\pi\alpha^2}{3}({\mathbf{s}}_1\cdot{\mathbf{s}}_2)\delta({\mathbf{r}}_{12})\\
\label{HOrbitOrbit}\hat{H}_{\textrm{OO}}&=&-\frac{\alpha^2}{2}\left(\frac{{\mathbf{p}}_1\cdot{\mathbf{p}}_2}{r_{12}}
+\frac{{\mathbf{r}}_{12}({\mathbf{r}}_{12}\cdot {\mathbf{p}}_1)\cdot{\mathbf{p}}_2}{r_{12}^3}\right)\\
\hat{H}_{\textrm{SO}}&=&\frac{\alpha^2Z}{2}\sum_i\frac{{\mathbf{\hat{l}}}_i\cdot{\mathbf{\hat{s}}}_i}{r_i^3}\\
\hat{H}_{\textrm{SOO}}&=&-\frac{\alpha^2}{2}\sum_{i\ne j}\left(\frac{{\mathbf{r}}_{ij}}{r_{ij}^3}\times{\mathbf{p}}_i\right)
\cdot({\mathbf{s}}_i+2{\mathbf{s}}_j)\\
\hat{H}_{\textrm{SS}}&=&\frac{\alpha^2}{r_{12}^3}\left({\mathbf{s}}_1\cdot {\mathbf{s}}_2
-\frac{3}{r_{12}^2}({\mathbf{s}}_1\cdot {\mathbf{r}}_{12})({\mathbf{s}}_2\cdot {\mathbf{r}}_{12})\right),\end{aligned}$$ where $i$ and $j$ can be 1 or 2, ${\mathbf{p}}_{i}$ and ${\mathbf{r}}_i$ are the momentum and position of the $i$th electron with respect to the nucleus, respectively, ${\mathbf{r}}_{12}$ is the vector pointing from the first electron to the second, and ${\mathbf{\hat{s}}}_i$ and ${\mathbf{\hat{l}}}_i$ are the one-electron spin and angular momentum operators of the $i$th electron, respectively. The last three Hamiltonian terms are zero for ${}^1$S states due to symmetry considerations.
There are many higher order terms (see Refs. [@Drake1999; @DrakeMorton2007; @DrakeYan2008; @PachuckiYerokhin2011]) but these are not considered here.
Mass polarization and relativistic correction calculations {#MassPolRelCorrCalcs}
==========================================================
The mass polarization and low order relativistic corrections to the nonrelativistic Hamiltonian have been known for some time [@BetheSalpeter1957]. The main challenge in calculating these terms is finding adequate unperturbed wave functions. Early calculations [@Kinoshita1957; @KabirSalpeter1957; @Sucher1958; @ArakiEtAl1959] were critical for comparing experimental and theoretical energies, confirming that Schrödinger’s equation is correct in the nonrelativistic limit for helium.
The development of computers enabled Pekeris and coworkers [@Pekeris1958; @Pekeris1959; @SchiffEtAl1965] and others [@Schwartz1961; @Schwartz1964; @Hambro1972; @LewisSerafino1978; @DavisChung1982] to reach theoretical uncertainties in the energy of about $10^{-2}$ cm$^{-1}$. Such precision and the resulting precision in the wave function allowed Lewis and Serafino [@LewisSerafino1978] to calculate the fine structure constant from experimental measurements of the $2^3$P splitting. They obtained $\alpha^{-1}=137.03608(13)$ with an estimated uncertainty only surpassed at the time by the measurements of the electron anomalous magnetic moment $(g-2)$ (by a factor of two) and the ac Josephson experiments (by a factor of four).
Drake and collaborators [@Drake1987; @DrakeYan1992; @YanDrake1995; @DrakeGoldman1999; @Drake1999; @Drake2002; @Drake2004] and Pachucki and collaborators [@Pachucki1998; @PachuckiSapirstein2000; @PachuckiSapirstein2002; @PachuckiSapirstein2003; @Pachucki2006; @Pachucki2006b; @Pachucki2006c; @PachuckiYerokhin2009; @PachuckiYerokhin2010; @PachuckiYerokhin2011; @PachuckiYerokhin2011b] have pushed relativistic corrections for regular helium up to order $\alpha^5$ and beyond using a Hylleraas [@Hylleraas1929] type basis. Drake [@Drake2002] matched theoretical and observed energy differences in the $J=0,1$ splitting of the $2^3$P state and determined $\alpha^{-1}=137.0359893(23)$. Drake cited a difference with the $g-2$ result $137.0359996(8)$ but agreement with the ac Josephson result $137.0359872(43)$ [@Drake2002]. However, a similar calculation of his using the observed $J=1,2$ splitting gives an unreasonable value [@Drake2002]. Pachucki and collaborators have resolved the issue by finding errors in $\alpha^5$ terms and by increasing the error estimate due to $\alpha^6$ terms. Their most recent determination is $\alpha^{-1}=137.03599955(64)(4)(368)$, where the first error is experimental, the second numerical, and the third is their estimated error from higher order terms [@PachuckiYerokhin2011]. This value agrees with the latest $g-2$ results but is not as precise [@PachuckiYerokhin2011].
An alternative approach is to use an even simpler basis, with surprisingly accurate results. Korobov and collaborators have used an exponential basis (see Refs. [@ThakkarSmith1977; @FrolovSmith1995]) to calculate very precise helium [@KorobovKorobov1999; @Korobov2000; @KorobovYelkhovsky2001; @Korobov2002; @Korobov2002b; @Korobov2004] (up to order $\alpha^4$) and anti-protonic helium [@KorobovEtAl1999; @KorobovBakalov2001; @Korobov2003; @Korobov2006; @KorobovTsogbayar2007; @Korobov2008; @KorobovZhong2009; @KorobovZhong2009b] (up to order $\alpha^5$) electronic energies. The latter calculations have been used for the CODATA06 [@MohrEtAl2008a; @MohrEtAl2008b] recommended value of the electron-to-(anti)proton mass ratio.
Expectation values {#ExpectationValues}
==================
![\[rSq\] (Color online). The logarithm base 10 of the error ($\Delta Q$) in the expectation values of operators that scale as $\rho^2$ for helium. The dark blue circles are for $\langle r_1^2\rangle$, the light red crosses are for $\langle r_{12}^2 \rangle$, and the green pluses are for $\langle r_1 r_2\cos\theta_{12}\rangle$ with dashed blue, dotted red, and dot-dashed green fits, respectively (see Tab. \[ConvTab\]).](rSq.eps){width="\linewidth"}
The aim of this section is to test the pseudospectral method’s ability to represent the wave function in different parts of configuration space and to compare the convergence rates of the errors with that of the energies and oscillator strengths. For a representative set of calculations consider the expectation values of the operators needed for leading order relativistic (Sec. \[RelativisticCorrections\]) and finite nuclear mass (Sec. \[FiniteNuclearMass\]) corrections, for the oscillator strength sum rules (Eqs. \[SumRulesA\]-\[SumRulesB\]), interparticle distances, $\langle \hat{V} \rangle$, and $\langle \hat{V}^2 \rangle$. These expectation values test different parts of the wave function as well as different types of operators. They are organized by the weighting of the wave function and used to draw inferences about local errors.
![\[r\] (Color online). The logarithm base 10 of the error ($\Delta Q$) in the expectation values of operators that scale as $\rho$ for helium. The dark blue circles are for $\langle r_1\rangle$ and the light red crosses are for $\langle r_{12}\rangle$ with dashed blue and dotted red fits, respectively (see Tab. \[ConvTab\]).](r.eps){width="\linewidth"}
Figure \[rSq\] displays results for expectation values related to sum rule $S(-1)$ (Eq. \[SumRulesA\]), i.e. quantities scaling like $\rho^2$. These calculations are somewhat more sensitive to the wave function at large separation than, say, the normalization integral. In addition, they focus on parts of coordinate space which have low resolution compared to the coverage near the singularities. High accuracy is found for all three cases.
Figure \[r\] displays results for expectation values of operators scaling like $\rho$ similar to the length form of the oscillator strength. Higher accuracy is obtained here than for the oscillator strength at equivalent resolutions. This can be explained by the smaller length scale set by the higher energy of the P state, which enters only into the oscillator strength calculations. So a greater resolution is needed for the same accuracy.
![\[rIn\] (Color online). The logarithm base 10 of the error ($\Delta Q$) in the expectation values of operators that scale as $1/\rho$ for helium. The dark blue circles are for $\langle 1/r_1\rangle$ and the light red crosses are for $\langle 1/r_{12} \rangle$ with dashed blue and dotted red fits, respectively (see Tab. \[ConvTab\]).](rIn.eps){width="\linewidth"}
Figure \[rIn\] displays results for expectation values related to the potential energy of charged particles, i.e. quantities scaling like $1/\rho$. This probes the treatment of the singularities. The high degree of accuracy is evidence that these singularities have been treated correctly.
![\[rSqIn\] (Color online). The logarithm base 10 of the error ($\Delta Q$) in the expectation values of operators that scale as $1/\rho^2$ for helium. The dark blue circles are for $\langle 1/r_1^2\rangle$, the light red crosses are for $\langle 1/r_{12}^2 \rangle$, the green pluses are for $\langle 1/r_1 r_2\rangle$, and the black stars are for $\langle 1/r_1r_{12}\rangle$ with dashed blue, dotted red, dot-dashed green, and solid black fits, respectively (see Tab. \[ConvTab\]).](rSqIn.eps){width="\linewidth"}
Figure \[rSqIn\] displays results for expectation values related to the square of the potential energy, i.e. quantities scaling like $1/\rho^2$. These operators emphasize the singularities even further. One may expect that at a high enough inverse power of $\rho$ that the effect of the Fock logarithm become important and slow down convergence, but no evidence of that effect is apparent.
Even the expectation values of delta functions, related to sum rule $S(2)$ (Eq. \[SumRulesB\]), the Darwin term $\hat{H}_\textrm{D}$ (Eq. \[HDarwin\]), and the spin-spin contact term $\hat{H}_\textrm{SSC}$ (Eq. \[HSpinSpinContact\]), which are most sensitive to the Kato cusp conditions [@Kato1957] have the same convergence properties (See Fig. \[Delta\]). This provides evidence that our choices of coordinates allowed the pseudospectral method to deduce and represent the solution in the vicinity of a cusp. It also shows that if one can handle the non-analyticities of the matrix element by hand, as is possible for delta functions (see appendix \[SecQuadrature\]), one can still have exponentially fast convergence.
![\[Delta\] (Color online). The logarithm base 10 of the error ($\Delta Q$) in the expectation values of delta function operators for helium. The dark blue circles are for $\langle\delta(r_1)\rangle$ and the light red crosses are for $\langle\delta(r_{12})\rangle$ with dashed blue and dotted red fits, respectively (see Tab. \[ConvTab\]).](DeltaFuncs.eps){width="\linewidth"}
The error in the mass polarization $\hat{H}_{\textrm{mp}}$ (Eq. \[HMassPolarization\]), used for the finite-nuclear mass correction and the calculation of the sum rule $S(1)$ (Eq. \[SumRulek1\]), and the orbit-orbit terms $\hat{H}_{\textrm{OO}}$ (Eq. \[HOrbitOrbit\]), i.e. quadratic momentum contributions, are shown in Fig. \[MPBP\]. Calculations of derivatives (needed to form the appropriate operators) appear to be just as accurate as the function values, even when they are most strongly weighted close to the electron-electron cusp, as is the case for the orbit-orbit interaction.
The exponential rate of convergence and the magnitude of the errors are roughly the same in all the calculations of expectation values in Figs. \[rSq\]-\[MPBP\]. This is reflected in the fits (see Tab. \[ConvTab\]).
![\[MPBP\] (Color online). The logarithm base 10 of the error ($\Delta Q$) in the expectation values of the mass polarization and the orbit-orbit interaction operators for helium. The dark blue circles are for $\langle{\mathbf{p}}_1\cdot{\mathbf{p}}_2\rangle$ and the light red crosses are for $\langle\hat{H}_{\textrm{OO}}\rangle/\alpha^2$ with dashed blue and dotted red fits, respectively (see Tab. \[ConvTab\]).](MPBP.eps){width="\linewidth"}
These errors decrease until they reach roughly the level of error produced by truncating the wave function (see Sec. \[MatrixMethods\]) at the highest resolutions. The only easily discernible differences are at low resolution for which the representation of the wave function at large $\rho$ is certainly poor. It is unsurprising that the expectation values that scale as $\rho^2$ and $\rho$ have larger errors at low resolution due to the scarcity of points in the asymptotic tail of the wave function.
$Q$ Figure $A$ $\beta$
---------------------------------------------------- ----------- ---------------------- ---------
$\langle r_1^2\rangle$ \[rSq\] $3.3\times 10^{-11}$ 0.54
$\langle r_{12}^2\rangle$ \[rSq\] $1.2\times 10^{-10}$ 0.53
$\langle {\mathbf{r_1}}\cdot{\mathbf{r_2}}\rangle$ \[rSq\] $4.9\times 10^{-11}$ 0.47
$\langle r_1\rangle$ \[r\] $1.2\times 10^{-11}$ 0.48
$\langle r_{12}\rangle$ \[r\] $7.5\times 10^{-11}$ 0.46
$\langle 1/r_1\rangle$ \[rIn\] $1.1\times 10^{-10}$ 0.37
$\langle 1/r_{12}\rangle$ \[rIn\] $1.2\times 10^{-10}$ 0.37
$\langle 1/r_1^2\rangle$ \[rSqIn\] $7.1\times 10^{-10}$ 0.36
$\langle 1/r_{12}^2\rangle$ \[rSqIn\] $3.1\times 10^{-10}$ 0.38
$\langle 1/r_1r_2\rangle$ \[rSqIn\] $3.9\times 10^{-10}$ 0.37
$\langle 1/r_1r_{12}\rangle$ \[rSqIn\] $2.6\times 10^{-10}$ 0.37
$\langle\delta(r_1)\rangle$ \[Delta\] $2.4\times 10^{-10}$ 0.36
$\langle\delta(r_{12})\rangle$ \[Delta\] $6.5\times 10^{-11}$ 0.36
$\langle{\mathbf{p}}_1\cdot{\mathbf{p}}_2\rangle$ \[MPBP\] $2.4\times 10^{-10}$ 0.39
$\langle\hat{H}_{\textrm{OO}}\rangle/\alpha^2$ \[MPBP\] $4.3\times 10^{-10}$ 0.38
: \[ConvTab\] The fit parameters to all the convergence plots of quantities $Q$ in this section.
All convergence data were fit to functions of the form $A\times 10^{-\beta(n-23)}$ using the same procedure as in Ref. [@GrabowskiChernoff2010]. The fit parameters are shown in Tab. \[ConvTab\]. The most striking feature is how similar the magnitudes of the errors are at $n=23$. Also, the exponential parameter $\beta$ is roughly the same for all expectation values and the energies and oscillator strengths (see Tab. \[ConvTabEOS\]) with the differences already discussed. Indeed, as one increases resolution one increases the accuracy of all expectation values or oscillator strengths by roughly the same amount.
The contributions to the total energy of the ground state of ${}^4$He are summarized in Tab. \[EnergyTotal\]. The values from both this work and Drake’s [@Drake1996] are given. For a wave function with a much lower precision in its eigenvalue (nine decimal places compared to fifteen), nearly the same precision is obtained for the corrections to this eigenvalue.
Energy This Work[^5] Drake [@Drake1996]
--------------------------------------------- -------------------------------- -----------------------------------
$\langle \hat{H}_0 \rangle$ -2.9037243764(8) -2.9037243770341195
$\langle \hat{H}_\textrm{mass} \rangle$[^6] $-7.2006570459(3)\times 10^{-4}$
$\langle \hat{H}_\textrm{OO} \rangle $ $-7.4069807(1)\times 10^{-6}$ $-7.40698061439(5)\times 10^{-6}$
$\langle \hat{H}_\textrm{D}\rangle $ $5.879572027(5)\times 10^{-4}$ $5.8795720265(4)\times 10^{-4}$
$\langle \hat{H}_\textrm{SSC}\rangle $ $3.55818982(1)\times 10^{-5}$ $3.558189840(7)\times 10^{-5}$
$\langle \hat{H}_\textrm{mp}\rangle $ $2.18103579(2)\times 10^{-5}$ $2.1810357753732\times 10^{-5}$
: \[EnergyTotal\] The energy contributions to the ground state of ${}^4$He. These data use values of the physical constants $1/\alpha = 137.035999679$ and $m_e/m_\alpha = 0.000137093355571$, where $\alpha$ is the fine-structure constant, $m_e$ is the mass of the electron, and $m_\alpha$ is the mass of an alpha particle [@MohrEtAl2008a; @MohrEtAl2008b]. The errors do not include the uncertainties in these values.
Conclusions {#Conclusions}
===========
We developed a general prescription for choosing coordinates and subdomains for a pseudospectral treatment of partial differential equations in the presence of physical and coordinate-related singularities. This prescription was applied to Schrödinger’s equation for helium to determine the fully correlated wave function. The treatment accounts for two-body but not three-body coalescences. Other problems with Coulomb singularities can now be tackled with this method.
We explored the fidelity of the pseudospectral method’s results. The method attained exponentially fast convergence for a wide selection of expectation values and matrix elements like the oscillator strength. Variational approaches minimize energy-weighted errors but generally do not yield comparable results for other operators. In contrast, we found that the pseudospectral method produced errors and convergence rates that were very similar for all the quantities studied including energy.
The approach should be widely applicable. No fine tuning was done to improve convergence other than ensuring non-analytic behavior was treated properly. The numerical method we developed was capable of solving the large matrix problems with modest computational resources. The calculations were pushed to the limits of double precision arithmetic. Higher precision floating point arithmetic will be necessary to go further.
This work generalized our previous treatment from S to P states and demonstrated the calculation of a variety of matrix elements. It can be further extended to higher angular momenta in a straightforward manner, albeit at larger computational cost.
The oscillator strength of the helium $1^1$S $\to 2^1$P transition was calculated to about the same accuracy as the most accurate value in the literature [@Drake1996] and was found to agree to the expected precision.
Bhatia and Temkin Hamiltonian {#BTAppendix}
=============================
Bhatia and Temkin [@BhatiaTemkin1964] derived and we checked the following explicit expressions that make up the Hamiltonian in their three-three splitting:
$$\begin{aligned}
\label{BhatiaHamiltonian}
\hat{H}_S&=& -\frac{1}{2}\sum_{i=1}^2\frac{1}{r_i^2}\left({\frac{\partial}{\partial{r_i}}}r_i^2{\frac{\partial}{\partial{r_i}}}+\frac{1}{\sin\theta_{12}}{\frac{\partial}{\partial{\theta_{12}}}}\sin\theta_{12}{\frac{\partial}{\partial{\theta_{12}}}}\right)+ \hat{V}\\
\hat{V}&=&-\frac{Z}{r_1}-\frac{Z}{r_2}+\frac{1}{r_{12}}\\
\hat{H}_{\nu,\kappa,-1}^\gamma&=& (1-\delta_{0\kappa}-\delta_{1\kappa}+(-1)^j\delta_{2\kappa})
h_\nu^\gamma B_{l\kappa,-1}
\left\{\begin{array}{ll}
\cot\theta_{12} & \textrm{ if }\nu=\gamma\\
(-1)^\nu & \textrm{ if }\nu\ne\gamma
\end{array}\right.\\
\hat{H}_{\nu\kappa 0}^\gamma&=& h_\nu^\gamma
\left\{\begin{array}{ll}
2\frac{l(l+1)-\kappa^2}{\sin\theta_{12}}+\kappa^2\sin\theta_{12}-\gamma\cot\theta_{12}l(l+1)\delta_{1\kappa}&\textrm{ if }\nu=\gamma\\
\nu\kappa(2\cos\theta_{12}+4\sin\theta_{12}{\frac{\partial}{\partial{\theta_{12}}}})-l(l+1)\delta_{1\kappa}&\textrm{ if }\nu\ne \gamma
\end{array}\right.\\
\hat{H}_{\nu\kappa1}^\gamma&=& (1-\nu \delta_{0\kappa})h_\nu^\gamma B_{l,\kappa+2,1}
\left\{\begin{array}{ll}
\cot\theta_{12} & \textrm{ if }\nu=\gamma\\
(-1)^\gamma & \textrm{ if }\nu\ne\gamma
\end{array}\right.\\
h_\nu^\gamma&=&\frac{1}{8\sin\theta_{12}}\left(\frac{1}{r_2^2}+\frac{\nu\gamma}{r_1^2}\right)\\
B_{l\kappa n}&=&(1+\delta_{2\kappa}(\sqrt{2}-1))^n\sqrt{(l-\kappa+1)(l-\kappa+2)(l+\kappa)(l+\kappa-1)}.\end{aligned}$$
Matrix methods {#MatrixMethods}
==============
Calculating matrix elements with Bhatia and Temkin’s radial functions {#BTOscStrength}
=====================================================================
Oscillator Strength
-------------------
In the Bhatia and Temkin three-three splitting [@BhatiaTemkin1964], the matrix elements for an ${}^1$S $\to {}^1$P oscillator strength transition are written: $$\sum_m|\langle {}^1\textrm{S}|\hat{{\mathbf{D}}}|{}^1\textrm{P} m \rangle|^2
=\left[\int d\tau g_{000}^0
\left(d_D^0g_{110}^0+d_D^1g_{110}^1\right)\right]^2,$$ where $d\tau = r_1^2r_2^2\sin\theta_{12}dr_1dr_2d\theta_{12}$, $\hat{{\mathbf{D}}}$ is one of the operators found inside the matrix elements of Eqs. \[OscStrengthEq\] and the operators $d_D^i$ are given by $$\begin{aligned}
d_{{\mathbf{R}}}^0&=&(r_1+r_2)\cos\frac{\theta_{12}}{2}\\
d_{{\mathbf{R}}}^1&=&(r_1-r_2)\sin\frac{\theta_{12}}{2}\\
d_{{\mathbf{P}}}^0&=&\frac{(r_1+r_2)(3+\cos\theta_{12})}{4r_1r_2\cos\frac{\theta_{12}}{2}} \nonumber\\
& &+\cos\frac{\theta_{12}}{2} \left({\frac{\partial}{\partial{r_1}}}+{\frac{\partial}{\partial{r_2}}}\right)\nonumber\\
& &-\frac{(r_1+r_2)\sin\frac{\theta_{12}}{2}}{r_1r_2}{\frac{\partial}{\partial{\theta_{12}}}}\\
d_{{\mathbf{P}}}^1&=&\frac{(r_1-r_2)(-3+\cos\theta_{12})}{4r_1r_2\sin\frac{\theta_{12}}{2}}\nonumber\\
& &+\sin\frac{\theta_{12}}{2}\left({\frac{\partial}{\partial{r_1}}}-{\frac{\partial}{\partial{r_2}}}\right) \nonumber\\
& &-\frac{(r_1-r_2)\cos\frac{\theta_{12}}{2}}{r_1r_2}{\frac{\partial}{\partial{\theta_{12}}}}\\
d_{{\mathbf{A}}}^0&=&\frac{Z(r_1^2+r_2^2)\cos\frac{\theta_{12}}{2}}{r_1^2r_2^2}\\
d_{{\mathbf{A}}}^1&=&\frac{Z(r_1^2-r_2^2)\sin\frac{\theta_{12}}{2}}{r_1^2r_2^2}.\end{aligned}$$
Expectation Values {#expectation-values}
------------------
Similarly, an expectation value for an S state is calculated by $$\label{ExpValFormula}
\langle {}^1\textrm{S}| \hat{{\mathbf{D}}} | {}^1 \textrm{S} \rangle = \int d\tau g^0_{000} d^0_D g^0_{000}.$$ Most of the operators $d^0_D$ used for expectation values in this article have trivial forms. We write here only the two most complicated ones: $$\begin{aligned}
d^0_{{\mathbf{p}}_1\cdot{\mathbf{p}}_2} &=& \frac{1}{r_1 r_2}
\left[ \sin\theta_{12}\left(r_1{\frac{\partial}{\partial{r_1}}}+r_2{\frac{\partial}{\partial{r_2}}}\right){\frac{\partial}{\partial{\theta_{12}}}} \right.\nonumber\\
& & -r_1 r_2 \cos\theta_{12}{\frac{\partial^2}{\partial{r_1}\partial{r_2}}}+\cos\theta_{12}{\frac{\partial^2}{\partial{\theta_{12}}^2}}\nonumber\\
& &\left. +\frac{1}{\sin\theta_{12}}{\frac{\partial}{\partial{\theta_{12}}}}\right] \\
d^0_{H_{OO}} &=& -\frac{\alpha^2}{2r_{12}^3}
\left[ \sin\theta_{12}\left(x_{12}{\frac{\partial}{\partial{r_1}}}+x_{21}{\frac{\partial}{\partial{r_2}}}\right){\frac{\partial}{\partial{\theta_{12}}}} \right. \nonumber\\
& &+r_1 r_2 z_+{\frac{\partial^2}{\partial{r_1}\partial{r_2}}}+z_-{\frac{\partial^2}{\partial{\theta_{12}}^2}}\nonumber\\
& & \left. +\frac{r_{12}^2}{r_1 r_2\sin\theta_{12}}{\frac{\partial}{\partial{\theta_{12}}}}\right],\end{aligned}$$ where $$x_{ij}=\frac{r_i^2+r_{12}^2-{\mathbf{r}}_1\cdot{\mathbf{r}}_2}{r_j}$$ and $$z_\pm = (1\pm 3)\cos\theta_{12}(\cos\theta_{12}-\rho^2/2r_1r_2)+\sin^2\theta_{12}.$$ All of these forms must be converted to the appropriate coordinates in each subdomain.
We thank Harald P. Pfeiffer for help in solving large pseudospectral matrix problems, Saul Teukolsky and Cyrus Umrigar for guidance and support, and Charles Schwartz for useful comments on the manuscript. This material is based upon work supported by the National Science Foundation under Grant No. AST-0406635 and by NASA under Grant No. NNG-05GF79G.
[^1]: The method is not limited to ground states. A trial wave function, exactly orthonormal to all lower energy states, has calculated energy which is an upper bound to the exact result for the excited state.
[^2]: The symbols used here are slightly different than those of [@BhatiaTemkin1964] so that the equations can be written in a simplified form.
[^3]: The full class of one dimensional problems for which pseudospectral methods converge exponentially fast is larger than this description. The method needs the solution to be smooth which is a weaker statement than that it be analytic. This distinction is not material for the singular points discussed here.
[^4]: A zero- or one-dimensional singularity can be made to look two-dimensional by a coordinate transformation. For example, in the previous example, which has used spherical coordinates, the Coulomb singularity appears at $r=0$. This point is approached on a two-dimensional sphere of constant radius by taking the limit as a single coordinate, the radius, approaches zero.
[^5]: Values come from the $n=23$ calculation. The errors are calculated by assuming an uncertainty five times greater than the fits given in Tab. \[ConvTab\] to account for the spread about these fits.
[^6]: Direct evaluation of the operators $p_i^4$ ($i=1,2$) on the ket yields delta function contributions which are unsuitable for direct numerical evaluation on the grid. So Eq. \[ExpValFormula\] cannot be used to produce an exponentially accurate expectation value. As is well known, instead applying $p_i^2$ to both the bra and ket produces well-behaved functions, but we do not carry out this calculation in this article.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We measure the neutral $D$ total forward cross section and the differential cross sections as functions of Feynman-$x$ ([$x_{\!F}$]{}) and transverse momentum squared for 500 GeV/$c$ $\pi^-$–nucleon interactions. The results are obtained from 88990$\pm$460 reconstructed neutral $D$ mesons from Fermilab experiment E791 using the decay channels [${\ensuremath{D^{\raise0.3ex\hbox{\scriptsize{\;\!0}}}}}\!\rightarrow\!K^-\pi^+$]{} and [${\ensuremath{D^{\raise0.3ex\hbox{\scriptsize{\;\!0}}}}}\!\rightarrow\!K^-\pi^+\pi^-\pi^+$]{} (and charge conjugates). We extract fit parameters from the differential cross sections and provide the first direct measurement of the turnover point in the [$x_{\!F}$]{} distribution, 0.0131$\pm$0.0038. We measure an absolute [$D^{\raise0.3ex\hbox{\scriptsize{\;\!0}}}$]{}+[$\overline{D}^{\lower0.3ex\hbox{\scriptsize{\;\!0}}}$]{} ([$x_{\!F}$]{}$>$0) cross section of $15.4 {{\ensuremath{\mathrel{\hbox{\rlap{\hbox{\lower2.4pt\hbox{\scriptsize{$-$}}}}\raise4.3pt\hbox{\scriptsize{$+$}}}}}}}{\ensuremath{\mathrel{\hbox{\rlap{\hbox{\lower2.4pt\hbox{\scriptsize{2.3}}}}\raise4.3pt\hbox{\scriptsize{1.8}}}}}}$ $\mu$barns/nucleon (assuming a linear $A$ dependence). The differential and total forward cross sections are compared to theoretical predictions and to results of previous experiments.'
address:
- 'Centro Brasileiro de Pesquisas F[í]{}sicas, Rio de Janeiro, Brazil'
- 'University of California, Santa Cruz, California 95064, USA'
- 'University of Cincinnati, Cincinnati, Ohio 45221, USA'
- 'CINVESTAV, 07000 Mexico City, DF Mexico'
- 'Fermilab, Batavia, Illinois 60510, USA'
- 'Illinois Institute of Technology, Chicago, Illinois 60616, USA'
- 'Kansas State University, Manhattan, Kansas 66506, USA'
- 'University of Massachusetts, Amherst, Massachusetts 01003, USA'
- 'University of Mississippi–Oxford, University, Mississippi 38677, USA'
- 'Princeton University, Princeton, New Jersey 08544, USA'
- 'Universidad Autonoma de Puebla, Mexico'
- 'University of South Carolina, Columbia, South Carolina 29208, USA'
- 'Stanford University, Stanford, California 94305, USA'
- 'Tel Aviv University, Tel Aviv 69978, Israel'
- 'Box 1290, Enderby, British Columbia V0E 1V0, Canada'
- 'Tufts University, Medford, Massachusetts 02155, USA'
- 'University of Wisconsin, Madison, Wisconsin 53706, USA'
- 'Yale University, New Haven, Connecticut 06511, USA'
author:
- 'E. M. Aitala'
- 'S. Amato'
- 'J. C. Anjos'
- 'J. A. Appel'
- 'D. Ashery'
- 'S. Banerjee'
- 'I. Bediaga'
- 'G. Blaylock'
- 'S. B. Bracker'
- 'P. R. Burchat'
- 'R. A. Burnstein'
- 'T. Carter'
- 'H. S. Carvalho'
- 'N. K. Copty'
- 'L. M. Cremaldi'
- 'C. Darling'
- 'K. Denisenko'
- 'S. Devmal'
- 'A. Fernandez'
- 'G. F. Fox'
- 'P. Gagnon'
- 'C. Gobel'
- 'K. Gounder'
- 'A. M. Halling'
- 'G. Herrera'
- 'G. Hurvits'
- 'C. James'
- 'P. A. Kasper'
- 'S. Kwan'
- 'D. C. Langs'
- 'J. Leslie'
- 'B. Lundberg'
- 'J. Magnin'
- 'S. MayTal-Beck'
- 'B. Meadows'
- 'J. R. T. de Mello Neto'
- 'D. Mihalcea'
- 'R. H. Milburn'
- 'J. M. de Miranda'
- 'A. Napier'
- 'A. Nguyen'
- 'A. B. d’Oliveira'
- 'K. O’Shaughnessy'
- 'K. C. Peng'
- 'L. P. Perera'
- 'M. V. Purohit'
- 'B. Quinn'
- 'S. Radeztsky'
- 'A. Rafatian'
- 'N. W. Reay'
- 'J. J. Reidy'
- 'A. C. dos Reis'
- 'H. A. Rubin'
- 'D. A. Sanders'
- 'A. K. S. Santha'
- 'A. F. S. Santoro'
- 'A. J. Schwartz'
- 'M. Sheaff'
- 'R. A. Sidwell'
- 'A. J. Slaughter'
- 'M. D. Sokoloff'
- 'J. Solano'
- 'N. R. Stanton'
- 'R. J. Stefanski'
- 'K. Stenson'
- 'D. J. Summers'
- 'S. Takach'
- 'K. Thorne'
- 'A. K. Tripathi'
- 'S. Watanabe'
- 'R. Weiss-Babai'
- 'J. Wiener'
- 'N. Witchey'
- 'E. Wolin'
- 'S. M. Yang'
- 'D. Yi'
- 'S. Yoshida'
- 'R. Zaliznyak'
- 'C. Zhang'
title: 'Total Forward and Differential Cross Sections of Neutral $D$ Mesons Produced in 500 GeV/$c$ $\pi^-$–Nucleon Interactions '
---
FERMILAB-Pub-99/185-E
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Charm hadroproduction is a convolution of short range processes that can be calculated in perturbative quantum chromodynamics (QCD) and long range processes that cannot be treated perturbatively and thus must be modeled using experimental measurements. The large theoretical uncertainties from both contributions are reflected in the relatively large number of input parameters that can be adjusted when comparing models to the results of experiments. A single measurement, no matter how precise, cannot unambiguously determine these parameters. However, the results of high statistics measurements like the ones reported here, when combined with other measurements of similar precision, can constrain such parameters as the charm quark mass, the intrinsic transverse momentum of the partons in the incoming hadrons, and the effective factorization and renormalization scales used in theoretical calculations.
We report here measurements of the differential cross sections versus the kinematic variables Feynman-$x$ ([$x_{\!F}$]{}) and transverse momentum squared ([$p_T^2$]{}), as well as the total forward cross section for the hadroproduction of neutral $D$ mesons. The relatively high pion beam momentum, 500 GeV/$c$, coupled with the good geometric acceptance of the Tagged Photon Laboratory (TPL) spectrometer, allows us to investigate a wide kinematic region that includes points at negative [$x_{\!F}$]{}. We are able to measure the shape of the differential cross section versus [$x_{\!F}$]{} with sufficient precision to confirm, for the first time, that the turnover in the cross section does occur at [$x_{\!F}$]{}$>$0, as expected for incident pions [@pinacoteca].
Combining data from two [$D^{\raise0.3ex\hbox{\scriptsize{\;\!0}}}$]{} decay modes, [${\ensuremath{D^{\raise0.3ex\hbox{\scriptsize{\;\!0}}}}}\!\rightarrow\!K\pi$]{} and [${\ensuremath{D^{\raise0.3ex\hbox{\scriptsize{\;\!0}}}}}\!\rightarrow\!K\pi\pi\pi$]{}, [^1] we extract a sample of 88990$\pm$460 (78730$\pm$430 at [$x_{\!F}$]{}$>$0) fully reconstructed charm decays to use for these measurements. In addition to the greater statistical significance, the use of two modes provides a means to better understand the systematic errors associated with the reconstruction of the decay products of these fully-charged decays.
The data were accumulated during the 1991/1992 Fermilab fixed-target run of experiment E791 [@exp791; @e791_pairs]. The experiment utilized the spectrometer built by the previous TPL experiments, E516 [@exp516], E691 [@exp691], and E769 [@exp769], with significant improvements. The experiment employed a 500 GeV/$c$ $\pi^-$ beam tracked by eight planes of proportional wire chambers (PWC’s) and six planes of silicon microstrip detectors (SMD’s). The beam impinged on one 0.52-mm thick platinum foil (1.6 cm in diameter) followed by four 1.56-mm thick diamond foils (1.4 cm in diameter), each foil center separated from the next by an average of 1.53 cm, allowing most charm particles to decay in air. The downstream spectrometer consisted of 17 planes of SMD’s for vertexing and tracking along with 35 planes of drift chambers, 2 PWC planes, and 2 analysis magnets (bending in the same direction) for track and momentum measurement. Two multicell threshold [Čerenkov]{} counters, an electromagnetic calorimeter, a hadronic calorimeter, and a wall of scintillation counters for muon detection provided particle identification. The trigger was generated using signals from scintillation counters as well as the electromagnetic and hadronic calorimeters. The beam scintillation counters included a beam counter 1.3 cm in diameter (14 cm upstream of the first target) and a large beam-halo veto counter with a 1.0 cm hole (8 cm upstream of the first target). The interaction counter was located 2.0 cm downstream of the last target and 0.6 cm upstream of the first SMD plane. The first-level trigger required a signal corresponding to at least 1/2 of that expected for a minimum ionizing particle (MIP) in the beam counter, no signal greater than 1/2 of a MIP in the beam halo counter, and a signal corresponding to greater than $\sim$4.5 MIP’s in the interaction counter (consistent with a hadronic interaction in one of the targets). The second-level trigger required more than 3 GeV of transverse energy in the calorimeters. Additional requirements eliminated events with multiple beam particles. A fast data acquisition system [@e791daq] collected data at rates up to 30 Mbyte/s with 50 $\mu$s/event deadtime. Over 2$\times$10$^{10}$ events were written to 24000 8mm magnetic tapes during a six-month period.
The raw data were reconstructed and filtered [@e791_pairs; @farms] to keep events with at least two separated vertices, consistent with a primary interaction and a charm particle decay. Following the event reconstruction and filtering, selection criteria for the [$D^{\raise0.3ex\hbox{\scriptsize{\;\!0}}}$]{} candidates were determined by maximizing $S / \sqrt{S+B}$ where $S$ is the (normalized) number of signal events resulting from a Monte Carlo simulation and $B$ is the number of background events appearing in the data sidebands of the reconstructed [$K\pi$]{} or [$K\pi\pi\pi$]{} mass distribution. Only selection variables that are well modeled by the Monte Carlo simulation were used. The final selection criteria varied by decay type ([$K\pi$]{} and [$K\pi\pi\pi$]{}) and by [$x_{\!F}$]{} region. The full range of a cut variation is given in the descriptions below. To eliminate generic hadronic interaction backgrounds as well as secondary interactions, the secondary vertex was required to be longitudinally separated from the primary vertex by more than 8-11 times the measurement uncertainty on the longitudinal separation ($\sim$400 $\mu$m) and to lie outside of the target foils. Backgrounds from the primary interaction were also reduced by requiring that the candidate decay tracks miss the primary vertex by at least 20-40 $\mu$m. To ensure a correctly reconstructed charm particle and primary vertex, the momentum vector of the [$D^{\raise0.3ex\hbox{\scriptsize{\;\!0}}}$]{} candidate was required to point back to within 35-60 $\mu$m of the primary vertex and to have a momentum component perpendicular to the line connecting the primary and secondary vertices of less than 350-450 MeV/$c$. Finally, the sum of the squares of the transverse momenta of the decay tracks relative to the candidate [$D^{\raise0.3ex\hbox{\scriptsize{\;\!0}}}$]{} momentum vector was required to be greater than 0.4 (GeV/$c$)$^2$ (0.15 (GeV/c)$^2$) for the [$K\pi$]{} ([$K\pi\pi\pi$]{}) candidates to favor the decay of a high-mass particle. All primary vertices were required to occur in the diamond targets. Thus, our results come from a light, isoscalar target. The [Čerenkov]{} information is not used in this analysis; all particle-identification combinations are tried. The inclusive [$K\pi$]{} and [$K\pi\pi\pi$]{} signals are shown in Fig. \[fig:yield\].
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The reconstructed data were split into 20 bins of [$x_{\!F}$]{}, integrating over all [$p_T^2$]{}, and 20 bins of [$p_T^2$]{}, integrating over [$x_{\!F}$]{}$>$0. To combine data from varied conditions ([e.g.]{}, using particles that pass through one and two magnets), the normalized mass ($m_n$) is constructed for each candidate using its calculated mass and error ($m$ and $\sigma_m$) and the measured mean mass ($m_D$): $m_n \equiv \frac{m - m_{D_{\null}}}{\sigma_m}$. Using the binned maximum likelihood method, the normalized mass distributions were fit to a simple Gaussian for the signal and linear or quadratic polynomials for the background.
The acceptance can be factorized into trigger efficiency ($\epsilon_{trig}$) and reconstruction efficiency ($\epsilon_{rec}$). Most of the trigger inefficiency is due to vetoes on multiple beam particles. These resulted in a (70.3$\pm$1.1$\pm$4.2)% trigger efficiency, where the first error is statistical and the second error is systematic. The interaction and transverse energy requirements were greater than 99% efficient for reconstructable hadronic charm decays. Writing and reading the data tapes was (97.6$\pm$1.0)% efficient. Combining these efficiencies gives $\epsilon_{trig} = (68.3\pm4.4)\%$, where the error is dominated by the systematic error. The reconstruction efficiency is obtained from a Monte Carlo simulation. The Monte Carlo simulation used [<span style="font-variant:small-caps;">Pythia/Jetset</span>]{} [@pythia] as a physics generator and models the effects of resolution, geometry, magnetic fields, and detector efficiencies as well as all analysis cuts. The efficiencies were separately modeled for five evenly spaced temporal periods during the experiment. This was motivated by a highly inefficient region of slowly increasing size in the center of the drift chambers caused by the 2 MHz pion beam. The Monte Carlo events were weighted to match the observed data distributions of [$x_{\!F}$]{}, [$p_T^2$]{}, and the summed [$p_T^2$]{} of all two-magnet charged tracks in the event other than those from the candidate $D$ meson. The resulting reconstruction efficiencies as a function of [$x_{\!F}$]{} and [$p_T^2$]{} are shown in Fig. \[fig:acc\].
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From the number of reconstructed [$D^{\raise0.3ex\hbox{\scriptsize{\;\!0}}}$]{} candidates, the reconstruction efficiency, the trigger efficiency, and the PDG branching fractions [@pdg], we obtain the number of [$D^{\raise0.3ex\hbox{\scriptsize{\;\!0}}}$]{} mesons produced in our experiment during the experiment livetime, $N_{prod}$. The cross section as a function of each variable $z$ (where $z$ = [$x_{\!F}$]{} or [$p_T^2$]{}), is: $$\sigma(\pi^-N \,\rightarrow\, {\ensuremath{D^{\raise0.3ex\hbox{\scriptsize{\;\!0}}}}}X ; z) \;\:=\;\:
\frac{N_{prod}({\ensuremath{D^{\raise0.3ex\hbox{\scriptsize{\;\!0}}}}}; z)}{T_N\: N_{\pi^-}}.
\label{eq:sigtot}$$ $T_N$ is the number of nucleons per area in the target (calculated from the target thickness) and is $(1.224\pm0.004) \times 10^{-6}$ nucleons/$\mu$b. $N_{\pi^-}$ is the number of incident $\pi^-$ particles during the experiment livetime. This is obtained directly from a scaler which counted clean beam particles ($>$1/2 MIP signals in the beam and interaction counters and no signal greater than 1/2 MIP in the beam-halo veto counter) during the experiment livetime. These are the only beam particles which could cause a first-level trigger. Using Eq. \[eq:sigtot\], we obtained the [$D^{\raise0.3ex\hbox{\scriptsize{\;\!0}}}$]{}+[$\overline{D}^{\lower0.3ex\hbox{\scriptsize{\;\!0}}}$]{} differential cross sections versus [$x_{\!F}$]{} and [$p_T^2$]{} shown in Figs. \[fig:xf\_fit\_d0\] and \[fig:pt\_fit\_d0\].
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The systematic errors are divided into two categories and incorporated in two stages. The *uncorrelated* systematic errors are determined individually for the [$K\pi$]{} and [$K\pi\pi\pi$]{} results. These systematic errors include uncertainties in the Monte Carlo modeling of the selection criteria, the background functions, and the widths used in the Gaussian signal functions. The *correlated* systematic errors are calculated for the combined [$D^{\raise0.3ex\hbox{\scriptsize{\;\!0}}}$]{} result, obtained from adding the [$K\pi$]{} and [$K\pi\pi\pi$]{} samples together, weighted by the inverse-square of the combined statistical and uncorrelated systematic errors. The correlated errors are associated with uncertainties in the [$D^{\raise0.3ex\hbox{\scriptsize{\;\!0}}}$]{} lifetime, the Monte Carlo production model, the Monte Carlo weighting procedure, and the run period weighting procedure. Finally, we compare our measured [$K\pi$]{} to [$K\pi\pi\pi$]{} branching ratio to the PDG [@pdg] value to estimate the residual tracking and vertexing efficiency modeling error. In addition to these errors, which can affect both the normalization and the shape of the differential cross sections, there are two errors which affect only the normalization: the uncertainties in the trigger efficiency and target thickness.
For the differential cross sections shown in Figs. \[fig:xf\_fit\_d0\] and \[fig:pt\_fit\_d0\], the systematic errors are factorized into shape and normalization parts. The error bars in the figures show the sum, in quadrature, of the statistical and all systematic errors after factoring out the normalization component. Although the relative importance varies bin-by-bin, the most important systematic errors generally come from uncertainties in the signal width and the Monte Carlo efficiency modeling. In all cases, the systematic error dominates. For the total forward cross section, all of the errors are summarized and summed in Table \[tab:totsig\].
\[tab:totsig\]
-- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ -------------------------------------------------
[$K\pi$]{} [$K\pi\pi\pi$]{} [$D^{\raise0.3ex\hbox{\scriptsize{\;\!0}}}$]{}
${{\ensuremath{\mathrel{\hbox{\rlap{\hbox{\lower4.4pt\hbox{\small{$-$}}}}\raise4.4pt\hbox{\small{$+$}}}}}}}{\ensuremath{\mathrel{\hbox{\rlap{\hbox{\lower4.4pt\hbox{\small{9.0}}}}\raise4.4pt\hbox{\small{4.7}}}}}}$ ${{\ensuremath{\mathrel{\hbox{\rlap{\hbox{\lower4.4pt\hbox{\small{$-$}}}}\raise4.4pt\hbox{\small{$+$}}}}}}}{\ensuremath{\mathrel{\hbox{\rlap{\hbox{\lower4.4pt\hbox{\small{11.0}}}}\raise4.4pt\hbox{\small{7.3}}}}}}$ ${{\ensuremath{\mathrel{\hbox{\rlap{\hbox{\lower4.4pt\hbox{\small{$-$}}}}\raise4.4pt\hbox{\small{$+$}}}}}}}{\ensuremath{\mathrel{\hbox{\rlap{\hbox{\lower4.4pt\hbox{\small{7.6}}}}\raise4.4pt\hbox{\small{4.7}}}}}}$
Statistics $\pm$1.0 $\pm$2.8
Selection criteria efficiency modeling ${{\ensuremath{\mathrel{\hbox{\rlap{\hbox{\lower4.4pt\hbox{\small{$-$}}}}\raise4.4pt\hbox{\small{$+$}}}}}}}{\ensuremath{\mathrel{\hbox{\rlap{\hbox{\lower4.4pt\hbox{\small{0.0}}}}\raise4.4pt\hbox{\small{0.1}}}}}}$ ${{\ensuremath{\mathrel{\hbox{\rlap{\hbox{\lower4.4pt\hbox{\small{$-$}}}}\raise4.4pt\hbox{\small{$+$}}}}}}}{\ensuremath{\mathrel{\hbox{\rlap{\hbox{\lower4.4pt\hbox{\small{0.0}}}}\raise4.4pt\hbox{\small{1.2}}}}}}$
MC background function ${{\ensuremath{\mathrel{\hbox{\rlap{\hbox{\lower4.4pt\hbox{\small{$-$}}}}\raise4.4pt\hbox{\small{$+$}}}}}}}{\ensuremath{\mathrel{\hbox{\rlap{\hbox{\lower4.4pt\hbox{\small{5.5}}}}\raise4.4pt\hbox{\small{0.0}}}}}}$ ${{\ensuremath{\mathrel{\hbox{\rlap{\hbox{\lower4.4pt\hbox{\small{$-$}}}}\raise4.4pt\hbox{\small{$+$}}}}}}}{\ensuremath{\mathrel{\hbox{\rlap{\hbox{\lower4.4pt\hbox{\small{6.0}}}}\raise4.4pt\hbox{\small{0.0}}}}}}$
[$D^{\raise0.3ex\hbox{\scriptsize{\;\!0}}}$]{} signal width in fits ${{\ensuremath{\mathrel{\hbox{\rlap{\hbox{\lower4.4pt\hbox{\small{$-$}}}}\raise4.4pt\hbox{\small{$+$}}}}}}}{\ensuremath{\mathrel{\hbox{\rlap{\hbox{\lower4.4pt\hbox{\small{4.1}}}}\raise4.4pt\hbox{\small{4.0}}}}}}$ ${{\ensuremath{\mathrel{\hbox{\rlap{\hbox{\lower4.4pt\hbox{\small{$-$}}}}\raise4.4pt\hbox{\small{$+$}}}}}}}{\ensuremath{\mathrel{\hbox{\rlap{\hbox{\lower4.4pt\hbox{\small{4.4}}}}\raise4.4pt\hbox{\small{3.9}}}}}}$
PDG [@pdg] branching ratio $\pm$2.3 $\pm$5.3
${{\ensuremath{\mathrel{\hbox{\rlap{\hbox{\lower4.4pt\hbox{\small{$-$}}}}\raise4.4pt\hbox{\small{$+$}}}}}}}{\ensuremath{\mathrel{\hbox{\rlap{\hbox{\lower4.4pt\hbox{\small{9.9}}}}\raise4.4pt\hbox{\small{6.9}}}}}}$
[$D^{\raise0.3ex\hbox{\scriptsize{\;\!0}}}$]{} lifetime
Monte Carlo kinematic weighting
Monte Carlo production model
Time dependent efficiency modeling
$\pm$4.6
$\pm$6.4
$\pm$0.3
${{\ensuremath{\mathrel{\hbox{\rlap{\hbox{\lower4.4pt\hbox{\small{$-$}}}}\raise4.4pt\hbox{\small{$+$}}}}}}}{\ensuremath{\mathrel{\hbox{\rlap{\hbox{\lower4.4pt\hbox{\small{14.8}}}}\raise4.4pt\hbox{\small{11.5}}}}}}$
-- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ -------------------------------------------------
: Sources and values of the uncertainties on the total forward [$D^{\raise0.3ex\hbox{\scriptsize{\;\!0}}}$]{}+[$\overline{D}^{\lower0.3ex\hbox{\scriptsize{\;\!0}}}$]{} cross section measurement. The total systematic error comes from the quadratic sum of the [$D^{\raise0.3ex\hbox{\scriptsize{\;\!0}}}$]{} systematic errors. Although some of the systematic errors are subdivided, each [$D^{\raise0.3ex\hbox{\scriptsize{\;\!0}}}$]{} systematic error is obtained directly (including correlations) and is not the quadratic sum of the components.
In the past, [$x_{\!F}$]{} distributions have been fit with $$\frac{d \sigma}{d {\ensuremath{x_{\!F}}}} \;\:=\;\: A (1 - |{\ensuremath{x_{\!F}}}|)^n .
\label{eq:xffun}$$ Fitting Eq. \[eq:xffun\] in the range $0.05\!<\!{\ensuremath{x_{\!F}}}\!<\!0.50$, we find $n=4.61\pm0.19$, as shown in Fig. \[fig:xf\_fit\_d0\]. This function does not provide a complete representation of our data. Although the [$\chi^2/dof$]{} is small (0.3), the value of $n$ is quite dependent on the range fitted and on the errors on the data points. Another function, which can be extended into the negative [$x_{\!F}$]{} region, is an extension of Eq. \[eq:xffun\] which uses a power-law function in the tail region and a Gaussian in the central region; that is, $$\frac{d \sigma}{d {\ensuremath{x_{\!F}}}} \;\:=\;\:
\begin{cases}
A (1 - |{\ensuremath{x_{\!F}}}- x_c|)^{n'} , & |{\ensuremath{x_{\!F}}}- x_{c_{\null}}| > x_b \\
A' \exp{\left[-\frac{1}{2}(\frac{{\ensuremath{x_{\!F}}}- x_c}{\sigma})^2\right]} , & |{\ensuremath{x_{\!F}}}- x_c| < x_b
\end{cases}
.
\label{eq:xfandfun}$$ Requiring continuous functions and derivatives allows us to write Eq. \[eq:xfandfun\] with one normalization parameter and three shape parameters: $n'$ gives the shape in the tail region, $x_c$ is the turnover point, and $x_b$ is the boundary between the Gaussian and power-law function. The fit parameters from this function are nearly independent of the fit range. Fitting our data in the range $-0.125$$<$[$x_{\!F}$]{}$<$0.50 gives $n'=4.68\pm0.21$, $x_c=0.0131\pm0.0038$, and $x_b=0.062\pm0.013$ with a [$\chi^2/dof$]{}=0.4, as shown in Fig. \[fig:xf\_fit\_d0\]. This is the first measurement of the turnover point $x_c$ in the charm sector. The fact that it is significantly greater than zero is consistent with a harder gluon distribution in the beam pions than in the target nucleons.
The functions which have been used in the past to fit the [$p_T^2$]{} distribution are: $$\frac{d \sigma}{d {\ensuremath{p_T^2}}} \;\:=\;\: A e^{-b {\ensuremath{p_T^2}}}
\label{eq:ptfunb}$$ at low [$p_T^2$]{} ([$p_T^2$]{}$\,<\,$4.0 (GeV/$c$)$^2$ for this analysis), $$\frac{d \sigma}{d {\ensuremath{p_T^2}}} \;\:=\;\: A e^{-b' {\ensuremath{p_T}}}
\label{eq:ptfunbp}$$ at high [$p_T^2$]{} ([$p_T^2$]{}$\,>\,$1.0 (GeV/$c$)$^2$ for this analysis), and $$\frac{d \sigma}{d {\ensuremath{p_T^2}}} \;\:=\;\: \left[\frac{A}{\alpha\,m_c^2 \:+\: {\ensuremath{p_T^2}}} \right]^{\beta}
\label{eq:ptfunman}$$ over all [$p_T^2$]{} with $m_c$ set to 1.5 GeV/$c$$^2$ [@frixione]. The results of fitting these equations to the data are shown in Fig. \[fig:pt\_fit\_d0\]. For the ranges given above, the fit results are:
- [$b$ = 0.83$\pm$0.02 with [$\chi^2/dof$]{} = 2.8,]{}
- [$b'$ = 2.41$\pm$0.03 with [$\chi^2/dof$]{} = 1.7, and]{}
- [$\alpha$ = 2.36$\pm$0.23 (GeV/$c$$^2$)$^{-2}$ and $\beta$ = 5.94$\pm$0.39 with [$\chi^2/dof$]{} = 0.3.]{}
Equation \[eq:ptfunb\] does not provide a good fit even over the very limited range to which it is applied. While the [$\chi^2/dof$]{} (1.7) of the fit to Eq. \[eq:ptfunbp\] is not good, it appears to be a reasonable fit to the data. Equation \[eq:ptfunman\] provides a very good fit to the data over the entire range of [$p_T^2$]{}. Unfortunately, using two free parameters (in addition to the normalization) makes it more difficult to compare to other experiments and theory since the parameters in this fit are highly correlated. This is reflected in the large (7-10%) errors on $\alpha$ and $\beta$ compared to the error on $b'$ (1%), as shown above.
Figures \[fig:xf\_theory\_d0\] and \[fig:pt\_theory\_d0\] show a comparison of our [$x_{\!F}$]{} and [$p_T^2$]{} distributions to theoretical predictions for charm quark and $D$ meson production. Although the data come from [$D^{\raise0.3ex\hbox{\scriptsize{\;\!0}}}$]{} mesons, the theoretical predictions for charm quark production are included for completeness. The theoretical curves are normalized to obtain the best fit (lowest [$\chi^2/dof$]{}) to our data. The theoretical predictions come from a next-to-leading order (NLO) calculation by Mangano, Nason, and Ridolfi (MNR) [@mnrtheory] and the [<span style="font-variant:small-caps;">Pythia/Jetset</span>]{} [@pythia] event generator. The MNR NLO charm quark calculation uses SMRS2 [@pdf_smrs] (HMRSB [@pdf_hmrsb]) NLO parton distribution functions for the pion (nucleon), a charm quark mass of 1.5 GeV/$c$$^2$, and an average intrinsic transverse momentum of the incoming partons ([$\sqrt{\langle k_t^2\rangle}$]{}) of 1.0 GeV/$c$. The value for [$\sqrt{\langle k_t^2\rangle}$]{} was suggested by M. L. Mangano [@mlmsug] and is independently motivated by the study of azimuthal angle correlations between two charm particles in the same event [@frixione]. The $D$ meson results are obtained by convoluting the charm quark results with the Peterson fragmentation function [@peterson] with $\epsilon = 0.01$. The low value for $\epsilon$ was also suggested by M. L. Mangano [@mlmsug] in response to a reanalysis of $D$ fragmentation in $e^+e^-$ collisions [@dfragee]. The [<span style="font-variant:small-caps;">Pythia/Jetset</span>]{} event generator uses leading order DO2 [@pdf_do2] (CTEQ2L [@pdf_cteq2]) parton distribution functions for the pion (nucleon), a charm quark mass of 1.35 GeV/$c$$^2$, [$\sqrt{\langle k_t^2\rangle}$]{} of 0.44 GeV/$c$, and the Lund string fragmentation scheme to obtain [$D^{\raise0.3ex\hbox{\scriptsize{\;\!0}}}$]{} results. Tables \[tab:comp\_xf\] and \[tab:comp\_pt\] show a comparison of our [$x_{\!F}$]{} and [$p_T^2$]{} fit results to theoretical predictions and to recent high-statistics charm experiments which used pion beams. The evident energy dependence of the shape parameters in Tables \[tab:comp\_xf\] and \[tab:comp\_pt\] are consistent with theoretical predictions [@frixione].
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\[tab:comp\_xf\]
-------------------------------------------------------- -------- ------------------------- ------------------- --------------- -----------------
Experi- Energy [$x_{\!F}$]{} Range$\!$ $n$ $n'$ $x_c$
ment (GeV)
E791 500 0.05–0.5 $\!$4.61$\pm$0.19 4.68$\pm$0.21 .0131$\pm$.0038
WA92[@wa92_1] 350 0.0–0.8 4.27$\pm$0.11
E769[@e769_2] 250 0.0–0.8 4.03$\pm$0.18
MNR NLO $c$ 500 0.05–0.5 4.68 5.06 0.0231
MNR NLO $D$ 500 0.05–0.5 5.53 6.00 0.0237
Pythia $c$ 500 0.05–0.5 5.01 5.12 0.0115
Pythia [$D^{\raise0.3ex\hbox{\scriptsize{\;\!0}}}$]{} 500 0.05–0.5 3.62 3.66 0.0041
-------------------------------------------------------- -------- ------------------------- ------------------- --------------- -----------------
: Comparison of [$x_{\!F}$]{} shape parameters to recent high-statistics pion-beam charm production experiments and to theory. E791 results are for [$D^{\raise0.3ex\hbox{\scriptsize{\;\!0}}}$]{} mesons; E769 (WA92) results are from a combined sample of [$D^{\raise0.3ex\hbox{\scriptsize{\;\!0}}}$]{}, $D^+$, and $D_s$ ([$D^{\raise0.3ex\hbox{\scriptsize{\;\!0}}}$]{} and $D^+$) mesons. The theoretical results are obtained using the default parameters (described in the text). The fit range for $n$ (Eq. \[eq:xffun\]) is in the table. The fit range for $n'$ and $x_c$ (Eq. \[eq:xfandfun\]) is $-0.125$$<$[$x_{\!F}$]{}$<$0.50.
\[tab:comp\_pt\]
------------------------------------------------------- ----------- ----------------------- ------------------ ------------------------ --------------- --
Experi- Energy $b$ $b'$ $\alpha$ $\beta$
ment $\!$(GeV) $\!\!$(GeV/$c$)$^2\!$ (GeV/$c$)$^{-2}$ $\!$(GeV/$c$)$^{-1}\!$
E791 500 0.83$\pm$0.02 2.41$\pm$0.03 2.36$\pm$0.23 5.94$\pm$0.39
WA92[@wa92_1]$\!$ 350 0.89$\pm$0.02
E769[@e769_2] 250 1.08$\pm$0.05 2.74$\pm$0.09 1.4$\pm$0.3 5.0$\pm$0.6
MNR NLO $c$ 500 0.57 1.88 6.20 8.68
MNR NLO $D$ 500 0.94 2.32 1.99 5.30
Pythia $c$ 500 0.77 2.09 2.32 5.14
Pythia [$D^{\raise0.3ex\hbox{\scriptsize{\;\!0}}}$]{} 500 1.06 2.58 1.55 5.07
------------------------------------------------------- ----------- ----------------------- ------------------ ------------------------ --------------- --
: Comparison of [$p_T^2$]{} shape parameters to recent high-statistics pion-beam charm production experiments and to theory. Data samples are as described in Table \[tab:comp\_xf\]. The fit range for $b$ (Eq. \[eq:ptfunb\]) is 0$<$[$p_T^2$]{}$<$4 (GeV/$c$)$^2$ except for WA92 which is 0$<$[$p_T^2$]{}$<$7 (GeV/$c$)$^2$. The functions used to extract $b'$ (Eq. \[eq:ptfunbp\]) and $\alpha, \beta$ (Eq. \[eq:ptfunman\]) are fit in the range [$p_T^2$]{}$>$1 (GeV/$c$)$^2$ and [$p_T^2$]{}$<$18 (GeV/$c$)$^2$, respectively.
We obtain the total forward cross section by summing the [$x_{\!F}$]{} differential cross section for [$x_{\!F}$]{}$>$0 and assuming the cross section for 0.8$<$[$x_{\!F}$]{}$<$1.0 is half that of the cross section for 0.6$<$[$x_{\!F}$]{}$<$0.8 but with the same error. Assuming a linear dependence on the atomic number [@e769_adep], we obtain the neutral $D$ total forward cross section, $\sigma({\ensuremath{D^{\raise0.3ex\hbox{\scriptsize{\;\!0}}}}}\!+{\ensuremath{\overline{D}^{\lower0.3ex\hbox{\scriptsize{\;\!0}}}}};{\ensuremath{x_{\!F}}}\!>0) \,=\, 15.4 {{\ensuremath{\mathrel{\hbox{\rlap{\hbox{\lower2.4pt\hbox{\scriptsize{$-$}}}}\raise4.3pt\hbox{\scriptsize{$+$}}}}}}}{\ensuremath{\mathrel{\hbox{\rlap{\hbox{\lower2.4pt\hbox{\scriptsize{2.3}}}}\raise4.3pt\hbox{\scriptsize{1.8}}}}}}$ $\mu$barns/nucleon. To obtain the total charm cross section, $\sigma({\ensuremath{c \overline{c}}})$, we multiply our [$D^{\raise0.3ex\hbox{\scriptsize{\;\!0}}}$]{}+[$\overline{D}^{\lower0.3ex\hbox{\scriptsize{\;\!0}}}$]{}cross section by 1.7. This accounts for three multiplicative effects: the relative production of charm quarks compared to [$D^{\raise0.3ex\hbox{\scriptsize{\;\!0}}}$]{} mesons (2.1), the conversion from [$x_{\!F}$]{}$>$0 to all [$x_{\!F}$]{} (1.6), and the conversion to the [$c \overline{c}$]{} cross section from the sum of charm plus anticharm cross sections (0.5) [@fmnr_hqp97]. We compare our total charm cross section to other experiments and to the NLO predictions as a function of pion-beam energy in Fig. \[fig:mnretot\]. All experimental results are obtained by multiplying the [$D^{\raise0.3ex\hbox{\scriptsize{\;\!0}}}$]{}+[$\overline{D}^{\lower0.3ex\hbox{\scriptsize{\;\!0}}}$]{}cross section by 1.7. The rise of the charm production cross section with energy is modeled reasonably well by the NLO theory, although the absolute value at any point depends greatly on the input parameters to the theory.
= =10.0cm
In this paper we have presented the total forward cross section and differential cross sections versus [$x_{\!F}$]{} and [$p_T^2$]{} for [$D^{\raise0.3ex\hbox{\scriptsize{\;\!0}}}$]{} mesons from Fermilab experiment E791 data. This analysis represents the first measurement of the [$D^{\raise0.3ex\hbox{\scriptsize{\;\!0}}}$]{} cross section for a 500 GeV/$c$ pion beam. The high statistics allows us to clearly observe a turnover point greater than zero ($x_c$ = 0.0131$\pm$0.0038) in the Feynman-$x$ distribution, providing evidence for a harder gluon distribution in the pion than in the nucleon.
We have compared our differential cross section results to predictions from the next-to-leading order calculation by Mangano, Nason, and Ridolfi [@mnrtheory] and to the Monte Carlo event generator [<span style="font-variant:small-caps;">Pythia</span>]{} by T. Sjöstrand [@pythia]. With suitable choices for the intrinsic [$k_t$]{} of the partons and the Peterson fragmentation function parameter, the NLO $D$ meson calculation provides a good match to the [$p_T^2$]{} spectra and a fair match to the [$x_{\!F}$]{} distribution. The string fragmentation scheme in [<span style="font-variant:small-caps;">Pythia</span>]{} softens the original charm quark [$p_T^2$]{} distribution too much, and hardens the [$x_{\!F}$]{} spectra too much in both directions. However, the [$D^{\raise0.3ex\hbox{\scriptsize{\;\!0}}}$]{} result does predict the flattening of the [$x_{\!F}$]{} cross section at high [$x_{\!F}$]{}. The many adjustable parameters in the theoretical models allow one to obtain distributions which are quite consistent with these data. Unfortunately, a given set of parameters is neither unique, nor does it necessarily provide a good match to other data. In conjunction with other charm production results from this and other recent high-statistics experiments, however, it may be possible to find a unique set of parameters. These results come from experiments with a variety of beam energies and types, and include measurements of differential cross sections [@wa92_1; @e769_1; @e781], production asymmetries [@wa92_1; @e769_1; @e791_asym; @e769_asym; @e781; @e687_asym], and correlations between two charm particles in the same event [@e791_pairs; @wa92_pairs; @e687_pairs].
Unlike the uncertainties in the theoretical calculations of the differential cross sections, the uncertainties in the theoretical calculation of the total cross section come mostly from the perturbative calculation. The relatively large uncertainties are due to the low mass of the charm quark, which results in a large (unknown) contribution from higher-order terms. The total forward [$D^{\raise0.3ex\hbox{\scriptsize{\;\!0}}}$]{}+[$\overline{D}^{\lower0.3ex\hbox{\scriptsize{\;\!0}}}$]{} cross section measured by E791 is $\sigma({\ensuremath{D^{\raise0.3ex\hbox{\scriptsize{\;\!0}}}}}\!+{\ensuremath{\overline{D}^{\lower0.3ex\hbox{\scriptsize{\;\!0}}}}}\;;{\ensuremath{x_{\!F}}}\!>0) \,=\, 15.4 {{\ensuremath{\mathrel{\hbox{\rlap{\hbox{\lower2.4pt\hbox{\scriptsize{$-$}}}}\raise4.3pt\hbox{\scriptsize{$+$}}}}}}}{\ensuremath{\mathrel{\hbox{\rlap{\hbox{\lower2.4pt\hbox{\scriptsize{2.3}}}}\raise4.3pt\hbox{\scriptsize{1.8}}}}}}$ $\mu$barns/nucleon, assuming a linear atomic number dependence. The cross section is consistent with the MNR NLO prediction.
We express our special thanks to S. Frixione, M. L. Mangano, P. Nason, and G. Ridolfi for the use, and help in the use, of their NLO QCD software. We gratefully acknowledge the assistance from Fermilab and other participating institutions. This work was supported by the Brazilian Conselho Nacional de Desenvolvimento Científico e Technológico, CONACyT (Mexico), the Israeli Academy of Sciences and Humanities, the U.S. Department of Energy, the U.S.-Israel Binational Science Foundation, and the U.S. National Science Foundation.
[99.]{} R. K. Ellis and C. Quigg, FERMILAB-FN-445 (January 1987). J. A. Appel, Ann. Rev. Nucl. Part. Sci. [**42**]{} (1992) 367; D. J. Summers , Proceedings of the [*XXVII$^{\,th}$ Rencontre de Moriond*]{}, Electroweak Interactions and Unified Theories, Les Arcs, France (15-22 March, 1992) 417. E791 Collaboration, E. M. Aitala , submitted to Eur. Phys. J. C, Fermilab-Pub-98-297-E, hep-ex/9809029 (September 1998). E516 Collaboration, K. Sliwa , (32,1053,1985). E691 Collaboration, J. R. Raab , (37,2391,1988). E769 Collaboration, G. A. Alves , (69,3147,1992). S. Amato , (A324,535,1992). S. Bracker , (43,2457,1996); F. Rinaldo and S. Wolbers, (7,184,1993). H.-U. Bengtsson and T. Sjöstrand, (82,74,1994); T. Sjöstrand, <span style="font-variant:small-caps;">Pythia</span> 5.7 and <span style="font-variant:small-caps;">Jetset</span> 7.4 Physics and Manual, CERN-TH.7112/93, 1995. Particle Data Group, (3,497,1998). S. Frixione, M. L. Mangano, P. Nason, and G. Ridolfi, (B431,453,1994). M. L. Mangano, P. Nason, and G. Ridolfi, (B373,295,1992). P. J. Sutton , (45,2349,1992). P. N. Harriman , (42,798,1990). M. L. Mangano, hep-ph/9711337 (1997); M. L. Mangano, private communication (April 1999). C. Peterson , (27,105,1983). M. Cacciari and M. Greco, (55,7134,1997). D. W. Duke and J. F. Owens, (30,49,1984). J. Botts , (B304,159,1993). BEATRICE Collaboration, M. Adamovich , (B495,3,1997). E769 Collaboration, G. A. Alves , (77,2392,1996). E769 Collaboration, G. A. Alves , (70,722,1993). S. Frixione, M. L. Mangano, P. Nason, and G. Ridolfi, Heavy Flavours II, Advanced Series on Directions in High Energy Physics, 1997, hep-ph/9702287. E653 Collaboration, K. Kodama , (B284,461,1992). LEBC-EHS Collaboration, M. Aguilar-Benitez , (B161,400,1985). ACCMOR Collaboration, S. Barlag , (39,451,1988). ACCMOR Collaboration, S. Barlag , (49,555,1991). E769 Collaboration, G. A. Alves , (77,2388,1996). E791 Collaboration, E. M. Aitala , (B371,157,1996); (B411,230,1997). E769 Collaboration, G. A. Alves , (72,812,1994). SELEX Collaboration, F G. Garcia, S. Y. Jun, , to be published in the proceedings of American Physical Society (APS) Meeting of the Division of Particles and Fields (DPF 99), Los Angeles, CA, 5-9 Jan 1999, hep-ex/9905003. WA89 Collaboration, M. I. Adamovich , submitted to Eur. Phys. J. C, CERN-EP/98-41, hep-ex/9803021 (March, 1998). E687 Collaboration, P. L. Frabetti , (B370,222,1996). WA82 Collaboration, M. Adamovich , (B305,402,1995). BEATRICE Collaboration, M. Adamovich , (B348,256,1995); (B495,3,1997); (B385,487,1996). E687 Collaboration, P. L. Frabetti , (B308,193,1993).
[^1]: Charge conjugates are always implied. We use [$D^{\raise0.3ex\hbox{\scriptsize{\;\!0}}}$]{} to represent the sum of [$D^{\raise0.3ex\hbox{\scriptsize{\;\!0}}}$]{} and [$\overline{D}^{\lower0.3ex\hbox{\scriptsize{\;\!0}}}$]{}. Similarly, [$K\pi$]{} ([$K\pi\pi\pi$]{}) includes [$K^-\pi^+$]{} ([$K^-\pi^+\pi^-\pi^+$]{}) and charge conjugate.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The benchmarks of the neoclassical transport codes based on the several local drift-kinetic models are reported here. Here, the drift-kinetic models are ZOW, ZMD, DKES-like, and global, as classified in \[Matsuoka et al., Physics of Plasmas 22, 072511 (2015)\]. The magnetic geometries of HSX, LHD, and W7-X are employed in the benchmarks. It is found that the assumption of $ \boldsymbol E \times \boldsymbol B $ incompressibility causes discrepancy of neoclassical radial flux and parallel flow among the models when $ \boldsymbol E \times \boldsymbol B $ is sufficiently large compared to the magnetic drift velocities. For example, $ \mathcal{M}_p \leq 0.4$ where $\mathcal{M}_p$ is the poloidal Mach number. On the other hand, when $\boldsymbol E \times \boldsymbol B$ and the magnetic drift velocities are comparable, the tangential magnetic drift, which is included in both the global and ZOW models, fills the role of suppressing unphysical peaking of neoclassical radial-fluxes found in the other local models at $E_r \simeq 0$. In low collisionality plasmas, in particular, the tangential drift effect works well to suppress such unphysical behavior of the radial transport caused in the simulations. It is demonstrated that the ZOW model has the advantage of mitigating the unphysical behavior in the several magnetic geometries, and that it also implements evaluation of bootstrap current in LHD with the low computation cost compared to the global model.'
author:
- 'B. Huang'
- 'S. Satake'
- 'R. Kanno'
- 'H. Sugama'
- 'S. Matsuoka'
date: 'Oct. 2016'
title: 'Benchmark of the Local Drift-kinetic Models for Neoclassical Transport Simulation in Helical Plasmas'
---
Introduction
============
The magnetic field geometry of a fusion device is given by the external coil system and the plasma current. One of the advantages of stellarator/heliotron configuration compared to axisymmetric tokamaks is that plasma current is not necessary to sustain the confinement magnetic field. However, owing to the geometry, the helical ripple enhances the neoclassical radial particle and energy transport. Therefore, optimization of the field geometry is required for minimizing the neoclassical transport together with stabilizing the magnetohydrodynamics (MHD) equilibrium and improving the fast particle confinement.[@Grieger1992] The future fusion device will be operated in the higher-beta and higher-temperature condition compared to that in the present experimental devices. In such a collisionless and high pressure gradient plasma, the neoclassical bootstrap current is supposed to increase enough to interact with the MHD equilibrium. A self-consistent algorithm is required to investigate both the optimization of the neoclassical transport and the MHD equilibrium. The algorithm must satisfy both the efficiency and the accuracy in order to evaluate a quantitative study for the design of a fusion reactor.
From this viewpoint, neoclassical transport in helical plasmas has been investigated by transport codes based on several local approximations, for example, DKES[@Rij_1989] [@Hirshman1986], GSRAKE[@Beidler_2001], EUTERPE[@JMGarcia_PPCF55_2013_074008], and NEO-2[@Kernbichler_2008] et al. A comprehensive cross-benchmark of several local codes has been presented in Ref. [@Beidler_nf2011] In the local models, the tangential grad-B and curvature drift on the flux surfaces is often assumed to be negligibly small compared to the parallel motion and $\boldsymbol E \times \boldsymbol B$ drift. Further, the mono-energy assumption is sometimes employed in the local neoclassical codes, in which the momentum conservation of the collision operator is broken because the Lorentz pitch-angle scattering operator is adopted. Note that momentum correction techniques by Taguchi[@Taguchi_1992], Sugama-Nishimura[@penta_theory_2002][@penta_code_2005][@spong_2005], and Maaßberg[@Maassberg_2009] have been devised to recover the parallel momentum balance. Several benchmarks have shown that the momentum correction affects the quantitative accuracy of neoclassical transport calculations in helical plasmas[@Maassberg_2009][@Tribaldos_2011], especially in the quasi-axisymmetric HSX plasma.[@Lore_2010][@Briesemeister_PPCF_55_014002_2013]
The recent studies indicate the contribution of the magnetic tangential drift[@Matsuoka2015][@Landreman_2014] [@sugama_2016] in the evaluation of radial neoclassical transport when the $\boldsymbol E\times \boldsymbol B$ drift velocity is slower than the magnetic drift. Matsuoka[@Matsuoka2015] has devised a way to include the tangential magnetic drift in the local drift-kinetic equation solver. There are also some global neoclassical codes which treat the full 3-dimensional guiding-center motion including both the radial and tangential magnetic drift term. However, only a few global neoclassical codes have been applied on helical plasmas.[@Satake_2008][@Murakami_gnet_nf2000][@Tribaldos_PPCF_2005_545] Compared with the local codes, the global codes are stricter solutions to evaluate the drift-kinetic equation with the finite magnetic drift effect, but it takes more computational resources than the local codes. Therefore, it is almost impossible to utilize the global codes to investigate the interaction between bootstrap current and MHD equilibrium because it requires iterations between neoclassical transport and MHD simulations. The local approximations are appropriate for the purpose, but this has not been thoroughly verified among the neoclassical local models with global ones to guarantee the quantitative reliability of the neoclassical radial flux and parallel flow obtained from these local drift-kinetic models.
In this paper, following the previous study by Matsuoka[@Matsuoka2015], the neoclassical transport is examined with four types of neoclassical transport codes in Large Helical Device(LHD), Helically Symmetric Experiment(HSX), and Wendelstein 7-X(W7-X). The series of numerical simulations are carried out by the $\delta$-f drift-kinetic equation solver FORTEC-3D[@Satake_2008]. In the beginning, FORTEC-3D was developed as a global neoclassical transport code; recently, it has been extended to treat several types of the local drift-kinetic models[@Matsuoka2015]. The following approximations are employed to evaluate the neoclassical transport. (a) The global model takes the minimum assumption, which considers both the tangential and radial magnetic drift on the convective derivative term on the perturbed distribution, $\boldsymbol{v}_{m} \cdot\nabla \delta f$. The global model solves the drift-kinetic equation in 5-dimensional phase space. (b) The zero orbit width model (ZOW) excludes the radial component of magnetic drift, and it becomes a local neoclassical model. The magnetic drift term is treated as $$\label{eq:hat_v_m}
\hat{\boldsymbol{v}}_{m} \equiv \boldsymbol{v}_{m} - ( \boldsymbol{v}_{m} \cdot \nabla \psi ) \boldsymbol{e}_{\psi},$$ where $\psi$ is a flux-surface label and $\boldsymbol{e}_{\psi} \equiv \partial \boldsymbol{X} / \partial \psi $. The [*local*]{} indicates the neglect of radial drift in the guiding-center equation of motion. Therefore, the ZOW model becomes a 4-dimensional model and reduces computational resources. However, the ZOW model breaks Liouville’s theorem in the phase space. The ZOW model requires a modification in the delta-f method to solve the model properly as will be explained in Sec. \[sec:delta\_f\_scheme\]. (c) The zero magnetic drift (ZMD) model takes a further approximation. The ZMD model ignores not only the radial magnetic drift but also the tangential magnetic drift from a particle orbit. Then, the magnetic drift term in the drift-kinetic equation is treated as $\boldsymbol{v}_{m}\cdot\nabla\delta f = 0$. Liouville’s theorem is satisfied in the ZOW. (d) The DKES model further employs mono-energetic assumption, i.e., $\dot{v}(\partial \delta f/\partial v ) =0$, and the incompressible $\boldsymbol E \times \boldsymbol B$ drift approximation.[@Rij_1989][@Hirshman1986] With the Lorentz pitch-angle scattering operator, the drift-kinetic equation in DKES model reduces to a 3-dimensional model.
The remainder of this paper is organized as follows. In Sec.\[sec:Local\_Drift\_kinetic\_Models\],the drift kinetic equations based on global, ZOW, ZMD and DKES models are described. The conservation properties of the phase-space volume of each model is also discussed in this section. Then, the numerical scheme of the $\delta f$ method is explained briefly in Sec.\[sec:delta\_f\_scheme\]. The particle, parallel momentum, and energy balance equations in each drift-kinetic model are examined in Sec.\[section:Drift-kinetic\_Equation\]. In Sec.\[sec:result\], the simulation results are presented. The drift-kinetic models are benchmarked by the neoclassical fluxes such as the radial particle flux, radial energy flux, and flux-surface average parallel mean flow. The effect of $\boldsymbol E \times \boldsymbol B$, the effect of magnetic drift, and the electron neoclassical transport are analyzed. Finally, the bootstrap current is presented. A summary is given in Sec.\[sec:summary\]. In Appendix \[AppendixA\], the property of the source/sink term is presented. In Appendix \[AppendixB\], the derivations of the second-order viscosity tersors $\boldsymbol{\Pi}_{2}$ for the local models are presented.
Local Drift-kinetic Models {#sec:Local_Drift_kinetic_Models}
==========================
The neoclassical transport simulations are carried out by the $\delta f$ method under the following transport ordering assumptions. The gyro-radius $\rho$ is small compared with the typical scale length $L$, i.e., $\rho/L\sim \mathcal{O} (\delta)$, where $\delta$ represents a small ordering parameter. It is assumed that the plasma time evolution is slow, $$\frac{\partial}{\partial t} \sim \mathcal{O} \left(\delta^2 \frac{v_{th}}{L}\right)$$ where $v_{th}=\sqrt{2T/m}$ is thermal velocity. The order of magnitude of the $\boldsymbol E\times \boldsymbol B$ drift velocity is assumed as $$\frac{v_{E} }{v_{th}} \sim \mathcal{O} \bigg ( \frac{ \rho }{L} \bigg ) \sim \mathcal{O} ( \delta),$$ where the $\boldsymbol E \times \boldsymbol B$ drift velocity is given as $$v_{E} \equiv \frac{ | \boldsymbol E \times \boldsymbol B | }{ B^2 }.$$ If the radial electric field satisfies the ambipolar conditions, its magnitude is assumed as $$\label{eq:Mp}
\mathcal{M}_p \equiv \frac{ v_{E}}{v_{th} } \frac{ B }{ B_p }
\sim \frac{ E_r }{ v_{th} B_{ax} } \frac{ q R_{ax} }{ r }
$$ where $B_p$ and $B_{ax}$ are the poloidal magnetic field strength and the magnetic field strength on the magnetic axis, respectively. $r$, $R_{ax}$, and $q$ denote the minor radius, the major radius of the magnetic axis, and the safety factor, respectively. In the present work, the order of magnitude $\mathcal{M}_p \sim 1$ for ions is allowed on the local drift-kinetic simulations because (a) the ion thermal velocity is much slower than the electron and (b) the order of the poloidal magnetic field magnitude is approximately $$B_p \sim \frac{ r }{ q R_{ax} }B_{ax} \sim \mathcal{O} ( \delta B).$$ Even though $\mathcal{M}_p\sim 1$ is allowed, it still assumes that the slow-flow ordering is valid, $v_{E} / v_{th} \ll 1$.
The guiding-center distribution function of species $a$ is denoted as $f_{a}(\boldsymbol{Z}, t)$. The guiding-center variable $\boldsymbol{Z}$ is chosen as $\boldsymbol{Z} \equiv (\boldsymbol{X}, v, \xi; t )$ with the guiding-center position $\boldsymbol{X}$, guiding-center velocity $v$, and the cosine component of parallel velocity pitch-angle $\xi \equiv v_{\parallel} / v$. The parallel velocity $v_{\parallel}$ is defined as $v_{\parallel} \equiv \boldsymbol{v} \cdot \boldsymbol{b}$ where $\boldsymbol{b} \equiv \boldsymbol{B} / | \boldsymbol{B} |$ is a unit vector of the magnetic field. In Boozer coordinates, the position vector $\boldsymbol{X}$ is assigned as $\boldsymbol{X} \equiv (\psi, \theta, \zeta)$, where $\psi$, $\theta$, and $\zeta$ are toroidal magnetic flux, poloidal angle, and toroidal angles, respectively. The magnetic field $\boldsymbol{B}$ is given as $$\begin{aligned}
\boldsymbol{B} &= \nabla \psi \times \nabla \theta + \iota (\psi) \nabla \zeta \times \nabla \psi \\ &= I(\psi) \nabla \theta + G(\psi) \nabla \zeta + {\beta}^{*} ( \psi, \theta , \zeta )\nabla \psi.\end{aligned}$$ where $\iota (\psi)$ is defined as rotational transform. The radial covariant component ${\beta}^{*} ( \psi, \theta , \zeta )$ is assumed to be negligible because it does not influence the drift equation of motion up to the standard drift ordering $ \mathcal{O} ({\rho}/ L)$.
The guiding center drift-kinetic equation of species $ a$ is given by $$\label{eq:d-k-1}
\frac{ \partial f_{a} }{\partial t}
+ \frac{ d {Z}_{i} }{ d t } \frac {\partial f_{a} } { \partial {Z}_{i} }
= \mathcal{C}_a + \mathcal{S}_a,$$ where $\mathcal{C}_a$ is Coulomb collision operator and $\mathcal{S}_a$ is a source/sink term. The conservation law in the phase-space or the Liouville’s theorem is presented as $$\label{eq:j-variable}
\frac{ \partial \mathcal{J} }{\partial t}
+\frac{ \partial }{ \partial {Z}_{i} }\bigg( \mathcal{J} \frac{ d {Z}_{i} }{ d t } \bigg)
= \mathcal{J} \mathcal{G}.$$ Here, $\mathcal{J}$ represents the Jacobian of the phase space. $\mathcal{G} = 0$ if the trajectory follows the guiding center Hamiltonian. As the recent studies showed, the local drift-kinetic models are derived from approximation of the guiding center motion but do not satisfy the Hamiltonian. Therefore, $\mathcal{G} = 0$ is not guaranteed in general. For some local neoclassical models, the approximated guiding-center equations of motion $d {Z}_{i} / dt$ are chosen ingeniously to maintain $\mathcal{G} = 0$. To consider a general case, $\mathcal{G} \neq 0$ is retained in the following derivation. Combining Eqs. and , the conservative form of drift-kinetic equation is obtained as $$\label{eq:d-k-1-0}
\frac{ \partial \big( \mathcal{J} f_{a} \big ) }{\partial t}
+\frac {\partial } { \partial {Z}_{i} }\bigg ( \mathcal{J} f_{a} \frac{ d {Z}_{i} }{ d t } \bigg )
= \mathcal{J} \big [ \mathcal{C}_a + \mathcal{S}_a \big ]
+ \mathcal{J} f_{a} \mathcal{G},$$ which is used in taking the moments of drift-kinetic equation in section \[section:Drift-kinetic\_Equation\].
Global Drift-kinetic Model {#sec:Global}
--------------------------
The original FORTEC-3D is a global drift-kinetic code of which guiding center motion satisfies the Hamiltonian. FORTEC-3D treats the drift-kinetic equation for the perturbed distribution function and equation as follows: the distribution function $f_{a} $ is decomposed into a Maxwellian $f_{a,M}$ and perturbation $f_{a,1}$ $$\label{eq:fff}
f_{a,1} (\boldsymbol{X}, v,\xi , t ) \equiv f_{a} (\boldsymbol{X}, v, \xi, t ) - f_{a,M} ( \psi, v ),$$ where the local Maxwellian $f_{a,M}$ is defined as $$\begin{aligned}
\label{eq:f_M_local}
f_{a,M} &= n_{a}(\psi) \Bigg \lgroup
\frac{ m_{a} }{ 2 \pi T_{a}(\psi) } \Bigg \rgroup ^{3/2}
\cdot
\mathrm{exp} \Bigg \lgroup - \frac{ m_a v^{2} }{ 2 T_{a} (\psi) } \Bigg \rgroup,\end{aligned}$$
$$\begin{aligned}
\label{eq:DKE0-1}
\bigg ( \frac{ \partial } { \partial t } & + \dot{\boldsymbol Z} \cdot \frac{ \partial }{ \partial \boldsymbol{ Z } } \bigg ) f_{a,1}
= \mathcal{S}_{a,0} + \mathcal{C}^L ( f_{a,1}) + \mathcal{S}_{a,1},\end{aligned}$$
where $ \dot{\boldsymbol Z} = \frac{d}{dt} ( \boldsymbol X, v,\xi ) $ and $\mathcal{C}^{L} ( f_{a,1})$ is a linearized Fokker-Planck collision operator $$\begin{aligned}
\label{eq:collision-0}
\mathcal{C}^L ( f_{a} ) &= \sum_{b}\mathcal{C} ( f_{a,M}, f_{b,1}) + \mathcal{C} ( f_{a,1}, f_{b,M}),\end{aligned}$$ and the source term $\mathcal{S}_{a,0}$ is defined as $$\label{eq:S0_f3d}
\mathcal{S}_{a,0} \equiv- \frac{d \boldsymbol Z }{dt} \cdot\frac{\partial}{\partial\boldsymbol Z} f_{a,M} = - \bigg ( {\dot v} \frac{\partial}{\partial \psi} + {\dot \psi} \frac{\partial}{\partial \psi} \bigg) f_{a,M}.$$ On the other hand, $\mathcal{S}_{a,1} $ is an additional source/sink term, which helps the numerical simulation to reach a quasi-steady state. The $\mathcal{S}_{a,1} $ is discussed in Sec. \[section:particle\_flux\] and \[sec:parall\_monentum\_balance\].
The guiding-center trajectory is given as follows:[@littlejohn1983]
\[eq:f3d-o1\] $$\begin{aligned}
{3}
\dot{ \boldsymbol X } =& v \xi \boldsymbol{b} + \frac{1}{ e_a B_{\parallel}^* } \boldsymbol{b} \times \bigg \{ m_a (v \xi)^2 \boldsymbol{b} \cdot \nabla \boldsymbol{b} + \mu \nabla B - e_a\boldsymbol{E}^* \bigg \},
\\
\frac{d v }{dt} =& \frac{ e_a}{m_a v} \dot{ \boldsymbol X } \cdot \boldsymbol{E}^{*}
+ \frac{ \mu }{m_a v} \frac{ \partial B }{ \partial t } ,
\\
\frac{d \xi }{dt} =& - \frac{ \xi}{ v }\frac{dv}{dt}
- \frac{ \boldsymbol{b} }{ m_a v} \cdot ( \mu \nabla B - e_{a}\boldsymbol{E}^* ) + \xi \frac{d \boldsymbol X}{dt} \cdot
\boldsymbol{\kappa},\end{aligned}$$
where, $m_a$ and $e_a$ denote the mass and charge of the species $a$, and
$$\begin{aligned}
{7}
& \mu \equiv \frac{m_a v^2}{2B} ( 1 - {\xi}^2 ),
\\
& \boldsymbol A^{*} \equiv \boldsymbol A + \frac {m_a v \xi}{e_a} \boldsymbol { b },
\\ \label{eq:E_ast}
& \boldsymbol E^{*} \equiv - \frac{ \partial \boldsymbol A^{*} }{ \partial t } - \nabla \Phi,
\\
& \boldsymbol B^{*} \equiv \nabla \times \boldsymbol A^{*},
\\ \label{eq:B_para_ast}
&B^{*}_{\parallel} \equiv \boldsymbol { b } \cdot \boldsymbol B^{*},
\\ \label{eq:kappa}
& \kappa \equiv \left ( { \boldsymbol b} \cdot \nabla \right ) { \boldsymbol b}.\end{aligned}$$
The trajectory is derived from Hamiltonian so that it satisfies the Liouville equation, i.e., $$\label{eq:f3d-g=0}
\mathcal{JG}=0$$ Note that the phase-space Jacobian in Boozer coordinates is $$\mathcal{J} = \frac{ 2 \pi B^{*}_{\parallel} v^{2} }{ B } \frac{ G + \iota I }{ B^{2} }.$$
Zero Orbit Width(ZOW) Model
---------------------------
The zero orbit width (ZOW) approximation[@Matsuoka2015] is a local drift-kinetic model, which ignores only the radial drift $\dot \psi ~ \partial f_{1} / \partial \psi$. The subscript of particle species is omitted here and hereafter unless it is necessary. The drift-kinetic equation Eq. becomes $$\begin{aligned}
\label{eq:zow-f1}
&
\bigg ( \frac{ \partial } { \partial t }
+ {\dot {\boldsymbol{Z}} }^{\text{zow}} \cdot\frac{ \partial }{ \partial \boldsymbol{Z} } \bigg ) f_{1}
= \mathcal{S}_{0} + \mathcal{C}^L ( f_{1}) + \mathcal{S}_{1}\end{aligned}$$ where $\dot { \boldsymbol{Z} }^{\text{zow}} = \frac{d}{dt} ( \theta, \zeta, v, \xi ) $. In the present study, stationary electromagnetic field approximation is assumed $$\label{eq:dBdt=dphidt=0}
\frac{ \partial B }{ \partial t } = \frac{ \partial \Phi }{ \partial t } = 0.$$ Thus, the electric field is approximated as $$\label{eq:dEdt=0}
\boldsymbol E^* \simeq - \nabla \psi \frac{d \Phi}{d \psi}$$ where $\Phi = \Phi (\psi) $ is the electrostatic potential, which is assumed to be a flux-surface function for simplicity. Other approximations employed in local models are $ \boldsymbol{B}^* \cdot { \boldsymbol b } \simeq B$ and $$\begin{aligned}
\label{eq:kappa-app}
\boldsymbol \kappa \simeq \frac{ {\nabla}_{\perp} B }{ B }.\end{aligned}$$ Here, the $\mathcal{O}(\delta)$ correction in $B^{*}_{\parallel}$ is neglected. When $\beta \equiv p/(B^2/2\mu_0) $, one has $$\begin{aligned}
\label{eq:kappa-app-1}
\boldsymbol \kappa \nonumber
&= { \boldsymbol b} \times \left ( \frac{ \nabla B \times \boldsymbol B }{ B^2 } - \frac{ \nabla \times \boldsymbol B }{ B} \right )
\\ \nonumber
&= \frac{1}{B} \left( \nabla B - { \boldsymbol b} \cdot \nabla B \right)
+ \mu_{0} \frac{ \boldsymbol J \times \boldsymbol B }{ B^2 }
\\ \nonumber
&= \frac{ {\nabla}_{\perp} B }{ B } + \mu_{0} \frac{ \boldsymbol J \times \boldsymbol B }{ B^2 }
\\
&= \frac{ {\nabla}_{\perp} B }{ B } + \frac { \mu_{0} \nabla p } { B^2 }
\simeq \frac{ {\nabla}_{\perp} B }{ B } + \mathcal{O} ( \beta )\end{aligned}$$ where $p = p(\psi)$ denotes the scalar pressure. The second term is negligible in low-$\beta$ approximation.
The particle trajectories ${\dot {\boldsymbol{Z}} }^{\text{zow}}$ are treated as if they are crawling on a specific flux surface and given as follows:
\[eq:zoworbit\] $$\begin{aligned}
{3}
\dot{ \boldsymbol{X} } =& v \xi { \boldsymbol{b} } + \boldsymbol{v}_{E} + \hat {\boldsymbol{v}}_{m},
\\
\dot{ v } =& \frac{ -e_{a} }{ m_{a} v } \boldsymbol{v}_{m} \cdot \nabla \psi \frac{ d \Phi }{ d \psi },
\\ \nonumber
\dot{ \xi } =& - \frac{ 1 - {\xi}^{2} }{ 2 B } \bigg ( v \boldsymbol{b} \cdot \nabla B \bigg) \nonumber
\\
& - \xi ( 1 - { \xi}^{2}) \frac{ d \Phi }{ d \psi } \frac{ \boldsymbol{B} \times \nabla B }{ 2 B^3 } \cdot \nabla \psi .\end{aligned}$$
Note that the radial magnetic drift ${\boldsymbol v}_{m} \cdot \nabla \psi$ is still kept in the time evolution of velocity $\dot v$. Even though the $\dot{\psi} \partial f_{1}/\partial \psi$ term is neglected in the LHS of Eq., the source/sink term $S_0 \propto \dot{\psi}$ in the RHS is the same as Eq. in the global model. The $\boldsymbol{E} \times \boldsymbol{B}$ drift is defined as $$\label{eq:vE}
\boldsymbol{v}_{E} \equiv \frac{ ~ d \Phi ~}{ d \psi} \frac{~ \boldsymbol{B} \times \nabla \psi ~}{ B^2}$$ and the magnetic drift is defined as $$\label{eq:vm-0}
\boldsymbol{v}_{m} \cdot \nabla \psi \equiv \frac{ m_{a}v^2 }{2 e_{a} B^3} \bigg ( 1+\xi^2 \bigg) \boldsymbol{B} \times \nabla B \cdot \nabla \psi.$$ The radial virtual drift velocity $\dot{\psi}$ in the local model is evaluated from $ \nabla \psi\cdot $ product of Eq. , and the tangential magnetic drift $\hat{\boldsymbol v}_m$ is defined as in Eq., $$\hat{\boldsymbol{v}}_{m} \equiv \boldsymbol{v}_{m} -\dot{\psi} \boldsymbol{e}_{\psi}.$$ The Jacobian of ZOW in phase space becomes $$\mathcal{J} = \frac{ 2 \pi B^{*}_{\parallel} v^{2} }{ B } \frac{ G + \iota I }{ B^{2} } \simeq 2 \pi v^{2} \frac{ G + \iota I }{ B^{2} }.$$
The guiding-center equations of motion Eq. does not include the radial drift term $ \dot \psi$ and disobeys Hamiltonian. As a result, $\dot{\boldsymbol{Z}}^{\text{zow}}$ is compressible on 4-dimensional phase space where $$\label{eq:gneq0}
\mathcal{G} = \nabla_z \cdot \dot{\boldsymbol Z }^{\text{zow}} =\frac{1}{\mathcal{J}}\frac{\partial}{\partial Z_i}\cdot
(\mathcal{J}\dot{Z}_i)\neq 0.$$ Here, $\nabla_z$ represents the divergence in the phase space. Following and , the variation of phase-space volume along the guiding-center trajectories is $$\begin{aligned}
\label{eq:zow-J}
\nabla_z \cdot \dot{\boldsymbol Z }^{\text{zow}}
& = \frac{ m v^{2} ( 1 + {\xi}^{2} ) }{2eB(G+\iota I)} \Bigg \{ \frac{3}{B} \frac{ \partial B}{ \partial \psi } \Bigg ( I \frac{ \partial B }{ \partial \zeta } - G \frac{ \partial B }{ \partial \theta } \Bigg ) \nonumber
\\ &+ \Bigg ( G \frac{\partial ^2 B }{\partial\psi \partial \theta} - I \frac{\partial ^2 B }{\partial\psi \partial \zeta} \Bigg ) \Bigg \}.\end{aligned}$$ This term affects the balance equation of particle number, parallel momentum, and energy, which will be discussed in Sec. \[section:Drift-kinetic\_Equation\].
Zero Magnetic Drift (ZMD) Model
-------------------------------
The zero magnetic drift (ZMD) model is similar to ZOW. It follows Eq. but it excludes all the magnetic drift term in $\dot{ \boldsymbol X } $. The particle trajectories of ZMD is given as the following:
\[eq:zmdorbit\] $$\begin{aligned}
{3}
\dot{ \boldsymbol{X} } =& v \xi { \boldsymbol{b} } + \boldsymbol{v}_{E},
\\
\dot{ v } =& \frac{ -e_{a} }{ m_{a} v } \boldsymbol{v}_{m} \cdot \nabla \psi \frac{ d \Phi }{ d \psi },
\\
\dot{ \xi } =& - \frac{ 1 - {\xi}^{2} }{ 2 B} \bigg ( v \boldsymbol{b} \cdot \nabla B \bigg) \nonumber
\\ & - \xi ( 1 - { \xi}^{2}) \frac{ d \Phi}{ d \psi } \frac{ \boldsymbol{B} \times \nabla B }{ 2 B^3 } \cdot \nabla \psi.\end{aligned}$$
Following the ZMD 4-dimensional guiding-center orbit, the incompressibility of the phase-space volume $\mathcal{G} = 0$ is still retained.
DKES-like Model {#sec:DKES-like}
---------------
The DKES-like model takes a further approximation on ZMD, that is, the mono-energetic assumption $\dot v = 0$. Then, the DKES-like model is reduced to be a 3-dimensional problem, in which $\dot{\boldsymbol{Z}}^{\text{dkes}} = d/dt(\psi, \theta, \zeta)$ on the LHS of the drift-kinetic equation. Following the trajectory Eq. and the mono-energetic particle approximation $\dot v = 0$, the phase space volume is not conserved: $$\label{eq:dles-J}
\nabla_z \cdot \dot { \boldsymbol{Z}}^{ \text{dkes} } = \frac{ 3 ( 1 + {\zeta}^2 ) }{ 2 B^3 \mathcal{J} } \bigg ( G \frac{ \partial B }{ \partial \theta} - I \frac{ \partial B }{ \partial \zeta } \bigg ) \frac{d \Phi}{d \psi}.$$ In order to maintain $\mathcal{G} = 0$, the electric potential $\nabla \Phi$ is replaced by $$\nabla \Phi \simeq \nabla \Phi \frac{B^2}{ \langle B^2 \rangle }$$ and the incompressible $\boldsymbol{E} \times \boldsymbol{B}$ drift is denoted as $$\label{eq:imcompressible}
\hat{ \boldsymbol v }_{E} \equiv \frac{ \boldsymbol{E} \times \boldsymbol{B} }{ \langle B^2 \rangle }.$$ In summary, the guiding-center trajectory in the DKES-like model is given as follows:
\[eq:deksorbit\] $$\begin{aligned}
{3}
\dot{ \boldsymbol{X} } =& v \xi { \boldsymbol{b} } + \hat{\boldsymbol{v}}_{E},
\\
\dot{ v } =& ~0,
\\
\dot{ \xi } =& - \frac{ (1 - {\xi}^{2} )v }{ 2 B} \boldsymbol{b} \cdot \nabla B,\end{aligned}$$
and the particle trajectory conserves the phase space volume, $ \mathcal{G} = \nabla_z \cdot \dot { \boldsymbol{Z}}^\text{dkes}=0$.
In the original DKES code, the collision operator is simplified by the Lorentz pitch-angle scattering operator $$\label{eq:pitch-angle_scattering}
\mathcal{L} f_a = \frac{ \nu_{ab} }{2} \frac{ \partial }{ \partial \xi } ( 1 - {\xi}^2 ) \frac{ \partial }{ \partial \xi } f_a,$$ where the particle does not change the velocity either by guiding-center motion or by collision. However, in the series of simulations in this paper, all models use the same linear collision operator to benchmark neoclassical transport. The linear collision operator includes the energy scattering term and field-particle part to maintain the conservation property of Fokker-Plank operator[@Satake_2008]. Between the original DKES and the DKES-like in the simulation, the effects of different collision operators appear in a quasi-symmetric geometry because of the conservation of momentum. It is essential to evaluate neoclassical transport as discussed in Sec. \[sec:large\_Er\].
Two-weight $\delta f$ Scheme {#sec:delta_f_scheme}
----------------------------
The two-weight $\delta f$[@Satake_2008][@Hu_1994] scheme is employed to solve the global and local drift-kinetic models in Section \[sec:Global\] - \[sec:DKES-like\]. Let us briefly explain the two-weight $\delta f$ scheme in the case of $\mathcal G \neq 0$. The weight functions ${\mathfrak{w}}$ and ${\mathfrak{p}}$ are given as follows:
\[eq:w\_p=gf\] $$\begin{aligned}
{2}
f_{1} ({\boldsymbol Z} ) &=g ({\boldsymbol Z} ) {\mathfrak{w}}({\boldsymbol Z} ) \label{eq:f1=gw}
\\
f_{M} ({\boldsymbol Z} ) &=g ({\boldsymbol Z} ) {\mathfrak{p}}({\boldsymbol Z} ) \label{eq:fM=gp}\end{aligned}$$
where $g( \boldsymbol Z)$ is the marker distribution function. Following Eq., an operator includes the total derivative along the particle trajectory and the test-particle collision is defined as $$\begin{aligned}
\label{eq:Df_1/Dt}
\frac{ D f_{1} }{ Dt } &\equiv \frac{ \partial f_{1} }{ \partial t } + \dot{\boldsymbol Z} \cdot \frac{ \partial f_{1} }{ \partial \boldsymbol Z } - \mathcal{C}_T ( f_{1} ) \nonumber
\\
&= \mathcal{S}_{0} + \mathcal{S}_{1} +\mathcal{C}_F\end{aligned}$$ Here, we employ the linearized collision operator decomposed into the test-particle part $ \mathcal{C}_T$ and the field-particle part $\mathcal{C}_F$[@Satake_2008][@Z_Lin_1995]. The former is implemented by the Monte Carlo method and the latter is constructed so as to satisfy the conservation properties of particle number, parallel momentum, and energy for like-species collisions. In the ion calculation, the ion-electron collision is neglected because of the large mass-ratio $m_e/m_i\ll 1$, while the electron-ion collision is approximated by the pitch-angle scattering operator Eq. with stationary background, that is, Maxwellian ions.
According to Eqs. and , the drift-kinetic equation of marker distribution $g( \boldsymbol Z)$ is obtained $$\label{eq:Dg/Dt}
\frac{ D g }{ Dt } = - g~\mathcal{G}.$$ Eq. is extended with Eq. $$\label{eq:extend_Df_1/Dt}
\frac{ D f_{1} }{ Dt } = {\mathfrak{w}} \frac{ D g }{ Dt } + g \frac{ D {\mathfrak{w}} }{ Dt }.$$ Following , , and , the time evolution of the weight function ${\mathfrak{w}}$ is obtained $$\begin{aligned}
\label{eq:w0}
\dot{{\mathfrak{w}}}&= \frac{1}{g} \frac{ D f_{1} }{ Dt } - \frac{{\mathfrak{w}}}{g} \frac{ D g }{ Dt }
\\ \nonumber
&= \frac{ {\mathfrak{p}} }{ f_{M} } \bigg ( \mathcal{S}_{0} + \mathcal{S}_{1} +\mathcal{C}_{F} ( f_{M} ) \bigg ) + {\mathfrak{w}} \mathcal{G}\end{aligned}$$ Similarly, the time evolution of the weight function $\mathfrak{p}$ is obtained as follows: $$\begin{aligned}
\label{eq:p0}
\mathfrak{\dot p} = \frac{ \mathfrak{p} }{ f_{M} } \bigg ( \dot{\boldsymbol{Z}} \cdot \frac{ \partial }{ \partial \boldsymbol Z } \bigg ) f_{M} + \mathfrak{p} \mathcal{G}.
$$ The time evolution of the weights $\mathfrak{w}$ Eq. and $\mathfrak{p}$ include $\mathcal{G}$ which is non-zero in the ZOW model only. The last term in Eqs. (35) and (36) is required so that the two-weight $\delta f$ scheme is applicable to the case in which the phase-space volume is not conserved. Note that $\dot{\boldsymbol Z}$ in RHS of Eq. depends on the drift-kinetic models. For the global model, it is denoted as $$\dot{\boldsymbol{Z}} \cdot \frac{ \partial f_{M} }{ \partial \boldsymbol Z } = - \mathcal{S}_{0};$$ for ZOW and ZMD, it is denoted as $$\dot{\boldsymbol{Z}} \cdot \frac{ \partial f_{M} }{ \partial \boldsymbol Z } = \dot{v} \frac{ \partial }{ \partial v } f_{M};$$ for the DKES-like, due to $\dot v = 0 $, $\dot \psi = 0$, and $\mathcal{G} = 0$, the weight function $\mathfrak{p}$ becomes constant as $$\mathfrak{\dot p} =0.$$
Moments of Drift-kinetic Equation {#section:Drift-kinetic_Equation}
=================================
In this section, the balance equations of particle number, parallel momentum, and energy are investigated for global and local models. The compressibility of phase space $\mathcal G$ and the approximations on guiding-center trajectories in each model are taken into account. The requirement of adaptive source-sink term $S_1$ is explained, which is essential for obtaining a steady-state solution in some models.
In order to take moments of Eq., consider an arbitrary function $\mathcal{A} ( \boldsymbol{X}, v, \xi, t ) $ which is independent of the gyro-phase. For density variable $\int d^3 v f \mathcal{A}$ in $\boldsymbol{X}$-space, the balance equation is yielded by multiplying $\mathcal{A}$ with Eq. and taking integral over the velocity-space. By partial integral, Eq. is rewritten as $$\begin{aligned}
\label{eq:d-k-A-1}
& \frac{ \partial }{ \partial t } \Bigg ( \int d^{3}v ~ f_{a} \mathcal{A} \Bigg )
+ \nabla \cdot \Bigg ( \int d^{3}v ~ f_{a} ~ \mathcal{A} \dot{ \boldsymbol{X} } \Bigg )
\nonumber \\
&= \int d^{3}v ~ \Bigg ( f_{a} \frac{ d \mathcal{A} }{ d t } + \big [ \mathcal{C}_a+\mathcal{S}_a \big ] \mathcal{A} \Bigg )
\nonumber \\
&+ \int d^3v ~ f_{a} \mathcal{G} \mathcal{A},\end{aligned}$$ where the integral of velocity-space is given as $$\int d^3v = 2 \pi \int d v v^2 \int d \xi \mathcal{J}.$$ Furthemore, the following equation is employed to derive Eq. $$\frac{ d \mathcal{A} }{ d t }
\equiv \Bigg ( \frac{\partial }{\partial t}
+ \dot{ \boldsymbol{Z} } \cdot \frac{\partial }{\partial \boldsymbol{Z}}
\Bigg ) \mathcal{A}.$$
The Particle and Energy Balance on the Local DKE Models {#section:particle_flux}
-------------------------------------------------------
In order to derive the conservation law of particle number, substituting $\mathcal{A} = 1$ into Eq. yields $$\begin{aligned}
&\frac{ \partial }{ \partial t } \Bigg ( \int d^{3}v ~ f_{a} \Bigg )
+ \nabla \cdot \Bigg ( \int d^{3}v~ f_{a} \dot{ \boldsymbol{X} } \Bigg )
\nonumber \\
& = \int d^{3}v ~ \mathcal{S}_{a} + \int d^3v ~ f_{a} \mathcal{G}\end{aligned}$$ where $\int d^3 v ~ \mathcal{C}_a =0$ is used. The continuity equation is obtained as $$\begin{aligned}
\label{eq:m0}
\frac{ \partial n_{a} }{ \partial t }
+ \nabla \cdot ( n_{a} \boldsymbol V_a )
= \int d^{3}v ~ \mathcal{S}_{a} + \int d^{3}v f_{a} \mathcal{G} .\end{aligned}$$ The density $n_a$ and the mean flow velocity $n_{a} \boldsymbol V_a$ are defined as
$$\begin{aligned}
{2}
&
n_a \equiv \int d^3 v ~ f_a,
\\
& n_a \boldsymbol V_a \equiv \int d^3 v ~ \dot{\boldsymbol X} f_a.
$$
The balance of kinetic energy is obtained by substituting $\mathcal{A}=\mathcal{K}$ into Eq., $$\begin{aligned}
\label{eq:m2-kinetic}
&\frac{ \partial }{ \partial t } \Bigg ( \int d^{3}v ~ f_{a} \mathcal{K} \Bigg )
+ \nabla \cdot \Bigg ( \int d^{3}v ~ f_{a} \mathcal{K} \dot{ \boldsymbol{X} } \Bigg )
\nonumber \\
&= \int d^{3}v ~ \Bigg ( f_{a} \frac { d \mathcal{K} }{dt} + \big [ \mathcal{C}_{a}+ \mathcal{S}_{a} \big ] \mathcal{K} \Bigg )
\nonumber \\
&+ \int d^3v ~ f_{a} \mathcal{G}\mathcal{K} .\end{aligned}$$ Here the kinetic energy $\mathcal{K}$ is defined as $$\mathcal{K} \equiv \frac{1}{2} m_{a}{v_{\parallel}}^2 + \mu B = \mathcal{E} - e_{a} \Phi$$ where $\mu$ is the magnetic momentum, $\mathcal{E}$ is the total energy, and $\Phi$ is the electrostatic potential. The time derivative of the kinetic energy is denoted as $$\begin{aligned}
\label{eq:K_dot}
\frac { d \mathcal{K} }{dt} &= \frac{ d \mathcal{E}}{dt} - e_{a} \frac{ d \Phi}{dt}
\nonumber \\
&= \mu \frac{ \partial B ( \boldsymbol X , t ) }{ \partial t} + e_{a} \boldsymbol{ E }^{*} \cdot \frac { d \boldsymbol X}{dt},\end{aligned}$$ where $ \boldsymbol{ E }^{*} $ is defined in Eq.. In the series of simulations, a stationary electromagnetic field approximation is employed, which are Eqs. and . Therefore, Eq. is approximated as $$\label{eq:dkdt}
\frac { d \mathcal{K} }{dt} \simeq - e_{a}\frac{ d \Phi}{dt} = - e_{a}\frac{ d \boldsymbol X }{ d t} \cdot \nabla \Phi.$$ According to Eqs., , and , if the Liouville theorem is violated, $\mathcal{G}$ affects the particle, momentum, and energy balance. The approximation trajectories and balance equations in the local models are presented in the following subsections.
The flux-surface-average is denoted as $$\langle \mathcal{A} \rangle \equiv \frac{ ~ \int d \theta d \zeta ~ \mathcal{J} \mathcal{A} ~ }{ \mathcal{V}'},$$ where $\mathcal{A}$ is an arbitrary function and $\mathcal{V}'$ is defined as $$\mathcal{V}' \equiv \frac{d \mathcal{V} }{ d \psi} = \int d \theta d \zeta ~ \mathcal{J}.$$ The particle density from $f_{a,1}$ is denoted as $$\label{eq:n1}
\mathcal{N}_{1} \equiv \int d^3 v ~ f_{1} ( \boldsymbol Z ).$$ According to continuity equation Eq., the time evolution of density is $$\begin{aligned}
\label{eq:dz1_dt}
& \frac{\partial \left \langle \mathcal{N}_{1} \right \rangle }{ \partial t} + \left \langle \nabla \cdot \bigg ( \mathcal{N}_{1} \boldsymbol{V} \bigg ) \right \rangle \nonumber
\\
&= \left \langle \int d^3 v ~ \mathcal{S}_{1} ~\right \rangle +
\left \langle \int d^3 v ~ f_{1}~ \mathcal{G} \right \rangle .\end{aligned}$$ After taking the flux-surface-average, the contribution of $\mathcal{S}_{0}$ is zero because the source/sink term is a Maxwellian Eq. with flux-surface functions $n$ and $T$. Note here that in the global model the particle flow $ \mathcal{N}_{1} \boldsymbol{V} $ contains the radial component and $\mathcal{G}=0$. Then, Eq. for the global model becomes $$\label{eq:global_n_a1}
\frac{ \partial \langle \mathcal{N}_{1} \rangle }{ \partial t } + \frac{ d }{ d \mathcal{V} } \left ( \Gamma^{\psi} \mathcal{V}' \right )= \left \langle \int d^3 v \mathcal{S}_{1} \right \rangle,$$ where the particle flux is calculated by $$\label{eq:p-flux}
\Gamma^\psi \equiv
\left \langle \int d^3 v ~ f_{1} {\dot \psi} ~ \right\rangle,$$ and the following identity is employed $$\left \langle \nabla \cdot \boldsymbol A \right \rangle
= \frac{d}{ d \mathcal{V} } \left \langle \boldsymbol A \cdot \nabla \mathcal{V} \right \rangle
= \frac{1}{ \mathcal{V}' }\frac{d}{ d \mathcal{\psi} } \left \langle \mathcal{V}' \boldsymbol A \cdot \nabla \psi \right \rangle.$$ The finite $d(\Gamma^{\psi} \mathcal{V}')/d \mathcal{\mathcal{V}}$ term is a corollary of global simulation in which the actual radial particle flux across a flux surface is solved. Therefore, it is essentially required to include the particle source to obtain a steady-state solution. On the other hand, in the three local models, the $\mathcal{N}_{1} \boldsymbol{V}$ term has only the tangential component to the flux surface. Therefore, the $ \langle \nabla \cdot ( \mathcal{N}_{1} \boldsymbol{V}) \rangle $ term vanishes in ZOW, ZMD and DKES-like models. However, for ZOW, the artificial source/sink term $\mathcal{S}_{1}$ is required because of the compressibility $\mathcal{G} \neq 0$ [@Matsuoka2015], $$\begin{aligned}
\label{eq:particle_number_ZOW}
\frac{\partial \left \langle \mathcal{N}_{1} \right \rangle }{ \partial t}
= \left \langle \int d^3 v ~ \mathcal{S}_{1} ~\right \rangle +
\left \langle \int d^3 v ~ f_{1}~ \mathcal{G} \right \rangle.\end{aligned}$$ According to Eq., the last term in Eq. is estimated as $\mathcal{O}(\delta^2)$. For ZMD and DKES-like, the particle density $\mathcal{N}_{1}$ is constant naturally without $\mathcal{S}_1$, as pointed out by Landreman,[@Landreman_2014] $$\begin{aligned}
\label{eq:particle_number_ZMD_DKES-like}
\frac{\partial \left \langle \mathcal{N}_{1} \right \rangle }{ \partial t}
= 0.\end{aligned}$$
The energy balance equation ~~of energy~~ for each model is derived similarly, as follows. The energy flux is introduced as $$\label{eq:energy-flux}
\boldsymbol{Q} \equiv \int d^3 v f_{1} \mathcal{K} \dot{\boldsymbol{X}},$$ and the flux-surface-average of radial energy flux is defined as $$\label{eq:radial_q-flux}
Q^{\psi} \equiv \left \langle \int d^3 v f_{1} \mathcal{K} \dot{\psi} \right \rangle.$$ The pressure perturbation on flux surface is given as $$\label{eq:p1}
P_{1} \equiv \frac{2}{3} \int d^3 v f_{1} \mathcal{K}.$$ According to balance of kinetic energy Eq., the time evolution of $P_{1}$ is rewritten as $$\begin{aligned}
\label{eq:m2-kinetic-1}
& \frac{3}{2} \frac{ \partial }{ \partial t } \left \langle P_{1} \right \rangle
+ \left \langle \nabla \cdot \boldsymbol Q \right \rangle \nonumber
\\ \nonumber
&= \left \langle \int d^{3}v ~ f_{1} ~ \frac { d \mathcal{K} }{dt} \right \rangle
+ \left \langle \int d^{3}v ~ \mathcal{S}_{1} ~ \mathcal{K} \right \rangle
\\
&+ \left \langle \int d^3v ~ f_{1} ~ \mathcal{G} ~\mathcal{K}~ \right \rangle.\end{aligned}$$ In Eq., the contribution from $\mathcal{S}_0$ vanishes again. The energy exchange by collision is omitted because we neglect the ion-electron collision and the electron-ion collision is approximated by pitch-angle scattering in the simulations. In the RHS of Eq., the time evolution of kinetic energy is approximated as $$\begin{aligned}
&\left \langle \int d^{3}v ~ f_{1} ~ \frac { d \mathcal{K} }{dt} \right \rangle \nonumber
\\
&\simeq e E_{\psi} \left \langle \int d^{3}v ~\dot{\psi} ~ f_{1} \right \rangle
= e E_{\psi} ~ \Gamma^{\psi}, \end{aligned}$$ which represents the work done by the radial current. For the global model, the finite $\langle \nabla \cdot \boldsymbol Q \rangle = d ( Q^{\psi} \mathcal{V}' ) / d \mathcal{V} $ remains as in Eq.. Therefore, an energy source $\mathcal{S}_1 \mathcal{K} $ is essentially required to reach a steady-state. On the other hand, the radial energy flux $Q_a^{\psi}$ vanishes in the local models. For the ZOW and ZMD models, $\mathcal{S}_{1} \mathcal{K} $ is required to satisfy the balance equation of energy because of $d \mathcal{K} / dt $ and $\mathcal{G}$ $$\begin{aligned}
\label{eq:en_ZOW}
\frac{ \partial }{ \partial t } \left \langle P_{1} \right \rangle
&= \left \langle \int d^{3}v ~ \mathcal{S}_{1} ~ \mathcal{K} \right \rangle \nonumber
\\
& + e E_{\psi} ~ \Gamma^{\psi} + \left \langle \int d^3v ~ f_{1} ~ \mathcal{G} ~\mathcal{K}~ \right \rangle\end{aligned}$$ where $\mathcal{G}$ appears only in the ZOW model. Eq. indicates that ZMD cannot maintain the conservation law on energy when $E_{\psi} \neq 0$, even if it holds the constant particle number in Eq.. Finally, DKES-like maintains the energy balance without $\mathcal{S}_{1} \mathcal{K}$ because of $d \mathcal{K} /dt = 0 $ and $\mathcal{G} = 0 $.
Recently, Sugama has derived another type of ZOW model[@sugama_2016] in which guiding-center variables are chosen as $(\boldsymbol{X},v_\parallel,\mathcal{K})$ and the tangential magnetic drift is defined as $$\label{eq:zow_sgm}
\hat{\boldsymbol{v}}_m=\boldsymbol{v}_m-\frac{(\boldsymbol{v}_m\cdot\nabla\psi)}{|\nabla\psi|^2}\nabla\psi.$$ In this model, the magnetic moment $\mu$ is allowed to vary in time so that the kinetic energy $\mathcal{K}$ is conserved. It is shown that the new local model satisfies both particle and energy balance relations without source/sink term. Although such a conservation property is desirable as a drift-kinetic model, we employ Matsuoka’s ZOW model here for two reasons. First, the definition of tangential magnetic drift as in Eq. requires the geometric factor $|\nabla\psi|^2$ on each marker’s position, which will increase the computation cost. Second, it is necessary to find a modified Jacobian with which the phase-space volume conservation is recovered in this local model. To obtain such a modified Jacobian, another differential equation as Eq. (84) in Ref. 18 is required to be solved. Instead, in this paper, we adopt the source/sink term in ZOW and ZMD models after the verification as discussed in Appendix \[AppendixA\]. The verification shows that the source/sink term does not affect the long-term time average value of neoclassical fluxes after the simulation reaches a quasi-steady state.
The Parallel Momentum Balance and Parallel Flow {#sec:parall_monentum_balance}
-----------------------------------------------
The parallel momentum balance equation is derived from Eq. with $\mathcal{A} = m_{a} v_{\parallel}$, [@sugama_2016] $$\begin{aligned}
\label{eq:m1-parallel}
&\frac{\partial}{ \partial t } ( n_{a} m_{a} V_{a, \parallel} ) + \boldsymbol{b} \cdot ( \nabla \cdot \boldsymbol{P}_{a} )
\nonumber \\
&= n_{a} e_{a} E_{\parallel} + F_{\parallel,a}
+ \int d^{3}v ~ \mathcal{S}_{a} m_{a} v_{\parallel}
\nonumber \\
&+ \int d^3v ~ f_{a} \mathcal{G} m_{a} v_{\parallel},\end{aligned}$$ where $E_{\parallel} = \boldsymbol b \cdot\boldsymbol E$ and $\boldsymbol{P}_{a}$ is the pressure tensor. The parallel friction of collision $F_{a, \parallel}$ is given as $$F_{a, \parallel} \equiv \boldsymbol b \cdot \sum_{b\neq a}\boldsymbol{F}_{ab}=\sum_{b\neq a}\int d^3 v ~ \mathcal{C}_{ab}(f_a ,f_b) m_{a} {v}_{\parallel}.$$ In order to derive Eq., the expression of the time derivative of the parallel velocity $\dot{v}_{\parallel}$ is required. For the global model, it is given as $$\label{eq:dot_v_para}
\dot{v}_{\parallel} = -\frac{ 1 }{ m } \boldsymbol b \cdot \left( \mu \nabla B - e \boldsymbol{E}^{*} \right) + {v}_{\parallel} \dot { \boldsymbol X } \cdot \boldsymbol \kappa$$ following the particle orbit Eqs. and . We substitute Eq. into the parallel momentum equation Eq.. The pressure tensor $\boldsymbol{P}$ includes the diagonal component, the Chew-Goldbeger-Low () tensor $\boldsymbol{P}_{\text{CGL}}$, and the $ \boldsymbol{\Pi}_{2} $ term, the viscosity tensor $\boldsymbol{\Pi}_{2}$. See Appendix \[AppendixB\] for the derivation. According to the $\delta f$ method, the viscosity tensors become
$$\begin{aligned}
{2}
& \boldsymbol {b} \cdot \nabla \cdot \boldsymbol P_{\text{CGL}} \label{eq:p_cgl_1} \nonumber
\\ &=
\boldsymbol {b} \cdot \nabla \cdot \left[ \int d^3 v ~ \left ( ~ ( ~ m v_{\parallel}^2 ~ \boldsymbol b \boldsymbol b + \mu B ~ ( \boldsymbol I - \boldsymbol b \boldsymbol b ) \right ) f_{1} \right ] ,
\\
& \boldsymbol {b} \cdot \nabla \cdot \boldsymbol \Pi_{2} \label{eq:pi_2_f1} \nonumber
\\ &=
\boldsymbol {b} \cdot \nabla \cdot \left[ \int d^3 v ~ m v_{\parallel} ~ \bigg ( \dot{\boldsymbol X}_{\perp} \boldsymbol b + \boldsymbol b \dot{\boldsymbol X}_{\perp} \bigg ) f_{1} \right ] ,\end{aligned}$$
where $ f_{a,0} $ is an even function but $ v_{\parallel} \dot{\boldsymbol X }_{\perp} $ is an odd function. Therefore, $f_{a,0}$ does not contribute to $\nabla \cdot \boldsymbol \Pi_{2}$. According to Eq., $\nabla \cdot \boldsymbol P_{\text{CGL}} $ does not explicitly depend on the approximations in $\boldsymbol v_{m}$ and $\boldsymbol v_{E}$. Multiplying Eq. with $B$, the flux-surface-average of the parallel momentum balance equation becomes $$\begin{aligned}
\label{eq:m1-parallel_f1}
&\left \langle \frac{\partial}{ \partial t } ( n m V_{\parallel} B ) \right \rangle + \left \langle \boldsymbol{B} \cdot \nabla \cdot ( \boldsymbol{P}_{\text{CGL}} + \boldsymbol{\Pi}_{2}) \right \rangle
\nonumber \\
&= \left \langle n e E_{\parallel} B \right \rangle + \left \langle F_{\parallel} B \right \rangle + \left \langle B \int d^{3}v ~ \mathcal{S}_{1} ~ m v_{\parallel} \right \rangle
.\end{aligned}$$
For the ZOW model, the parallel momentum balance equation is calculated with $ \dot{\boldsymbol X}_{\perp} = \boldsymbol v_{E} + \boldsymbol {\hat{v}}_{m} $ and the time derivative of parallel velocity $$\begin{aligned}
\label{eq:dot_v_para-zow}
\dot{v}_{\parallel}
&= -\frac{ \mu }{ m } \boldsymbol b \cdot \nabla B + {v}_{\parallel} \boldsymbol v_{E} \cdot \frac{ \nabla_{\perp} B }{ B }
\nonumber \\
&= -\frac{ \mu }{ m } \boldsymbol b \cdot \nabla B
+ {v}_{\parallel} \dot { \boldsymbol X }_{\perp} \cdot \boldsymbol \kappa
+ {v}_{\parallel} \left( \frac { \dot { \psi } } { B } \frac{ \partial B }{ \partial \psi } \right),\end{aligned}$$ following the particle orbit Eq.. Then, the parallel momentum balance equation becomes $$\begin{aligned}
\label{eq:m1-parallel_f1-ZOW}
&\left \langle \frac{\partial}{ \partial t } ( n m V_{\parallel} B ) \right \rangle + \left \langle \boldsymbol{B} \cdot \nabla \cdot ( \boldsymbol{P}_{\text{CGL}} + \boldsymbol{\Pi}_{2,\text{ZOW}}) \right \rangle
\nonumber \\
&= \left \langle F_{\parallel} B \right \rangle
+ \left \langle B \int d^{3}v ~ \mathcal{S}_{1} ~ m v_{\parallel} \right \rangle
\nonumber \\
&+ \left\langle B \int d^3v ~ f ~ \mathcal{G} ~ mv_{\parallel}\right\rangle .\end{aligned}$$ For the ZOW model, the $ \boldsymbol{P}_{\text{CGL}} $ term is the same form as Eq. and the $ \boldsymbol{\Pi}_{2} $ term, Eq., is rewritten as $$\begin{aligned}
\label{eq:pi_2_ZOW}
& \langle \boldsymbol {B} \cdot \nabla \cdot \boldsymbol \Pi_{2, \text{ZOW} } \rangle
\nonumber \\
&= \left \langle \boldsymbol {B} \cdot \nabla \cdot \bigg[ m n {V}_{\parallel}( \boldsymbol b \boldsymbol v_{E} + \boldsymbol v_{E} \boldsymbol b ) \bigg ] \right \rangle \nonumber
\nonumber \\
&+ \left \langle \boldsymbol {B} \cdot \nabla \cdot \left [ \int d^3 v ~ m v_{\parallel} ~ \bigg ( \hat {\boldsymbol v}_{m} \boldsymbol b + \boldsymbol b \hat {\boldsymbol v}_{m} \bigg ) f_{1} \right ]
\right \rangle
\nonumber \\
&+ \left \langle \int d^{3}v ~ m {v}_{\parallel} B \left( \frac { \dot { \psi } } { B } \frac{ \partial B }{ \partial \psi } \right) f_{1} \right \rangle \end{aligned}$$ where $\hat {\boldsymbol v}_{m}$ is defined by Eq.. Eq. shows that $\langle \nabla \cdot \boldsymbol \Pi_{2, \text{ZOW} } \rangle $ of the ZOW model includes not only the $\boldsymbol E \times \boldsymbol B$ drift but also the partial magnetic drift. In the ZOW model, there is an extra term of viscosity in Eq., $$\begin{aligned}
\label{eq:extra_viscosity_ZOW}
\left \langle \int d^{3}v ~ m {v}_{\parallel} B \left( \frac { \dot { \psi } } { B } \frac{ \partial B }{ \partial \psi } \right) f_{1} \right \rangle\end{aligned}$$ which comes from the last term of Eq. and is estimated as $\mathcal O (\delta^2 )$. Actually, the $\partial B/\partial \psi$ is $\mathcal{O} (\delta)$ terms in MHD-equilibrium of helical devices, and Eq. becomes $\mathcal O (\delta^3 )$. The symmetry of $ \boldsymbol \Pi_{2, \text{ZOW} }$ is broken because of the third term in Eq.. Furthermore, there is an additional term on the RHS in Eq., $$\label{eq:G_ZOW}
\left\langle B \int d^3v ~ f ~ \mathcal{G} ~ mv_{\parallel}\right\rangle$$ which is estimated as $\mathcal O (\delta^2 )$. The effect of Eq. on the parallel flow will be discussed in Sec.\[sec:incompressibility\] below. Following the order of magnitude, the contribution of Eq. and are comparable in the parallel momentum equation Eq.. The parallel electric field $E_{\parallel}$ and its contribution to the parallel momentum balance are neglected in the local models for simplicity.
For the ZMD model, the parallel momentum balance equation is calculated with $ \dot{\boldsymbol X}_{\perp} = \boldsymbol v_{E}$ and the time derivative of parallel velocity $$\begin{aligned}
\label{eq:dot_v_para-zmd}
\dot{v}_{\parallel}
&= -\frac{ \mu }{ m } \boldsymbol b \cdot \nabla B + {v}_{\parallel} \boldsymbol v_{E} \cdot \frac{ \nabla_{\perp} B }{ B }
\nonumber \\
&= -\frac{ \mu }{ m } \boldsymbol b \cdot \nabla B
+ {v}_{\parallel} \dot { \boldsymbol X }_{\perp} \cdot \boldsymbol \kappa, \end{aligned}$$ following the particle orbit Eq.. If the scalar pressure is assumed as a function of $p = p (\psi)$, $\nabla_{\perp} B / B \cdot \boldsymbol v_{E} $ is rewritten as $\dot { \boldsymbol X }_{\perp} \cdot \boldsymbol \kappa $, according to Eq.. Then, the parallel momentum balance equation becomes $$\begin{aligned}
\label{eq:m1-parallel_f1-ZMD}
&\left \langle \frac{\partial}{ \partial t } ( n m V_{\parallel} B ) \right \rangle + \left \langle \boldsymbol{B} \cdot \nabla \cdot ( \boldsymbol{P}_{\text{CGL}} + \boldsymbol{\Pi}_{2,\text{ZMD} } ) \right \rangle
\nonumber \\
&= \left \langle F_{\parallel} B \right \rangle
+ \left \langle B \int d^{3}v ~ \mathcal{S}_{1} ~ m v_{\parallel} \right \rangle.\end{aligned}$$ Equation for ZMD is rewritten as $$\begin{aligned}
\label{eq:ZMD_Pi}
&\langle \boldsymbol {B} \cdot \nabla \cdot \boldsymbol \Pi_{2, \text{ZMD} } \rangle
= \left \langle \boldsymbol {B} \cdot \nabla \cdot \left [ m n V_\parallel
( \boldsymbol b \boldsymbol v_{E} + \boldsymbol v_{E} \boldsymbol b ) \right] \right\rangle\end{aligned}$$ where $\boldsymbol v_{m}$ does not exist in $\boldsymbol \Pi_{2}$. This equation shows that ZMD maintains not only $\mathcal{G} = 0 $ but also the symmetry of $\langle \boldsymbol {B} \cdot \nabla \cdot \boldsymbol \Pi_{2, \text{ZMD}} \rangle$. For the DKES model, the parallel momentum balance equation is calculated with $ \dot{\boldsymbol X}_{\perp} = \hat {\boldsymbol v}_{E}$ from Eq. and the time derivative of parallel velocity $$\begin{aligned}
\label{eq:dot_v_para_dkes}
\dot{v}_{\parallel} = -\frac{ \mu }{ m } { \boldsymbol b } \cdot \nabla B,\end{aligned}$$ following the particle orbit Eq.. Then, the parallel momentum balance equation becomes $$\begin{aligned}
\label{eq:m1-parallel_f1-dkes}
&\left \langle \frac{\partial}{ \partial t } ( n m V_{\parallel} B ) \right \rangle + \left \langle \boldsymbol{B} \cdot \nabla \cdot ( \boldsymbol{P}_{\text{CGL}} + \boldsymbol{\Pi}_{2,\text{DKES} } ) \right \rangle
\nonumber \\
&= \left \langle F_{\parallel} B \right \rangle
\nonumber \\
&+ \left \langle B \int d^{3}v ~ \mathcal{S}_{1} ~ m v_{\parallel} \right \rangle.\end{aligned}$$ With the incompressible $\boldsymbol E \times \boldsymbol B$ flow, Eq. is rewritten as $$\begin{aligned}
\label{eq:DKES_Pi}
&\langle \boldsymbol {B} \cdot \nabla \cdot \boldsymbol \Pi_{2,\text{DKES} } \rangle
\nonumber \\
&= \left \langle { \boldsymbol B } \cdot \nabla \cdot
\left [ \frac{m n V_\parallel}{\langle B^2\rangle} ( { \boldsymbol b} \boldsymbol E \times \boldsymbol B + \boldsymbol E \times \boldsymbol B { \boldsymbol b} ) \right ] \right \rangle
\nonumber \\
& + \left \langle B n m V_{\parallel} \hat {\boldsymbol v}_{E} \cdot \boldsymbol \kappa \right \rangle .\end{aligned}$$ DKES maintains $\mathcal{G} = 0$ and the symmetry is broken in viscosity $\langle \boldsymbol {B} \cdot \nabla \cdot \boldsymbol \Pi_{2,\text{DKES} } \rangle$.
The viscosity tensors are different among the ZOW, ZMD, and DKES-like models because of the approximation of incompressible $\boldsymbol E \times \boldsymbol B $ drift. The effect of incompressibility is discussed in Sec.\[sec:large\_Er\] below.
For the parallel momentum balance in all of the global and local models, the constraint imposed on the source/sink term $ \mathcal{S}_{1} $ is that its contribution to parallel momentum should vanish; $$\begin{aligned}
\int d^{3}v ~ \mathcal{S}_1 m v_{\parallel} = 0.\label{eq:para_mom_src}\end{aligned}$$ In fact, unlike the particle or energy balance relation, the drift-kinetic simulation reaches a steady state of parallel flow without any additional source/sink term. Note that the parallel momentum source vanishes not by flux-surface averaging, but is set to be zero anywhere on a flux surface. The effect of parallel friction $F_{\parallel}$ and the finite-$\mathcal{G}$ terms on the parallel momentum are discussed in the next section.
Simulation Result and Discussion {#sec:result}
================================
A series of simulations are carried out to benchmark the local and the global drift-kinetic models. We compare the neoclassical radial particle flux $\Gamma^\psi_a$ Eq., radial energy flux Eq., and the flux-surface average parallel mean flow multiplied by $B$, $$\label{eq:vb}
\langle {V}_{a,\parallel} B \rangle \equiv
\Bigg \langle \int d^3 v ~ f_{a,1} v_{a,\parallel} B( \psi, \theta, \zeta ) \Bigg \rangle.$$ To see the radial fluxes and the heat fluxes in the units \[1/m$^2$s\] and \[W/m$^2$\], respectively, these are redefined as $$\Gamma_a \equiv \frac{dr}{d\psi}\Gamma_a^\psi,\quad Q_a\equiv \frac{dr}{d\psi}Q_a^\psi,$$ where $r=a\sqrt{\psi/\psi_{edge}}$ and $a$ is the effective minor radius of the plasma boundary, $\psi=\psi_{edge}$. $a$ and $\psi_{edge}$ are given from VMEC MHD equilibrium calculation code[@VMEC_1991]. Note that in the local models even though $f_1$ does not contribute to radial fluxes in the particle and energy balance equations in Sec.\[section:particle\_flux\], $\Gamma_a$ and $Q_a$ are evaluated by the virtual radial displacement $\boldsymbol{v}_m \cdot \nabla r$-term in the local approximations.
The plasma parameters are given as TABLE \[tb:para\]. Two types of normalized ion collisionality $\nu_{i}^*$ are given in the table : $\nu_{i,PS}^*\equiv qR_{ax}\nu_{ii}/v_{thi}=1$ represents the Plateau - Pfirsch–Schlüter boundary and $\nu_{i,B}^* \equiv \nu_{i,PS}^* /(r/R_{ax})^{1.5}=1 $ is the Banana-Plateau boundary. For LHD, the inward-shift configuration is employed, in which the neoclassical radial transport is expected to be suppressed compared to that in a standard configuration. For W7-X, the magnetic geometry is adjustable by the coil current system. Here, the standard configuration [@Gieger_PPCF(2015)_014004] in the zero-$\beta$ limit is employed. For HSX, the quasi-helically symmetric configuration is employed. The magnetic field configurations of both W7-X and HSX are chosen so as to reduce the radial guiding center excursion of trapped particles, while W7-X also aims at reducing the bootstrap current[@Gieger_PPCF(2015)_014004][@HSX_1995] The artificial density and temperature profiles are given in the LHD and W7-X investigations so that the plasmas are in $1 / \nu$ regime around $|E_r| \sim 0$. The HSX kinetic profile is the diagnostic data from HSX experiment.[@Briesemeister_2013] Compared to the other devices, the collisionality of the HSX plasma is high in terms of $\nu_{iB}^*$ because of very low $T_i$. In TABLE \[tb:para\], the ambipolar $E_r$ of the LHD and HSX simulations are shown, which have been evaluated by GSRAKE and DKES/PENTA, respectively.
In the following benchmarks, there are three types of DKES models, namely DKES, DKES-like, and DKES/PENTA. First, DKES is the original code with the pitch angle scattering collision operator. Thus, it does not guarantee the conservation of momentum. Second, DKES-like is the solver of Eq. with the $\delta f$ method and the linearized collision operator as ZOW and ZMD. The test-particle portions of collision operator include both the pitch-angle and energy scattering terms. The field-particle term maintains the conservation of particle numbers, parallel momentum, and energy in the simulation.[@Satake_2008] The third model, DKES/PENTA, is the numerical result from DKES and with momentum correction by Sugama-Nishimura method.[@penta_theory_2002][@spong_2005] For LHD, local models are also benchmarked with GSRAKE code[@Beidler_2001], which solves the mono-energy and the ripple-averaged drift-kinetic equations. GSRAKE is similar to DKES but the magnetic field spectrum in GSRAKE is approximated.[@Beidler_2001] It should be emphasized that the $\boldsymbol E \times \boldsymbol B$ drift term in GSRAKE is compressible, although this point has not been clearly mentioned in previous studies. [@Beidler_2001][@Beidler_nf2011] The original GSRAKE code is made so that it can include the tangential magnetic drift term. However, the term is omitted in the present benchmarks because the magnetic drift term is found to make the simulation result unstable[@Satake_2006].
LHD W7-X HSX
---------------------------- ----------- ----------- ------------
r/a 0.7375 0.7500 0.3100
$\iota$ 0.740 0.886 1.051
$R_{ax}/a$ 3.60/0.64 5.51/0.51 1.21/0.126
$n_{i}$ \[$10^{18}/m^3$\] 3.10 0.406 3.83
$T_{i}$ \[$keV$\] 0.891 0.350 0.061
$T_{e}$ \[$keV$\] 0.891 0.350 0.544
$B_{ax}$ \[$T$\] 2.99 2.77 1.00
${\nu}^{*}_{i,B}$ 0.0368 0.0910 17.3
${\nu}^{*}_{i,PS}$ 0.0017 0.0017 0.101
ambipolar $E_{r}$ \[kV/m\] -1.73 N/A 3.47
$\mathcal{M}_p$ -0.015 N/A 0.95
: Simuation parameters on each configurations.
\[tb:para\]
Effect of $ \bold E \times \bold B $ Compressibility {#sec:large_Er}
------------------------------------------------------
The radial electric field $E_r$ is given as a parameter in this series of investigations. In Figs. \[fig:p-flux\] and \[fig:q-flux\], the ion radial particle and energy fluxes among the different approximations are presented on LHD, W7-X, and HSX, respectively. The figures of parallel flow simulation are shown in Fig. \[fig:vb\]. The global simulations are carried out for LHD only because the global simulation requires much more computational resources than the local to reach a steady-state solution of $\langle V_{i,\parallel}B\rangle$. In Figs. \[fig:p-flux\] - \[fig:vb\], the good agreements appear among the local models in $\Gamma_i$, $Q_i$, and $\langle V_{i,\parallel}B\rangle$ if the radial electric field amplitude is moderate in terms of the poloidal Mach number, that is, $0\ll |M_p|\ll 1$.
Let us first focus on the difference which appears on the neoclassical fluxes at large-$E_r$ values. When the amplitude of $E_r$ rises, the discrepancies increase between DKES-like and the other local models. As shown in Figs. \[fig:p-flux\](a), \[fig:q-flux\](a), and \[fig:vb\](a), the LHD radial and parallel fluxes of the ZOW, ZMD, and GSRAKE models agree with the global model well. Thus, the discrepancies comes from the incompressibility approximation of the $\boldsymbol E \times \boldsymbol B$ drift on DKES-like according to Eq.. According to Figs. \[fig:p-flux\]-\[fig:vb\], the $\boldsymbol E\times \boldsymbol B$ compressibility effect is expected to be significant when $|M_p|>0.4$.
The $E_r$-dependence of $\Gamma_i$, $Q_i$, and $\langle V_{i,\parallel}B\rangle$ found in the HSX case need more explanations. First, in Fig.\[fig:p-flux\](d), all the cases, except for the original DKES, show a good agreement. The disagreement between the DKES model and the others is also found in the ion energy flux Fig. \[fig:q-flux\](c) and parallel flow Fig. \[fig:vb\](c). Recall that our DKES-like simulation uses the collision operator which ensures the conservation of parallel momentum in ion-ion collisions. The simulation result suggests that the momentum conservation property of the collision operator is essential for neoclassical transport calculation on quasi-symmetric devices like HSX. Secondly, as $E_r$ increases, the neoclassical fluxes of all the models disagree with one another. As in the LHD and W7-X cases, the $\boldsymbol E\times \boldsymbol B$ compressibility is supposed to be the main cause of the disagreement. However, it should be pointed out that the ion parallel flow in HSX becomes supersonic at $\mathcal{M}_p > 1$ as shown in Fig.\[fig:vb\]. Here, the parallel Mach number is defined as $$\label{eq:m_para}
\mathcal{M}_\parallel\equiv \frac{\langle V_{\parallel}B\rangle}{v_{th}B_{ax}}.$$ In the paper, the drift-kinetic models are constructed under the assumption $\mathcal{M}_{\parallel} \ll 1$ because we just takes the zeroth order distribution as the Maxwellian without the mean flow. See Eqs. and . The parallel flow dependence on $E_r$ in HSX is contrastive to that in W7-X, in which parallel mean flow remains very slow compared to thermal velocity, as in Fig. \[fig:vb\](b). Both HSX and W7-X configurations aim at reducing radial neoclassical flux. However, the magnetic configuration of W7-X is chosen to reduce the parallel neoclassical flow, too. This leads to the different dependence of parallel flow on $E_r$ in these two devices. Note also that $T_e \gg T_i$ in HSX [@Lore_2010] while $T_i = T_e$ in LHD and W7-X cases. In such a $T_e\gg T_i$ plasma, $\mathcal{M}_p$ of $ \boldsymbol E\times \boldsymbol B$ flow by ambipolar-$E_r$ can be $\mathcal{O}(1)$ because of the slow ion thermal velocity $v_{th,i}$. For example, under the ambipolar condition, $\mathcal{M}_p \simeq -0.015$ and $E_{r} \simeq -1.73$ kV/m on LHD by GSRAKE, while $\mathcal{M}_p \simeq 0.95$, and $E_{r} \simeq 3.47$ kV/m on HSX by DKES/PENTA. Such a large $M_p$ with the quasi-symmetric configuration of HSX results in $\mathcal{M}_{\parallel} \sim\mathcal{O}(1)$. When $\mathcal{M}_{\parallel}> 1$, all the drift-kinetic models violate the assumption of the slow-flow-ordering. Therefore, although $\mathcal{M}_p\sim \mathcal{O}(1)$ $\boldsymbol E\times \boldsymbol B$ flow is allowed in ZOW and ZMD models, the validation of the drift-kinetic models at $\mathcal{M}_{\parallel} \sim \mathcal{O}(1)$ has to be reconsidered by taking account of the centrifugal force and potential variation along the magnetic field lines[@sugama-horton_pop1997b]. This problem is beyond the scope of the present study.
![Ion particle fluxes $\Gamma_i$ of (a) LHD, (c) W7-X, and (d) HSX , respectively. (b) is an enlarged view of (a) around $E_r \sim 0$. The multiple numerical results of DKES model with the different collision operators are shown in (d). The vertical line shows the value of poloidal Mach number $\mathcal{M}_p$ defined in Eq..[]{data-label="fig:p-flux"}](./1-a.eps "fig:"){width="45.00000%"} ![Ion particle fluxes $\Gamma_i$ of (a) LHD, (c) W7-X, and (d) HSX , respectively. (b) is an enlarged view of (a) around $E_r \sim 0$. The multiple numerical results of DKES model with the different collision operators are shown in (d). The vertical line shows the value of poloidal Mach number $\mathcal{M}_p$ defined in Eq..[]{data-label="fig:p-flux"}](./1-b.eps "fig:"){width="45.00000%"} ![Ion particle fluxes $\Gamma_i$ of (a) LHD, (c) W7-X, and (d) HSX , respectively. (b) is an enlarged view of (a) around $E_r \sim 0$. The multiple numerical results of DKES model with the different collision operators are shown in (d). The vertical line shows the value of poloidal Mach number $\mathcal{M}_p$ defined in Eq..[]{data-label="fig:p-flux"}](./1-c.eps "fig:"){width="45.00000%"} ![Ion particle fluxes $\Gamma_i$ of (a) LHD, (c) W7-X, and (d) HSX , respectively. (b) is an enlarged view of (a) around $E_r \sim 0$. The multiple numerical results of DKES model with the different collision operators are shown in (d). The vertical line shows the value of poloidal Mach number $\mathcal{M}_p$ defined in Eq..[]{data-label="fig:p-flux"}](./1-d.eps "fig:"){width="45.00000%"}
![Ion energy flues $Q_i$ of (a) LHD, (b) W7-X, and (c) HSX, respectively.[]{data-label="fig:q-flux"}](./2-a.eps "fig:"){width="45.00000%"} ![Ion energy flues $Q_i$ of (a) LHD, (b) W7-X, and (c) HSX, respectively.[]{data-label="fig:q-flux"}](./2-b.eps "fig:"){width="45.00000%"} ![Ion energy flues $Q_i$ of (a) LHD, (b) W7-X, and (c) HSX, respectively.[]{data-label="fig:q-flux"}](./2-c.eps "fig:"){width="45.00000%"}
![Ion parallel flow of (a) LHD, (b) W7-X, and (c) HSX, respectively. (d) presents the enlarged details around $E_r \sim 0$ for the LHD and W7-X cases. The vertical axis represents the parallel Mach number $M_\parallel$ as defined in Eq. .[]{data-label="fig:vb"}](./3-a.eps "fig:"){width="45.00000%"} ![Ion parallel flow of (a) LHD, (b) W7-X, and (c) HSX, respectively. (d) presents the enlarged details around $E_r \sim 0$ for the LHD and W7-X cases. The vertical axis represents the parallel Mach number $M_\parallel$ as defined in Eq. .[]{data-label="fig:vb"}](./3-b.eps "fig:"){width="45.00000%"} ![Ion parallel flow of (a) LHD, (b) W7-X, and (c) HSX, respectively. (d) presents the enlarged details around $E_r \sim 0$ for the LHD and W7-X cases. The vertical axis represents the parallel Mach number $M_\parallel$ as defined in Eq. .[]{data-label="fig:vb"}](./3-c.eps "fig:"){width="45.00000%"} ![Ion parallel flow of (a) LHD, (b) W7-X, and (c) HSX, respectively. (d) presents the enlarged details around $E_r \sim 0$ for the LHD and W7-X cases. The vertical axis represents the parallel Mach number $M_\parallel$ as defined in Eq. .[]{data-label="fig:vb"}](./3-d.eps "fig:"){width="45.00000%"}
Effect of Magnetic Drift and Collisionality {#sec:magnetic_drift}
-------------------------------------------
Let us turn to the simulations around $E_r=0$. In Figs. \[fig:p-flux\]-\[fig:q-flux\], there are the very large peaks of $\Gamma_i$ and $Q_i$ at $E_r=0$ in the LHD and W7-X cases by the ZMD and the DKES-like models. On the contrary, the global and the ZOW models show the reduction of radial fluxes at $E_r\simeq 0$ and the peaks shift to negative-$E_r$ side. This tendency, which has been found in the previous study[@Matsuoka2015], greatly modifies the neoclassical transport in $1/\nu$-regime, especially in the LHD case. For the HSX case, however, such a peak at $E_r=0$ is not found in the ZMD and DKES-like models. What causes the reduction of $\Gamma_i$ and $Q_i$ in ZOW, and what makes the configuration dependence? Here we consider the problem by analytical formulation.
In a simple stellarator/heliotron magnetic configuration like LHD, the amplitude of the magnetic field is given approximately $$\label{eq:B_approx}
| \boldsymbol B | \approx B_{0} [ 1 - \epsilon_{h} \cos ( l \theta - m \zeta ) - \epsilon \cos \theta ],$$ where ${\epsilon}_{h}$ and $\epsilon$ are helical and toroidal magnetic field modulations, respectively. $l$ is the helical field coil number and $m$ is the number of toroidal periods. Once a particle is trapped by the helical ripples, its orbit drifts across the magnetic surface and contributes to the radial flux. The estimation of particle flux is roughly given as[@miyamoto2012plasma] $$\label{eq:nVr}
\Gamma \sim -\left\langle \int \frac{ \nu_{ \text{eff} } }{ ( \nu_{ \text{eff} })^2 + ( \omega_{h} + \omega_{E} )^2 } V_{\perp}^2 \frac{ \partial f_{M} }{ \partial r } d^3v\right\rangle .$$ Here, $\nu_{ \text{eff} }$ is the effective collision frequency of trapped particle and defined as $\nu_{ \text{eff} } \equiv \nu/ \epsilon_{h}$. $\omega_h$ and $\omega_E$ represent the poloidal precession frequency of the trapped particles by the magnetic drift and $\boldsymbol E\times \boldsymbol B$ drift, respectively. $V_\perp$ denotes the radial drift velocity. For trapped particles, they are estimated as [@Beidler_PPCF1995] $$\begin{split}\label{eq:vperp}
V_\perp\sim \frac{v_d}{\epsilon_tB_0}\frac{\partial B}{\partial \theta}&\sim v_d\frac{\epsilon}{\epsilon_t},\\
\omega_h\sim \frac{v_d}{\epsilon_tB_0}\frac{\partial B}{\partial r},& \quad\omega_E\sim \frac{E_r}{rB_0},
\end{split}$$ where $v_d\equiv {\mathcal{K}}/{eB_0R_0}$ and $\epsilon_t=r/R_0$. If $( \nu_{ \text{eff} })^2 \gg ( \omega_{h} + \omega_{E} )^2$, then Eq. indicates that the particle transport is inversely proportional to the collision frequency. The diffusion coefficient in $1/\nu$-regime is approximated as[@Beidler_PPCF1995] $$\label{eq:diffusion_coefficient}
D_{h} \approx{ \epsilon_{h} }^{1/2} ( \Delta_{h} )^2 \nu_{ \text{eff} }
\sim { \epsilon_{h} }^{3/2} \bigg ( \frac{ T }{ e B_0 R_0 } \frac{ \epsilon }{ \epsilon_t }\bigg )^2 \frac{1}{ \nu },$$ where $\Delta h=V_\perp/\nu_{\text{eff}}$ is the estimation of the radial step size of helically trapped particles. Approximating $\omega_h \rightarrow 0$ in Eq. corresponds to ZMD and DKES models. Then, around $E_r=0$, $\Gamma_i$ shows $1/[\nu_{\text{eff}}(1+x^2)]$-type dependence where $x=(\omega_E/\nu_{\text{eff}})^2$. The $\omega_E$ is common for all the particles on a flux surface so that it makes a strong resonance at $\omega_E=0$. Once the finite $\omega_h$ is considered, the peak of $\Gamma_i$ appearing at the poloidal resonance condition $\omega_h+\omega_E=0$ becomes blurred because of $\omega_h$ dependence on $\boldsymbol{v},\theta,$ and $\zeta$. This explains the difference between the ZOW and the ZMD models in the LHD case.
The analytic model of the $1/\nu$-type diffusion infers that the strong resonance of trapped-particles at $E_r=0$ in ZMD and DKES-like models is damped by Coulomb collisions. To demonstrate this, the 10 times larger density simulations are carried out for the LHD case as shown in Fig. \[fig:LHD\_n10\]. It is found that the strong peak in $\Gamma_i$ and $\langle V_{i,\parallel}B\rangle$ at $E_r=0$ in ZMD and DKES-like calculations are diminished, and the difference from the ZOW result is small. It is concluded that the tangential magnetic drift is more important for neoclassical transport calculation in the lower collisionality case and when $|\omega_E| < |\omega _h|$.
Quasisymmetric HSX can be regarded as the $\epsilon\rightarrow 0$ limit of Eq. .[@HSX_1995] The bounce-average radial drift $\langle V_\perp\rangle$ vanishes in the quasisymmetric limit $\epsilon/\epsilon_t=0$ so that HSX shows the low radial particle transport at $E_r\simeq0$ as in Fig.\[fig:p-flux\](d) in all local models. The radial flux is of comparable level to that in equivalent tokamaks. However, it should be noted that the collisionality of the present HSX case is in plateau-regime. Then, the discussion on the radial transport level in HSX using Eq. is inadequate. Following the previous benchmark study on local neoclassical simulations[@Beidler_nf2011], there are tiny magnetic ripples in the actual HSX magnetic field made by the discrete modular coils, which causes $1/\nu$-type diffusion coefficient at very low-collisionality, $\nu_{PS}^*<10^{-3}$, in the DKES calculation. Therefore, we benchmarked the local drift-kinetic models in HSX with $100$ times smaller plasma density ($\nu_{PS}^*\simeq 1.0\times 10^{-3}$) at $E_r=0$. The results are shown in Table \[tb:HSX\]. The radial flux in very low-collisionality regime in HSX shows discrepancy among ZOW, ZMD, and DKES-like models, as found in the LHD and W7-X cases. Though the $1/\nu$-regime appears from lower $\nu^*$ value in HSX than LHD, the effect of the tangential magnetic drift on neoclassical transport appears in the same way.
Concerning the W7-X case, the magnetic field spectrum is much more complicated than the simple model Eq. . It is generally expressed in a Fourier series as follows: $$B ( \psi,\theta, \zeta ) = B_{0} \sum_{ m, n } b_{m,n} ( \psi ) \cos ( m\theta - 5n \zeta ).$$ Compared with LHD, W7-X has good modular coil feasibility to adjust $b_{m,n}$ [@Grieger1992] where the helical $b_{1,1}$ and toroidal $b_{1,0}$ magnetic field modulations are equal to $\epsilon_h$ and $\epsilon$ respectively in Eq.. One of the neoclassical optimizations is performed by the reduction of average toroidal curvature $b_{1,0}/\epsilon_t\sim 0.5$[@Gieger_PPCF(2015)_014004] compared to that in LHD, $\epsilon/\epsilon_t\simeq 1$. According to Eq. , this partially explains the smallness of $1/\nu$-regime transport in W7-X. However, the magnetic spectrum of W7-X contains other Fourier components which are comparable to $b_{1,0}$ and $b_{1,1}$. Thus, the simple analytic model, such as Eqs. and , is insufficient to explain its optimized neoclassical transport level.
The quasi-isodynamic concept of the neoclassical optimized stellarator configuration is as follows: the trapped particles in the toroidal magnetic mirrors $b_{0,1}$ precess in the poloidal direction while their radial displacements are small and return to the same flux surface after they circulate poloidally. The trapped particle trajectory in quasi-isodynamic W7-X configuration has been analyzed using the second adiabatic invariant[@Gori_Lotz_Nuhrenberg-ISPP17] $$\begin{aligned}
\mathfrak{J}_{\parallel} &
= \int dl~ v_{\parallel}
= \int d\zeta ~ \frac{ \sqrt{ ( 2 \mathcal{K} - 2 \mu B )/m } }{ \boldsymbol B \cdot \nabla \zeta }
\nonumber \\
& \propto \int d\zeta \frac{ \sqrt{B_{ref}-B} }{B^2} ,\end{aligned}$$ where $B_{ref}$ represents the magnetic field strength at the reflecting point of a trapped particle. Deeply-trapped particles move along the $\mathfrak{J}_{\parallel}=$const. surfaces. Then, if the constant- $\mathfrak{J}_{\parallel}$-contours on a poloidal cross-section are near a flux-surface function and if the contours are closed, the radial transport of the trapped particles are suppressed. However, the standard configuration, which we investigate, is not fully optimized as is the quasi-isodynamic configuration. The $\mathfrak{J}_{\parallel}=$ constant surfaces in the standard configuration have small deviation from the flux surfaces[@Gori_Lotz_Nuhrenberg-ISPP17]. Therefore, in the limit $\omega_E+\omega_h=0$, the deeply-trapped particles drift radially along the $\mathfrak{J}_{\parallel}$ contours. Consequently, the radial flux in W7-X solved with ZMD and DKES-like models shows the strong peak at $E_r=0$. As expected from the form of Eq. , either by increasing the collision frequency or by taking account of finite $\omega_h$ as in the ZOW model results in decreasing the radial transport at $E_r=0$. In Fig. \[fig:W7X\] we have examined the radial and parallel flux in 10 times larger density W7-X plasma than those in Figs. \[fig:p-flux\](c) and \[fig:vb\](b). As found in the LHD case, the difference among the ZOW, ZMD, and DKES-like models at $E_r=0$ diminished in the higher collisionality W7-X case. It is worthwhile to note that it has already been pointed out that the improvement of collisionless particle confinement in W7-X configuration is realized not only in quasi-isodynamic geometry but also by enhancing the poloidal magnetic drift in finite-$\beta$ W7-X plasma because $\partial b_{0,0}/\partial r \propto \omega_h$ increases as the plasma-$\beta$.[@Yokoyama_NF2001]
Concerning the parallel flows, Fig.\[fig:vb\] shows that all models agree with each other well at $ 0 \ll \mathcal{M}_p \ll 1 $. Compared to the radial flux, the magnetic drift does not influence the parallel flow strongly at $E_r \sim 0$, even in the low-collisionality LHD and W7-X cases. On the other hand, the discrepancies of parallel flows at large-$|\mathcal{M}_p|$ appear as clearly as that of the radial flux.
In the simulations, steady-state solution of parallel flow is obtained when the parallel momentum balance relation Eq. is satisfied. As explored in Sec. \[sec:parall\_monentum\_balance\], in the parallel momentum balance relation, the differences among the drift-kinetic models includes four parts: (1) the explicit difference of the tangential drift velocities in $\langle \boldsymbol B \cdot \nabla \cdot \boldsymbol{\Pi}_{2}\rangle$, (2) the implicit difference of $\langle \boldsymbol B \cdot \nabla \cdot \boldsymbol{P}_{\text{CGL}}\rangle$ through $f_1$, (3) the extra term Eq. which breaks the symmetry of $\Pi_{2}$ in ZOW, and (4) the term \[sec:incompressibility\] related to $\mathcal G = \nabla_z \cdot \dot{\boldsymbol Z}^{\text {ZOW} } \neq 0$. $\langle \boldsymbol B \cdot \nabla \cdot \boldsymbol{\Pi}_{2}\rangle$ in DKES-like and ZMD models do not contain $\hat{\boldsymbol v}_{m}$. These models disagree with each other gradually as $E_r$ increases. This indicates that the discrepancy between Eqs. and on the compressibility of $\boldsymbol E \times \boldsymbol B$ affects the evaluation of parallel flow. Meanwhile, the ZMD and ZOW tendencies are similar in the wide range of $E_r$ in Figs.\[fig:vb\]. As a result, the two extra parallel-viscosity terms appearing in the ZOW model do not influence the parallel flow. In Fig.\[fig:vb\](d), there are small peaks at $E_r = 0$. When $E_r = 0$, the poloidal resonance leads to the extra large radial fluxes in Fig.\[fig:p-flux\](a) and \[fig:p-flux\](c). Equations and suggest that $f_1$ becomes very large at the resonance. However, the resonance occurs on trapped particles, which cannot contribute to parallel flow. The influence of resonance is passed to the passing particles via collisions to change the momentum balance through $\langle \boldsymbol B \cdot \nabla \cdot \boldsymbol{P}_{\text{CGL}}\rangle$. The parallel flows peak at $E_r=0$ is much less than the radial flux peaks because it is driven by this indirect mechanism.
In summary, as long as the collisionality is low enough to present the $1/\nu$-type diffusion at the condition $|\omega_E|<|\omega_h|$, the ZMD and DKES-like models, which ignore the tangential magnetic drift term, tend to overestimate the neoclassical flux at $E_r\rightarrow 0$ in all three helical configurations in this work. The ZOW model reproduces the similar trend as the global simulation in which the finite $\omega_h$ term results in reducing the $1/\nu$-type diffusion. The strong poloidal resonance $\omega_E=0$ without $\omega_h$ term in these local models results in the strong modification in the perturbed distribution function $f_1$, and it indirectly affects the evaluation of parallel flow $\langle V_\parallel B\rangle$, too.
![(a) The radial particle flux and (b) ion parallel flow of high collision frequency test of LHD. The normalized collision frequency is 10 times higher than $\nu^*$ on LHD in Table \[tb:para\].[]{data-label="fig:LHD_n10"}](./4-a.eps "fig:"){width="45.00000%"} ![(a) The radial particle flux and (b) ion parallel flow of high collision frequency test of LHD. The normalized collision frequency is 10 times higher than $\nu^*$ on LHD in Table \[tb:para\].[]{data-label="fig:LHD_n10"}](./4-b.eps "fig:"){width="45.00000%"}
![(a) The radial particle flux and (b) ion parallel flow of higher collision frequency test on W7-X. The normalized collision frequency is 10 times higher than $\nu^*$ of W7-X in Table \[tb:para\]. []{data-label="fig:W7X"}](./5-a.eps "fig:"){width="45.00000%"} ![(a) The radial particle flux and (b) ion parallel flow of higher collision frequency test on W7-X. The normalized collision frequency is 10 times higher than $\nu^*$ of W7-X in Table \[tb:para\]. []{data-label="fig:W7X"}](./5-b.eps "fig:"){width="45.00000%"}
Model $\Gamma$ \[$1/m^2s$\]
----------- -----------------------
ZOW $1.72 \times 10^{15}$
ZMD $2.15 \times 10^{16}$
DKES-like $2.10 \times 10^{16}$
: The particle flux of HSX at $E_r = 0$ with $0.01$ times density than that in Table \[tb:para\].[]{data-label="tb:HSX"}
Electron Neoclassical Transport and Bootstrap Current
-----------------------------------------------------
In order to benchmark the bootstrap current calculation at ambipolar condition among the local models, the electron neoclassical transport simulations were carried out for the LHD case. The results are shown in Figs. \[fig:LHD\_e\]. In the entire range of $E_r$, it is found that the differences of $\Gamma_e$, $Q_e$, and $\langle V_{e,\parallel} B\rangle$ between the two groups, i.e., (global, ZOW) and (ZMD, DKES-like), are smaller than those in the ion calculations. As the electron thermal velocity is much faster than the ions, the poloidal Mach number for electrons is always regarded as $\mathcal{M}_{p,e} \sim \mathcal{O}(\delta)$. Therefore, the $\boldsymbol E\times \boldsymbol B$-compressibility is not important for the electron calculation. Moreover, compared with Fig.\[fig:p-flux\](a), Fig.\[fig:LHD\_e\](a) does not present any obviously unphysical peak of the radial particle transport at $E_r \simeq 0$. There is the same feature in the energy flux. ( See Fig.\[fig:p-flux\](a) and \[fig:LHD\_e\](b).) Even though the normalized collision frequencies $\nu_{*,B}$ (or $\nu_{*,PS}$) are the same in the ion and electron simulations, it seems that the collision effect is stronger in electrons than ions to blur the tangential magnetic drift effect around $E_r=0$. Note that the precession drift frequency by the magnetic drift is also the same order between ions and electrons. See Eq.. The difference of the tendency at $E_r\simeq 0$ between ions and electrons is considered as follows. The collision frequency of particle species $a$ is proportional to [@Braginskii_1965] $$\nu_{a} \propto \frac{ ( e_a )^4 n_{a} }{ m_a^2 v_a^3 }.$$ For the LHD simulations, the temperature is set as $T_e = T_i$. Thus, the ratio of collision frequency between the electron and ion is $$\label{eq:nu_00}
\frac{ \nu_e }{ \nu_i } \propto \bigg ( \frac{ m_i }{ m_e } \bigg )^{ \frac{1}{2} } \gg 1$$ because of $Z_i=Z_e=1$ in this work. On the other hand, the normalized collision frequency $\nu_{*}(=\nu_{*,PS})$ is defined as $\nu_{*,a} \equiv q R_{ax} \nu_{a}/v_{th,a}$. Therefore, the ratio between the normalized electron and ion collision frequency is $$\label{eq:normalized_nu_01}
\frac{ \nu_{*,i} }{\nu_{*,e}} \propto \bigg ( \frac{ m_i }{ m_e } \bigg )^{ \frac{1}{2} } ~
\frac{ v_{th, e} }{ v_{th, i} }.$$ Eqs. and suggest that $\nu_{e} \gg \nu_{i}$ though $\nu_{*,e} =\nu_{*,i}$. In Eq., it is not the normalized collision frequency but the real collision frequency that appears in the form $\nu_{a, \text{eff}}=\nu_{a}/\epsilon_h$. $\omega_h$ and $\omega_E$ are the same order between ions and electrons so that the ratio between these terms in the denominator of Eq. , $$(\omega_h+\omega_E)^2/\nu_{\text{eff}}^2$$ is smaller for electrons than that for ions. Therefore, the finite-$\omega_h$ effect in the ZOW model is not as important for electrons than as for ions.
The LHD bootstrap current is investigated among the drift-kinetic models. In Fig.\[fig:BC\], the bootstrap current is estimated by ion and electron parallel flows as $$J_{BC} = e ( Z_i\langle v_{i, \parallel} B \rangle n_{i} - \langle v_{e, \parallel} B \rangle n_{e}) /B_{ax}.$$ It is found that the discrepancy of bootstrap current among the models increases when $E_r$ rises. This indicates that the gap mainly comes from the effect of $\boldsymbol E \times \boldsymbol B$ compressibility on the ion parallel flow as it is found in Fig.\[fig:vb\](a). The local drift-kinetic models are divided into two groups, DKES-like and the others. In the following discussion, the two extra terms in the ZOW model, Eqs. and , are ignored because it is found that the difference caused from these two terms is negligible among the ZMD and the ZOW models. Neglecting the $ n e E_{\parallel} B $ term in Eq., the parallel momentum balance in a steady-state is written as $$\begin{aligned}
\langle \boldsymbol{B} \cdot \nabla \cdot ( \boldsymbol{P}_{\text{CGL}} + \boldsymbol{\Pi}_{2} ) \rangle_a
= \langle B F_{a,\parallel} \rangle.\end{aligned}$$ The friction $ F_{\parallel}$ is estimated as follows: For ion, the friction between ions and electrons is ignored because of large mass ratio. And, the parallel momentum balance depends only on $\boldsymbol{P}_{\text{CGL}}$ and $ \boldsymbol{\Pi}_{2}$: $$\begin{aligned}
\label{eq:tensor_and_friction01}
\langle \boldsymbol{B} \cdot \nabla \cdot ( \boldsymbol{P}_{\text{CGL}} + \boldsymbol{\Pi}_{2} ) \rangle_{i} = 0.\end{aligned}$$ For electrons, not only the viscosity but also the electron-ion parallel friction $F_{ei,\parallel}$ are considered. And the parallel friction is approximated by $$\label{eq:friction_model}
\langle F_{ei,\parallel} \rangle \simeq \nu_{ei,\parallel} m_{e} n_{e} ( V_{i,\parallel} - V_{e,\parallel} ),$$ where $\nu_{ei,\parallel}$ is the parallel momentum-transfer frequency. The friction force acting on ions is ignored, $F_{ie,\parallel}=-F_{ei,\parallel}$ so that the total parallel momentum is not conserved in the simulation. Moreover, as explained in section \[sec:delta\_f\_scheme\], the electron-ion collision in the simulation is simplified by the pitch-angle scattering operator Eq. where ion mean flow is ignored. Therefore, in the present simulation models, the electron parallel momentum balance is approximated as $$\label{eq:tensor_and_friction02}
\langle \boldsymbol{B} \cdot \nabla \cdot ( \boldsymbol{P}_{\text{CGL}} + \boldsymbol{\Pi}_{2} ) \rangle_{e}
= -\nu_{\parallel,ei} m_{e} n_{e} V_{e,\parallel} B.$$ In Eqs. and , the viscosity $\boldsymbol \Pi_{2}$ is directly influenced by the treatment of the guiding center motion tangential to the flux surface. See Eqs., , and . $J_{BS}$ in the DKES-like model deviates from that in the ZOW and the ZMD model. This shows that the incompressible-$\boldsymbol E \times \boldsymbol B$ assumption in $\boldsymbol{\Pi}_{2}$ mainly causes the difference in parallel momentum balance. Meanwhile, the contribution of the tangential magnetic drift $ \hat {\boldsymbol{v}}_{m} $ is minor in the parallel momentum balance equation because the difference is negligible between the ZMD and the ZOW models in Fig.\[fig:BC\]. It should be noted that the approximation in the $F_{\parallel, ei}$ in our simulation is valid when $ |V_{\parallel,e}| \gg |V_{\parallel,i}|$. Actually, the electron and ion parallel flows can become comparable. For a more quantitative evaluation of bootstrap current, the effect should be considered when ion mean flow dominates the bootstrap current, for example, when $J_{BS}$ is at $E_r > 30 kV/m$ in Fig.\[fig:BC\]. The present work is to investigate neoclassical transport among the local drift-kinetic models so that the rigorous treatment of the parallel friction is left for future work.
In this section, the dependence of neoclassical transport on radial electric field was studied. The obvious difference appears at $E_r\simeq 0$ or $\mathcal{M}_{p} \sim 1$ among the drift-kinetic models. For the practical application on helical devices, it is important for evaluating the neoclassical fluxes at the ambipolar condition. The LHD ambipolar condition is investigated by searching the $E_r$ value where $Z_i \Gamma_i = \Gamma_e$. As shown in Table \[tb:ambi\], the ambipolar-$E_r$ values from different models are located between $-2.6$ and $-1.5$ \[$kV/m$\]. The amplitude of electric field, radial flux, and bootstrap current at the ambipolar condition are obtained by the interpolation as shown in Table \[tb:ambi\]. The ambipolar-$E_r$ magnitude of the ZMD model is close to the DKES-like and GSRAKE magnitudes, while the ZOW model predicts closer $E_r$ to the global simulation. Around the ambipolar condition, the bootstrap current amplitudes are just minor differences among the drift-kinetic models. Owing to $T_i \sim T_e$, the ambipolar condition is on the ion-root. In the present case, the finite $E_r$ on the ion-root is sufficient to suppress the poloidal resonance but insufficient to make an obvious gap by the $\boldsymbol E \times \boldsymbol B$ compressibility. The present case does not show any obvious advantage of the ZOW model compared to the other local models. If the tangential magnetic drift $\hat{v}_{m}$ increases or if the plasma collisionality is lower, the ZOW model will perhaps be more reliable than the other models in predicting the ambipolar-$E_r$, bootstrap current, and radial fluxes. The result of the ZOW model is close to the global simulation values so that the code requires less computation resources than the global. For example, in the LHD case, the ZOW model takes about $20\%$ computational resources compared to a global calculation with the same number of radial flux surfaces. In local simulation, one can choose a proper time step size according to the local parameters. On the other hand, in a global code, the time step size is a common parameter for all the markers. The step size must be small enough to resolve the fast guiding-center motion in the core, but it is much too fine for the markers in the low-temperature peripheral region. Another advantage of local simulation is fewer time steps to finish a calculation than a global one. For a local model, the calculation can be stopped after the time evolution converges on a single flux surface. For a global model, the calculation has to be continued untill the whole the plasma reaches a steady state.
![(a) The electron radial particle flux, (b) the energy flux, and (c) the parallel flow in the LHD case which are shown in Table \[tb:para\]. []{data-label="fig:LHD_e"}](./6-a.eps "fig:"){width="45.00000%"} ![(a) The electron radial particle flux, (b) the energy flux, and (c) the parallel flow in the LHD case which are shown in Table \[tb:para\]. []{data-label="fig:LHD_e"}](./6-b.eps "fig:"){width="45.00000%"} ![(a) The electron radial particle flux, (b) the energy flux, and (c) the parallel flow in the LHD case which are shown in Table \[tb:para\]. []{data-label="fig:LHD_e"}](./6-c.eps "fig:"){width="45.00000%"}
{width="100.00000%"}
$E_{r}$ \[$kV/m$\] $\Gamma$ \[$10^{19}/m^{2}s$\] $J_{BS}$ \[$kA/m^{2}$\] $Q_i$\[$kW/m^2$\] $Q_e$\[$kW/m^2$\]
-------- -------------------- ------------------------------- ------------------------- ------------------- -------------------
Global -2.34 0.057 2.88 0.272 0.343
ZOW -2.59 0.089 3.23 0.614 0.515
ZMD -1.55 0.123 3.22 0.880 0.819
DKES -1.66 0.125 3.55 0.595 0.807
GSRAKE -1.73 0.089 N/A 0.519 0.628
summary {#sec:summary}
=======
A series of neoclassical transport benchmarks have been presented among the drift-kinetic models in helical plasmas. The two-weight $\delta f$ scheme is employed to carry out the calculations of particle flux, energy flux, and parallel flow. The $\delta f$ formulation in this work allows the violation of Liouville’s theorem in a local drift-kinetic approximation as in the ZOW model. The treatments of the convective derivative term $(\boldsymbol v_E+\boldsymbol v_m)\cdot\nabla f_{a,1}$ are different among the local drift-kinetic models. For example, the ZOW model maintains the tangential magnetic drift $\hat{ \boldsymbol v }_{m}$ which results in the compressible phase-space flow, $ \mathcal{G} \neq 0$. On the contrary, in the ZMD and the DKES models, the magnetic drift is completely neglected, but instead the phase-space volume is conserved. The finite $\mathcal{G}$ term in ZOW brings $\mathcal{O}( \delta^2 )$-correction in the particle, parallel momentum, and energy balance equations. The simulation results have demonstrated that the ZOW and the ZMD models agree with each other well in the wide range of $E_r$ value. This indicates that the $\mathcal{O}( \delta^2 )$-correction term is negligible in neoclassical transport calculation. The only exception is around $\boldsymbol v_E\simeq 0$, where the ZMD and DKES-like models show the very large peaks of neoclassical flux. Owing to the tangential magnetic drift $\hat{ \boldsymbol v }_{m}$, the ZOW simulation evaluates the radial fluxes and parallel flows around $E_r \simeq 0$ which are much more smoothly dependent on $E_r$ and similar to those obtained from the global calculations.
Effects of the tangential magnetic drift $\hat{v}_{m}$ because stronger under the following conditions. First, according to the simulations, the tangential magnetic drift $\hat{ v }_{m} $ is more obvious in LHD than W7-X and HSX. In W7-X and HSX, the magnetic configuration is chosen so as to reduce the radial drift of trapped particles and remains the neoclassical transport in $1/\nu$-regime. This reduces the peak value of $\Gamma_i$ at the poloidal resonance, $\omega_E+\omega_h=0$ in Eq. , and results in the small gap between the ZMD and the ZOW models in these machines compared to LHD. Second, the effect is obvious in the low collisional plasma. At $E_r \simeq 0$, the tangential magnetic drift is required to avoid the poloidal resonance. Otherwise, the artificially strong $1/\nu$-type neoclassical transport will occur. Third, the ZOW, ZMD, and DKES-like models agree with one another in a series of electron simulations. The discrepancies occur more clearly on the ions. This suggests that the conventional local drift-kinetic models are sufficient for electron simulation.
The difference in the treatment of the $\boldsymbol E\times \boldsymbol B$ drift term has also been found to cause a large error in neoclassical transport calculation. The assumption of incompressible $E \times B$ drift in the DKES-like model results in the miscalculation of the neoclassical transport for the larger poloidal Mach number of $\mathcal{M}_{p} > 0.4$. Due to the mass dependency of $\mathcal{M}_{p} > \propto v_E/v_{th,a} \sim \sqrt{m_a}$, the heavier ion $\mathcal{M}_{p}$ such as He and W increases. Therefore, the parameter window in which the incompressible-$\boldsymbol E\times \boldsymbol B$ approximation is valid will be narrower for heavier species.
Regarding the practical application, the neoclassical flux and bootstrap current are evaluated at the ambipolar condition. The ion-root usually exists when $T_i \simeq T_e$; the electron-root appears when $T_i \ll T_e$[@yokoyama_CPP_2010]. The peak of $\Gamma_i$ at $E_r=0$ is an artifact of the ZMD and the DKES-like models. It suggests that the $T_e/T_i$ is the threshold of transition between the ion-root and the electron-root. Therefore, the magnitude of $T_e/T_i$ will be less/lower in the global and the ZOW models than in the ZMD and the DKES models. The neoclassical transport varies drastically if the ambipolar-$E_r$ switches from an ion-root to an electron-root. Therefore, the introduction of tangential magnetic drift term in a local code plays an importance role for the investigation of the ambipolar-root transition. Figure \[fig:BC\] indicates that the $\hat{v}_{m}$ term slightly affects the bootstrap current evaluation. Furthermore, the sign of the bootstrap current may change when the ambipolar-$E_r$ transits from a negative to a positive root. This will be also related to the study on the bootstrap current effect on MHD equilibrium.
On the basis of the present study, the particle flux, energy flux, and bootstrap current of FFHR-d1 will be studied in the future. The investigation will be carried out by iteration between the MHD equilibrium and the bootstrap current calculations in order to collect data for the design of FFHR-d1. The FFHR-d1 magnetic configuration is similar to LHD so that the present study on an LHD configuration provides useful insight on the magnetic drift effect on the neoclassical transport in FFHR-d1. The effect of the bootstrap current on the MHD equilibrium will play a more important role in FFHR-d1 than that in present LHD operations because the central $\beta$ will be about $ 5 \% $[@goto_nf2015].
It is found that the $\hat{v}_m$ term does not only decrease the height of the peak of $\Gamma_i$ but also changes the value of $E_r$ at which $\Gamma_i(r,E_r)$ peaks. The approximated amount of the shift in $E_r$ in LHD can be estimated by the bounce-averaged poloidal precession drift[@Beidler_2001] of thermal ions as in Eq.. The bounce-averaged magnetic drift for deeply-trapped particles is approximated as $$\begin{split}\label{eq:bave}
\omega_h & \sim \frac{ v_d }{\epsilon_t B_0 } \frac{ \partial B_{2,10} (\rho) }{ \partial r }
\left\langle \cos ( m \theta - n \zeta )\right\rangle_b
\\ & \sim - \frac{ 4v_d}{ a }
\end{split}$$ where $\rho\equiv r /a$ and $\langle\cdots\rangle_b$ denotes the bounce-average over a particle trajectory trapped in a helical magnetic ripple. The radial dependence of the helical component is approximated as $B_{2,10} (\rho) \simeq 2 (a/R_0) B_0 \rho^2 $ according to the tendency found in the MHD equilibrium for LHD plasma. In Eq., $(\theta, \zeta) = (0,\pi/10)$ is chosen because this is the bottom position of both toroidal and helical ripples. Substituting the parameters $B_0, a, \epsilon_t$, and $v_d$ for the LHD case, the shift of the $\Gamma_i$-peak is estimated as $$\label{eq:Er-shift}
E_r \simeq - \frac{ 4 T_i \rho}{ e_i R_0}$$ at which poloidal resonance $\omega_E+\omega_h=0$ occurs. Eq. agrees with the tendency of the peak shift in $\Gamma_i$ from the ZOW and the global models, which are Figs.8-10 in Matsuoka et al.[@Matsuoka2015] Since high-temperature discharge $T_i>10 keV$ is planned in FFHR-d1, it is anticipated that the peak of $\Gamma_i$ in the ZOW model will appear more negative-$E_r$ which can be close to the ion-root $E_r$ value. In such a case, the difference between the ZOW and ZMD models becomes significant in evaluating the neoclassical transport level in the ambipolar condition.
The authors would like to thank Dr. J. M. García Regaña for the W7-X configuration data, and Mr. Jason Smoniewski for the DKES and PENTA numerical results of HSX. The simulations are carried out by Plasma Simulator, National Institute for Fusion Science. This work was supported in part by Japan Ministry of Education, Culture, Sports, Science and Technology (Grant No. 16K06941) and in part by the NIFS Collaborative Research Programs (NIFS16KNST092 and NIFS16KNTT035).
Source and Sink term in FORTEC-3D {#AppendixA}
=================================
As explained in Sec. \[section:particle\_flux\] and \[sec:parall\_monentum\_balance\], an adaptive source and sink term is introduced in the global and local FORTEC-3D codes. Thus, the flux-surface averaged density and pressure perturbation from the $f_{1}$ part, which are defined by Eqs. and , become negligible compared to the background density and pressure, i.e., $\langle\mathcal{N}_{1}\rangle\ll n$ and $\langle P_{1}\rangle \ll nT$. Such a source/sink term is constructed according to the following considerations.
First, the source/sink term acts to reduce the flux-surface average perturbations $\langle\mathcal{N}_1 \rangle$ and $\langle P_1\rangle$. It is considered that the source/sink term should **not** smoothen the spatial variation of them on the flux surface, because the non-uniform distribution reflects the compressible flow on the flux surface. Therefore, the source-sink term is constructed to reduce $\langle \mathcal{N}_1\rangle$ and $\langle P_1\rangle$, while it maintains the fluctuation patterns on the flux surface, $\mathcal{N}_1- \langle\mathcal{N}_1\rangle$ and ${P}_1-\langle P_1\rangle$. Second, the source-sink term should be adaptive. The strength of the source-sink term is proportional to $\langle \mathcal{N}_1\rangle$ and $\langle P_1\rangle$ so that the users do not have to control the strength of the source-sink term. Third, the source/sink term does not contribute as a parallel momentum source as shown in Eq., because the steady-state parallel momentum balance can be found without giving an artificial source/sink term.
In the drift-kinetic equation for $f_{1}$ , the source-sink term $S_1$, which satisfies the conditions explained above, is given in the form $S_1=s(\psi,v,\xi,t)f_M$ with the following constraints: $$\begin{aligned}
\label{eq:ss_constraint}
\int d^3v~ sf_{M}&=&-\nu_S \langle \mathcal{N}_1\rangle,\nonumber\\
\int d^3v~ m_av_\parallel sf_{M}&=&0,\\
\int d^3v~ \frac{m_av^2}{2}sf_{M}&=&- \frac{3}{2} \nu_S \langle P_1\rangle,\nonumber\end{aligned}$$ where $\nu_S$ is a numerical factor to control the strength of the adaptive source-sink term. There is arbitrariness to make a source/sink term which satisfies Eq.. The examples of the adaptive source/sink terms can be found in the references[@Landreman_2014][@Jolliet_NF_2012_023026]. In FORTEC-3D code, the source/sink term is implemented by diverting the field-particle collision operator $\mathcal{C}_Ff_M$. The field-particle operator is made so as to satisfy the following conservation laws for the like-particle linearized collision term[@satake_Comp_Phys_Comm2010], $$\begin{aligned}
\label{eq:ctcf}
\int d^3v~ \mathcal{C}_Ff_M&=&-\int d^3v~ \mathcal{C}_T(f_1),\nonumber\\
\int d^3v~ mv_\parallel \mathcal{C}_Ff_M&=&-\int d^3v~ mv_\parallel\mathcal{C}_T(f_{1}),\\
\int d^3v~ \frac{mv^2}{2}\mathcal{C}_Ff_M&=&-\int d^3v~ \frac{mv^2}{2}\mathcal{C}_T(f_{1}).\nonumber\end{aligned}$$ By comparing Eqs. and , one can see that operator $\mathcal{C}_F$ can be directly used to implement the source/sink term. In FORTEC-3D, the source/sink term is operated in the $(\theta,\zeta)$ cells on a flux-surface which is the same as those prepared for the collision terms. In this simulation, $20 \times 10 ( 20 \times 20 )$ cells on a $(\theta,\zeta)$-plane are employed. The strength of the source/sink term $\nu_S$ is varied case by case because the growth rate of $\langle\mathcal{N}_1\rangle$ and $\langle P_1 \rangle$ depends on the drift-kinetic model, magnetic configuration, and parameters such as $E_\psi$. See Eqs. and . In most cases, the moderate strength $\nu_S=0.5\sim 1.0\times \nu_{i}$ is enough to suppress $\mathcal{N}_1$ and $P_1$ to $\mathcal{O}(10^{-2})$, where $\nu_i$ is the ion-ion collision frequency. As demonstrated in Fig. \[fig:apd1\] for the ZOW and ZMD simulations in the LHD case, it is confirmed that the final steady-state solutions of the neoclassical fluxes are not affected by the strength of the source/sink term nor the timing from when the source/sink term is turned on. It is obvious that without the source/sink term the ZMD model does not conserve $\langle P_1\rangle$. The $\langle\mathcal{N}_1\rangle$ and $\langle P_1 \rangle$ both continue to change in the ZOW model, as expected from the particle and energy balance relations in Sec. \[section:particle\_flux\]. In the series of simulations without source/sink, the neoclassical fluxes $\Gamma_i$ and $\langle V_\parallel B\rangle$ continue evolving and one cannot obtain a quasi-steady state solution. By adopting $\nu_S= 0.5$ or $1.0$, the ZOW and ZMD models both converge to a quasi-steady state at which one can take a time average. It is observed that the pattern of the fluctuations on the flux surface, $\mathcal{N}_1- \langle\mathcal{N}_1\rangle$ and ${P}_1-\langle P_1\rangle$, are sustained before and after turning on the source/sink term. This scheme works well in the global, ZOW, and ZMD models. For the DKES-like model, the source/sink term is not necessary because it preserves the total particle number and energy ideally. However, the weak source/sink was given in the DKES-like model in this work to reduce the numerical error accumulation in $\mathcal{N}_1$ and $P_1$.
![The time evolution of (a) the density perturbation $\langle \mathcal{N}_1\rangle$, (b) the pressure perturbation $\langle P_1\rangle$, (c) the neoclassical particle flux $\Gamma_i$ and (d) the parallel flow $\langle V_\parallel B\rangle$ are the LHD ion case shown in Sec. \[sec:large\_Er\]. Furthermore, Figs.(a) and (b) are normalized by background density and pressure, respectively. In Fig.(d), the parallel flows of the ZMD model are plotted offset by $-4$. The source/sink term is turned on at $t=1.6\tau_i$ or $2.7\tau_i$. The numbers after “SS” in the legend indicate the strength of the source/sink term, $\nu_S$. []{data-label="fig:apd1"}](./fig-apdx_a.eps "fig:"){width="0.9\columnwidth"} ![The time evolution of (a) the density perturbation $\langle \mathcal{N}_1\rangle$, (b) the pressure perturbation $\langle P_1\rangle$, (c) the neoclassical particle flux $\Gamma_i$ and (d) the parallel flow $\langle V_\parallel B\rangle$ are the LHD ion case shown in Sec. \[sec:large\_Er\]. Furthermore, Figs.(a) and (b) are normalized by background density and pressure, respectively. In Fig.(d), the parallel flows of the ZMD model are plotted offset by $-4$. The source/sink term is turned on at $t=1.6\tau_i$ or $2.7\tau_i$. The numbers after “SS” in the legend indicate the strength of the source/sink term, $\nu_S$. []{data-label="fig:apd1"}](./fig-apdx_b.eps "fig:"){width="0.9\columnwidth"} ![The time evolution of (a) the density perturbation $\langle \mathcal{N}_1\rangle$, (b) the pressure perturbation $\langle P_1\rangle$, (c) the neoclassical particle flux $\Gamma_i$ and (d) the parallel flow $\langle V_\parallel B\rangle$ are the LHD ion case shown in Sec. \[sec:large\_Er\]. Furthermore, Figs.(a) and (b) are normalized by background density and pressure, respectively. In Fig.(d), the parallel flows of the ZMD model are plotted offset by $-4$. The source/sink term is turned on at $t=1.6\tau_i$ or $2.7\tau_i$. The numbers after “SS” in the legend indicate the strength of the source/sink term, $\nu_S$. []{data-label="fig:apd1"}](./fig-apdx_c.eps "fig:"){width="0.9\columnwidth"} ![The time evolution of (a) the density perturbation $\langle \mathcal{N}_1\rangle$, (b) the pressure perturbation $\langle P_1\rangle$, (c) the neoclassical particle flux $\Gamma_i$ and (d) the parallel flow $\langle V_\parallel B\rangle$ are the LHD ion case shown in Sec. \[sec:large\_Er\]. Furthermore, Figs.(a) and (b) are normalized by background density and pressure, respectively. In Fig.(d), the parallel flows of the ZMD model are plotted offset by $-4$. The source/sink term is turned on at $t=1.6\tau_i$ or $2.7\tau_i$. The numbers after “SS” in the legend indicate the strength of the source/sink term, $\nu_S$. []{data-label="fig:apd1"}](./fig-apdx_d.eps "fig:"){width="0.9\columnwidth"}
Derivation of Viscosity Tensor {#AppendixB}
==============================
The parallel moment equation is derived from Eq. with $\mathcal{A} = m v_{\parallel}$, $$\begin{aligned}
\label{eq:d-k-A-A1}
&\frac{\partial}{\partial t} \left( \int d v^3 f m v_{\parallel} \right) + \nabla \cdot \left( \int d v^3 f m v_{\parallel} \dot{ \boldsymbol X } \right)
\nonumber \\
=& \left( \int d v^3 f m \dot{ v }_{\parallel} \right)
+ \left( \int d v^3 f m [ \mathcal{S} + \mathcal{C} ] \right)
\nonumber \\
+& \int d v^3 f m v_{\parallel} \mathcal{G}.\end{aligned}$$ With $\dot {v}_{\parallel}$ in the global model, Eq., we have the following relation $$\begin{aligned}
\label{eq:d-k-A-A2}
\nonumber
&\nabla \cdot \left( \int d v^3 f m v_{\parallel} \dot{ \boldsymbol X } \right)
- \left( \int d v^3 f m \dot{ v }_{\parallel} \right)
\nonumber \\
& = \nabla \cdot \left( \int d v^3 f m v_{\parallel}^2 \boldsymbol b \right)
+ \int d v^3 f \boldsymbol b \cdot \left( \mu \nabla B - e_a \boldsymbol E \right)
\nonumber \\
& + \nabla \cdot \left( \int d v^3 f m v_{\parallel} \dot{ \boldsymbol X }_{\perp} \right)
- \int d v^3 f m v_{\parallel} \dot{ \boldsymbol X }_{\perp} \cdot \boldsymbol \kappa
\nonumber \\
& = \boldsymbol b \cdot \left\lbrace \nabla \cdot \left( \int d v^3 f \left[ m v_{\parallel}^2 \boldsymbol b \boldsymbol b + \mu B \left( \boldsymbol I - \boldsymbol b \boldsymbol b \right) \right] \right)
\right\rbrace
\nonumber \\
& + \boldsymbol b \cdot \left\lbrace \nabla \cdot \left[ \int d v^3 f m v_{\parallel} \left( \boldsymbol b \dot{ \boldsymbol X }_{\perp} + \dot{ \boldsymbol X }_{\perp} \boldsymbol b \right) \right]
\right\rbrace
\nonumber \\
& + e_a E_{\parallel} \int d v^3 f.\end{aligned}$$ Then, Eq. is obtained by rewriting Eq., $$\begin{aligned}
&\frac{\partial}{ \partial t } ( n m V_{ \parallel} ) + \boldsymbol{b} \cdot ( \nabla \cdot \boldsymbol{P} )
\nonumber \\
&= n e_{a} E_{\parallel} + F_{\parallel}
+ \int d^{3}v ~ \mathcal{S} m v_{\parallel}
+ \int d^3v ~ f \mathcal{G} m v_{\parallel},\end{aligned}$$ where
$$\begin{aligned}
{4}
& \boldsymbol P \equiv \boldsymbol P_{CGL} + { \boldsymbol \Pi }_{2},
\\
& \boldsymbol P_{CGL} \equiv \int d^3 v ~ [ ( m v_{\parallel}^2 \boldsymbol b \boldsymbol b + \mu B ( \boldsymbol I - \boldsymbol b \boldsymbol b ) ] f, \label{eq:P_cgl}
\\
& \boldsymbol \Pi_{2} \equiv \int d^3 v ~ m v_{\parallel} \bigg ( \dot{ \boldsymbol X }_{\perp} \boldsymbol b + \boldsymbol b \dot{ \boldsymbol X }_{\perp} \bigg ) f\label{eq:pi_2}
.\end{aligned}$$
It should be noted that the $\dot{ \boldsymbol X}_{\perp} \cdot \boldsymbol \kappa$ term in Eq. is involved in the symmetry of the $\boldsymbol \Pi_{2}$ tensor. On the other hand, Eq. is independent of the explicit form of $\dot{\boldsymbol X}_{\perp}$.
For the ZOW model, the parallel momentum balance equation is calculated with ${\dot { \boldsymbol X }}_{\text ZOW} = v_{\parallel} \boldsymbol b + \boldsymbol{v}_{E} + \hat{\boldsymbol v}_{m} $ and $$\begin{aligned}
\label{eq:dot_v_para-zow-1}
\dot{v}_{\parallel}
&= -\frac{ 1 }{ m } { \boldsymbol b } \cdot \left( \mu \nabla B \right)
+ {v}_{\parallel} \boldsymbol v_{E} \cdot \frac{ \nabla_{\perp} B }{ B }
\nonumber \\
&= -\frac{ 1 }{ m } { \boldsymbol b } \cdot \left( \mu \nabla B \right)
+ {v}_{\parallel} {\dot { \boldsymbol X }}_{\text ZOW} \cdot \boldsymbol \kappa
\nonumber\\
& - {v}_{\parallel} \left(
{\dot { \boldsymbol X }}_{\text ZOW} \cdot \boldsymbol \kappa
-\boldsymbol v_{E} \cdot \frac{ \nabla_{\perp} B }{ B }
\right).\end{aligned}$$ Then, the last term in Eq. is rewritten as $$\begin{aligned}
\label{eq:last_zow}
& {\dot { \boldsymbol X }}_{\text ZOW} \cdot \boldsymbol \kappa - \boldsymbol v_{E} \cdot \frac{ \nabla_{\perp} B }{ B }
\nonumber \\
& = \left( \hat{\boldsymbol v}_{m} + \boldsymbol{v}_{E} \right) \cdot \boldsymbol \kappa - \boldsymbol v_{E} \cdot \frac{ \nabla_{\perp} B }{ B }
\nonumber \\
& = \left( \hat{\boldsymbol v}_{m} + \boldsymbol{v}_{E} \right) \cdot \left( \frac{ {\nabla}_{\perp} B }{ B }
+ \frac{ \mu_{0} \boldsymbol J \times \boldsymbol B }{ B^2 } \right)
- \boldsymbol v_{E} \cdot \frac{ \nabla_{\perp} B }{ B }
\nonumber \\
& = \left[ \boldsymbol {v}_{m} \cdot \left( \boldsymbol I - \nabla \psi \boldsymbol{e}_{\psi} \right) \right] \cdot \left( \frac{ {\nabla}_{\perp} B }{ B }
+ \frac{ \mu_{0} \nabla p }{ B^2 } \right)
+ \boldsymbol v_{E} \cdot \frac{ \mu_{0} \nabla p }{ B^2 }
\nonumber \\
& = -\frac{ 1 }{ B } \frac{ \partial B }{ \partial \psi } \dot{\psi} .\end{aligned}$$ Therefore, Eq. is obtained. Using this $\dot{v}_{\parallel}$ for the ZOW model, Eq. is rewritten as $$\begin{aligned}
\nonumber
&\frac{\partial}{\partial t} \left( \int d v^3 f m v_{\parallel} \right)
+ \boldsymbol b \cdot \left\lbrace \nabla \cdot \left( \int d v^3 f m v_{\parallel}^2 \boldsymbol b \boldsymbol b \right)
\right\rbrace
\nonumber \\
& + \boldsymbol b \cdot \nabla \cdot \left[ \boldsymbol P_{CGL} + \boldsymbol \Pi_{2.\text{ZOW}} \right]
\nonumber \\
& = \left( \int d v^3 f m v_{\parallel} \mathcal{S} \right) + \left( \int d v^3 f m v_{\parallel} \mathcal{G} \right)
,\end{aligned}$$ where $$\begin{aligned}
\label{eq:B_pi_zow}
&\boldsymbol b \cdot \nabla \cdot \boldsymbol \Pi_{2, \text{ZOW}}
\nonumber \\
& = \boldsymbol b \cdot \left\lbrace \nabla \cdot \left[ \int d v^3 f m v_{\parallel} \left( \boldsymbol b \dot{ \boldsymbol X }_{\perp, {\text ZOW}} + \dot{ \boldsymbol X }_{\perp,{\text ZOW}} \boldsymbol b \right) \right] \right\rbrace
\nonumber \\
&- \left( \int d v^3 {v}_{\parallel} \frac{ 1 }{ B } \frac{ \partial B }{ \partial \psi } \dot{\psi} \right).\end{aligned}$$ The second term in Eq. breaks the symmetry of the $\Pi_{2}$ tensor.
For the ZMD model, the parallel momentum balance equation is calculated with ${\dot { \boldsymbol X }}_{\text ZMD} = v_{\parallel} \boldsymbol b + \boldsymbol{v}_{E}$ and $$\begin{aligned}
\label{eq:dot_v_para-zmd-1}
\dot{v}_{\parallel}
= -\frac{ 1 }{ m } { \boldsymbol b } \cdot \left( \mu \nabla B \right)
+ {v}_{\parallel} \boldsymbol v_{E} \cdot \frac{ \nabla_{\perp} B }{ B }.\end{aligned}$$ Because of the difference of $\dot{\boldsymbol X}_{\perp}$ between ZOW and ZMD, one finds that $$\begin{aligned}
& \boldsymbol v_{E} \cdot \frac{ \nabla_{\perp} B }{ B } - {\dot { \boldsymbol X }}_{\text ZMD} \cdot \boldsymbol \kappa
\nonumber \\
& = \boldsymbol v_{E} \cdot \frac{ \nabla_{\perp} B }{ B } - \left( \frac{ {\nabla}_{\perp} B }{ B }
+ \boldsymbol v_{E} \cdot \frac{ \mu_{0} \boldsymbol J \times \boldsymbol B }{ B^2 } \right)
\nonumber \\
& = 0.\end{aligned}$$ Therefore, $\boldsymbol \Pi_{2, \text{ZMD}}$ becomes $$\begin{aligned}
\label{eq:B_pi_zmd}
&\boldsymbol b \cdot \nabla \cdot \boldsymbol \Pi_{2, \text{ZMD}}
\nonumber \\
& = \boldsymbol b \cdot \left\lbrace \nabla \cdot \left[ \int d v^3 f m v_{\parallel} \left( \boldsymbol b \dot{ \boldsymbol X }_{\perp, {\text ZMD}} + \dot{ \boldsymbol X }_{\perp,{\text ZMD}} \boldsymbol b \right) \right] \right\rbrace
\nonumber \\
&= \boldsymbol b \cdot \left\lbrace \nabla \cdot \left[ n m V_{\parallel} \left( \boldsymbol b { \boldsymbol v }_{E} + {\boldsymbol v}_{E} \boldsymbol b \right) \right] \right\rbrace.\end{aligned}$$ Note that Eq. is equivalent to Eq.(33) in Ref[@Landreman_2014].
For the DKES model, the parallel momentum balance equation is calculated with ${\dot { \boldsymbol X }}_{\text DKES} = v_{\parallel} \boldsymbol b + \hat{\boldsymbol{v}}_{E}$ and $$\begin{aligned}
\label{eq:dot_v_para-dkes-1}
\dot{v}_{\parallel}
= -\frac{ 1 }{ m } { \boldsymbol b } \cdot \left( \mu \nabla B \right),\end{aligned}$$ which lacks in the $ {\dot { \boldsymbol X }}\cdot \boldsymbol \kappa $ term. Therefore, $\boldsymbol \Pi_{2, \text{DKES}}$ becomes $$\begin{aligned}
\label{eq:B_pi_dkes}
\boldsymbol b \cdot \nabla \cdot \boldsymbol \Pi_{2, \text{DKES}}
& = \boldsymbol b \cdot \left\lbrace \nabla \cdot \left[ \int d v^3 f m v_{\parallel} \left( \boldsymbol b \hat{ \boldsymbol v }_{E} + \hat{ \boldsymbol v }_{E} \boldsymbol b \right) \right] \right\rbrace
\nonumber \\
&+ n m V_{\parallel} \hat{ \boldsymbol v }_{E} \cdot \boldsymbol \kappa \end{aligned}$$ which is equivalent to Eq.(34) in Ref[@Landreman_2014]. The symmetry of Eq. is broken. Note that in the derivations shown in Appendix \[AppendixB\], we use assumptions $p = p(\psi)$, $\boldsymbol J \times \boldsymbol B = \nabla p $, and $\boldsymbol E = - \nabla \Phi (\psi)$.
In conclusion, the symmetry of viscosity tensor $\boldsymbol \Pi_{2}$ depends on the form of $\dot{ \boldsymbol X } \cdot \boldsymbol \kappa$ term in $\dot{v}_{\parallel}$ in each local model.
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---
abstract: 'We propose a general scheme for the “logic” of elementary propositions of physical systems, encompassing both classical and quantum cases, in the framework given by Non Commutative Geometry. It involves Baire\*-algebras, the non-commutative version of measurable functions, arising as envelope of the $C$\*-algebras identifying the topology of the (non-commutative) phase space. We outline some consequences of this proposal in different physical systems. This approach in particular avoids some problematic features appearing in the definition of the state of “initial conditions” in the standard ($W^*$-)algebraic approach to classical systems.'
---
[**Quantum Logic and Non-Commutative Geometry**]{}
P.A. Marchetti
*Dipartimento di Fisica, Università degli Studi di Padova,*
and
Istituto Nazionale di Fisica Nucleare, Sezione di Padova,
Via F. Marzolo, 8, 35131 Padova, Italia
e-mail: [email protected]
R. Rubele
3.0truecm Keywords: Quantum Logic, Non-Commutative Geometry, Baire\*-algebras 1.0truecm
6 mm
Introduction
============
0.2truecm
In many respects Non-Commutative Geometry (NCG) [@Connes1] appears as the most complete mathematical setting for a unified description of quantum and classical physical systems, besides being a source of some highly imaginative ideas in the attempt of constructing a unified theory of fundamental forces including gravity (see e.g.[@doplicher; @connes2; @froehlich; @witten] and references therein).
In this paper we propose a characterisation of the lattice of elementary propositions, i.e. the “logic”, of quantum and classical systems which appears to fit naturally in the framework of NCG and solves some problematic feature of the more standard $W$\*-algebraic approach (see e.g. [@haag; @primas; @redei; @thirring]). In order to keep the paper reasonably self-contained some basic notions concerning $C$\*- and $W$\*-algebras used throughout the text are given in the Appendix.
A root of Non-Commutative Geometry is the idea that one can generalise many branches of standard functional analysis, such as measure theory, topology and differential geometry, by replacing the commutative algebras of functions over some space $X$ by a suitable non-commutative algebra which may in a sense be interpreted as the “algebra of functions over a non-commutative space".
In the commutative case one can consider various degrees of regularity of the functions ranging from measurable, to continuous, to smooth. The non-commutative analogue of the algebra of complex bounded continuous functions is a $C$\*-algebra, whereas spaces of complex essentially bounded measurable functions ($\mathscr{L}^\infty$) are generalized by von Neumann algebras or, in abstract form, $W$\*-algebras. Algebraic generalizations of spaces of smooth functions are pre-$C$\*-algebras, i.e. $\ast$-subalgebras of a $C$\*-algebra closed under the holomorphic functional calculus. Probability measures on spaces of continuous functions find a non-commutative generalization in the concept of algebraic states, henceforth simply states: the linear positive normalized functionals on a $C^*$-algebra; in particular Dirac measures with support on one point are generalized by pure states, i.e. states that cannot be written as convex combinations of other states. (Notice that since a $C^*$-algebra is a Banach space, states are elements of its dual as they are continuous being bounded.)
A link with quantum theory appears when quantum mechanics is interpreted as a “mechanics over a non-commutative phase space" in the spirit of Heisenberg and Dirac. If we consider a quantum non-relativistic elementary particle with classical analogue i.e. without internal degrees of freedom, the appropriate algebra of “smooth functions, or observables, in phase space" is the Weyl algebra generated by the bounded version,
$$e^{i \vec\alpha\cdot \vec q} e^{i \vec\beta \cdot \vec p} =
e^{i\vec\beta \cdot \vec p} e^{i \vec\alpha\cdot \vec q} e^{i
\hbar \vec\alpha\cdot \vec \beta \over 2},$$
of the celebrated Heisenberg commutation relations:
$$q_i p_j - p_j q_i = i \hbar \delta_{ij},$$
where $\alpha_i, \beta_i \in \mathbb{R}$ and $\{ q_i
\}^3_{i=1}$ and $ \{p_i\}^3_{i=1}$ are the “coordinates" of the “non-commutative phase space" corresponding respectively to canonical coordinates of the underlying commutative classical configuration space and their conjugate momenta. It turns out that the corresponding $C$\*-algebra of “continuous bounded observables" is isomorphic to the $C$\*-algebra ${\mathfrak{K}}({\mathscr H})$ of compact operators on an infinite dimensional separable complex Hilbert space, ${\mathscr H}$, and the $W$\*-algebra of “bounded measurable observables" is isomorphic to the algebra ${\mathfrak{B}}({\mathscr H})$ of all bounded operators on ${\mathscr H}$. (The qualification “continuous bounded” used above is meant to evocate the analogy with the commutative case and is not referred to norm continuity of operators on Hilbert spaces, which of course is equivalent to boundedness.)
A relation with quantum logic then appears as follows. It has been recognised in the seminal work of Birkhoff and von Neumann [@vonneumann], that the system of elementary physical propositions corresponding to yes-no experiments of quantum mechanics can be represented as the complete orthomodular[^1] lattice of closed subspaces of a separable Hilbert space ${\mathscr H}$. (Actually, orthomodularity was not introduced in [@vonneumann], but by Piron [@piron1]; for an historical comment see [@redei].) Such lattice can be characterized also algebraically in terms of the associated orthogonal projectors, $p$, in ${\mathscr H}$, with the well known definitions of orthocomplement ${}^\bot$, meet $\wedge$ and join $\vee$ operations: $p^\bot ={\bf
1} -p$, $p_1 \wedge p_2 =$lim$_{n \rightarrow \infty}(p_1 p_2)^n =
$lim$_{n \rightarrow \infty}(p_2 p_1)^n$, $p_1 \vee p_2 =(p_1^\bot
\wedge p_2^\bot)^\bot$ and partial ordering defined by $p_1 \leq
p_2$ iff $p_1=p_1 \wedge p_2$. In turn, the projectors are the self-adjoint elements of the von Neumann algebra ${\mathfrak{B}}({\mathscr
H})$ satisfying $p^2=p$. The set of projectors of any $W$\*-algebra has the structure of a complete orthomodular lattice with lattice operations defined algebraically as above. Therefore it has been proposed to identify as a model for the propositional lattice of physical systems the lattice of projectors of a $W$\*-algebra.
A classical system in this setting is given in terms of a commutative $W$\*-algebra; the corresponding lattice of propositions is therefore distributive, i.e. a Boolean algebra. Hence the transition from the classical to the quantum level corresponds to the elimination of the commutativity postulate, due to the existence of the universal constant $\hbar$, which is replaced by $0$ in classical mechanics. More precisely, for a classical particle the $W$\*-algebra generated by the commutation relation (1) with $\hbar =0$ is taken to be $\mathscr{L}^\infty
(\Omega, \omega_L)$ where $\Omega$ is the phase space and $\omega_L$ is the Lebesgue measure on $\Omega$ which coincides with the Liouville measure given in terms of the symplectic form.
Although the $W$\*-approach has the great virtue of describing classical and quantum systems and the related logics in a unifying canonical scheme, it reveals some drawbacks in the definition of states at the classical level. In the algebraic approach the states describe the “states of knowledge" of the observable quantities and pure states correspond to maximal knowledge. However in classical systems points in phase space are of zero $\omega_L$-measure and hence “invisible" to $\mathscr{L}^\infty (\Omega, \omega_L)$. Therefore, as already noticed by von Neumann, it is not naturally defined in this setting the most fundamental state of classical mechanics corresponding to a single point in phase space selecting “initial conditions” of the system; see [@halvorson] for a more refined and recent analysis of the problem. Although this fact could be attributed to the practical impossibility of a precise measurement, it is at least philosophically somewhat unnatural. For a related problem e.g. Teller [@teller] argued that *“if we believe that systems possess exact values for continuous quantities, classical theory contains the descriptive resources for attributing such values to the system, whether or not measurements are taken to be imprecise in some sense”.*
Instead, points in phase space can be taken as support of Dirac measures and these are naturally defined as states on $C_0 (\Omega)$, the $C$\*-algebra of bounded continuous functions on $\Omega$ vanishing at infinity, generated by the commutation relations (1) with $\hbar =0$. However $C_0 (\Omega)$ does not contain non-trivial projectors, since these are characteristic functions which are not continuous. Analogously the $C$\*-algebra of compact operators on a separable Hilbert space, ${\mathfrak{K}}({\mathscr
H})$, generated by (1) with $\hbar \not =0$, does not contain a lattice of projectors even $\sigma$-complete, i.e. stable under a countable number of meet and join operations, and this is the weakest reasonable completeness to require in a logic, excluding “unsharp” approaches, see e.g. [@dallachiara] (we use the word “complete” to denote stability under an arbitrary, even non countable, number of meet and join operations). On the other hand the pure states on ${\mathfrak{K}}({\mathscr
H})$ are exactly in correspondence with the rays of ${\mathscr
H}$, as required on physical grounds. In fact the dual of ${\mathfrak{K}}({\mathscr
H})$ is isomorphic to the space of trace-class operators on ${\mathscr
H}$, the condition of positivity and normalization then identifies the states as the “statistical matrices”. The pure states correspond to one dimensional projectors hence to rays, but this correspondence does not hold for the pure states on ${\mathfrak{B}}({\mathscr H})$, which include also unphysical “improper states”. ( To save the physically required correspondence in this case one has to restrict to the normal states, i.e those which are completely additive.)
Hence in a NCG setting as a natural framework to embed an algebraic model of elementary propositions one is naturally looking for a “space” in general larger then the $C$\*-algebra of “continuous bounded observables" ${\mathfrak{A}}$, but smaller than the $W$\*-algebra of “essentially bounded measurable observables", and containing a $\sigma$-complete orthomodular lattice of projectors. Furthermore one would like this space still to be some “closure" of the $C$\*-algebra ${\mathfrak{A}}$, which in the NCG approach identifies the topology of the non-commutative phase space and is taken as the basic algebra, identifying the space of physical states.
This “space” in fact exists, it is called Baire\*-algebra and can be described in the above terminology as the $C$\*-algebra of (Baire) measurable bounded functions or observables on a generally “non-commutative" space, and it is generated by ${\mathfrak{A}}$, as a suitable enveloping algebra. We denote it by $\mathscr{B}({\mathfrak{A}})$.
We propose to identify the lattice of projectors of $\mathscr{B}({\mathfrak{A}})$, denoted by $\mathbb{P}(\mathscr{B}({\mathfrak{A}}))$, as a model for the lattice of elementary propositions of the physical systems described by ${\mathfrak{A}}$, and to identify the logical states $\phi_L$ \[see next section\] as the restriction to $\mathbb{P}(\mathscr{B}({\mathfrak{A}}))$ of the lift $\tilde\phi$ to $\mathscr{B}({\mathfrak{A}})$ of states $\phi$ on ${\mathfrak{A}}$. If $a \in
\mathscr{B}({\mathfrak{A}})$, then $\tilde\phi(a)$ is the expectation value of the measurable observable $a$ in the state $\tilde\phi$ and in particular if $p$ is a projector in $\mathscr{B}({\mathfrak{A}})$, then $\tilde\phi(p)= \phi_L(p) \in [0,1]$ yields the probability that the proposition represented by $p$ is true in the logical state $\phi_L$.
As it will be discussed in section 3, this setting solves the above quoted difficulty of the $W$\*-algebra approach. The scheme can be summarized by means of the following commutative diagram
1truecm
(100,120)(-52,190) (47,268)[${\mathbb P} ({\mathscr B} ({\mathfrak{A}}))$]{} (130,268)[${\mathscr B} ({\mathfrak{A}})$]{} (210,268)[${\mathfrak{A}}$]{} (95,206) (174,206)[${\mathbb
C}$]{}
(93,243)[${\phi_L}$]{} (168,243)[$\widetilde{\phi}$]{} (187,243)[$\phi$]{}
(103,268)[$\stackrel{i}{\longrightarrow}$]{} (170,268)[$\stackrel{j}{\longleftarrow}$]{} (135,206)[$\stackrel{k}{\longrightarrow}$]{}
(210,260)[(-3,-4)[28]{}]{} (75,260)[(3,-4)[28]{}]{} (150,260)[(3,-4)[28]{}]{}
\
where $i,j$ and $k$ are the obvious injections. We remark that a consequence of this proposal is that the lattice of elementary propositions of a physical quantum system, although always orthomodular $\sigma$-complete it is not always complete, nor atomic, nor Hilbertian (i.e. isomorphic to all the orthogonal subspace of a separable Hilbert space). These specific features are encoded in the $C$\*-algebra of “continuous bounded observables" ${\mathfrak{A}}$ of the system. More obviously, for classical systems ${\mathfrak{A}}$ is abelian and this implies a distributive property for the lattice of propositions.
In the rest of this paper we will make mathematically precise the setting described above. Although the mathematical results presented here are not original the overall scheme and its degree of generality to the best of our knowledge are novel.
0.2truecm
Logical States
==============
0.2truecm
Let $\mathscr{L}$ be the orthomodular $\sigma$-complete lattice assumed to describe the set of elementary propositions of a physical system. A logical state (in the sense of Mackey-Jauch-Piron [@Mackey; @jauch; @piron]) $\phi_L$ is a $\sigma$-orthoadditive map from ${\mathscr L}$ to \[0,1\]; more explicitly, if $P$ is a proposition in ${\mathscr L}$, then $\phi_L(P^\bot)=1-\phi_L(P)$, and if $\{P_i\}_{i \in I}$ is a countable number of propositions pairwise orthogonal, i.e. $P_i \leq P_j^\bot$ for $i \neq j$, then $\phi_L(\vee_i P_i)= \sum_i \phi_L(P_i)$. $\phi_L(P)$ is the probability that the proposition $P$ is true in the state $\phi_L$. A logical state $\phi_L$ is “pure” if it cannot be written as a convex combination of other logical states i.e. if for any two logical states $\phi_1$ and $\phi_2$ the equation $\phi_L= \alpha \phi_1 + (1- \alpha) \phi_2, 0< \alpha <1,$ implies $\phi_L=\phi_1=\phi_2$. Pure logical states correspond to the maximal knowledge attainable on the propositional system. A logical state is called “normal” if is completely orthoadditive.
In the $W$\*-algebraic approach we have the following:
0.2truecm **Theorem 2.1.** [@cirelli] *Identifying as a model for ${\mathscr L}$ the lattice of projectors of a $W$\*-algebra ${\mathfrak{M}}$, the restriction to $\mathbb{P}({\mathfrak{M}})$ of normal states on ${\mathfrak{M}}$ are normal logical states; furthermore pure logical states corresponds to restriction of pure states.* 0.2truecm
As discussed in the introduction the proposal to identify the logical states as restriction of normal states on $W$\*-algebras excludes the states corresponding to single points in phase space in classical systems unless the phase space is discrete in view of the following:
0.2truecm **Theorem 2.2.** *A state on a $W$\*-algebra ${\mathfrak{M}}$ is normal iff it is an element of its predual ${\mathfrak{M}}_*$.*
0.2truecm Since for classical systems ${\mathfrak{M}}=
\mathscr{L}^\infty (\Omega, \omega_L)$ and ${\mathfrak{M}}_* = \mathscr{L}^1
(\Omega, \omega_L)$, this excludes the Dirac measures concentrated on one point of $\Omega$ as they do not belong to $\mathscr{L}^1 (\Omega,
\omega_L)$.
On the other hand the requirement of normality is perfectly suited for a standard (i.e. without, or at least with, a countable set of superselection sectors) quantum mechanical system with finite dynamical degrees of freedom, where we know that physical states are “statistical matrices", which are positive trace 1 elements of $J_1 ({\mathscr
H})$, the space of trace-class operators on the separable Hilbert space of physical vector states ${\mathscr H}$. In this case, in fact, ${\mathfrak{M}}= {\mathfrak{B}}({\mathscr H})$ and ${\mathfrak{M}}_* =
J_1 ({\mathscr H})$.
The choice of a $W$\* or von Neumann algebra as foundational in a $C$\* approach is also mathematically not entirely natural in NCG, as Connes [@connes3] pointed out: *“It is true, and at first confusing, that any von Neumann algebra is a $C$\*-algebra, but not an interesting one because it is usually not norm separable. For instance let $(X,\mu)$ be a diffuse probability space (every point $p \in X$ is $\mu$-negligible), then $\mathscr{L}^\infty (X, \mu)$ is a von Neumann algebra but it is not norm separable and its spectrum* \[see definition after Theorem 3.1 and comment after Definition 3.2\] *as a $C$\*-algebra is a pathological space that has little to do with the original standard Borel space $X$”.*
The somewhat unsatisfactory situation outlined above is avoided if we introduce the notion of Baire\*-algebra.
0.2truecm
Baire\*-algebras
================
0.2truecm
To put in a proper perspective the definition of a Baire\*-algebra it is convenient to recall some basic notions of the theory of Baire functions, whose space is the commutative version of a Baire\*-algebra.
Let $(X, \Sigma$) be a measure space, where $\Sigma$ denotes a $\sigma$-algebra of subsets of $X$. A real or complex function is $\Sigma$-measurable if $f^{-1} (B) \in \Sigma$ for every $B$ borelian in $\mathbb{R}$ or in $\mathbb{C}$, respectively. A class $\mathscr{F}$ of real functions over $X$ is called monotonically sequentially complete if every limit of a monotonic sequence of functions of $\mathscr{F}$ belongs to $\mathscr{F}$. The class of real $\Sigma$-measurable functions is an algebra monotonically complete $\sigma$-stable under the lattice operations of meet and join.
Let $X$ be a locally compact topological space. A compact set of $X$ is of type $G_\delta$ if it is a countable intersection of open sets of $X$. The class of $G_\delta$ compacts generates the $\sigma$-algebra $B_X$ of the Baire sets of $X$. This is the smallest $\sigma$-algebra from which one can reconstruct the topology of $X$ [@halmos].
A real function on $X$ is called a Baire function if it is $B_X$-measurable; a complex function is a Baire function if both its real and imaginary part are Baire functions. The class of real Baire functions is the smallest class including all continuous function in $X$ and the limit of every bounded monotone sequence of them. The class of complex Baire functions on $X$ will be denoted by ${\mathscr B} (X)$. If $X$ is a metric space then the $\sigma$-algebra of Baire sets coincides with the $\sigma$-algebra of Borel sets, generated by the open sets of $X$, and the Baire functions are Borel functions. For this reason Baire\*-algebras were called Borel\*-algebras in [@pedersen]. To each point $p
\in X$ is associated a Dirac measure $d \mu_p$ on $\mathscr{B}(X)$ with support $\{p\}$ and mass 1.
To discuss the generalization to a non-commutative setting we need some preliminary definitions which extend to such a setting the basic notions involved in the constructions outlined above. A $C$\*-algebra ${\mathfrak{A}}$ is called monotonically sequentially complete if every bounded monotone sequence of the self-adjoint part of ${\mathfrak{A}}$, ${\mathfrak{A}}_{sa}$, possesses a limit in ${\mathfrak{A}}_{sa}.$ A state $\phi$ over a monotonically sequentially complete $C$\*-algebra ${\mathfrak{A}}$ is called $\sigma$-normal if for every bounded monotone sequence $\{x_n \}_{n \in {\bf N}}$ in ${\mathfrak{A}}_{sa}$ we have $$\phi (\bigvee_n x_n) = \bigvee_n \phi (x_n).$$
0.2truecm **Definition 3.1.** [@pedersen] *A $C$\*-algebra ${\mathscr B}$ is called a Baire\*-algebra if it is monotonically sequentially complete ad it admits a separating family of $\sigma$-normal states.*
0.2truecm Notice that, as discussed below, in the commutative case ${\mathscr B} (X)$ is a Baire\*-algebra with separating family of $\sigma$-normal states generated by the Dirac measures $\{d \mu_p \}_{p \in X}$.
An important result connecting Baire\* and $W$\*-algebras is the following:
0.2truecm **Theorem 3.1.** *If a Baire\*-algebra has a faithful representation in a separable Hilbert space, then it is isomorphic to a $W$\*-algebra.*
0.2truecm There is a natural “closure" of a $C$\*-algebra to obtain a Baire\*-algebra. To present this construction we need some preliminary definitions. Given a $C$\*-algebra ${\mathfrak{A}}$, let $\hat{\mathfrak{A}}$, be its spectrum, i.e. the set of (equivalence classes of unitarily equivalent) irreducible representations of ${\mathfrak{A}}$. Let $\phi$ be a (representative) pure state corresponding to a point of $\hat{\mathfrak{A}}$, and by $\pi_p$ the corresponding representation. The atomic representation of ${\mathfrak{A}}$ is given by $\pi_a = \oplus_{\phi\in \hat{\mathfrak{A}}} \pi_\phi$ and it is a faithful representation of ${\mathfrak{A}}$.
Then we have the following:
0.3truecm **Definition 3.2.** (Baire\* enveloping algebra)[@pedersen] *Given a $C$\*-algebra, ${\mathfrak{A}}$, and a subset $M \subset {\mathfrak{A}}_{sa}$, we define the monotone sequential closure of $M$, ${\mathscr B} (M)$, as the smallest subset of the atomic representation $\pi_a ({\mathfrak{A}}_{sa})$, containing $\pi_a(M)$ and the limit of every monotone sequence of elements of $\pi_a (M)$. The Baire\* enveloping algebra of ${\mathfrak{A}}$, is given by $${\mathscr B} ({\mathfrak{A}}) \equiv {\mathscr B}({\mathfrak{A}}_{sa}) + i {\mathscr
B}({\mathfrak{A}}_{sa}).$$ ${\mathscr B} ({\mathfrak{A}})$ is a Baire\*-algebra with the family of $\sigma$-normal states given by the unique extension of the states on ${\mathfrak{A}}$ to ${\mathscr B}
({\mathfrak{A}})$.*
0.2truecm To better understand the meaning of the Baire\* enveloping algebra notice that if ${\mathfrak{A}}$ is commutative and separable, then by the Gel’fand isomorphism (see e.g. [@pedersen; @thirring]), the spectrum $\hat{\mathfrak{A}}$ is a locally compact Hausdorff space and ${\mathfrak{A}}$ is isomorphic to $C_0
(\hat{\mathfrak{A}})$, the space of continuous function in $\hat{\mathfrak{A}}$ vanishing at infinity (if $\hat{\mathfrak{A}}$ is non-compact). Therefore ${\mathscr B} ({\mathfrak{A}}) = {\mathscr B} (\hat{\mathfrak{A}})$, i.e. the enveloping Baire\*-algebra is exactly the algebra of complex Baire functions on $\hat{\mathfrak{A}}$. Conversely if ${\mathfrak{A}}=C(X)$ with $X$ locally compact, $\hat{\mathfrak{A}}\simeq X$ as a topological space and $\widehat{\mathscr B} ({\mathfrak{A}}) \simeq X$ as a Borel space. The irreducible representations correspond to pure states given by the normalised Dirac measures $\{d\mu_p \}_{p \in \hat{\mathfrak{A}}}.$
Notice that since ${\mathscr B} ({\mathfrak{A}})$ has no faithful representations on a separable Hilbert space unless $\hat{\mathfrak{A}}$ is discrete, then in general the commutative Baire\*-algebra ${\mathscr B} ({\mathfrak{A}})$ is not a $W$\*-algebra. However we have the following result refining the previous one:
0.2truecm **Theorem 3.2.** [@davies] *If ${\mathfrak{A}}$ has a faithful representation $\pi$ on a separable Hilbert space then ${\mathscr B}(\pi({\mathfrak{A}})) \simeq
\pi({\mathfrak{A}})''$ i.e. it is isomorphic to the von Neumann algebra generated by $\pi({\mathfrak{A}})$ and its $\sigma$-normal states are the normal states of the von Neumann algebra.*
0.2truecm For the logical interpretation, the crucial property of Baire\*-algebras is the following:
0.2truecm **Theorem 3.3.** *The set of projectors ${\mathbb P}({\mathscr B})$ of a Baire\*-algebra ${\mathscr B}$ is an orthomodular $\sigma$-complete lattice.*
0.2truecm Furthermore, since the extensions to ${\mathscr B} ({\mathfrak{A}})$ of the states on ${\mathfrak{A}}$ are $\sigma$-normal, we have:
0.2truecm **Proposition.** *The restriction of the $\sigma$-normal states of the Baire\*-enveloping algebra ${\mathscr B}({\mathfrak{A}})$ to ${\mathbb P}({\mathscr B}({\mathfrak{A}}))$ are logical states.* 0.2truecm The identification of Baire\*-algebras as the abstract setting for bounded measurable observables is the one that makes it transparent the interpretation of quantum mechanics as a “theory of quantum probability". Although there is a high amount of papers written on this topic, it seems that a framework like the one we are outlining here is not considered. As an example, in a quite recent general review on the subject [@streater], R. F. Streater pointed out that: *“Though the classical axioms were yet to be written down by Kolmogorov, Heisenberg, with help of the Copenhagen interpretation, invented a generalisation of the concept of probability, and physicists showed that this was the model of probability chosen by atoms and molecules."* However, the algebraic ($W^*$-)approach envisaged therein appears less close than ours to the standard treatment of probability on topological measure spaces, where the Borel or Baire structure is determined by the topology, as ${\mathscr B}({\mathfrak{A}})$ is determined by ${\mathfrak{A}}$.
We end this section with a
0.2truecm **Remark.** In the definition of enveloping Baire\*-algebra we can replace the atomic representation $\pi_a$ with the universal representation $\pi_u = \oplus_{\phi\in
S({\mathfrak{A}})} \pi_\phi$, where $S ({\mathfrak{A}})$ is the set of states on ${\mathfrak{A}}$ and the corresponding ${\mathscr B} ({\mathfrak{A}})$ is isomorphic to the one defined via $\pi_a$. Then ${\mathscr B} ({\mathfrak{A}}) \subset
\pi_u ({\mathfrak{A}})''$, which is the universal enveloping von Neumann algebra of ${\mathfrak{A}}$.
Therefore the $\sigma$-complete orthomodular lattice of ${\mathscr B} ({\mathfrak{A}})$ describing the elementary propositions of the system characterized by ${\mathfrak{A}}$ can be embedded in the complete orthonormal lattice of $\pi_u ({\mathfrak{A}})'' $; for the relevance of the existence of the embedding from the logical point of view see [@dallachiara].
0.3truecm
Consequences for the logic of physical systems
==============================================
0.2truecm
Using the notions introduced in the previous section one can make precise the scheme outlined in the Introduction. At the foundational level one considers the algebra of “continuous bounded observables" of the physical system, described by a $C$\*-algebra ${\mathfrak{A}}$, possibly given as the closure of a pre-$C$\*-algebra of “smooth observables”, and the states on ${\mathfrak{A}}$ giving the expectation values of the observables. The algebraic realization of the lattice of elementary propositions corresponding to yes-no experiments, concerning the system described by ${\mathfrak{A}}$ is given by the $\sigma$-complete orthomodular lattice of the projectors of the Baire\* enveloping algebra ${\mathscr B}( {\mathfrak{A}})$, i.e. ${\mathbb P} ({\mathscr
B}({\mathfrak{A}}))$. Logical states are given by the restriction to ${\mathbb P}({\mathscr B})$ of the lift to ${\mathscr B}({\mathfrak{A}})$ of the algebraic states on ${\mathfrak{A}}$. Then pure logical states describing maximal knowledge correspond to pure states on ${\mathfrak{A}}$; notice that in general they are not pure states of ${\mathscr B}(
{\mathfrak{A}})$.
Let us comment on some implications of the above scheme for the logic of elementary propositions of physical systems.
0.3truecm **1) Systems in classical mechanics.**
0.2truecm
If the phase space $\Omega$ of the system is a locally compact Hausdorff space, then ${\mathfrak{A}}= C_0 (\Omega)$ and ${\mathscr
B}({\mathfrak{A}}) = {\mathscr B} (\Omega)$. The states on ${\mathfrak{A}}$ are the regular Borel probability measures which have a unique extension to ${\mathscr B} (\Omega)$. Pure states are Dirac measures $\{d\mu_p\}_{p\in \Omega}$ with support on one point in phase space , hence solving the problem outlined in the Introduction.
0.2truecm **Remark.** This solution was first envisaged in [@davies; @plymen] where instead of Baire\* enveloping algebras, $\Sigma$\* enveloping algebras were used, roughly speaking replacing monotone sequential closure with weak sequential closure. In particular in the abelian case the two concepts coincide.
0.2truecm The lattice of propositions ${\mathbb
P} ({\mathscr B} (\Omega))$ is both atomic and distributive. As always in the algebraic setting, there is a direct correspondence between the abelian structure of the algebra of observables characterising their classical nature and the distributive property of the lattice of elementary propositions.
0.3truecm **2) Quantum mechanical system with countable superselection sectors.**
0.2truecm
The algebra ${\mathfrak{A}}$ is the $C$\*-algebra generated by the Weyl commutation relations (1) and it is isomorphic to ${\mathfrak{K}}({\mathscr H})$ with ${\mathscr H}$ separable infinite dimensional; in view of Theorem 3.2, ${\mathscr B}({\mathfrak{A}}) \simeq
{\mathfrak{K}}({\mathscr H})'' \simeq {\mathfrak{B}}({\mathscr H})$; the $\sigma$-normal states correspond to the statistical matrices. ${\mathbb P} ({\mathscr B}({\mathfrak{A}}))$ is atomic and Hilbertian. In this specific example it is also irreducible, in correspondence with the absence of superselection sectors. Notice that in the Baire approach for classical system naturally appear the Dirac measures excluded in the $W$\* approach, whereas in quantum mechanics are naturally excluded the singular, i.e. non-normal, states of the above approach. By the way, our approach also provides a natural justification for the choice made e.g. in [@duvenhage] (see also [@clifton] for a variant) to discuss information theory in the algebraic setting using measurable functions in classical mechanics and bounded operators in quantum mechanics.
0.2truecm **Remark.** The Baire approach permits also to avoid a problematic feature appearing in the definition of states in the temporal logic approach proposed in [@primas], where, motivated by ontological considerations (which of course one may not agree with), a distinction is made between “ontic” states and “epistemic” logical states. Let ${\mathscr L}$ be the orthomodular $\sigma$-complete lattice assumed to describe the set of proposition of a physical system. An “ontic” state is a lattice ortho-homomorfism $\rho$ of a maximal orthomodular sublattice ${\mathscr T}$ of ${\mathscr L}$ into ${\mathscr B}_2$, the Boolean algebra of truth values. The requirement on ${\mathscr T}$ to be maximal means that it does not exist an orthomodular sublattice ${\mathscr T}^\prime$, containing properly ${\mathscr T}$, to which $\rho$ can be extended as ortho-homomorphism in ${\mathscr B}_2$. This requirement corresponds to the physical intuition of a state with “maximal information" and in the algebraic approach these are the pure algebraic states. An “ontic” state is called normal if $\rho$ is a $\sigma$-homomorphism. In this approach an “ontic” state refers to “actualized” properties the system has (at some time). States which refer to our knowledge are called “epistemic”. On this basis, if ${\mathscr L}$ is the lattice of projectors of a $W$\*-algebra ${\mathfrak{M}}$, ontic states are identified with (arbitrary, even non normal) pure states and epistemic states with normal states. Therefore “ontic" states are not a subset of “epistemic” states. Furthermore only for normal states it has been proved that every ontic states on ${\mathbb P}({\mathfrak{M}})$ has a unique extension to a pure state of ${\mathfrak{M}}$ and every pure state on ${\mathfrak{M}}$ defines a unique ontic state. For non normal states the situation appear obscure, in particular for $W$\*-algebras that do not admit pure normal states! Instead in the Baire approach, i.e. if ${\mathscr L}= {\mathbb P} ({\mathscr B}({\mathfrak{A}}))$, one could simply identify “epistemic” states with the $\sigma$-normal states and the “ontic” would be those corresponding to the lift of the pure states on ${\mathfrak{A}}$, thus a subset of the epistemic.
0.3truecm **3) Quantum mechanical system with non countable superselection sectors.**
0.2truecm
The algebra ${\mathfrak{A}}$ is the $C$\*-algebra generated by the Weyl commutation relations $$e^{i n \varphi} e^{i \beta p} = e^{i \beta p} e^{i n \varphi} e^{{i \bar h\over 2}n \beta}$$
where $\varphi$ is the angle parametrizing the circle $S^1, n \in {\mathbb Z}, \beta \in [0, 2 \pi]/\hbar$.
Inequivalent irreducible representations are labelled by an angle $\theta \in [0, 2\pi)$ and the corresponding Hilbert space will be denoted by ${\mathscr H}_\theta$, see e.g. [@thirring]. These are the so-called $\theta$-sectors and they arise physically e.g. in models where the particle is charged and coupled to a vector potential whose magnetic field strength is supported in a region in the interior of the disk bounded by circle $S^1$, in the region forbidden for the particle motion.
A magnetic flux $\Phi$ through the disk induces a representation of ${\mathfrak{A}}$ labelled by $\theta = \Phi$ mod $2\pi$. Hence ${\mathfrak{A}}\simeq \oplus_\theta {\mathfrak{K}}({\mathscr H}_\theta) \simeq C(S^1, {\mathfrak{K}}({\mathscr H}))$ with ${\mathscr H}_\theta$ and ${\mathscr H}$ separable infinite dimensional. ${\mathscr B} ({\mathfrak{A}}) \simeq
{\mathscr B} (S^1, {\mathfrak{B}}({\mathscr H})),$ the Baire (or Borel) functions on $S^1, {\mathfrak{B}}({\mathscr H})$-valued.
${\mathbb P} ({\mathscr B}({\mathfrak{A}}))$ is atomic, coincides with the lattice of closed subspace of $\oplus_\theta {\mathscr H}$, but is not the usual Hilbert lattice of Hilbert Quantum Logic, since $\oplus_\theta {\mathscr H}_\theta$ is not separable, so that in particular the lattice is not complete.
0.3truecm **4) Local observable algebras in massive RQFT.**
0.2truecm The algebraic description of (massive) Relativistic Quantum Field Theory (RQFT) is based on the following structure [@haag]: an inclusion preserving map ${\mathscr O}
\rightarrow {\mathscr A}({\mathscr O})$ assigning to each finite contractible open region (or alternatively open double cone) ${\mathscr O}$ in Minkowski space-time, ${\bf M}_4$, the abstract $C$\*-algebra of observables measurable in ${\mathscr O}$. The $C$\*-algebra generated by the net $\{{\mathscr A}({\mathscr
O})\}_{{\mathscr O} \subset {\bf M}_4}$ via inductive limit and norm closure is denoted by ${\mathscr A}$ and is called the algebra of quasi-local observables. Locality holds: if ${\mathscr
O}_1$, ${\mathscr O}_2$ are spacelike separated, then ${\mathscr
A}({\mathscr O}_1)$ commute with ${\mathscr A}({\mathscr O}_2)$ elementwise.
0.2truecm **Remark.** It would be interesting to translate the causal structure underlying the observable net, due to a universal maximal velocity of propagation of information, i.e. $c \neq \infty$, purely in logical terms, like the non-distributivity of the propositional lattice in quantum systems reflects the limitations imposed by $\hbar \neq
0$. Relevant steps in this direction can be found in [@haag; @mundici].
0.2truecm The elements of the Poincaré group ${\mathscr P^\uparrow_+}$ act as automorphisms on the net preserving the local structure. Among the irreducible representations of ${\mathscr A}$ on a separable Hilbert space in which the Poincaré group is unitarily implemented, there is one, $\pi_0$, called the vacuum representation (for simplicity assumed unique) containing a ray, the vacuum, invariant under the unitary representation of ${\mathscr P^\uparrow_+}$. In infinite systems, as the one considered in RQFT, it appears in concrete examples that physically one should not consider the set of all the representations, but only a subset of “physically realizable” ones. The properties of RQFT at zero temperature and density are discussed in terms of the net $\{{\mathfrak{A}}({\mathscr
O})=\pi_0({\mathscr A}({\mathscr O}))\}_{{\mathscr O} \subset {\bf
M}_4}$. ${\mathfrak{A}}({\mathscr O})$ can be identified as the “space of bounded continuous observables in the vacuum representation measurable in ${\mathscr O}$” . In view of Theorem 3.2, ${\mathscr
B} ({\mathfrak{A}}({\mathscr O})) \simeq \pi_0({\mathscr A}({\mathscr
O}))''$ (and are these concrete algebras that appear in the constructive approach to RQFT in low dimensions [@glimm]); since these algebras are von Neumann algebras, ${\mathbb
P}({\mathscr B} ({\mathfrak{A}}({\mathscr O})))$ is a complete lattice. A deep result of RQFT with mass gap is that $\pi_0({\mathscr
A}({\mathscr O}))''$ for ${\mathscr O}$ a double cone is a type III${}_1$ von Neumann algebra [@fredenhagen], conjectured on physical grounds to be a factor [@haag]. Hence the associated lattice of propositions is non-atomic, the projectors having Murray-von Neumann dimensions only [0, $\infty$]{}. In the Baire approach the $\sigma$-normal states are the normal states of $\pi_0({\mathscr A}({\mathscr
O}))''$, however a factor III${}_1$ does not possess pure normal states. Nevertheless in our approach pure logical states corresponding to maximal knowledge on the proposition lattice of the local system are naturally defined, as they are obtained from lifts of states on $\pi_0({\mathscr A}({\mathscr O}))$, which being a $C$\*-algebra with unity has a separating family of pure states.
0.3truecm
Conclusions
===========
0.2truecm
Summarizing, in this paper we propose that the lattice of elementary propositions of physical systems is completely encoded in the $C$\*-algebra ${\mathfrak{A}}$ of “continuous bounded functions or observables” on a generally “non-commutative phase space $X$” in the sense of Non Commutative Geometry. The propositional lattice can be represented as the $\sigma$-complete orthomodular lattice of projectors of the space of “(Baire) measurable bounded observables on $X$”, which can be obtained as a suitable closure, via the Baire envelope, of ${\mathfrak{A}}$. Hence the propositional logic depends on the physical system, but it captures only a very “coarse grained” structure of it. For example it is able to identify the classical or quantum nature of the system and it is sensible to the related “completeness” or “incompleteness” through the verification of the validity of the Lindenbaum property [@giuntini] in the corresponding logic. But it is also able to distinguish more refined features of quantum systems e.g. the presence of a countable from a non-countable set of superselection sectors or the “dimension” in the sense of Murray-von Neumann of the sectors.
#### Acknowledgements.
We gratefully acknowledge R. Nobili for many illuminating discussions and for a critical reading of a preliminary version of the manuscript. This work is supported in part by the European Community’s Human Potential Programme under contract HPRN-CT-2000-00131 Quantum Spacetime.
0.2truecm
0.3truecm
Appendix
========
0.2truecm
**$\mathbf{C}$\*-algebra.** A $C$\*-algebra ${\mathfrak{A}}$ is an algebra over $\mathbb{C}$, with an involution \* and a norm $|| \cdot||$; in this norm ${\mathfrak{A}}$ is complete, i.e. Banach, and $\forall a,b \in {\mathfrak{A}}$ $||ab|| \leq
||a|| ||b||;||a^*||=||a||$; the key property linking the algebraic and the topological structure holds: $||a^*a||=||a||^2$ and if the unity ${\bf 1} \in {\mathfrak{A}}$ then ${\mathfrak{A}}$ is called unital and $ ||{\bf 1}||=1$. Every $C^*$ algebra without unity ${\mathfrak{A}}$ can be canonically embedded in a unital $C^*$ algebra $\tilde{\mathfrak{A}}$ as an ideal satisfying $\tilde{\mathfrak{A}}/{\mathfrak{A}}\simeq {\mathbb
C}$; in the following if ${\mathfrak{A}}$ is not unital ${\bf 1}$ is referring to $\tilde{\mathfrak{A}}$. An element $a \in
{\mathfrak{A}}$ is called [*self-adjoint*]{} or hermitian iff $a^*=a$; [*projector*]{} iff $a^2=a=a^*$; unitary iff $aa^*=a^*a={\bf 1}$; positive iff there exists $b \in {\mathfrak{A}}$ such that $a=b^*b$; an element $b \in {\mathfrak{A}}$ is called the inverse of $a$ iff $ab=ba={\bf
1}$ and then denoted by $a^{-1}$. The [*spectrum*]{} of $a \in
{\mathfrak{A}}$ is the set $Sp(a)=\mathbb{C} \, \backslash \{z \in
\mathbb{C} , (z-a)^{-1} \in {\mathfrak{A}}\} $. The norm of a $C^*$-algebra can be uniquely algebraically defined as $||a||=$sup$\{|z|, z \in
Sp(a^*a)\}^{1/2}$.
In a $C$\* approach the bounded physical observable quantities of a physical system are described by the self-adjoint elements of a $C$\*-algebra and the possible results of a measurement on the physical observable described by $a$ are given by the spectrum of $a$.
0.2truecm **State.** An algebraic state (here simply called state) on ${\mathfrak{A}}$ is a positive linear functional $\phi$ on ${\mathfrak{A}}$, normalized by $\phi({\bf 1})=1$. Convex combinations of states are states. States that cannot be written as convex combination of other states are called [*pure*]{}. A family $F$ of states is called [*separating*]{} if $\phi(a)=0$ for all $\phi \in F$ implies $a=0$ for all positive $a
\in {\mathfrak{A}}$. Every unital $C$\*-algebra has a separating family of pure states.
In a $C$\* approach the (algebraic) states describe the “states of knowledge" of the observable quantities and pure states correspond to maximal knowledge. The expectation value of the measures performed on the physical observable described by $a$ in the state of knowledge described by $\phi$ is given by $\phi(a)$.
0.2truecm **Representation.** Let ${\mathfrak{A}}$ be a $C$\*-algebra, ${\mathscr H}$ a Hilbert space and ${\mathfrak{B}}({\mathscr
H})$ the $C$\*-algebra of bounded operators on ${\mathscr H}$. A [*representation*]{} $\pi$ on ${\mathscr H}$ is a homomorphism of ${\mathfrak{A}}$ into ${\mathfrak{B}}({\mathscr H})$ preserving the involution. If ${\mathscr S}$ is a $\ast$-subalgebra of ${\mathfrak{B}}({\mathscr H})$, ${\mathscr S}'$ denotes its commutant, i.e. the set of elements of ${\mathscr B}({\mathscr H})$ commuting with all the elements of ${\mathscr S}$. A representation $\pi$ is called [*faithful*]{} iff $\pi(a)=0$ implies $a=0$; [*irreducible*]{} if the commutant $\pi({\mathfrak{A}})'$ contains only multiples of the unity; two representations $\pi_1$ on ${\mathscr H}_1$ and $\pi_2$ on ${\mathscr H}_2$ are called [*unitarily equivalent*]{} if there exists an isometry $u$ of ${\mathscr H}_1$ onto ${\mathscr H}_2$ such that $u \pi_1(a) u^* =\pi_2(a), \forall a \in {\mathfrak{A}}$.
0.2truecm **von Neumann algebra.** A weakly closed $\ast$-subalgebra ${\mathfrak{M}}$ of ${\mathfrak{B}}({\mathscr H})$ is called a von Neumann algebra. The von Neumann double commutant theorem states that ${\mathfrak{M}}= {\mathfrak{M}}''$; more generally if ${\mathscr S}$ is a $*$-subalgebra of ${\mathscr B}({\mathscr H})$, then ${\mathscr
S}''$ is called the [*von Neumann algebra generated by*]{} ${\mathscr S}$. A von Neumann algebra ${\mathfrak{M}}$ is called a [*factor*]{} iff the centre ${\mathfrak{M}}\cap {\mathfrak{M}}'$ contains only multiples of the unity.
0.2truecm **$\mathbf{W}$\*-algebra.** A $W$\*-algebra ${\mathfrak{M}}$ is a $C$\*-algebra which in addition is the dual of a Banach space, called its [*predual*]{} and denoted by ${\mathfrak{M}}_*$. The dual space ${\mathfrak{M}}^*$ of linear functionals on ${\mathfrak{M}}$ is larger then the predual, hence the set of states on ${\mathfrak{M}}$ have a distinguished subset contained in the predual; these are the [*normal states*]{}; they are completely additive on projectors of ${\mathfrak{M}}$. Every $W$\*-algebra ${\mathfrak{M}}$ admits a faithful representation as a von Neumann algebra in some Hilbert space ${\mathscr H}$; ${\mathscr H}$ can be taken separable iff the predual ${\mathfrak{M}}_*$ is norm separable.
0.2truecm **Murray- von Neumann dimension.** Two projectors $p_1$ and $p_2$ in a factor ${\mathfrak{M}}$, projecting onto subspaces ${\mathscr H}_1$ and ${\mathscr H}_2$ of ${\mathscr H}$ are said equivalent iff there exists a partial isometry $V \in {\mathfrak{M}}$ from ${\mathscr H}_1$ to ${\mathscr H}_2$, i.e $p_1=V^*V, p_2=VV^*$ and then we write $p_1 \sim p_2$. One can order the equivalence class of projectors by setting $p_1 < p_2$ iff $p_1 \nsim p_2$ and there exists a proper subspace of ${\mathscr H}_1$ whose associated projector is equivalent to $p_2$. A projector $p_1$ is called finite iff $p
\leq p_1$ and $p \sim p_1$ implies $p = p_1$. There exists a positive function on the equivalence classes of projectors, the [*Murray-von Neumann dimension*]{} $d$, satisfying $d(0)=0$, $d(p_1)=d(p_2)$ iff $p_1 \sim p_2$, $d(p_1) < d(p_2)$ iff $p_1 <
p_2$ and, if $p_1p_2=0$, $d(p_1+p_2)=d(p_1)+d(p_2)$. For factors with separable predual the following alternatives exists: a factor is of type I if it contains atoms, i.e. minimal nonzero projectors, whose von Murray-von Neumann dimension is 1 and the range of $d$ is a subset of ${\mathbb N}$, in particular it is called of type I$_n$ if $n$ is the maximal value in the range of $d$; of type II if it is atom-free and it contains some nonzero finite projector; of type III if it does not contain any nonzero finite projector and then $d$ takes only the values $0$ and $\infty$.
[9]{} Birkhoff G., von Neumann J. \[1936\]: “The logic of Quantum Mechanics”, [*Annals of Mathematics 37*]{}: 823-843. Cirelli R., Gallone F. \[1973\]: “Algebra of observables and quantum logic”, [*Annales Institute Henri Poincaré A19*]{}: 297-331. Clifton R., Bub J., Halvorson H. \[2003\]: “Characterizing Quantum Theory in Terms of Information-Theoretic Constraints”, [*Foundations of Physics 33*]{}: 1561-1591. Connes A. \[1993\]: “Non Commutative Geometry and Physics” IHES/M/93/32. Connes A. \[1994\]: [*Non Commutative Geometry*]{} (Academic Press, San Diego). Connes A., Douglas M., Schwarz A. \[1998\]: “Non Commutative Geometry and Matrix theory: compactification on tori”. [*JHEP 9802*]{}:003. Dalla Chiara M.L., Giuntini R. \[2001\]: “Quantum Logic”, e-Print Archive: quant-ph/01001028. Davies E.B. \[1968\]: “On the Borel structure of $C$\*-Algebras”, [*Commununications in Mathematical Physics 8*]{}: 147-163. Doplicher S., Fredenhagen K., Roberts J.E. \[1995\]: “The quantum structure of space-time at the Planck scale and quantum fields” [*Communications in Mathematical Physics 172*]{}: 187-220. Duvenhage R. \[2002\]: “The Nature of Information in Quantum Mechanics”, [*Foundations of Physics 32*]{}: 1399-1417. Fredenhagen K. \[1985\]: “On the modular structure of local algebras of observables”, [*Commununications in Mathematical Physics 97*]{}: 79-89. Froehlich J., Grandjean O., Recknagel A. \[1995\] “Supersymmetric quantum theory, noncommutative geometry and gravitation” in [*Les Houches 1995, Quantum symmetries*]{} 221-385. e-Print Archive: hep-th/9706132. Giuntini R. \[1987\]: “Quantum logics and Lindenbaum property” [*Studia Logica 46*]{}: 17-35. Glimm J., Jaffe A. \[1987\]: [*Quantum physics. A functional integral point of view*]{} (Springer, Berlin). Haag R. \[1992\] [*Local Quantum Physics: Fields, Particles, Algebras*]{} (Springer, Berlin). Halmos P. \[1950\] [*Measure Theory*]{} (van Nostrand, Princeton). Halvorson H. \[2001\]:“On the nature of continuous physical quantities in classical and quantum mechanics” [*Journal of Philosophical Logic 30*]{}: 27-50 . Jauch J.M. \[1968\]: [*Foundations of Quantum Mechanics*]{} (Addison-Wesley, Reading). Mackey,G.W. \[1963\]: [*Mathematical Foundations of Quantum Mechanics*]{} (W. A. Benjamin, New York). Mundici D. \[1984\]: “Abstract Model theory and nets of $C^*$-algebras: noncommutative interpolation and preservation properties”in [*Lecture Notes in Math. 1103*]{}: 351-377 Springer. Pedersen G.K. \[1979\]: [*$C$\*-Algebras and their Automorphism Groups*]{} (Academic Press, London). Piron C. \[1964\]: “Axiomatique quantique”, [*Helvetica Physica Acta 37*]{}: 439-468. Piron C. \[1976\]: [*Foundation of Quantum Physics*]{} (W. A. Benjamin, Reading). Plymen R.J. \[1968\]: “$C$\*-Algebras and Mackey’s Axioms”, [*Communications in Mathematical Physics 8*]{}: 132-146. Primas H. \[1983\]: [*Chemistry, Quantum Mechanics and Reductionism*]{} (Springer, Berlin). Redei M. \[1998\]: [*Quantum Logic in Algebraic Approach*]{} (Kluwer, Dordrecht). Streater R.F. \[2000\]: “Classical and quantum probability”, [*Journ. Math. Phys. 41*]{}: 3556-3603. Teller P. \[1995\] [*An Interpretative Introduction to Quantum Field Theory*]{} (Princeton University Press, Princeton). Thirring W. \[1981\] [*Quantum Mechanics of Atoms and Molecules. A Course in Mathematical Physics 3*]{} (Springer, New York). Witten E. \[2001\]: “Overview of K theory applied to strings”, e-Print Archive: hep-th/0007175 [*International Journal of Modern Physics A16*]{}: 693-706.
[^1]: In an orthocomplemented lattice $\mathscr{L}$ with partial order $\leq$ the orhomodularity can be expressed as: if $a,b \in \mathscr{L}$ and $a \leq b$, then $b= a
\vee (a^\bot \wedge b)$.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Abstract reasoning refers to the ability to analyze information, discover rules at an intangible level, and solve problems in innovative ways. Raven’s Progressive Matrices (RPM) test is typically used to examine the capability of abstract reasoning. In the test, the subject is asked to identify the correct choice from the answer set to fill the missing panel at the bottom right of RPM (, a 3$\times$3 matrix), following the underlying rules inside the matrix. Recent studies, taking advantage of Convolutional Neural Networks (CNNs), have achieved encouraging progress to accomplish the RPM test problems. Unfortunately, simply relying on the relation extraction at the matrix level, they fail to recognize the complex attribute patterns inside or across rows/columns of RPM. To address this problem, in this paper we propose a Hierarchical Rule Induction Network (HriNet), by intimating human induction strategies. HriNet extracts multiple granularity rule embeddings at different levels and integrates them through a gated embedding fusion module. We further introduce a rule similarity metric based on the embeddings, so that HriNet can not only be trained using a tuplet loss but also infer the best answer according to the similarity score. To comprehensively evaluate HriNet, we first fix the defects contained in the very recent RAVEN dataset and generate a new one named Balanced-RAVEN. Then extensive experiments are conducted on the large-scale dataset PGM and our Balanced-RAVEN, the results of which show that HriNet outperforms the state-of-the-art models by a large margin.'
author:
- |
Sheng Hu^\*^, Yuqing Ma^\*^, Xianglong Liu^$\dagger$^, Yanlu Wei, Shihao Bai\
State Key Laboratory of Software Development Environment, Beihang University, Beijing, China\
[husheng\[email protected], {mayuqing,xlliu}@nlsde.buaa.edu.cn, {weiyanlu,16061167}@buaa.edu.cn]{}
title: Hierarchical Rule Induction Network for Abstract Visual Reasoning
---
Introduction
============
Abstract reasoning, also known as inductive reasoning, refers to the ability to analyze information, discover rules at an intangible level, and solve problems in innovative ways. This type of reasoning, as the foundation for human intelligence, helps human understand the world. It has been generally regarded and pursued as a critical component to the development of artificial intelligence during the past decades, and has attracted increasing attention in recent years.
![An example of RPM question and its solution. The underlying rule on the number of circles could be *Progression* (2-1=3-2) or *Arithmetic* (1+2=3) along row 1, and *Arithmetic* (2+3=5) along row 2. Therefore the dominant rule is *Arithmetic*. Apply it to the third row to figure out the answer (2+2=4). Besides, no viable rule can be found along the columns.[]{data-label="human-stra"}](example.pdf){width="1\linewidth"}
Raven’s Progressive Matrices (RPM) test [@Raven1938; @kunda2013computational; @raven2000raven; @strannegaard2013anthropomorphic] is one of the highly accepted and well-studied tools to examine the ability of abstract reasoning, which is believed as a good estimate of the real intelligence [@Carpenter1990What]. An illustration of RPM is shown in Figure \[human-stra\], where usually the test-taker is presented with a 3$\times$3 matrix with the bottom right panel left blank. The goal is to choose one image from an answer set of eight candidates to complete the matrix correctly, namely satisfying the underlying rules in the matrix. Subjects accomplish this by looking into the first two rows/columns and inducing the dominant rules which govern the attributes in those panels. The obtained rules can then be applied to the last row/column to figure out which answer belongs to the blank panel.
Computational models for RPM in the cognitive science community access symbolic representations of the images [@LovettModeling; @lovett2010structure; @Lovett2010Solving]. Recently there has been some success with end-to-end learning methods trying to accomplish abstract reasoning on RPM test [@hoshen2017iq; @BarrettMeasuring; @ZhangRAVEN; @zhanglearning], inspired by the progress of computer vision tasks [@he2016deep; @krizhevsky2012imagenet; @zeiler2014visualizing; @simonyan2014very; @szegedy2015going]. Barrett [@BarrettMeasuring] proposed the Procedurally Generated Matrices (PGM) dataset constructed with relation-object-attribute tuples, which was automatically generated by a computer program. They also designed the Wild Relational Network (WReN) to learn a probability score for each multiple-choice panel infilled. Zhang [@ZhangRAVEN] built another large-scale dataset Relational and Analogical Visual rEasoNing (RAVEN) with structure annotations. RAVEN contains diverse rule instantiations, structures, and figure configurations, making it more comprehensive compared with the PGM dataset. Unfortunately, previous deep learning-based models simply relied on the relation extraction at the matrix level, and thus failed to recognize the complex attribute patterns inside or across rows/columns of RPM.
In this paper, we develop a novel architecture called Hierarchical Rule Induction Network (HriNet) inspired by human induction strategies. HriNet induces the underlying rules from the two given rows/columns, by extracting multiple granularity rule embeddings at different levels, including cell-wise, individual-wise, and ecological embeddings. The cell-wise hierarchy focuses on the attributes inside each panel, such as size, type, . The individual-wise hierarchy further takes the relationships inside each row/column into consideration. The ecological hierarchy comprehensively handles the correlations among all the panels within the two given sequences. These hierarchical embeddings are also fused in a hierarchical way using gate functions, to induce the shared rule embeddings between the two inputs. In order to determine the fitness of the candidate answer according to the extracted rule, we further introduce a rule similarity metric, based on which HriNet can not only be well trained using a tuplet loss but also quickly infer the best answer.
To fairly evaluate the capability of abstract reasoning, it is fundamental to build an unbiased RPM-style dataset. However, by taking a close look at the recently published RAVEN dataset [@ZhangRAVEN], we find that there exist severe defects (or obvious patterns) among the answer set, where the correct one can be easily found without considering the context panels. A neural network trained with only the eight multiple-choice panels as input can surprisingly achieve 90.1% test accuracy. Such inappropriate setting has caused misleading results in the recent research [@ZhangRAVEN; @zhanglearning]. To fix the defects of RAVEN, we propose a new way of generating the answer set and name the unbiased dataset Balanced-RAVEN. Finally, we extensively evaluate our HriNet on the popular PGM dataset and our Balanced-RAVEN. The experimental results show that HriNet outperforms state-of-the-art methods by a large margin, 63.9% accuracy compared to the second best 44.3% on Balanced-RAVEN.
Related work
============
Computational models for solving RPM in the cognitive science community was based on an oversimplified assumption that computer programs had access to symbolic inputs of images and the operations of rules [@Carpenter1990What; @LovettModeling; @lovett2010structure; @Lovett2010Solving]. Another research branch [@little2012bayesian; @mcgreggor2014confident; @mcgreggor2014fractals; @mekik2018similarity; @shegheva2018structural] explored RPM through measuring the similarity between images. Hoshen [@hoshen2017iq] first trained a CNN-based model, trying to resolve RPM problems from raw pixels on a simplified RPM-style dataset. Wang and Su [@wang2015automatic] proposed an automatic method to generate RPM questions using a computer program. Barrett [@BarrettMeasuring] borrowed the insight from [@wang2015automatic] and introduced the Procedurally Generating Matrices (PGM) dataset. They also designed the Wild Relational Network (WReN) for RPM which took the pair-wise relationships among panels into consideration. Hill [@HillLearning] proposed a training strategy to learn analogies by contrasting abstract relational structure (LABC). Zhang [@ZhangRAVEN] adopted Attributed Stochastic Image Grammar (A-SIG) [@fu1974syntactic; @lin2009stochastic; @park2015attributed; @wu2007compositional; @zhu2015reconfigurable; @zhu2007stochastic] as the hierarchical image syntax to represent RPM questions and introduced another RPM-style dataset named Relational and Analogical Visual rEasoNing (RAVEN). Based on the rich annotations provided by A-SIG for each problem instance, they further designed a plug and play module called Dynamic Residual Tree (DRT), trying to improve the performance on RPM using the annotations of image structure. However, there are some unexpected defects contained in the RAVEN dataset which we will discuss in details in Section \[exp:dataset\]. They ulteriorly discussed the order-invariant characteristic of RPM and proposed CoPINet in [@zhanglearning]. However, the reported results conducted on the biased RAVEN dataset can not be used as reference.
Our approach
============
In this section, we first give a formal definition of the abstract reasoning task on the RPM test. Then we introduce the motivation from the human reasoning strategies, and subsequently present our Hierarchical Rule Induction Network (HriNet) for this task. Finally, we demonstrate the learning and inference process of the proposed model.
Preliminary {#sec:preliminary}
-----------
For a common RPM problem, usually a 3$\times$3 matrix $\mathbf{M}^{-}$ is given, with bottom right context panel left blank. $\Omega$ denotes the answer set with $N$ multiple-choice panels, where typically $N$=8. The dominant rules governing the features inside the matrix could be inducted from the first two intact rows/columns. The goal is to select a multiple-choice panel $\omega \in \Omega$ to complete the context matrix $\mathbf{M}^{-}$, maintaining the dominant rule inside of the context matrix.
We define the completed matrix with a multiple-choice panel $\omega$ infilled as $\mathbf{M}$, where $\mathbf{M}_i$ is denoted as the $i$-th row, and $\mathbf{m}_{ij}$ indicates the panel in $i$-th row and $j$-th column. Intuitively, $\mathbf{M}$ is almost the same as $\mathbf{M}^{-}$, except for $\mathbf{m}_{33}=\omega$ while the corresponding element missing in $\mathbf{M}^{-}$. In fact, whether rules exist in rows or columns is uncertain. Therefore, our framework induces both the row-wise rule representation and the column-wise representation in the same way. In order to simplify the notation, we only take the induction of the row-wise rule representation as example.
The reasoning framework {#sec:framework}
-----------------------
We develop a novel abstract reasoning architecture named Hierarchical Rule Induction Network (HriNet), inspired by hierarchical induction strategies of human. As shown in Figure \[human-stra\], given a Raven’s Progressive Matrix, human strategies can be simplified into five key steps:
**S1**: look into each panel, including context panel and multiple-choice panel, to recognize the basic attributes of the graphical elements, , type, size, color, position.\
**S2**: compare panels in the same row to figure out the plausible rules inside it.\
**S3**: compare panels in two rows to figure out the shared relationships between the two rows.\
**S4**: scan the first two rows and induce the dominant rules, integrating hierarchical information from previous 3 step.\
**S5**: fill each multiple-choice panel in the matrix, infer the rules with hierarchical information as S4 did according to each currently completed matrix, and determine the correct answer which adheres to the dominant rule.
Given two input rows $\mathbf{M}_i, \mathbf{M}_j$, the proposed framework adopts similar strategies as humans do. It embeds the input into the multiple granularity embeddings at different levels using a hierarchical rule embedding module $\mathbb{E}$. Inspired by human reasoning strategy and the general information processing mechanism in the biological organization [@parent1996living], $\mathbb{E}$ consists of three hierarchies, namely cell-wise network $\mathbb{E}_{\text{cell}}$, individual-wise network $\mathbb{E}_{\text{ind}}$, and ecological network $\mathbb{E}_{\text{eco}}$, which respectively look into the matrix from different hierarchies, focusing on the attribute and pattern discovery from cell-wise hierarchy as S1, individual-wise hierarchy as S2, and ecological hierarchy as S3.
With the multiple granularity rule embeddings, the gated embedding fusion module $\mathbb{G}$ will integrate these hierarchical features and induce the final rule embedding $\mathbf{r}^{(3)}_{ij}$ of the two input sequences $\mathbf{M}_i$ and $\mathbf{M}_j$. The embedding representation of the rules preserves the semantic distances among rules, namely keep that of similar rules close and dissimilar rules far in the embedding space. Therefore, we further introduce a rule similarity metric $\mathcal{D}$ to estimate the similarity between the rule representations. As a result, the correct answer can be predicted by choosing the multiple-choice panel within the shortest distance to the dominant rule generated by the first two rows in the matrix, like S4 and S5 in the human reasoning process.
![The architecture of HriNet, consisting of a hierarchical rule embedding module and a gated embedding fusion module. Given two row sequences as input, it outputs the rule embedding.[]{data-label="network-HriNet"}](fram.pdf){width="1.0\linewidth"}
Hierarchical Rule Induction Network
-----------------------------------
Now we introduce the carefully designed hierarchical modules in our framework in details.
### Hierarchical rule embedding
As we all know, organization of behaviour into a nested hierarchy of tasks is characteristic of purposive cognition in humans. The prevalent Convolution Neural Network inspired by the human visual system, is a hierarchical model itself, with the projection from each layer showing the hierarchical nature of features. The bottom layers extract low-level features, such as texture, edge, , while the top layers abstract high-level semantic information from the low-level information transmitted from the bottom layers.
However, without specifying information from different levels, it is hard for CNN to figure out different hierarchies, and thus fail to obtain robust and representative features. Therefore, it would be better to feed the input of different hierarchies explicitly and extract rule representations from different granularity with artificial guidance. Motivated by that, we deploy a hierarchical rule embedding module, consisting of cell-wise hierarchy, individual-wise hierarchy, and ecological hierarchy.
**Cell-wise hierarchy** The network of the cell-wise hierarchy $\mathbb{E}_{\text{cell}}$ takes each panel as input and recognize the attributes of inside graphical elements. It handles each panel independently without considering the difference or correlations among panels inside the matrix. Therefore, it observes the information from the most detailed perspective. We obtain the cell-wise rule representation for each input panel: $$\mathbf{x}_{ij}=\mathbb{E}_{\text{cell}}(\mathbf{m}_{ij}).$$
**Individual-wise hierarchy** Moreover, the network of individual hierarchy takes each row as input. It begins to take the correlations among panels of the same row into consideration, and encode the entire row with a compact embedding, rather than simply combining each panel. In this way, the rule embedding process for each panel is coupled and interacts with each other. Intuitively, each row may contain multiple rules, such as color, number, . In this hierarchy, the framework extracts intermediate rule embedding for each row individually, which still ignores the comprehensive information from the matrix perspective, especially the correlations across rows. The individual-wise rule embedding $\mathbf{y}_{i}$ is denoted as: $$\mathbf{y}_{i}=\mathbb{E}_{\text{ind}}(\mathbf{M}_i).$$
**Ecological hierarchy** Furthermore, the network of the ecological hierarchy takes the two rows together as input and jointly learns the rule patterns underlying the two rows. As we mentioned before, in the individual hierarchy, the framework extracts intermediate rule embedding for each row, without considering the interaction between two rows. The rule that exists in one row may not lie in another. Therefore, to obtain the shared rule patterns between the two rows, it is essential to put these two rows together and jointly learn the features from an ecological level. Thus the shared rule embedding is obtained as follows: $$\mathbf{z}_{ij}=\mathbb{E}_{\text{eco}}([\mathbf{M}_i, \mathbf{M}_j]),$$ where $[\cdot, \cdot]$ denotes the concatenating operation.
### Gated embedding fusion
Since the rule embeddings at different levels focus on different attributes or patterns, to generate one discriminative representation for the rule, we should aggregate the multiple granularity embeddings. However, due to the requirement that the aggregation should preserve the order of cell-wise rule embeddings and be invariant to the order of the individual-wise ones, it is impracticable to receive all the rule embeddings simultaneously relying on a single fully connected network. Therefore we propose a hierarchical rule embedding learning method named gated embedding fusion module, which is responsible for hierarchically and gradually aggregating the multiple granularity embeddings. Specifically, we define a gate function $\varphi$ to fuse the rule embeddings from different hierarchies. It concatenates all the inputs and encodes into a single embedding using fully connected layers. The gate function is similar to the attention mechanism, which detects and concentrates on the useful features according to the task. Even for the same attribute, they may focus on different facets. More details could be found in supplementary materials. Based on the gate function, our gated embedding fusion module could regulate the flow of rule embeddings into the framework and make the utmost of their complementary information.
At the cell level, after obtaining cell-wise rule embeddings for panels in each row $\mathbf{M}_i$, the module aggregates them to infer a row-wise rule embedding $\mathbf{r}_{i}^{(1)}$: $$\mathbf{r}_{i}^{(1)}=\varphi_1(\mathbf{x}_{i1}, \mathbf{x}_{i2}, \mathbf{x}_{i3}),$$ Similarly, we obtain $\mathbf{r}_{j}^{(1)}$ for the $j$-th row $\mathbf{M}_j$. The fused embedding integrates different types of information in the panels such as type and size.
At the individual level, intuitively both $\mathbf{r}^{(1)}_{i}$ and $\mathbf{y}_{i}$ are the row-wise embeddings corresponding to the $i$-th row, but convey the different granularity rule information. We further fuse them, and jointly mine the shared rules contained in the $i$-th and $j$-th row : $$\mathbf{r}^{(2)}_{ij} = \varphi_2(\mathbf{r}_{i}^{(1)}, \mathbf{y}_{i}, \mathbf{r}_{j}^{(1)}, \mathbf{y}_{j}).$$
At the ecological level, similarly we can further combine hierarchically fused embedding $\mathbf{r}^{(2)}_{ij}$ and $\mathbf{z}_{ij}$ using the gate fusion function, abstracting the final rule embedding: $$\mathbf{r}_{ij}^{(3)}=\varphi_3(\mathbf{r}^{(2)}_{ij}, \mathbf{z}_{ij}).$$ In practice, to make sure the framework is order-invariant to the input rows, we can simply exchange the concatenation order between the two input rows and average their rule embeddings. This invariance ensures that, the induced rule embedding respects the property of RPM and thus distills the representative information of the relations existing in the inputs.
On the whole, the HriNet can be formulated in its simplest form as follows: $$\begin{aligned}
\mathbf{r}_{ij}^{(3)} =& \text{HriNet}(\mathbf{M}_i, \mathbf{M}_j) \\
=& \mathbb{G}(\mathbf{x}_{i}, \mathbf{x}_{j},\mathbf{y}_{i}, \mathbf{y}_{j},\mathbf{z}_{ij}),
\end{aligned}$$ where $\mathbf{r}_{ij}^{(3)}$ is the shared rule embedding of the $\mathbf{M}_i$ and $\mathbf{M}_j$. An illustration of HriNet is shown in Figure \[network-HriNet\].
Learning and inference
----------------------
With HriNet framework, the question turns to how we train the network, and apply it to infer the correct answer to RPM test. The key to address the question lies in the similarity measure between two rule embeddings, based on which we can define the loss function for HriNet training, and meanwhile determine the best choice during inference. **Similarity function** We first introduce similarity function $\mathcal{D}$ to measure the closeness between two rules in the embedding space. There are a number of candidate functions:
1. *Cosine similarity:* $$\mathcal{D}(\mathbf{r}, \mathbf{r'})=\frac{\mathbf{r}^{\text{T}}\mathbf{r'}}{\parallel \mathbf{r} \parallel \parallel \mathbf{r'} \parallel},$$
2. *Euclidean similarity:* $$\mathcal{D}(\mathbf{r}, \mathbf{r'})=-\parallel \mathbf{r}-\mathbf{r'}\parallel_2^2,$$
3. *Inner product similarity:* $$\mathcal{D}(\mathbf{r}, \mathbf{r'})= \mathbf{r}^{\text{T}}\mathbf{r'} .$$
In this paper, we simply adopt inner product similarity.
*Cosine similarity:* $$\mathcal{D}(\mathbf{r}, \mathbf{r'})=\frac{\mathbf{r}^{\text{T}}\mathbf{r'}}{\parallel \mathbf{r} \parallel \parallel \mathbf{r'} \parallel}$$
*Euclidean Similarity:* $$\mathcal{D}(\mathbf{r}, \mathbf{r'})=-\parallel \mathbf{r}-\mathbf{r'}\parallel_2^2$$
*Inner product similarity:* $$\mathcal{D}(\mathbf{r}, \mathbf{r'})= \mathbf{r}^{\text{T}}\mathbf{r'}$$
**Training** For a given RPM problem, the first two rows $\mathbf{M}_1, \mathbf{M}_2$ are fed into our proposed HriNet and produce the shared rule embedding ${\mathbf{g}}$: $$\mathbf{g}=\mathbf{r}_{12}^{(3)} = \text{HriNet}(\mathbf{M}_1, \mathbf{M}_2),$$ which represents the dominant pattern of the matrix.
Intuitively, the rule extracted from the first two rows can be treated as the reference rule, and we name it the dominant rule in the matrix. Subsequently, the correct answer can be found by checking whether its corresponding rule embedding is similar to the dominant rule. Specifically, given a multiple-choice panel $\omega_k \in \Omega$, where $k \in \{1,..., N\}$, we denote $\overline{\mathbf{r}}_k$ as the new rule embedding inside $\mathbf{M}$ caused by the $k$-th multiple-choice panel: $$\overline{\mathbf{r}}_k = \frac{1}{2}\left(\mathbf{r}_{13}^{(3)}+\mathbf{r}_{23}^{(3)}\right).$$ This procedure is illustrated in Figure \[network-structure\]. In practice, we generate the column-wise rule representation just as the row-wise one, and concatenate the two representations together as the final representation.
For the rule embedding $\overline{\mathbf{r}}^{*}$ generated by rows/columns infilled with correct answer, the desirable HriNet should enforce it to be more similar to the dominant rule ${\mathbf{g}}$, compared to the other rules $\overline{\mathbf{r}}_k$ corresponding to the wrong answers, where $\overline{\mathbf{r}}_k\not=\overline{\mathbf{r}}^*$. Subsequently, the generated rules of $N$ candidates, alongside with the dominant rule, form a tuple containing $N$+1 elements. Based on the similarity function, the ($N$+1)-tuplet loss [@sohn2016improved] can be defined for HriNet training:
$$\mathcal{L} = \text{log}(1+\sum_{k=1, \overline{\mathbf{r}}_k\not=\overline{\mathbf{r}}^*}^{N}\text{exp}(\mathcal{D}({\mathbf{g}}, \overline{\mathbf{r}}_k) - \mathcal{D}({\mathbf{g}}, \overline{\mathbf{r}}^{*}))),$$
which means the HriNet can be trained in a fully end-to-end manner. The architecture of the HriNet (Figure \[network-HriNet\]) is well matched
![The similarity score for a candidate answer. A multiple-choice panel from the answer set is infilled in the blank panel (row 3), generating a rule embedding $\overline{\mathbf{r}}_k$ through HriNet. The similarity score for the candidate answer can be estimated based on $\overline{\mathbf{r}}_k$ and the dominant rule embedding $\mathbf{g}$ extracted from row 1 and 2.[]{data-label="network-structure"}](structure.pdf){width="1.0\linewidth"}
to the problem of abstract reasoning, because it leverages human strategies and explicitly generates the rules governing the matrix.
**Inference** Once the training of HriNet is finished, we could make the inference of the newly given RPM problem. Initially, the intact rows/columns of the RPM are fed into the framework to get the dominant rule $\mathbf{g}$. After that, each multiple-choice panel is filled to the blank position to complete the matrix, and the framework will generate the rule embeddings $\overline{\mathbf{r}}_k$ for all candidate answers, given the current completed matrix. We can accomplish the abstract reasoning by choosing the correct multiple-choice as follows: $$k^{*}=\mathop{\arg\max}_{k} \mathcal{D}(\mathbf{g}, \overline{\mathbf{r}}_k).$$ Note that since we investigate each panel independently, the above inference framework promises that our model matches the nature of RPM that the answer should be invariant to the order of multiple-choice panels.
Experiments
===========
Datasets
--------
To comprehensively evaluate our model, we choose the recently proposed RAVEN [@ZhangRAVEN] and PGM [@BarrettMeasuring] datasets. Next, we first give a brief review of the two datasets, then we will demonstrate the defects of the original RAVEN and introduce an improved dataset named Balanced-RAVEN.
### RAVEN and PGM datasets {#sec:dataset}
**RAVEN** [@ZhangRAVEN] It consists of 70,000 RPM questions, distributed in 7 different figure configurations (`Center`, `2x2Grid`, `3x3Grid`, `Out-InCenter`, `Out-InGrid`, `Left-Right`, and `Up-Down`). Panels are constructed with 5 attributes (`Number`, `Position`, `Type`, `Size`, and `Color`). Each attribute is governed by one of 4 rules (`Constant`, `Progression`, `Arithmetic`, and `Distribute Three`) and takes a value from a predefined set. Rules are applied only row-wise in RAVEN.
**PGM** [@BarrettMeasuring] It contains 1.42M RPM questions. Rules in a matrix are composed with 1 to 4 relation-object-attribute tuples and can be applied along the rows or columns. For a fair comparison with the state-of-the-art abstract reasoning methods, we randomly sample 70,000 questions which is of the same size as RAVEN for experiments according to the underlying relations, making sure that it covers all 29 relations in this dataset. We denote the dataset as **PGM-70K**.
Making use of these different configurations, three generalization regimes were introduced where a model is trained on a configuration and tested on similar ones:
1. Train on `Center` and test on `Out-InCenter`, `Left-Right` and `Up-Down`
2. Train on `Left-Right` and test on `Up-Down`, and vice versa.
3. Train on `2x2Grid` and test on `3x3Grid`, and vice versa.
![Comparison between RAVEN and Balanced-RAVEN. []{data-label="ASR-sample"}](Balanced-RAVEN.pdf){width="1.0\linewidth"}
PGM [@BarrettMeasuring] dataset is an RPM-style dataset, consisting of 1.42M questions. It features with 7 extra generalization regimes (Interpolation, Extrapolation, Held-out `shape-colour`, Held-out `line-type`, Held-out Triples, Held-out Pairs of Triples and Held-out Attribute Pairs) in which the training and testing sets are different in a purposely designed manner. Different regimes are generated by holding out specific relation-object-attribute tuples in the training data. Rules in PGM can be applied along the rows or columns.
We randomly sample 70,000 questions for experiments, which coincides with RAVEN. This subset is large enough to cover all 29 underlying rules in PGM.
### Balanced-RAVEN {#exp:dataset}
After carefully examining the data in RAVEN, we find that there is unexpected bias among the eight multiple-choice panels. Each distractor in the answer set is generated by randomly modifying one attribute of the correct answer (see Figure \[ASR-sample\](a)). As a consequence, the panel with the most common values for each attribute will be the correct answer. This means that the correct answer can be found by simply scanning the answer set without considering the context images. An example is also shown on the right of Figure \[ASR-sample\](a). Among the answer set, the most common `Color` and `Type` are black (No. 1, 3, 4, 5, and 7) and pentagon (No. 1, 2, 3, 4, 6, and 8). Besides, multiple-choice panel 1, 2, 5, 6, 7, and 8 are in the same `Size`. Therefore, multiple-choice panel 1 is the panel with the most common attribute values, which is indeed the correct answer to the RPM test.
More severely, as shown in Table \[ASR-result\], such underlying patterns can also be easily detected by a neural network. We simply train two models including a normal abstract reasoning model based on ResNet classifier (as detailed in Section \[experimental setup\]) and a context-blind [@BarrettMeasuring] ResNet model. The context-blind ResNet model is trained with only eight multiple-choice panels as input, without considering the context. It is very surprising that the context-blind model can get close (even slightly better) performance to the normal ResNet. It is worth noting that, here we adopt a simple data augmentation method that shuffles candidate images during training. This augmentation method is essential especially for models taking the whole answer set as input. Therefore, our result is much higher than the accuracy (53.43%) reported in [@ZhangRAVEN].
[ Model]{} [ RAVEN]{} [ Balanced-RAVEN]{}
---------------------- ------------ ---------------------
ResNet 89.2% 40.3%
Context-blind ResNet 90.1% 12.5%
: Test on RAVEN and Balanced-RAVEN.[]{data-label="ASR-result"}
To fix the defects of RAVEN, we design an algorithm to generate the unbiased answer set, forming an improved dataset named Balanced-RAVEN. Figure \[ASR-sample\](b) demonstrates the generating process using a tree structure. Each node indicates a multiple-choice panel, and the root of the tree structure is the correct answer. Different levels indicate different iterations, where nodes of this level are the candidate answers of current answer set. The generating process flows in a top-down manner. For each iteration, only one attribute will be modified. At each level, a node has two children nodes, where one node remains the same with the father node, the other changes the value of the attribute sampled for this iteration of the father node. Finally, at the bottom level, we could obtain the whole answer set. Algorithm \[alg:dataset\] summarizes the key steps of the answer generating process.
Since the attribute modification is well balanced, no clue can be found to guess the answer only depending on the answer set. The right column in Table \[ASR-result\] shows that the performance of context-blind ResNet trained on Balanced-RAVEN is almost at a random guess level (12.5%), while the normal ResNet model further relying on the context can obtain much better performance. This observation proves that our improved dataset is more rigorous and fair for evaluating the capability of abstract reasoning.
[ Model]{} [ PGM-70K]{} [ Balanced-RAVEN]{} [ Center]{} [ 2$\times$2G]{} [ 3$\times$3G]{} [ O-IC]{} [ O-IG]{} [ L-R]{} [ U-D]{}
--------------- -------------- --------------------- ------------- ------------------ ------------------ ----------- ----------- ---------- ----------
LSTM 20.3 18.9 26.2 16.7 15.1 21.9 21.1 14.6 16.5
ResNet 21.7 40.3 44.7 29.3 27.9 46.2 35.8 51.2 47.4
ResNet+DRT — 40.4 46.5 28.8 27.3 46.0 34.2 50.1 49.8
Wild ResNet 26.6 44.3 50.9 33.1 30.8 50.9 38.7 53.1 52.6
WReN 29.1 23.8 29.4 26.8 23.5 22.5 21.5 21.9 21.4
WReN (ResNet) 27.0 42.6 75.7 45.9 39.0 37.2 34.8 31.2 34.8
HriNet **48.9** **63.9** **80.1** **53.3** **46.0** **71.0** **49.6** **72.8** **74.5**
the correct answer $\omega^*$ Initialize the answer set $\Omega = \{\omega^*\}$ Sample $3$ attributes $a_1, a_2, a_3$ according to $\omega^*$ Sample new value $v_i$ for each $a_i$ Initialize $\Gamma = \{\}$ $\gamma \leftarrow$ modifying attribute $a_i$ of $\omega_k$ with $v_i$ $\Gamma \leftarrow \Gamma \bigcup \{\gamma\}$ $\Omega \leftarrow \Omega \bigcup \Gamma$ the answer set $\Omega$ ($|\Omega| = 2^3 = 8$)
Experimental setup {#experimental setup}
------------------
\[models\] With PGM and Balanced-RAVEN, we first compare our method with several state-of-the-art models suited for RPM, including LSTM [@hochreiter1997long], ResNet-based [@he2016deep] image classifier (ResNet), ResNet with DRT [@ZhangRAVEN], Wild ResNet [@BarrettMeasuring], WReN [@BarrettMeasuring], and CoPINet [@zhanglearning]. Then we analyze the effects of different components in our HriNet.
We adopt the public implementations of LSTM, ResNet, and DRT in [@ZhangRAVEN]. Eight context panels and eight multiple-choice panels are stacked and passed through the ResNet to output an 8-dimensional probability score. DRT is a plug and play module which could be deployed in any model. However, it cannot be applied to PGM-70K for the lack of structure annotations. Wild ResNet takes one multiple-choice panel, along with the eight context panels as input. It is designed to provide a score value for each multiple-choice panel, independent of the other multiple-choice panels. WReN, which takes the same input as Wild ResNet, applies a Relation Network [@santoro2017simple] to obtain pairwise relationships among panels. We implement two versions of WReN, with its original 4-layer CNN or a ResNet-18 as the feature extractor. We haven’t managed to implement CoPINet and test it on our Balanced-RAVEN, since it was published in the very recent past, and thus we only compare with its accuracy on PGM reported in [@zhanglearning]. **LSTM** We implement a standard LSTM module which is fed into features extracted by a 4-layer CNN from context panels and multiple-choice panels sequentially. The final hidden state of the LSTM is passed through a linear layer to produce logits for the softmax cross entropy loss. **ResNet and its variants** We explore standard ResNet and its expansions, including ResNet with DRT and Wild ResNet. The standard ResNet model uses a ResNet as the feature extractor, followed by a MLP. As for ResNet with DRT, it passes features extracted by ResNet to the DRT module before MLP. Both ResNet and ResNet with DRT’s input are composed of 8 context images and 8 multiple-choices, while the Wild ResNet takes 8 context panels and 1 candidate image as input and each candidate’s score is predicted independently.
**WReN** WReN adopts the Relation Network (RN) [@santoro2017simple] to explore the inter-panel relationships. Each image feature extracted by a 4-layer CNN is paired with each other and passed through a RN. The output is the classification score for each candidate. Moreover, we test its performance when replacing its original 4-layer backbone with a ResNet.
For our HriNet, we adopt three ResNet-18 as the embedding networks for the three hierarchies, by only modifying the input channels. The gate fusion $\varphi_1$ and $\varphi_2$f are 2-layer fully connected networks, while $\varphi_3$ is a 4-layer fully connected network with dropout [@srivastava2014dropout] of 0.5 applied on the last layer. We adopt stochastic gradient descent using ADAM [@kingma2014adam] optimizer. The exponential decay rate parameters are $\beta_1=0.9$, $\beta_2=0.999$, $\epsilon=10^{-8}$ and the learning rate is $10^{-4}$. On both datasets, 20-fold validation is performed to evaluate model performance, and the accuracy is averaged over 20 folds.
Comparisons with state-of-the-art methods {#sec:comparisons}
-----------------------------------------
Table \[result-table\] lists the test accuracy of different models trained on PGM-70K and Balanced-RAVEN. From the table, it is obvious that our proposed HriNet outperforms other methods by a large margin on both datasets (18.7% and 19.6% accuracy increases respectively). Besides, we observe that models benefit from considering each multiple-choice panel independently, including the competitive Wild ResNet, WReN, and our HriNet. Such weight-sharing mechanism across panels can not only make a model invariant to the order of input multiple-choice panels, but also encourage to explore the underlying rules. Moreover, by comparing the results of two versions of WReN, we find that a deeper CNN backbone may improve the performance for abstract reasoning owing to the capability of extracting more complex patterns in the image. The very recent method CoPINet [@zhanglearning] achieved an accuracy of 32.39% when trained on a subset of PGM with 75,000 questions. Compared to CoPINet, our HriNet performs significantly better, when trained using a similar number of training data.
The performance of WReN (ResNet) is similar to WReN on PGM-70K. However, on the more comprehensive dataset Balanced-RAVEN, WReN (ResNet) have the advantage over WReN due to the strong ability of ResNet in extracting features. Similar trend could be seen in the contrast between LSTM and ResNet, where ResNet has a slight improvement compared to the LSTM on PGM-70K, and doubles the performance of LSTM on Balanced-RAVEN. Contradicting to previous results [@ZhangRAVEN], there is no performance boost after deploying the DRT module in ResNet on the improved dataset Balanced-RAVEN.
The LSTM has poor performance on both datasets. The ResNet has a slight improvement compared to the LSTM on PGM-70K, and doubles the performance of LSTM on Balanced-RAVEN. Contradicting to previous results [@ZhangRAVEN], there is no performance boost after deploying the DRT module in ResNet on the improved dataset Balanced-RAVEN. Contrastly, models which take the independence of each multiple-choice panel into consideration, such as Wild ResNet, WReN and our proposed model, achieves better performance on both datasets. The performance of WReN (ResNet) is similar to WReN on PGM-70K. However, on the more comprehensive dataset Balanced-RAVEN, WReN (ResNet) have the advantage over WReN due to the strong ability of ResNet in extracting features.
For more detailed comparison, Table \[result-table\] also reports the accuracy on seven figure configurations of Balanced-RAVEN. We can observe that accuracy on different configurations is not uniform, possibly due to the difficulty of configurations. But compare to other models, our HriNet consistently achieves the best performance on all the configurations, which proves that our model can work stably and robustly, even facing diverse conditions and complex rules. We will further interpret the reason in Section \[sec:interpretability\].
Also, we notice that accuracy across different configurations is not uniform.
HriNet achieves lower accuracy on figure configurations consisting of multiple graphical elements (`2x2Grid`, `3x3Grid` and `Out-InGrid`), and performs better on configurations with simpler structures (`Center`, `Out-InCenter`, `Left-Right` and `Up-Down`).
**Test on generalization.** The abstract reasoning skills do not only include the ability of inducing, but also the capability of deducing. This could be an evaluation of models’ generalization capacity, where in the test stage models will face RPM problems with dominant rules that are unseen in the training process. PGM and RAVEN introduce different generalization regimes as shown in Table \[Generalization on PGM\] and \[Generalization on Balanced-RAVEN\]. Each table lists the results both on validation set and test set of different generalization regimes, which respectively indicates the inducing ability and deducing ability. First, we can observe that our HriNet achieves the best performance in almost every generalization regime, except for Held-out `shape-colour` where all the performance are very close. Second, it is worth noticing that all the method witness the drop from the results of validation set to the corresponding ones of test set. That is to say, although state-of-the-art models make some progress in inducing the rules, it is still hard for models to deduce the rules facing the unseen conditions.
For PGM, different regimes are generated by holding out specific relation-object-attribute tuples in the training data. For RAVEN, a model is trained on a figure configuration and tested on similar one.
Interpolation is the easiest regime. After Interpolation, models perform relatively good in regimes where testing questions contain unseen combinations of seen triples in the training set (Held-out Triple Pairs and Held-out Attribute Pairs). Although Held-out `line-type` and Held-out `shape-color` are similar, their performance is quite different. This can be put down to the fact that `line` and `shape` are more difficult to infer compared with `type` and `color`, which we will further discuss in Section \[section-rules\]. Held-out Triples and Extrapolation are rather problematic, in which all models fail to generalize knowledge learned on the training set and transfer it to new circumstances.
For Balanced-RAVEN, HriNet outperforms all the competitors in every regime.
HriNet’s strong generalizability can be attributed to its innovative architecture which explicitly generates the rules. This structure helps the model when presented with a unseen question.
Ablation study
--------------
As aforementioned, our method mainly gains from the hierarchical architecture intimating human strategies. To validate this point, we study the effects of different hierarchies in abstract visual reasoning. Specifically, we set the rule embedding of certain hierarchy as a zero vector before gate function $\varphi$. Thus, the gate function regulates the flow of features into the gated embedding fusion module.
Table \[ablation-table\] lists the result of different choices of hierarchies. First, there is no doubt that our full model achieves the best performance compared to the other combinations, which indicates that all hierarchies contribute to our framework. Second, without considering the relationships among the panels, the performance of the cell-wise hierarchy is unsatisfactory, but still outperforms other state-of-the-art models (as shown in Table \[result-table\]). That is to say, our strategy that explicitly induces the rule representation and then compares with the dominant rule is totally reasonable.
One more interesting observation is that a simple combination of two arbitrary hierarchies does not always achieves better performance than one hierarchy, due to the fact that they may focus on the same or mutually-exclusive attributes, since different hierarchies focus on different attributes. Figure \[single-level-attribute\](a) also supports this observation as well. We conduct experiments only utilizing single-hierarchy rule embeddings from $\mathbb{E}_{\text{cell}}, \mathbb{E}_{\text{ind}}, \mathbb{E}_{\text{eco}}$, on Balanced-RAVEN, with respect to three attributes (`Type`, `Size` and `Color`). As shown in Figure \[single-level-attribute\](a), $\mathbb{E}_{\text{cell}}$ has strong capacity to infer attributes `Type` and `Size`, but struggles to distinguish attribute `Color`. By contrast, $\mathbb{E}_{\text{ind}}$ and $\mathbb{E}_{\text{eco}}$ have modest ability to infer attributes `Type` and `Size`, and are efficient for attribute `Color`.
A model chooses the correct answer on `Center` if and only if it figures out all three attribute values correctly. The accuracy across each attribute is shown in Figure \[single-level-attribute\](a). Cell-wise hierarchy has strong capacity to infer attributes `Type` and `Size`, but struggles to distinguish attribute `Color`. By contrast, individual hierarchy and ecological hierarchy have modest ability to infer attributes `Type` and `Size` and are efficient for attribute `Color`. The complementation of three hierarchies can be put down to their different ability to capture various attributes.
[ Model]{} [ PGM-70K]{} [ Balanced-RAVEN]{}
------------------------------------------------------------------------------------ -------------- ---------------------
$\mathbb{E}_{\text{cell}}$ 34.1 36.7
$\mathbb{E}_{\text{ind}}$ 42.2 48.7
$\mathbb{E}_{\text{eco}}$ 41.9 51.6
$\mathbb{E}_{\text{cell}}$ + $\mathbb{E}_{\text{ind}}$ 44.8 57.8
$\mathbb{E}_{\text{cell}}$ + $\mathbb{E}_{\text{eco}}$ 40.6 52.9
$\mathbb{E}_{\text{ind}}$ + $\mathbb{E}_{\text{eco}}$ 42.0 57.0
$\mathbb{E}_{\text{cell}}$ + $\mathbb{E}_{\text{ind}}$ + $\mathbb{E}_{\text{eco}}$ **48.9** **63.9**
: HriNet ($\mathbb{E}_{\text{cell}}$+$\mathbb{E}_{\text{ind}}$+$\mathbb{E}_{\text{eco}}$) and the results of eliminating different hierarchies.[]{data-label="ablation-table"}
[lccc]{} & [**Type**]{} & [**Size**]{} & [**Color**]{}\
\
$E_1$ & **97.40** & **98.75** & 60.85\
$E_2$ & 75.00 & 83.20 & 93.75\
$E_3$ & 67.60 & 81.20 & **99.30**\
The interpretability of rule embeddings {#sec:interpretability}
---------------------------------------
![Accuracy with respect to the different attributes or rule across each attribute in different level modules. It can be seen from the above figure, the cell level module has strong attribute type and size inference ability, while the individual level and ecological level module have strong attribute color inference ability.[]{data-label="single-table"}](Table6.pdf){width="1.0\linewidth"}
![Twelve relationships in PGM dataset. Our model’s performance on each single relationship is separated by relationship type, property type, and object type. Similar types are represented by similar colors. For example, line and shape are both object types, so they are represented by similar purple colors.[]{data-label="separate-rules"}](Table7.pdf){width="1.0\linewidth"}
In the real RPM test, it is not clear whether the rule exists in rows or columns. However, it is important to check whether the proposed model can discover the knowledge without any guidance. Therefore, in this part we further interpret the reasoning behaviors of the model.
Rule induction for columns is normally left out when trained on Balanced-RAVEN, given the prior knowledge that rules are applied only row-wise. In order to test the ability of distinguishing whether the rules are applied along rows or columns, we train a HriNet model on Balanced-RAVEN which the induction for column rules is reintegrated into. As a result, there is only a bit drop in accuracy (from 63.9% to 59.6%). This indicates that our model can neglect the distraction brought by columns on its own.We further analyse the contribution of columns and rows by calculating the ratio of column-wise similarity scores and row-wise similarity scores. The average absolute value of this ratio is shown in Table \[row-column-contribution\]. Columns have almost half contribution to classification score compared with rows. The ratio is smaller when the prediction is correct, which is in line with our expectations.
[lc]{} & [**HriNet**]{}\
\
OR & 61.1\
AND & 58.5\
XOR & 54.3\
consistent union & 47.9\
progression & 38.1\
position & 92.3\
type & 73.8\
color & 40.0\
number & 16.6\
size & 15.9\
line & 75.5\
shape & 30.2\
All Single Relations & 52.6\
Furthermore, since our model could induce the rule embeddings, we visualize these representations using the t-SNE [@maaten2008visualizing] scatter. The scatter indicates whether our model can extract the semantic relations and encode this information in the rule embeddings. Naturally, we simply select certain PGM questions with only one dominant rule inside the matrix. Figure \[embedding-plot\] respectively shows dominant rule representations of the matrix that are predicted right and wrong. Besides, to clearly understand our HriNet over different rule types, we also investigate the performance with respect to the relation type, attribute type and object type on PGM-70K dataset in Figure \[single-level-attribute\](b), where similar types (e.g., line and shape) are represented by similar colors.
From Figure \[single-level-attribute\](b), we get the consistent observation with that in Table \[result-table\], namely, the performance of our proposed model is imbalanced with respect to the difficulty of different configurations. Based on the observation, and further by comparing Figure \[embedding-plot\](a) with Figure \[embedding-plot\](b), we could further reach a conclusion that the performance of our model is positively related to the discrimination ability of the rule representations. For rules with good performance, such as `line type`, `shape position` and `line color`, they scatter closer as a dense cluster than those with poor performance. This further indicates that well-induced rule representations are helpful to find the correct answer to RPM test, and our HriNet owns strong capability of extracting discriminative rule embeddings.
This indicates that well-learned embedding representations help to find the correct answer, which is in line with our expectations.
We observe that the accuracy of each rule is highly correlated with its representations. For rules with good performance, such as `line type`, `shape position` and `line color`, their representations cluster closer than those with poor performance. This in turn indicates that well-learned embedding representations help to find the correct answer, which is in line with our expectations.
As we mentioned in Section \[sec:comparisons\], Similar phenomenon is observed on PGM dataset as well. We test the performance of our proposed method on PGM questions with single dominant rule inside, to measure its. The results are separated based on the relation type, attribute type and object type. As is shown in Figure \[single-level-attribute\](b), `OR`, `AND` and `XOR` are similar relation types with similar performance, followed by two more difficult relation types (`consistent union` and `progression`). Performance among different attribute types is quite various. Matrices involving position (92.3%) are the easiest , by contrast those involving size (15.9%) are the hardest. PGM questions involving line (75.5%) are significantly easier than those involving shape (30.2%).
Conclusion
==========
In this paper, we proposed a novel Hierarchical Rule Induction Network for abstract visual reasoning task intimating human inducing strategies, which could extract multiple granularity rule embeddings at different level and integrate them through a gated embedding fusion module. A rule similarity metric was further introduced based on the embeddings, so that HriNet can not only be trained using a tuplet loss but also infer the best answer according to the similarity score. We also designed an algorithm to fix the defects of the very recent proposed dataset RAVEN, and generated a more rigorous dataset based on the algorithm. Extensive experiments conducted on PGM-70K dataset and our improved dataset Balanced-RAVEN proved that, our proposed framework could significantly outperform other state-of-the-art approaches. Moreover, we studied the effects of each component of our proposed model and evaluated the interpretability of our induced rule embeddings. Although existing learning based models show promising performance in abstract reasoning, they mainly rely on the abundance of training data, and struggle to transfer the reasoning ability to RPM questions with unseen rules. In the future, we will introduce meta-learning strategies into our framework to improve both the inducing and deducing abilities at the same time.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
In this paper we extend the refined second-order Poincaré inequality in [@BassBerr] from a one-dimensional to a multi-dimensional setting. Its proof is based on a multivariate version of the Malliavin-Stein method for normal approximation on Poisson spaces. We also present an application to partial sums of vector-valued functionals of heavy-tailed moving averages. The extension we develop is not only in the co-domain of the functional, but also in its domain. Such a set-up has previously not been explored in the framework of stable moving average processes. It can potentially capture probabilistic properties which cannot be described solely by the one-dimensional marginals, but instead require the joint distribution.
*Key words*: Central limit theorem, heavy-tailed moving average, Lévy process, Malliavin-Stein method, Poisson random measure, second-order Poincaré inequality
*AMS 2010 subject classifications*: 60F05, 60G10, 60G15, 60G52, 60G55, 60H07
author:
- 'Ehsan Azmoodeh[^1]'
- 'Mathias Mørck Ljungdahl[^2]'
- 'Christoph Thäle[^3]'
bibliography:
- 'References.bib'
title: |
**Multi-Dimensional Normal Approximation\
of Heavy-Tailed Moving Averages**
---
Introduction
============
In recent decades the combination of Malliavin calculus and Stein’s method for normal approximation has led to a plethora of Gaussian limit theorems in fields ranging from stochastic geometry, over cosmology to statistics. Classically, the assumptions require third or fourth moment conditions which makes the Malliavin-Stein method unsuitable for distributions with heavier tails. However, in [@BassBerr] a careful differentiation between small and large values has led to a refined so-called second-order Poincaré inequality for Poisson functionals, which allows to circumvent these difficulties to a certain extent. Based on the approach in [@PeccMult] the principal goal of this paper is to obtain a multivariate extension of the central results in [@BassBerr]. This opens the possibility to capture properties of the underlying process not accessible solely by the one-dimensional marginal distributions.
We shall now define the heavy-tailed moving average model to which we are going to apply our general multivariate central limit theorem. Let $L = (L_t)_{t \in {\mathbb{R}}}$ be a two-sided Lévy process with no Gaussian component and Lévy measure $\nu$. We assume that the latter admits a Lebesgue density $w : {\mathbb{R}}\to {\mathbb{R}}$ such that $$\label{eq:stable-levy-measure}
\abs{w(x)} \leq C \abs{x}^{- 1 - \beta}$$ for all $x \neq 0$, some $\beta \in (0, 2)$ and a constant $C >
0$. Hence, the distribution of $L_1$ exhibits $\beta$-stable tails, see [@WataAsym]. Consider then for each $i \in \Set{1, \ldots, m}$, $m \in {\mathbb{N}}$, the process $$\label{eq:MA}
X_t^i \coloneqq \int_{{\mathbb{R}}} g_i(t - s) {{\,}{\textup{d}}}L_s \qquad (t \in {\mathbb{R}})$$ for some measurable function $g_i : {\mathbb{R}}\to {\mathbb{R}}$. Necessary and sufficient conditions for the integral to exists are given in [@RajpSpec] and if $L$ is symmetric around zero, i.e., if $-L_1$ and $L_1$ are identically distributed, then we mention that a sufficient condition is $\int_{{\mathbb{R}}} \abs{g_i(s)}^{\beta} {{\,}{\textup{d}}}s < \infty$.
The main examples of kernels $g_i$ we consider satisfy a power-law behaviour around zero and at infinity. Henceforth we shall assume for all $i \in \Set{1, \ldots, m}$ the existence of a constant $K > 0$ together with exponents $\alpha_i > 0$ and $\kappa_i \in {\mathbb{R}}$ such that $$\label{eq:kernel-ass}
\abs{g_i(x)} \leq K (x^{\kappa_i} {\mathds{1}}_{[0, a_i)}(x) + x^{-\alpha_i} {\mathds{1}}_{[a_i, \infty)}(x))$$ for all $x \in {\mathbb{R}}$, where $a_i>0$ are suitable splitting points, which may alter the constant $K$. Without loss of generality we choose $a_i = 1$ for all $i \in \Set{1, \ldots, m}$ and let $K$ stand for the corresponding constant. Note that this in particular implies that $g_i(x) = 0$ for $x < 0$, consequently we will only consider *casual* moving averages.
The main objects of interest in this paper are rescaled partial sums of multi-dimensional functionals of the joint distribution $X_s = (X_s^1, \ldots, X_s^m)$, namely $$\label{eq:main-statistic}
V_n(X; f) = \frac{1}{\sqrt{n}} \sum_{s = 1}^{n} (f(X_s^1, \ldots, X_s^m) - \Expt{f(X_0^1, \ldots, X_0^m)}),$$ where $f : {\mathbb{R}}^m \to {\mathbb{R}}^d$ is a suitable Borel-measurable function, with $d$ being some positive integer. Observe that $V_n(X; f)$ is a $d$-dimensional random vector and for convenience we shall denote by $V_n^i(X; f)$ its $i$th coordinate. We remark that in the one dimensional case $d = m = 1$ the distributional convergence of $V_n(X; f)$, as $n \to \infty$, is studied for general functions $f$ in [@BassOnLi] and here the so-called Appell rank of $f$ is seen to play an important role. The results in that paper also imply that one cannot in general expect convergence in distribution after rescaling with the factor $\sqrt{n}$ as in \[eq:main-statistic\] or a Gaussian limiting distribution if the memory of the processes are too long, i.e., if the $\alpha_i$ are too close to $0$. We shall see that if the tails are not too heavy and the memory is not too long, which in our case means that $\alpha_i \beta > 2$, we do in fact have convergence in distribution of $V_n(X; f)$ to a Gaussian random variable and we shall discuss the *speed* of this convergence by considering an appropriate metric on the space of probability laws on ${\mathbb{R}}^d$, see \[sec:limit-theory\] below. To conclude such a result, we could also in principle rely on a multivariate second-order Poincaré inequality for random vectors of Poisson functionals in [@SchuMult]. But as already observed in the one-dimensional case, the existing bounds are not suitable for the application to Lévy driven moving averages just described. In fact, in this specific situation the bounds in [@SchuMult] do not even tend to zero, as $n$ increases. Against this background, we will develop in this paper a refined multivariate second-order Poincaré inequality for general random vectors of Poisson functionals, which is more adapted to our situation and allows us to distinguish carefully between small and large values. We believe that this result is of independent interest as well. This eventually paves the way to the central limit theory for the random vectors $V_n(X; f)$.
One motivation for the extension of the theory from [@BassBerr] to a multivariate set-up is the fact that important properties of random processes, such as self-similarity, are determined by the finite dimensional distributions of $X$, but not by the one-dimensional marginals. The one-dimensional theory, i.e., the case $m = d = 1$, could so-far capture only probabilistic properties of the distribution of $X_1$. In the case of the linear fractional stable motion this made the estimation of the Hurst parameter problematic if jointly estimated with the scale parameter and the stability index $\beta$, see [@LjunAMin]. We will come back to such applications of the results we develop in this paper in a separate work. However, we would like to mention finally that the case $m = 1$ and general $d$ has been considered in the seminal paper [@PipiCent].
Main results {#sec:limit-theory}
============
A refined multivariate second-order Poincaré inequality {#sec:Poincare}
-------------------------------------------------------
Consider a measurable space $(S, {\mathcal{S}})$ equipped with a $\sigma$-finite measure $\mu$. Let $\eta$ be a Poisson process on $(S, {\mathcal{S}})$ with intensity measure $\mu$. This means that $\eta$ is a collection of random variables of the form $\eta(B)$, $B \in {\mathcal{S}}$, with the properties that
for each $B \in {\mathcal{S}}$ with $\mu(B) < \infty$ the random variable $\eta(B)$ is Poisson distributed with mean $\mu(B)$,
for $m \in {\mathbb{N}}$ and pairwise disjoint $B_1, \ldots, B_m \in {\mathcal{S}}$ with $\mu(B_1), \ldots, \mu(B_m) < \infty$ the random variables $\eta(B_1), \ldots, \eta(B_m)$ are independent.
We can and will regard $\eta$ as a random function from an underlying probability space $(\Omega, {\mathcal{F}}, {\mathbb{P}})$ to ${\mathcal{N}}$, the space of all integer-valued $\sigma$-finite measures on $(S, {\mathcal{S}})$. The set ${\mathcal{N}}$ is equipped with the evaluation $\sigma$-algebra, i.e. the $\sigma$-algebra generated by the evaluation mappings $\mu \mapsto \mu(A)$, $A \in {\mathcal{S}}$.
To each Poisson process $\eta$ we associate the Hilbert space ${\mathcal{L}}^2_\eta({\mathbb{P}})$ consisting of all square integrable Poisson functionals $F$, i.e., those random variables for which there exists a function $\phi : {\mathcal{N}}\to {\mathbb{R}}$ such that almost surely $F = \phi(\eta) \in {\mathcal{L}}^2({\mathbb{P}})$. Finally, we introduce the notion of the Malliavin derivative in a Poisson setting, which is also known as the add-one-cost operator. For each $z \in S$ and $F = \phi(\eta) \in {\mathcal{L}}^2_\eta({\mathbb{P}})$ we define $D_z F$ as $$D_z F \coloneqq \phi(\eta + \delta_z) - \phi(\eta)$$ and note that $D F$ is a bi-measurable map from $\Omega \times S$ to ${\mathbb{R}}$. In a straightforward way this definition extends to vector-valued Poisson functionals. Indeed, consider $F = (F_1, \ldots, F_d)$ where each $F_i$ lies in ${\mathcal{L}}^2_\eta({\mathbb{P}})$, then the Malliavin derivative $D_z F$ at $z \in S$ is given by $$D_z F = (D_z F_1, \ldots, D_z F_d).$$ Similarly to $D_zF$ we may introduce the iterated Malliavin derivative $D^2 F$ of $F$ by putting $$D^2_{z_1, z_2} F \coloneqq D_{z_1}(D_{z_2}F)
= D_{z_2}(D_{z_1}F), \qquad z_1, z_2 \in S.$$ For further background material on Poisson processes we refer to the treatments in [@LastLect; @LastPois; @PeccStoc], for the Malliavin formalism on Poisson spaces we refer to \[subsec:Malliavin\] below.
To measure the distance between (the laws of) two random vectors $X$ and $Y$ taking values in ${\mathbb{R}}^d$ we use the so-called $d_3$-distance, see [@PeccMult]. To introduce it, assume that $\Expt{\norm{X}^2_{{\mathbb{R}}^d}}, \Expt{\norm{Y}^2_{{\mathbb{R}}^d}} < \infty$, where $\norm{}_{{\mathbb{R}}^d}$ stands for the Euclidean norm in ${\mathbb{R}}^d$. The $d_3$-distance between (the laws of the) random vectors $X$ and $Y$, denoted by $d_3(X, Y)$, is given by $$d_3 (X, Y) \coloneqq \sup_{\varphi \in {\mathcal{H}}_3} \abs[\big]{\Expt{\varphi(X)} - \Expt{\varphi(Y)}},$$ where the class ${\mathcal{H}}_3$ of test functions indicates the collection of all thrice differentiable functions $\varphi : {\mathbb{R}}^d \to {\mathbb{R}}$ (i.e., $\varphi \in {\mathcal{C}}^3({\mathbb{R}}^d, {\mathbb{R}})$) such that $\norm{\varphi''}_{\infty} \leq 1$ and $\norm{\varphi'''}_{\infty} \leq 1$, where $$\norm{\varphi''}_\infty \coloneqq \max_{1 \leq i, j \leq d} \sup_{x \in {\mathbb{R}}^d} \abs[\Big]{\frac{\partial^2}{\partial x_i \partial x_j} \varphi(x)}, \qquad
\norm{\varphi'''}_\infty \coloneqq \max_{1 \leq i, j, k \leq d} \sup_{x \in {\mathbb{R}}^d} \abs[\Big]{\frac{\partial^3}{\partial x_i \partial x_j\partial x_k} \varphi(x)}.$$
We can now formulate our multivariate second-order Poincaré inequality, which generalises [@BassBerr Theorem 3.1] and refines [@SchuMult Theorem 1.1]. Its proof, which is given in \[sec:ProofThm1\] below, is based on the Malliavin-Stein technique for normal approximation of random vectors of Poisson functionals. For two Poisson functionals $F, G \in {\mathcal{L}}^2_\eta({\mathbb{P}})$ we define the quantities $$\begin{aligned}
\gamma_1^2(F,G) &\coloneqq 3 \int_{S^3} \Expt[\big]{(D^2_{z_1, z_3} F)^2 (D^2_{z_2, z_3} F)^2}^{1/2}
\Expt[\big]{(D_{z_1} G)^2 (D_{z_2} G)^2 }^{1/2}
{\,}\mu^3({\textup{d}}z_1, {\textup{d}}z_2, {\textup{d}}z_3),
\\
\gamma_2^2(F,G) &\coloneqq \int_{S^3} \Expt[\big]{(D^2_{z_1, z_3} F)^2 (D^2_{z_2, z_3} F)^2}^{1/2} \Expt[\big]{(D^2_{z_1, z_3} G)^2 (D^2_{z_2, z_3} G)^2}^{1/2} {\,}\mu^3({\textup{d}}z_1, {\textup{d}}z_2, {\textup{d}}z_3).\end{aligned}$$ Moreover, for $x, y \in {\mathbb{R}}$ we denote by $x \wedge y = \min \Set{x, y}$ the minimum of $x$ and $y$.
\[thm:wasserstein-dist\] Let $d \geq 1$ and assume that $F_1, \ldots, F_d \in {\mathcal{L}}^2_\eta({\mathbb{P}})$ satisfy $DF_i \in {\mathcal{L}}^2({\mathbb{P}}\otimes \mu)$ and $\Expt{F_i} = 0$ for all $i \in \Set{1, \ldots, d}$. Let $\sigma_{i k} \coloneqq \Expt{F_i F_k}$ and define the covariance matrix $\Sigma^2 = (\sigma_{ik})_{i,k = 1}^{d}$. Let $Y \sim N_d(0, \Sigma^2)$ be a centred Gaussian random vector with covariance matrix $\Sigma^2$ and put $F \coloneqq (F_1, \ldots,
F_d)$. Then $$d_3(F, Y) \leq \sum_{i, k = 1}^{d} (\gamma_{1}(F_i,F_k) + \gamma_{2}(F_i,F_k))
+ \gamma_3,$$ where the term $\gamma_3$ is defined as $$\label{eq:gamma3}
\gamma_3
\coloneqq \sum_{i, j, k = 1}^{d}
\int_S \Expt[\big]{\abs{D_z F_j D_z F_k}^{3/2} \wedge \norm{D_z F}_{{\mathbb{R}}^d}^{3/2}}^{2/3} \Expt[\big]{\abs{D_z F_i}^3}^{1/3} {\,}\mu({\textup{d}}z).$$
\[rem:CLT\]
The difference between \[thm:wasserstein-dist\] and [@SchuMult Theorem 1.1] lies in the term $\gamma_3$. We emphasise that the bound in [@SchuMult] does not lead to a meaningful error bound in the application to heavy-tailed moving averages we consider in the next section as the corresponding $\gamma_3$-term in [@SchuMult] would diverge. Similarly to the univariate case, the bound provided by \[thm:wasserstein-dist\] is much more suitable for our purposes as it leads to a reasonable error bound, which tends to zero, as the number of observations $n$ there tends to infinity.
It is in principal possible to derive error bounds as in \[thm:wasserstein-dist\] for probability metrics different from the $d_3$-metric. Namely, assuming in addition that the covariance matrix $\Sigma^2$ is *positive definite*, one can deal with the $d_2$-distance used in [@PeccMult] and even with the convex distance introduced and studied in [@SchuMult]. Since the corresponding error bounds for these notions of distance become rather long and technical, we refrain from presenting results in this direction. Moreover, in our application in the next section it seems in general rather difficult to check whether or not the covariance matrix is positive definite. This is another reason for us considering only the $d_3$-distance.
We would like to point out that quantitative central limit theorems for random vectors of Poisson functionals having a *finite* Wiener-Itô chaos expansion with respect to the $d_3$-distance were obtained [@LastMome]. Specifically, random vectors of so-called Poisson U-statistics were considered in [@LastMome] together with applications in stochastic geometry to Poisson process of $k$-dimensional flat in ${\mathbb{R}}^n$.
Asymptotic normality of multivariate heavy-tailed moving averages
-----------------------------------------------------------------
Here, we present our application of the refined multivariate second-order Poincaré inequality formulated in the previous section. For this recall the set-up described in the introduction. Especially, recall the definition of the random processes $(X_t^i)_{t \in {\mathbb{R}}}$, $i \in \Set{1, \ldots, m}$ from \[eq:MA\]. Also recall that the exponents $\alpha_i$ control the memory of the processes $X^i$. Given the limit theory for heavy-tailed moving averages as developed in [@LjunALim] it comes as no surprise that the smallest such $\alpha_i$ will be of dominating importance. Hence, we define $$\underline{\alpha} = \min \Set{\alpha_1, \ldots, \alpha_m}.$$ Finally, by ${\mathcal{C}}_b^2({\mathbb{R}}^m, {\mathbb{R}}^d)$ we denote the space of bounded functions $f : {\mathbb{R}}^m \to {\mathbb{R}}^d$ which are twice continuously differentiable and have all partial derivatives up to order two bounded by some constant.
\[thm:MA-clt\] Fix $d, m \geq 1$. Let $(X_t^i)$, $i = 1, \ldots, m$, be moving averages as in \[eq:MA\] with Lévy measure having density $w$ satisfying \[eq:stable-levy-measure\] for some $\beta \in (0, 2)$ and kernels $g_i$ which satisfy \[eq:kernel-ass\] with $\alpha_i \beta> 2$ and $\kappa_i > -1/\beta$. Let a function $f = (f_1, \ldots, f_d) \in {\mathcal{C}}_b^2({\mathbb{R}}^m, {\mathbb{R}}^d)$ be given and consider $V_n(X; f)$ as at \[eq:main-statistic\] based on $f$ and $X = (X^1, \ldots, X^m)$. Let $\Sigma_n = \operatorname{Cov}(V_n(X; f))^{1/2}$ denote a non-negative definite square root of the covariance matrix $\operatorname{Cov}(V_n(X; f))$ of the $d$-dimensional random vector $V_n(X;
f)$. Then $\Sigma_n \to \Sigma = (\Sigma_{i,j})_{i, j = 1}^d$, as $n \to \infty$, where, for $i,j \in \Set{1, \ldots, d}$, $$\label{eq:asymp-cov}
\begin{aligned}
\Sigma^2_{i, j} &= \sum_{s = 0}^{\infty} \operatorname{Cov}(f_i(X_s^1, \ldots, X_s^m), f_j(X_0^1, \ldots, X_0^m))
\\
&\qquad+ \sum_{s = 1}^{\infty} \operatorname{Cov}(f_i(X_0^1, \ldots, X_0^m), f_j(X_s^1, \ldots, X_s^m)).
\end{aligned}$$ Moreover, $V_n(X; f)$ converges in distribution, as $n \to \infty$, to a $d$-dimensional centred Gaussian random vector $Y \sim N_d(0, \Sigma^2)$ with covariance matrix $\Sigma^2$. More precisely, there exists a constant $C > 0$ such that $$d_3(V_n(X; f), Y) \leq C
\begin{cases}
n^{-1/2}, & \text{if $\underline{\alpha} \beta > 3$,}
\\
n^{-1/2} \log(n), & \text{if $\underline{\alpha} \beta = 3$,}
\\
n^{(2 - \underline{\alpha} \beta)/2}, & \text{if
$2 < \underline{\alpha} \beta < 3$.}
\end{cases}$$
We remark that in the special case $d = m = 1$ the order for the $d_3$-distance provided by \[thm:wasserstein-dist\] is precisely the same as that for the Wasserstein distance in [@BassBerr].
Even for particular functions $f = (f_1, \ldots, f_d)$, such as trigonometric functions, it seems to be a rather demanding task to check whether the covariance matrix $\Sigma^2$ is positive definite or not. Note in this context that even in the one-dimensional case $d = m = 1$ the question of whether the asymptotic variance constant is strictly positive or not is generally difficult. This is the reason why we are working with the $d_3$-distance in this paper, since more refined probability metric usually require positive definiteness of the covariance matrix, see \[rem:CLT\].
It is straightforward to modify the proof of \[thm:MA-clt\] to the situation where $X = (X_1, \ldots, X_m)$ for some fixed moving average $(X_t)_{t \in {\mathbb{R}}}$ as in \[eq:MA\] and where the kernel $g$ satisfy $$\abs{g(x)} \leq K (x^{\kappa} {\mathds{1}}_{[0, a)}(x) + x^{-\alpha} {\mathds{1}}_{[a, \infty)}(x))$$ for some constants $a, \alpha, K > 0$ and $\kappa \in {\mathbb{R}}$ such that $\alpha \beta > 2$ and $\kappa > - 1/\beta$. In this case the kernel of $X^i = X_i$ is simply $g_i = g(i + {\,\cdot\,})$. Choosing an appropriate functional $f$ in $V_n(X; f)$, such as the empirical characteristic function of $X$, opens up the possibility of inference on $(X_t)_{t \in {\mathbb{R}}}$ based on not only the marginal distribution $X_1$ as in much of the previous literature, but also on the joint distribution $(X_1, \ldots, X_m)$.
As in [@BassBerr], \[thm:wasserstein-dist\] can be applied to particular processes $(X_t^i)$. We mention here the linear fractional stable noises, which may be regarded as heavy-tailed extensions of a fractional Brownian motion. Let $L$ be a $\beta$-stable Lévy process with $\beta \in (0,2)$ and put $$X_t^i \coloneqq Y_t - Y_{t - 1} \qquad\text{for}\qquad
Y_t^i \coloneqq \int_{-\infty}^t \bigl[(t - s)_+^{H_i - 1/\beta} - (-s)_+^{H_i - 1/\beta} \bigr] {{\,}{\textup{d}}}L_s,$$ where $H_1, \ldots, H_m \in (0,1)$ (if $\beta = 1$ we additionally suppose that $L$ is symmetric). In this case, $\alpha_i = 1 - H_i + 1/\beta$ for all $i \in \Set{1, \ldots, m}$ and the condition $\underline{\alpha} \beta > 2$ translates into $\beta \in (1, 2)$ and $\max \Set{H_1, \ldots, H_m} < 1 - \frac{1}{\beta}$. Note that since $\beta > 1$ we automatically have that $\underline{\alpha} \beta <
3$. In this set-up the bound in \[thm:wasserstein-dist\] reads as follows: $$d_3(V_n(X; f), Y) \leq C \, n^{1/2 - \beta(1 - \max \{H_1, \ldots, H_m \}) / 2}.$$ As a second application we mention a stable Ornstein–Uhlenbeck process. Again, for a $\beta$-stable Lévy process $L$ with $\beta \in (0, 2)$ define for $i \in \{1, \ldots, m\}$, $$X_t^i \coloneqq \int_{-\infty}^{t} e^{-\lambda_i(t - s)} {{\,}{\textup{d}}}L_s,$$ where $\lambda_1, \ldots, \lambda_m > 0$. In this case, the parameters $\alpha_1, \ldots, \alpha_m$ may be arbitrary and the error bound in \[thm:wasserstein-dist\] reduces to $$d_3(V_n(X; f), Y) \leq C \, n^{-1/2}.$$ In a similar spirit, one my consider multivariate quantitative central limit theorems for functionals of linear fractional Lévy noises or of stable fractional ARIMA processes, see [@BassBerr] for the corresponding one-dimensional situations.
Background material
===================
Malliavin calculus on Poisson spaces {#subsec:Malliavin}
------------------------------------
To take advantage of the powerful Malliavin-Stein method we need to recall some background material regarding the Malliavin formalism on Poisson spaces. For further details we refer to [@LastLect; @LastPois; @NualIntr].
Throughout this section $\eta$ denotes a Poisson process with intensity measure $\mu$ defined on some measurable space $(S, {\mathcal{S}})$ and over some probability space $(\Omega, {\mathcal{F}}, {\mathbb{P}})$. We start by recalling that any $F \in {\mathcal{L}}^2_\eta({\mathbb{P}})$ admits a chaos expansion (with convergence in ${\mathcal{L}}^2({\mathbb{P}})$). That is, $$\label{eq:chaos-exp}
F = \sum_{n = 0}^{\infty} I_n(f_n),$$ where $I_n$ denotes the $n$th order Wiener-Itô integral with respect to the compensated Poisson process $\eta - \mu$ and the kernels $f_n \in {\mathcal{L}}^2(\mu^n)$ are symmetric functions (i.e., they are invariant under permutations of its variables). Especially, $I_0(c) = c$ for all $c \in {\mathbb{R}}$.
The Kabanov-Skorohod integral $\delta$ is defined for a subclass of random processes $u \in {\mathcal{L}}^2({\mathbb{P}}\otimes \mu)$ having chaotic decomposition $$u(z) = \sum_{n = 0}^{\infty} I_n(h_n({\,\cdot\,}, z)),$$ where for each $z\in S$ the function $h_n({\,\cdot\,}, z)$ is symmetric and belongs to ${\mathcal{L}}^2(\mu^n)$. Denoting by $\tilde{h}$ the canonical symmetrisation of a function $h : S^n \to {\mathbb{R}}$, i.e., $$\tilde{h}(z_1, \ldots, z_n) = \frac{1}{n!} \sum_{\sigma \in S_n} h(z_{\sigma(1)}, \ldots, z_{\sigma(n)}),$$ with $S_n$ being the group of all permutations of $\Set{1, \ldots, n}$, we put $$\delta(u) \coloneqq \sum_{n = 0}^{\infty} I_{n + 1}(\tilde{h}_n),$$ whenever $\sum_{n = 0}^{\infty} (n + 1) \norm{\tilde{h}_n}_{\smash{{\mathcal{L}}^2(\mu^{n
+ 1})}}^2 < \infty$ (we indicate this by writing $u \in \operatorname{dom}\delta$), where $\norm{}_{{\mathcal{L}}^2(\mu^{n + 1})}$ denotes the usual ${\mathcal{L}}^2$-norm with respect to $\mu^{n + 1}$.
Next, we shall define the two operators $L : \operatorname{dom}L \to {\mathcal{L}}^2_\eta({\mathbb{P}})$ and $L^{-1} : {\mathcal{L}}^2_\eta({\mathbb{P}}) \to {\mathcal{L}}^2_\eta({\mathbb{P}})$, where $\operatorname{dom}L$ denotes the class of Poisson functionals $F \in {\mathcal{L}}^2_\eta({\mathbb{P}})$ with chaos expansion as in \[eq:chaos-exp\] satisfying $\sum_{n = 1}^{\infty} n^2 n! \norm{f_n}_{{\mathcal{L}}^2(\mu^n)}^2 <
\infty$. Then, we define $$LF \coloneqq - \sum_{n = 1}^{\infty} n I_n(f_n).$$ Similarly, the pseudo-inverse $L^{-1}$ of $L$ acts on centred $F \in {\mathcal{L}}_\eta^2({\mathbb{P}})$ with chaotic expansion \[eq:chaos-exp\] as follows: $$L^{-1} F \coloneqq \sum_{n = 1}^{\infty} \frac{1}{n} I_n(f_n).$$ Finally, we recall that for $F \in {\mathcal{L}}_\eta^2({\mathbb{P}})$ with chaotic expansion \[eq:chaos-exp\] satisfying $\sum_{n = 0}^{\infty} (n + 1)! \norm{f_n}_{{\mathcal{L}}^2(\mu^n)}^2 < \infty$ the Malliavin derivative admits the representation $$D_z F = \sum_{n = 1}^{\infty} n I_{n - 1}(f_n({\,\cdot\,}, z)), \qquad z \in S.$$
Using these definitions and representations, one may prove the following crucial formulas and relationships of Malliavin calculus, which also play a prominent role in our approach:
\[it:op-rules:1\] $L L^{-1} F = F$ if $F$ is centred.
\[it:op-rules:2\] $LF = -\delta DF$ for $F \in \operatorname{dom}L$.
\[it:op-rules:3\] $\Expt{F \delta(u)} = \Expt{\int_S (D_z F) u(z) {\,}\mu({\textup{d}}z)}$, when $u \in \operatorname{dom}\delta$.
Multivariate normal approximation by Stein’s method
---------------------------------------------------
Stein’s method for multivariate normal approximation is a powerful device to prove quantitative multivariate central limit theorems. The proof of \[thm:wasserstein-dist\] is based on the following result, which is known as Stein’s Lemma (see [@NourNorm Lemma 4.1.3]). To present it, let us recall that the Hilbert-Schmidt inner product between two $d \times d$ matrices $A = (a_{i k})$ and $B = (b_{i k})$ is defined as $$\iprod{A, B}_{{\textup{HS}}} = \operatorname{Tr}(B^{\top} A)
= \sum_{i, k = 1}^{d} b_{k i} a_{k i}.$$ Moreover, for a differentiable function $\varphi : {\mathbb{R}}^d \to {\mathbb{R}}$ we shall write $\nabla \varphi$ for the gradient and $\nabla^2 \varphi$ for the Hessian of $\varphi$. Also, we let $\iprod{}_{{\mathbb{R}}^d}$ denote the Euclidean scalar product in ${\mathbb{R}}^d$.
\[lem:stein\] Let $\Sigma^2 \in {\mathbb{R}}^{d \times d}$ be a positive semi-definite matrix and $Y$ be a $d$-dimensional random vector. Then $Y \sim N_d(0, \Sigma^2)$ if and only if it for all twice continuously differentiable functions $\varphi : {\mathbb{R}}^d \to {\mathbb{R}}$ with bounded derivatives one has that $$\Expt{\iprod{Y, \nabla \varphi(Y)}_{{\mathbb{R}}^d} - \iprod{\Sigma^2, \nabla^2 \varphi(Y)}_{{\textup{HS}}}} = 0.$$
Proof of Theorem \[thm:wasserstein-dist\] {#sec:ProofThm1}
=========================================
By definition of the $d_3$-distance we need to prove that $$\abs{\Expt{\varphi(Y)} - \Expt{\varphi(F)}}
\leq \sum_{i, k = 1}^{d} (\gamma_{1}(F_i,F_k) + \gamma_{2}(F_i,F_k)) + \gamma_3$$ for every function $\varphi \in {\mathcal{H}}_3$. For this, we may assume that $Y$ and $F$ are independent. We start out by applying the interpolation technique already demonstrated in [@PeccMult]. Consider the function $\Psi : [0, 1] \to {\mathbb{R}}$ given by $$\Psi(t) \coloneqq \Expt{\varphi(\sqrt{1 - t} F + \sqrt{t} Y)}, \qquad t \in [0, 1].$$ Note that from the mean value theorem it follows that $$\abs{\Expt{\varphi(Y)} - \Expt{\varphi(F)}} = \abs{\Psi(1) - \Psi(0)}
\leq \sup_{t \in (0, 1)} \abs{\Psi'(t)}.$$ Hence it is enough to consider $\Psi'$, which is given by $$\Psi'(t) = \Expt[\big]{\iprod{\nabla \varphi(\sqrt{1 - t} F + \sqrt{t} Y), \tfrac{1}{2 \sqrt{t}} Y - \tfrac{1}{2 \sqrt{1 - t}} F}_{{\mathbb{R}}^d}}
\eqqcolon \frac{1}{2 \sqrt{t}} T_1 - \frac{1}{2 \sqrt{1 - t}} T_2.$$ We consider the two terms $T_1$ and $T_2$ separately. For $T_1$ it follows first by independence of $F$ and $Y$ and Stein’s Lemma (used on the function $y \mapsto \varphi(\sqrt{1 - t} a + \sqrt{t} y)$ and then dividing by $\sqrt{t}$) that $$\begin{aligned}
T_1 &= \Expt{\iprod{\nabla \varphi(\sqrt{1 - t} F + \sqrt{t} Y), Y}_{{\mathbb{R}}^d}}
\\
&= \Expt[\big]{\Expt{\iprod{\nabla \varphi(\sqrt{1 - t} a + \sqrt{t} Y), Y}_{{\mathbb{R}}^d}} \mid_{a = F}}
\\
&= \sqrt{t}\, \Expt[\big]{\Expt{\iprod{\Sigma^2, \nabla^2 \varphi(\sqrt{1 - t} a + \sqrt{t} Y)}_{{\textup{HS}}}} \mid_{a = F}}.\end{aligned}$$ Let $\partial_i f$ denote the derivative of $f$ in the $i$th coordinate. We have by independence of $F$ and $Y$ and the Malliavin rules rephrased at the end of \[subsec:Malliavin\] that $$\begin{aligned}
T_2 &= \Expt{\iprod{\nabla \varphi(\sqrt{1 - t} F + \sqrt{t} Y), F}_{{\mathbb{R}}^d}}
= \sum_{i = 1}^{d} \Expt[\big]{\Expt{\partial_i \varphi(\sqrt{1 - t} F + \sqrt{t} a) F_i} \mid_{a = Y}}
\\
&= \sum_{i = 1}^{d} \Expt[\big]{\Expt{\partial_i \varphi(\sqrt{1 - t} F + \sqrt{t} a) L (L^{-1} F_i) }\mid_{a = Y}}
\\
&= -\sum_{i = 1}^{d} \Expt[\big]{\Expt{\partial_i \varphi(\sqrt{1 - t} F + \sqrt{t} a) \delta (D L^{-1} F_i)} \mid_{a = Y}}
\\
&= \sum_{i = 1}^{d} \Expt[\big]{\Expt{\iprod{D \partial_i \varphi(\sqrt{1 - t} F + \sqrt{t} a), -D L^{-1} (F_i)}_{{\mathcal{L}}^2(\mu)} } \mid_{a = Y}}.\end{aligned}$$ Consider now the function $\varphi_i^{t, a} : {\mathbb{R}}^d \to {\mathbb{R}}$ defined by $$\varphi_i^{t, a}(x) \coloneqq \partial_i \varphi(\sqrt{1 - t} x + \sqrt{t} a).$$ By Taylor expansion we can write $$D_z \varphi_i^{t, a}(F)
= \sum_{k = 1}^{d} \partial_k \varphi_i^{t, a}(F) (D_z F_k) + R_i^a(D_z F)$$ for any $z \in {\mathbb{R}}^d$, where the remainder term $R_i^a(D_z F) = \sum_{j, k = 1}^{d} R_{i, j, k}^a(D_z F_k, D_z F_j)$ satisfies the estimate $$\label{eq:remainder-1}
\begin{aligned}
\abs{R_{i, j, k}^a(x, y)} &\leq \frac{1}{2} \abs{x y} \max_{k, l} \sup_{x \in {\mathbb{R}}^d} \abs[\big]{\partial_{k, l} \varphi_i^{t, a}(x)}
\\
&\leq \frac{1}{2} \abs{x y} (1 - t) \max_{k, l} \sup_{x \in {\mathbb{R}}^d} \abs[\big]{\partial_{i, k, l} \varphi(\sqrt{1 - t} x + \sqrt{t} a)}
\\
&\leq \frac{1}{2} (1 - t) \abs{x y}.
\end{aligned}$$ Here, we have used the definition of the class ${\mathcal{H}}_3$. On the other hand, the remainder term also satisfies the inequality $$\begin{aligned}
\label{eq:remainder-2}
\abs[\Big]{D_z \varphi_i^{t, a}(F) - \sum_{k = 1}^{d} \partial_k \varphi_i^{t, a}(F) (D_z F_k)}
&\leq \abs{D_z \varphi_i^{t, a}(F)} + \abs{\iprod{\nabla \varphi_i^{t, a}(F), D_z F}_{{\mathbb{R}}^d}}
\\
&\leq 2 \norm{\nabla \varphi_i^{t, a}(F)}_{{\mathbb{R}}^d} \norm{D_z F}_{{\mathbb{R}}^d}
\\
&\leq 2 \sqrt{1 - t} \norm{D_z F}_{{\mathbb{R}}^d},
\end{aligned}$$ where we used again the mean value theorem and the Cauchy-Schwarz inequality. We may thus rewrite $T_2$ as $$\begin{aligned}
T_2 &= \sum_{i, k = 1}^{d} \Expt[\big]{\Expt{\iprod{\partial_k \varphi_i^{t, a}(F) (D F_k), - D L^{-1} (F_i)}_{{\mathcal{L}}^2(\mu)}} \mid_{a = Y}}
\\
&\qquad + \sum_{i = 1}^{d} \Expt[\big]{\Expt{\iprod{R_i^a(D F), - D L^{-1}(F_i)}_{{\mathcal{L}}^2(\mu)}} \mid_{a = Y}}
\\
&= \sqrt{1 - t} \sum_{i, k = 1}^{d} \Expt[\big]{ \partial_{k, i} \varphi(\sqrt{1 - t} F + \sqrt{t} Y) \iprod{ D F_k, - D L^{-1} (F_i)}_{{\mathcal{L}}^2(\mu)}}
\\
&\qquad + \sum_{i = 1}^{d} \Expt[\big]{\Expt{\iprod{R_i^a(D F), - D L^{-1}(F_i)}_{{\mathcal{L}}^2(\mu)} } \mid_{a = Y}}.\end{aligned}$$ From this together with the Cauchy-Schwarz inequality and the bounds \[eq:remainder-1\] and \[eq:remainder-2\] it follows that $$\begin{aligned}
\MoveEqLeft \abs{\Expt{\varphi(Y)} - \Expt{\varphi(F)}} \leq \sup_{t \in (0, 1)} \abs{\Psi'(t)}
\\
&\leq \sup_{t \in (0, 1)} \frac{1}{2} \sum_{i, k = 1}^{d} \Expt[\Big]{\abs[\big]{\partial_{i, k} \varphi(\sqrt{1 - t} F + \sqrt{t} X)} \abs[\big]{\sigma_{ik} - \iprod{D F_k, -D L^{-1} (F_i)}_{{\mathcal{L}}^2(\mu)}}}
\\
&\qquad + \sup_{t \in (0, 1)} \frac{1}{2 \sqrt{1 - t}} \sum_{i = 1}^{d} \Expt[\big]{{\abs{\iprod{R_i^a(D F), - D L^{-1}(F_i)}_{{\mathcal{L}}^2(\mu)}} } \mid_{a = Y}}
\\
&\leq \frac{1}{2} \sum_{i, k = 1}^{d} \Expt[\big]{\abs{\sigma_{ik} - \iprod{D F_k, - D L^{-1} F_i}_{{\mathcal{L}}^2(\mu)}}}
\\
&\qquad + \sum_{i, j, k = 1}^{d} \int_S \Expt{(\abs{D_z F_j D_z F_k} \wedge \norm{D_z F}_{{\mathbb{R}}^d}) \abs{D_z L^{-1} F_i}} {\,}\mu({\textup{d}}z).\end{aligned}$$ Applying now Proposition 4.1 in [@LastNorm] to the first of these terms yields the inequality $$\sum_{i, k = 1}^{d} \Expt[\Big]{\abs[\big]{\sigma_{i k} - \iprod{D F_k, - D L^{-1} F_i}_{{\mathcal{L}}^2(\mu)}}}
\leq 2 \sum_{i, k = 1}^{d} (\gamma_{1, i, k} + \gamma_{2, i, k}).$$ For the remainder term we deduce by Hölder’s inequality with exponents $3$ and $3 / 2$ that $$\begin{aligned}
\MoveEqLeft \int_S \Expt[\big]{(\abs{D_z F_j D_z F_k} \wedge \norm{D_z F}_{{\mathbb{R}}^d}) \abs{D_z L^{-1} F_i}} {\,}\mu({\textup{d}}z)
\\
&\leq \int_S \Expt[\big]{(\abs{D_z F_j D_z F_k} \wedge \norm{D_z F}_{{\mathbb{R}}^d})^{3/2}}^{2/3} \Expt[\big]{\abs{D_z L^{-1} F_i}^3}^{1/3} {\,}\mu({\textup{d}}z)
\\
&\leq \int_S \Expt[\big]{\abs{D_z F_j D_z F_k}^{3/2} \wedge \norm{D_z F}_{{\mathbb{R}}^d}^{3/2}}^{2/3} \Expt[\big]{\abs{D_z F_i}^3}^{1/3} {\,}\mu({\textup{d}}z),\end{aligned}$$ where we also used the contraction inequality $\Expt{\abs{D_z L^{-1} F_i}^p} \leq \Expt{\abs{D_z F_i}^p}$ from [@LastNorm Lemma 3.4], which is valid for all $p \geq 1$ and $z \in {\mathbb{R}}^d$. This completes the proof of \[thm:wasserstein-dist\].$\Box$
Proof of Theorem \[thm:MA-clt\]
===============================
In order to apply \[thm:wasserstein-dist\] we need to ensure that the processes $(X_t^i)$ can be represented in terms of a Poisson process. Indeed, following [@RosiRepr] and [@BassBerr] we can represent $X^i$ as the integral $$X_t^i = \int_{{\mathbb{R}}^2} g_i(t - s) x \bigl(\eta({\textup{d}}s, {\textup{d}}x) - \tau(g_i(t - s) x) {{\,}{\textup{d}}}s {\,}\nu({\textup{d}}x) \bigr)
+ \tilde{b}_i,$$ with $$\tilde{b}_i \coloneqq \int_{{\mathbb{R}}} \Bigl( g_i(s) b + \int_{{\mathbb{R}}} (\tau(x g_i(s)) - g_i(s) \tau(x)) {\,}\nu({\textup{d}}x) \Bigr) {{\,}{\textup{d}}}s,$$ and where $\eta$ is a Poisson process on ${\mathbb{R}}^2$ with intensity measure $\mu({\textup{d}}s, {\textup{d}}x) \coloneqq {\textup{d}}s {\,}\nu({\textup{d}}x)$. Here, $\nu$ is the Lévy measure of $L$, $b$ the shift parameter in the characteristic triple for $L_1$ and $\tau$ is a truncation function, cf. (8.3)–(8.4) in [@SatoLevy].
In what follows, $C$ will denote a strictly positive constant whose value might change from occasion to occasion.
Estimating the Malliavin derivative
-----------------------------------
We start out by deriving simple estimates on the Malliavin derivative. By definition of the terms $\gamma_1$, $\gamma_2$, $\gamma_3$ introduced in \[sec:Poincare\] it is sufficient to consider the Malliavin derivatives of each of the coordinates of $f = (f_1, \ldots, f_d)$ separately. So, let $i \in \Set{1, \ldots, d}$ and $z_j = (x_j, t_j) \in {\mathbb{R}}^2$ for $j \in \Set{1, 2}$ be given. Define for $z = (x, t) \in {\mathbb{R}}^2$ the vector $\delta_s(z)$, for $s \in {\mathbb{R}}$, as $$\label{eq:delta-def}
\delta_s(z) \coloneqq x(g_1(s - t), \ldots, g_m(s - t)) \in {\mathbb{R}}^m.$$ The mean value theorem together with the Cauchy-Schwarz inequality and the assumption that $f_i \in {\mathcal{C}}_b^2({\mathbb{R}}^m, {\mathbb{R}})$ then yield the existence of a constant $C > 0$ such that $$\label{eq:malliavin-ineq:1}
\begin{aligned}
\abs{D_{z_1} f_i(X_s^1, \ldots, X_s^m)}
&= \abs{f_i((X_s^1, \ldots, X_s^m) + \delta_s(z_1)) - f_i(X_s^1, \ldots, X_s^m)}
\\
&\leq C (1 \wedge \norm{\delta_s(z_1)}_{{\mathbb{R}}^m}).
\end{aligned}$$ Similarly, we deduce again by the mean value theorem and boundedness of $f_i$ and its derivatives the following inequality for the iterated Malliavin derivative: $$\label{eq:malliavin-ineq:2}
\begin{aligned}
\abs{D^2_{z_1, z_2} f_i(X_s^1, \ldots, X_s^m)}
&= \bigl\lvert f_i((X_s^1, \ldots, X_s^m) + \delta_s(z_1) + \delta_s(z_2))
\\
&\qquad - f_i((X_s^1, \ldots, X_s^m) + \delta_s(z_1))
\\
&\qquad - f_i((X_s^1, \ldots, X_s^m) + \delta_s(z_2))
+ f_i(X_s^1, \ldots, X_s^m) \bigr\rvert
\\
&\leq C (1 \wedge \norm{\delta_s(z_1)}_{{\mathbb{R}}^m}) (1 \wedge \norm{\delta_s(z_2)}_{{\mathbb{R}}^m}).
\end{aligned}$$ Note that the estimates \[eq:malliavin-ineq:1,eq:malliavin-ineq:2\] are purely deterministic and allow us to replace stochastic terms by deterministic estimates of the underlying kernels. This confirms in another context that many properties of moving averages can be deduced solely from the driving spectral density, see, for example, [@KonoSelf].
Analysing the asymptotic covariance matrix
------------------------------------------
Define for each $k \in {\mathbb{Z}}$ and $i, j \in \Set{1, \ldots, m}$ the integral $$\label{eq:rho-and-mu}
\rho_{i, j, k} \coloneqq \int_{{\mathbb{R}}} \abs{g_i(x) g_j(x + k)}^{\beta/2} {{\,}{\textup{d}}}x$$ and observe that $\rho_{i, j, k} = \rho_{j, i, -k}$. Now, $\rho_{i, j, k}$ is closely related to the asymptotic covariances, which motivates the following technical lemma, which in turn leads to our assumption that $\alpha_i \beta > 2$ for any $i \in \Set{1, \ldots, m}$. In what follows we write $x \vee y \coloneqq \max\Set{x, y}$ for the maximum of $x, y \in {\mathbb{R}}$.
\[lem:cov-ineq\] Let $k \in {\mathbb{N}}$ and $i, j \in \Set{1, \ldots, m}$. Then there is a constant $C>0$ such that $$\rho_{i, j, k} \leq C k^{-(\alpha_i \wedge \alpha_j) \beta / 2}.$$
The same technique as in the proof of [@BassBerr Lemma 4.1] yields the bound $\rho_{i, j, k} \leq C k^{-\alpha_j \beta /2}$ and to obtain a bound symmetric in $i$ and $j$ observe that obviously $$\rho_{i, j, k} \leq C (k^{-\alpha_i \beta / 2} \vee k^{-\alpha_j \beta / 2})
= C k^{- (\alpha_i \wedge \alpha_j) \beta / 2}.$$ This completes the argument.
\[prop:cov-conv\] The series defining $\Sigma_{i, j}^2$ in \[eq:asymp-cov\] is absolutely convergent and we have that $\Sigma_n^2 \to \Sigma^2$, as $n \to \infty$. In particular, $\Sigma_n \to \Sigma$.
First, we prove that the series in converges absolutely. By symmetry it is enough to show that $$\sum_{s = 1}^{\infty} \abs{\operatorname{Cov}(f_i(X_s^1, \ldots, X_s^m), f_j(X_0^1, \ldots, X_0^m))} < \infty
\quad \text{for all $i, j \in \{1, \ldots, d\}$.}$$ To this end, we let $\widetilde{\eta}$ be a Poisson process on $[0,1]\times{\mathbb{R}}^2$ with intensity measure ${\textup{d}}u {\,}\mu({\textup{d}}z)$ with the property that $\eta=\widetilde{\eta}([0,1]\times
{\,\cdot\,})$. Using now the covariance identity for Poisson functionals from [@LastPois Theorem 5.1] we conclude that $$\begin{aligned}
\MoveEqLeft\operatorname{Cov}(f_i(X_s^1, \ldots, X_s^m), f_j(X_0^1, \ldots, X_0^m))
\\
&= \Expt[\Big]{\int_{0}^1 \Bigl( \int_{{\mathbb{R}}} \Expt{D_z f_i(X_s^1, \ldots, X_s^m) {}{\mathcal{G}}_u} \Expt{D_z f_j(X_0^1, \ldots, X_0^m) {}{\mathcal{G}}_u} {\,}\mu({\textup{d}}z) \Bigr) {{\,}{\textup{d}}}u},
\end{aligned}$$ where ${\mathcal{G}}_u$ is the $\sigma$-algebra generated by the restriction of the Poisson process $\widetilde{\eta}$ to $[0,u]\times{\mathbb{R}}^2$. Applying the Cauchy-Schwarz inequality, our assumption on $\nu$ and \[eq:malliavin-ineq:1\] implies that $$\begin{aligned}
\MoveEqLeft\abs{\operatorname{Cov}(f_i(X_s^1, \ldots, X_s^m), f_j(X_0^1, \ldots, X_0^m))}
\\
&\leq \int_{0}^1 \Bigl( \int_{{\mathbb{R}}^2} \Expt[\big]{\abs{\Expt{D_z f_i(X_s^1, \ldots, X_s^m) {}{\mathcal{G}}_u} \Expt{D_z f_j(X_0^1, \ldots, X_0^m) {}{\mathcal{G}}_u} } } {\,}\mu({\textup{d}}z) \Bigr) {{\,}{\textup{d}}}u
\\
&\leq \int_{{\mathbb{R}}^2} \Expt[\big]{\abs{D_z f_i(X_s^1, \ldots, X_s^m)}^2 }^{1/2} \Expt[\big]{ \abs{D_z f_j(X_0^1, \ldots, X_0^m)}^2 }^{1/2} {\,}\mu({\textup{d}}z)
\\
&\leq C \int_{{\mathbb{R}}} \Bigl( \int_{{\mathbb{R}}} (1 \wedge \abs{x}^2 \norm{(g_{\ell}(s - t))_{\ell = 1}^{m}}_{{\mathbb{R}}^m}
\norm{(g_{\ell}(-t))_{\ell = 1}^{m}}_{{\mathbb{R}}^m}) \abs{x}^{-1 - \beta} {{\,}{\textup{d}}}x \Bigr) {{\,}{\textup{d}}}t
\\
&= C\int_{{\mathbb{R}}} \norm{(g_\ell(s - t))_{\ell = 1}^{m}}_{{\mathbb{R}}^m}^{\beta/2}
\norm{(g_\ell(- t))_{\ell = 1}^{m}}_{{\mathbb{R}}^m}^{\beta/2} {{\,}{\textup{d}}}t
\\
&\leq C \sum_{k, \ell = 1}^{m} \int_{{\mathbb{R}}} \abs{g_\ell(s - t) g_k( - t)}^{\beta/2} {{\,}{\textup{d}}}t
\\
&= C \sum_{k, \ell = 1}^{m} \rho_{k, \ell, s}
\\
& \leq C s^{-\underline{\alpha} \beta / 2},
\end{aligned}$$ where the last inequality follows from \[lem:cov-ineq\]. Since $\underline{\alpha} \beta > 2$ by assumption the series in \[eq:asymp-cov\] converges absolutely. To deduce the convergence $\Sigma_n^2 \to \Sigma^2$ we use the stationarity of the sequence $(X_t^1, \ldots, X_t^m)$, $t \in {\mathbb{R}}$, to see that, for any $i, j \in \Set{1, \ldots, d}$, $$\begin{aligned}
&\operatorname{Cov}(V_n^i(X; f), V_n^j(X; f))
\\
&= n^{-1} \sum_{s, t = 1}^{n} \operatorname{Cov}(f_i(X_s^1, \ldots, X_s^m), f_j(X_t^1, \ldots, X_t^m))
\\
&= n^{-1} \sum_{\substack{s, t = 1 \\ s \geq t}}^{n} \operatorname{Cov}(f_i(X_{s - t}^1, \ldots, X_{s - t}^m), f_j(X_0^1, \ldots, X_0^m))
\\
&\qquad + n^{-1} \sum_{\substack{s, t = 1 \\ s < t}}^{n} \operatorname{Cov}(f_i(X_0^1, \ldots, X_0^m), f_j(X_{t - s}^1, \ldots, X_{t - s}^m))
\\
&= \sum_{k = 0}^{n - 1} (1 - \tfrac{k}{n}) \operatorname{Cov}(f_i(X_k^1, \ldots, X_k^m), f_j(X_0^1, \ldots, X_0^m))
\\
&\qquad + \sum_{k = 1}^{n - 1} (1 - \tfrac{k}{n}) \operatorname{Cov}(f_i(X_0^1, \ldots, X_0^m), f_j(X_k^1, \ldots, X_k^m))
\longrightarrow \Sigma_{i, j}^2,
\end{aligned}$$ as $n \to \infty$, where the convergence follows by Lebesgue’s dominated convergence theorem together with the absolute convergence of the series defining the limit $\Sigma_{i, j}^2$. Finally, the last claim simply follows by continuity of the square root.
Bounding $d_3(V_n, Y)$
----------------------
Recall for $i,k\in\{1,\ldots,m\}$ the definition of the quantities $\gamma_{1}(F_i,F_k)$ and $\gamma_{2}(F_i,F_k)$ from \[sec:Poincare\], which are applied with $F_i = V^i_n(X; f)$ and $F_k = V^k_n(X;f)$. According to \[thm:wasserstein-dist\] we have that for any $n \in {\mathbb{N}}$, $$d_3(V_n(X; f), Y)
\leq \sum_{i, k = 1}^{d} (\gamma_{1}(F_i,F_k) + \gamma_{2}(F_i,F_k))
+ \gamma_3,$$ where $\gamma_3$ is defined at \[eq:gamma3\]. We consider each of these terms separately in the following three lemmas. Let us point to the fact that the sum will converge at a speed of order $1/\sqrt{n}$, whereas the $\gamma_3$-term will generally converge at a lower speed, depending on the parameters $\underline{\alpha}$ and $\beta$. It is also this last term that requires the stronger assumption \[eq:kernel-ass\] rather than just $\sum_{u = 0}^{\infty} \rho_{i, j, u} < \infty$ for all $i, j \in \Set{1, \ldots, m}$. Indeed, as a product, in $\gamma_3$ we carefully have to distinguish between small and large values, where the latter are non-negligible for heavy-tailed moving averages.
\[lem:gamma1\] There exists a constant $C > 0$ such that $\gamma_1(F_i,F_k) \leq C n^{-1/2}$ for any $i, k \in \Set{1, \ldots, m}$.
To simplify the notation put $V_n^i \coloneqq V_n^i(X; f)$ and recall that $$\begin{aligned}
\gamma_{1}^2(F_i,F_k)&= 3 \int_{({\mathbb{R}}^2)^3} \Expt[\big]{(D_{z_1, z_3}^2 V_n^i)^2 (D_{z_2, z_3}^2 V_n^i)^2}^{1/2}
\\
&\hspace{3cm}\times\Expt[\big]{(D_{z_1} V_n^k)^2 (D_{z_2} V_n^k)^2}^{1/2} {\,}\mu^3({\textup{d}}z_1, {\textup{d}}z_2, {\textup{d}}z_3).
\end{aligned}$$ If $z_i = (x_i, t_i) \in {\mathbb{R}}^2$ for $i \in \Set{1, 2, 3}$, the integrand can be bounded using \[eq:malliavin-ineq:1\] and \[eq:malliavin-ineq:2\] as follows: $$\begin{aligned}
\MoveEqLeft[0] \Expt[\big]{(D_{z_1, z_3}^2 V_n^i)^2 (D_{z_2, z_3}^2 V_n^i)^2}^{1/2} \Expt[\big]{(D_{z_1} V_n^k)^2 (D_{z_2} V_n^k)^2}^{1/2}
\\
&\leq \frac{C}{n^2} \Bigl(\sum_{s_1 = 1}^{n} (1 \wedge \norm{\delta_{s_1}(z_1)}_{{\mathbb{R}}^m}) (1 \wedge \norm{\delta_{s_1}(z_3)}_{{\mathbb{R}}^m}) \Bigr)
\\
&\hspace{3cm}\times \Bigl(\sum_{s_2 = 1}^{n} (1 \wedge \norm{\delta_{s_2}(z_2)}_{{\mathbb{R}}^m}) (1 \wedge \norm{\delta_{s_2}(z_3)}_{{\mathbb{R}}^m}) \Bigr)
\\
&\hspace{3cm} \times \Bigl(\sum_{s_3 = 1}^{n} (1 \wedge \norm{\delta_{s_3}(z_1)}_{{\mathbb{R}}^m}) \Bigr)
\Bigl(\sum_{s_4 = 1}^{n} (1 \wedge \norm{\delta_{s_4}(z_2)}_{{\mathbb{R}}^m}) \Bigr)
\\
&\leq \frac{C}{n^2} \sum_{s_1, \ldots, s_4 = 1}^{n} \bigl[ (1 \wedge \norm{\delta_{s_1}(z_1)}_{{\mathbb{R}}^m} \norm{\delta_{s_3}(z_1)}_{{\mathbb{R}}^m}) (1 \wedge \norm{\delta_{s_2}(z_2)}_{{\mathbb{R}}^m} \norm{\delta_{s_4}(z_2)}_{{\mathbb{R}}^m})
\\
&\hspace{3cm} \times (1 \wedge \norm{\delta_{s_1}(z_3)}_{{\mathbb{R}}^m} \norm{\delta_{s_2}(z_3)}_{{\mathbb{R}}^m}) \bigr]
\\
&\leq \frac{C}{n^2} \sum_{s_1, \ldots, s_4 = 1}^{n} \sum_{j_1, \ldots, j_6 = 1}^{m} (1 \wedge x_1^2 \abs{g_{j_1}(s_1 - t_1) g_{j_2}(s_3 - t_1)})
\\
&\hspace{3cm} \times (1 \wedge x_2^2 \abs{g_{j_3}(s_2 - t_2) g_{j_4}(s_4 - t_2)}) (1 \wedge x_3^2 \abs{g_{j_5}(s_1 - t_3) g_{j_6}(s_2 - t_3)}).
\end{aligned}$$ Using the substitution $u_i = x_i^2 y_i$ for $y_i > 0$, $i \in \Set{1, 2, 3}$, one easily verifies the relation $$\int_{{\mathbb{R}}^3} (1 \wedge x_1^2 y_1) (1 \wedge x_2^2 y_2) (1 \wedge x_3^2 y_3) \abs{x_1 x_2 x_3}^{-1 - \beta} {{\,}{\textup{d}}}x_1 {{\,}{\textup{d}}}x_2 {{\,}{\textup{d}}}x_3 = C y_1^{\beta / 2} y_2^{\beta / 2} y_3^{\beta / 2},$$ for $\beta \in (0, 2)$. This yields the bound $$\begin{aligned}
\gamma_{1}^2(F_i,F_k) &\leq \frac{C}{n^2} \sum_{j_1, \ldots, j_6 = 1}^{m} \sum_{s_1, \ldots, s_4 = 1}^{n} \Bigl( \int_{{\mathbb{R}}} \abs{g_{j_1}(s_1 - t_1) g_{j_2}(s_3 - t_1)}^{\beta / 2} {{\,}{\textup{d}}}t_1
\\
&\hspace{1cm} \times \int_{{\mathbb{R}}} \abs{g_{j_3}(s_2 - t_2) g_{j_4}(s_4 - t_2)}^{\beta / 2} {{\,}{\textup{d}}}t_2
\int_{{\mathbb{R}}} \abs{g_{j_5}(s_1 - t_3) g_{j_6}(s_2 - t_3)}^{\beta / 2} {{\,}{\textup{d}}}t_3 \Bigr)
\\
&= \frac{C}{n^2} \sum_{j_1, \ldots, j_6 = 1}^{m} \sum_{s_1, \ldots, s_4 = 1}^{n} \rho_{j_1, j_2, s_3 - s_1} \rho_{j_3, j_4, s_4 - s_2} \rho_{j_5, j_6, s_2 - s_1}
\\
&\leq \frac{C}{n} \sum_{j_1, \ldots, j_6 = 1}^{m} \Bigl(\sum_{u = -n}^{n} \rho_{j_1, j_2, u} \Bigr) \Bigl(\sum_{u = -n}^{n} \rho_{j_3, j_4, u} \Bigr) \Bigl(\sum_{u = -n}^{n} \rho_{j_5, j_6, u} \Bigr)
\leq \frac{C}{n},
\end{aligned}$$ where the penultimate inequality follows by substitution and the last inequality is due to \[lem:cov-ineq\], and where we used that $\sum_{u = 0}^{\infty} \rho_{j, \ell, u} < \infty$ for all $j, \ell \in \Set{1, \ldots, m}$.
\[lem:gamma2\] There exists a constant $C > 0$ such that $\gamma_2(F_i, F_k) \leq C n^{-1 / 2}$ for all $i, k \in \Set{1, \ldots, m}$ and $n \in {\mathbb{N}}$.
Using \[eq:malliavin-ineq:2\] we conclude that the integrand in the definition of $\gamma_2(F_i, F_k)$ is bounded as follows: $$\begin{aligned}
\MoveEqLeft \Expt[\big]{(D_{z_1, z_3}^2 V_n^i)(D_{z_2, z_3}^2 V_n^i)}^{1/2} \Expt[\big]{(D_{z_1, z_3}^2 V_n^k)(D_{z_2, z_3}^2 V_n^k)}^{1/2}
\\
&\leq \frac{C}{n^2} \sum_{s_1, \ldots, s_4 = 1}^{n} (1 \wedge \norm{\delta_{s_1}(z_1)}_{{\mathbb{R}}^m} \norm{\delta_{s_3}(z_1)}_{{\mathbb{R}}^m}) (1 \wedge \norm{\delta_{s_2}(z_2)}_{{\mathbb{R}}^m} \norm{\delta_{s_4}(z_2)}_{{\mathbb{R}}^m})
\\
&\hspace{3cm} \times (1 \wedge \norm{\delta_{s_1}(z_3)}_{{\mathbb{R}}^m} \norm{\delta_{s_2}(z_3)}_{{\mathbb{R}}^m} \norm{\delta_{s_3}(z_3)}_{{\mathbb{R}}^m} \norm{\delta_{s_4}(z_3)}_{{\mathbb{R}}^m})
\\
&\leq \frac{C}{n^2} \sum_{s_1, \ldots, s_4 = 1}^{n} \sum_{j_1, \ldots, j_8 = 1}^{m} (1 \wedge x_1^2 \abs{g_{j_1}(s_1 - t_1) g_{j_2}(s_3 - t_1)})
\\
&\hspace{3cm} \times (1 \wedge x_2^2 \abs{g_{j_3}(s_2 - t_2) g_{j_4}(s_4 - t_2)})
\\
&\hspace{3cm} \times (1 \wedge x_3^4 \abs{g_{j_5}(s_1 - t_3) g_{j_6}(s_2 - t_3) g_{j_7}(s_3 - t_3) g_{j_8}(s_4 - t_3)}).
\end{aligned}$$ Moreover, as in the proof of the previous lemma we have that $$\int_{{\mathbb{R}}^3} (1 \wedge x_1^2 y_1) (1 \wedge x_2^2 y_2) (1 \wedge x_3^4 y_3) \abs{x_1 x_2 x_3}^{-1 - \beta} {{\,}{\textup{d}}}x_1 {{\,}{\textup{d}}}x_2 {{\,}{\textup{d}}}x_3
= C y_1^{\beta / 2} y_2^{\beta / 2} y_3^{\beta / 4}$$ for $\beta \in (0, 2)$ and real numbers $y_1, y_2, y_3 > 0$. This implies that $$\begin{aligned}
\gamma_{2}^2(F_i,F_k) &\leq \frac{C}{n^2} \sum_{j_1, \ldots, j_8 = 1}^{m} \sum_{s_1, \ldots, s_4 = 1}^{n} \int_{{\mathbb{R}}} \abs{g_{j_1}(s_1 - t_1) g_{j_2}(s_3 - t_1)}^{\beta / 2} {{\,}{\textup{d}}}t_1
\\
&\hspace{1cm} \times \int_{{\mathbb{R}}} \abs{g_{j_3}(s_2 - t_2) g_{j_4}(s_4 - t_2)}^{\beta / 2} {{\,}{\textup{d}}}t_2
\\
&\hspace{1cm} \times \int_{{\mathbb{R}}} \abs{g_{j_5}(s_1 - t_3) g_{j_6}(s_2 - t_3) g_{j_7}(s_3 - t_3) g_{j_8}(s_4 - t_3)}^{\beta / 4} {{\,}{\textup{d}}}t_3
\\
&\leq \frac{C}{n^2} \sum_{j_1, \ldots, j_8 = 1}^{m} \sum_{s_1, \ldots, s_4 = 1}^{n} \rho_{j_1, j_2, s_3 - s_1} \rho_{j_3, j_4, s_4 - s_2} (\rho_{j_5, j_6, s_2 - s_1} + \rho_{j_7, j_8, s_4 - s_3})
\\
&\leq \frac{C}{n},
\end{aligned}$$ where the last inequality follows as in \[lem:gamma1\] and the penultimate inequality follows immediately from the fact that $\abs{x y} \leq x^2 + y^2$ for all $x,y\in{\mathbb{R}}$.
Finally, we consider the crucial term $\gamma_3$.
\[lem:gamma3\] There exists a constant $C > 0$ such that, for all $n \in {\mathbb{N}}$, $$\gamma_3 \leq C
\begin{cases}
n^{-1/2}, & \text{if $\underline{\alpha} \beta > 3$,}
\\
n^{-1/2} \log(n), & \text{if $\underline{\alpha} \beta = 3$,}
\\
n^{(2 - \underline{\alpha} \beta) / 2}, & \text{if
$2 < \underline{\alpha} \beta < 3$.}
\end{cases}$$
Recall the definition of $\delta_s(z) = (\delta_s^1(z), \ldots, \delta_s^m(z))$ from \[eq:delta-def\] and define for $i\in\{1,\ldots,m\}$, $$A_n^i(z) \coloneqq \frac{1}{\sqrt{n}} \sum_{s = 1}^{n} (1 \wedge \abs{\delta_s^i(z)}).$$ By \[eq:malliavin-ineq:1\] and the sub-additivity of the minimum it follows that $$\begin{aligned}
\gamma_3 &\leq C \int_{{\mathbb{R}}^2} \Bigl(n^{-1 / 2} \sum_{s = 1}^{n} 1 \wedge \norm{\delta_s(z)}_{{\mathbb{R}}^m} \Bigr)^2 \wedge
\Bigl(n^{-1 / 2} \sum_{s = 1}^{n} 1 \wedge \norm{\delta_s(z)}_{{\mathbb{R}}^m} \Bigr)^3 {\,}\mu({\textup{d}}z)
\\
&\leq \sum_{i, j = 1}^{m} \int_{{\mathbb{R}}^2} (A_n^i(z)^2 \wedge A_n^j(z)^3 ){\,}\mu({\textup{d}}z).
\end{aligned}$$ From this point on, we can literally follow the proof of Lemma 4.6 in [@BassBerr]. In fact, this shows that for any $p \in [0, 2]$, $q > 2$ and $i, j \in \Set{1, \ldots, m}$, one has that $$\int_{{\mathbb{R}}^2} (A_n^i(z)^p \wedge A_n^j(z)^q) {\,}\mu({\textup{d}}z) \leq C
\begin{cases}
n^{1 - q/2}, & \text{if $\underline{\alpha} \beta > q$,}
\\
n^{1 - q/2} \log(n), & \text{if $\underline{\alpha} \beta = q$,}
\\
n^{(2 - \underline{\alpha} \beta)/2}, & \text{if $2 < \underline{\alpha} \beta < 3$.}
\end{cases}$$ Indeed, these bounds rely solely on the tail behaviour (in terms of the $\alpha_i$’s) of the kernels $g_i$, where $\underline{\alpha}$ reflects the *weakest* behaviour, and the power behaviour (in terms of the $\kappa_i$’s) around $0$, all of which satisfies the condition $\kappa_i > - 1/\beta$. This completes the argument.
According to \[thm:wasserstein-dist\] we have that for any $n\in{\mathbb{N}}$, $$d_3(V_n(X; f), Y)
\leq \sum_{i, k = 1}^{d} (\gamma_{1}(F_i,F_k) + \gamma_{2}(F_i,F_k))
+ \gamma_3.$$ Using now Lemmas \[lem:gamma1\], \[lem:gamma2\] and \[lem:gamma3\] we see that $$\begin{aligned}
d_3(V_n(X; f), Y)
&\leq C(n^{-1/2} + n^{-1/2}) + C
\begin{cases}
n^{-1/2}, & \text{if $\underline{\alpha} \beta > 3$,}
\\
n^{-1/2} \log(n), & \text{if $\underline{\alpha} \beta = 3$,}
\\
n^{(2 - \underline{\alpha} \beta) / 2}, & \text{if
$2 < \underline{\alpha} \beta < 3$,}
\end{cases}
\\
&\leq C
\begin{cases}
n^{-1/2}, & \text{if $\underline{\alpha} \beta > 3$,}
\\
n^{-1/2} \log(n), & \text{if $\underline{\alpha} \beta = 3$,}
\\
n^{(2 - \underline{\alpha} \beta) / 2}, & \text{if
$2 < \underline{\alpha} \beta < 3$.}
\end{cases}\end{aligned}$$ This completes the argument.
Acknowledgement {#acknowledgement .unnumbered}
===============
We would like to thank Mark Podolskij for initiating this collaboration. MML acknowledges financial support from the project financed by the Villum Fonden.
[^1]: Department of Mathematical Sciences, University of Liverpool, E-mail: [email protected]
[^2]: Department of Mathematics, Aarhus University, E-mail: [email protected].
[^3]: Faculty of Mathematics, Ruhr University Bochum, E-mail: [email protected].
|
{
"pile_set_name": "ArXiv"
}
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---
abstract: 'We obtain the constraints on the diffuse neutrino background from primordial black hole evaporations using the recent data of KamLAND experiment setting an upper limit on the antineutrino flux from the Sun.'
address: ' Institute for Nuclear Research, Russian Academy of Sciences, Moscow 117312, Russia '
author:
- 'E. V. Bugaev and K. V. Konishchev'
date:
title: Search for solar antineutrinos and constraints on the neutrino background from PBHs
---
Initial PBH mass spectrum
=========================
According to Press-Schechter model, the fraction of mass in the universe contained in gravitationally bounded object more massive than $M$ is [@PS] $$\label{PS}
\frac{1}{\rho_i}\int\limits_M^{\infty} M\frac{dn}{dM}dM = P (\delta_R>\delta_c, M, t_i).$$ Here, $\frac{dn }{dM} dM$ is the number density of objects of mass in $(M,M+dM)$, and the integral on the left-hand side gives the total mass density in overdense regions more massive than $M$. $P(\delta_R, M, t_i)$ is the probability that a randomly chosen point in the universe has density contrast $\delta_R> \delta_c$ at time $t_i$, and $M$ is the smoothing scale of $\delta_R$ ($R$ is the radius of the spherical overdense region). The value $\delta_c$ is some critical value of the density contrast which corresponds to the beginning of the non-linear regime. In a case of a gaussian field of density fluctuations this probability is $$\label{__2}
P=\frac{1}{2}erfc\left(\frac{\delta_c}{\sigma_R\sqrt{2}}\right),$$ where $\sigma_R^2=\sigma_R^2(M,t)=<\delta_R^2>$ is the variance of the smoothed density. The mass distribution of gravitationally bounded objects is [@PS] $$\label{__3}
\frac{dn }{dM}=\frac{2\rho_i}{M}\frac{\partial P}{\partial M}.$$ The variance $\sigma_R^2$ is given by the relation [@ARLDHL] $$\label{__4}
\sigma_R^2(M,t)=\int\limits_0^{\infty}\frac{k^4}{(aH)^4}\delta_H^2(k,t)W^2(kR)T^2(k)\frac{dk}{k},$$ where $W(kR)$ is the smoothing window function, $\delta_H (k,t)$ is the horizon crossing amplitude, $T(k)$ is the transfer function.
For the calculation of the mass spectrum of PBHs one must use the connection between the PBH mass $M_{BH}$ and characteristics of the overdense region. In the model of the near-critical collapse proposed in ref.[@16] one has the relation: $$\label{__5}
M_{BH}= M_{i}^{1/3}M^{2/3}k(\delta_H^R - \delta_c)^{\gamma_k},$$ where $M_i$ is the horizon mass at time $t_i$ when the density fluctuations develop, and $\delta_H^R$ is the smoothed density contrast at the moment when the fluctuation crosses horizon, $$\label{__6}
\delta_H^R=(M/M_i)^{2/3}\delta_R.$$
In the approach of ref.[@PH] all PBHs have masses roughly equal to the horizon mass at a moment of the formation, independently of the perturbation size. The approximate connection between the PBH mass and the horizon mass in such a model is very simple: $$\label{__7}
M_{BH}=\gamma^{1/2}M_i^{1/3}M^{2/3}.$$ where $\gamma$ is the ratio of the pressure to energy density ($\gamma=1/3$ in the radiation dominated era).
Using these formulas one obtains the final expression for the PBH mass spectrum [@1]
$$\begin{aligned}
\label{1_20}
n_{BH}(M_{BH})=\sqrt{\frac{2}{\pi}} {\rho_i} \int\frac{1}{M\sigma_H (M_h)}
\left|\left\{
\frac{2}{3M}-
\left[
\frac{n'}{2}ln \frac{k_{fl}}{k_0}+
\frac{n(k_{fl})-1}{2}\frac{1}{k_{fl}}
\right]
\cdot\frac{\partial k_{fl}}{\partial M}
\right\}\right. \times\nonumber\\
\\
\left.\left(\frac{(\delta_H^R)^2}{\sigma_H^2 (M_h)}-1\right)\right|
e^{-\frac{(\delta_H^R)^2}{2 \sigma_H^2 (M_h) }}\frac{d\delta_H^R}{d f(M,\delta_H^R)/dM}\;.
\nonumber\end{aligned}$$
Here we use the general notation for the connection between $M_{BH}$, $M$ and $\delta_H^R$, $$\label{__8}
M_{BH}=f(M,\delta_H^R).$$ Particular cases of this connection are given by Eqs.(\[\_\_5\],\[\_\_7\]).
Substituting the expression for the window function, $$\label{__9}
W(kR)= 3\left(\frac{\sin{kR}}{(kR)^3} -\frac{\cos{kR}}{(kR)^2} \right),$$ and for the transfer function [@_1], $$\label{__10}
T(k)\approx W(\frac{kR}{\sqrt{3}}),$$ and the parametrization $$\label{__12}
\delta_H(k)=\delta_H(k_0)\left(\frac{k}{k_0}\right)^{\frac{n(k)-1}{2}}$$ for the horizon crossing amplitude into Eq.(\[\_\_4\]) for the variance $\sigma_R^2$ one obtains, approximately, $$\label{__13}
%\sigma_R^2(M)\approx\frac{k_{fl}^4}{(aH)^4}C^2 \delta_H^2(k_{fl})\equiv\frac{k_{fl}^4}{(aH)^4}
%\sigma_H^2(M_h).
\sigma_R^2(M)=\left(\frac{M}{M_i} \right)^{-4/3}\sigma_H^2(M)\approx
\frac{k_{fl}^4}{(aH)^4}C^2 \delta_H^2(k_{fl}).$$ The exact value of the coefficient $C^2$ depends on the spectral index (for $n=1.3$ $C^2=6.82$). In Eq.(\[\_\_13\]) $k_{fl}$ is the comoving wave number, characterizing the perturbed region, $k_{fl}=\frac{1}{R}$, and $M_h$ is the fluctuation mass at the moment when the perturbed region enters the horizon at the radiation dominated era. The value of $M_h$ is simply connected with $k_{fl}$: $$\label{__15}
k_{fl}=a_{eq}H_{eq}\left(\frac{M_{eq}}{M_h}\right)^{1/2}.$$ At the same time, an initial value of the fluctuation mass, $M$, is proportional to $R^3$, so $M_h\sim M^{2/3}$. The exact connection is $$\label{__16}
M_h=M_i^{1/3}M^{2/3}.$$ The denominator $(aH)^4$ in Eq.(\[\_\_13\]) must be taken at $t=t_i$, and one can easily see that $$\label{__17}
\frac{k_{fl}^2}{(a_iH_i)^2}=\left(\frac{M}{M_i}\right)^{-2/3}.$$ At last, in the spectrum formula (\[1\_20\]) the following notation is used $$\label{__18}
n'=\left.\frac{d n(k)}{dk}\right|_{k=k_{fl}}.$$ The limits of integration in Eq.(\[1\_20\]) are determined using the conditions $$\label{1_19}
\delta^R_H{}_{min}=\delta_c\;\;\;\;\;\;,\;\;\;\;\; M_{min}=M_{h}^{min}=M_i\;\;\;.$$
We have, for calculations of $n_{BH}(M_{BH})$ two free parameters: the spectral index $n$, giving the perturbation amplitude through the normalization on COBE data on large scales, and $t_i$, the moment of time just after reheating, from which the process of PBH formation started. The value of $t_i$ is connected, in our approach, with a value of the reheating temperature, $$\label{7_}
t_i=0.301 g_*^{-1/2}\frac{M_{pl}}{T_{RH}^2}$$ ($g_*\sim 100$ is a number of degrees of freedom in the early Universe). The initial horizon mass $M_i$ is expressed through $t_i$: $$\label{7a_}
M_i\cong \frac{4}{3}\pi t_i^3\rho_i.$$
In the case of Page-Hawking [@PH] collapse the minimum mass in the PBH mass spectrum is given by the simple formula $$\label{__20}
M_{BH}^{min}=\gamma^{1/2}M_i$$ which follows from Eq.(\[\_\_7\]). Correspondingly, the minimum value of PBH mass is determined entirely by the reheating temperature. In the case of the near-critical collapse there is no minimum value of the PBH mass in the spectrum because $\delta^R_H$ can be almost equal to $\delta_c$ (see Eq.(\[\_\_5\])). In this case the PBH mass spectrum has a broad maximum at $M_{BH}\sim M_i$.
Neutrino diffuse background from PBHs
=====================================
Evolution of a PBH mass spectrum due to the evaporation leads to the approximate expression for this spectrum at any moment of time: $$\label{8}
n_{BH}(m,t)=\frac{m^2}{(3\alpha t + m^3)^{2/3}} n_{BH} \left((3\alpha t +
m^3)^{1/3}\right),$$ where $\alpha$ accounts for the degrees of freedom of evaporated particles and, strictly speaking, is a function of a running value of the PBH mass $m$. In all our numerical calculations we use the approximation $$\label{9}
\alpha=const=\alpha (M_{BH}^{max}),$$ where $M_{BH}^{max}$ is the value of $M_{BH}$ in the initial mass spectrum corresponding to a maximum of this spectrum or, in the case of Page-Hawking collapse, to a minimum value of PBH mass in the spectrum. Special study shows that errors connected with such an approximation are rather small. The detailed calculations of the function $\alpha(m)$ were carried out in the works [@22]. Here we use the simplified approach in which $\alpha(m)$ is represented by a series of step functions [@11]
The expression for a spectrum of the background radiation is [@11] $$\begin{aligned}
\label{10}
S(E)=\frac{c}{4\pi}\int dt \frac{a_0}{a}\left(\frac{a_i}{a_0}\right)^{3}
\int dm \frac{m^2}{(3\alpha t + m^3)^{2/3}} n_{BH} \left[(3\alpha t +
m^3)^{1/3}\right]
\cdot f(E\cdot (1+z),m)e^{-\tau(E,z)}\nonumber\\
\\
\equiv \int F(E,z)d \log_{10} (z+1).\nonumber\end{aligned}$$
In this formula $a_i$, $a$ and $a_0$ are cosmic scale factors at $t_i$, $t$ and at present time, respectively, and $f(E,m)$ is a total instantaneous spectrum of the background radiation (neutrinos or photons) from the black hole evaporation. It includes the pure Hawking term and contributions from fragmentations of evaporated quarks and from decays of pions and muons (see [@11] and earlier papers [@22; @23; @24] for details).
In the last line of Eq.(\[10\]) we changed the variable $t$ on $z$ using the flat model with $\Omega_{\Lambda}=0$ for which $$\begin{aligned}
\label{11}
\frac{dt}{dz}=-\frac{1}{H_0 (1+z)}\left(\Omega_m (z+1)^{3}+\Omega_r
(z+1)^{4}\right)
^{-1/2},\nonumber\\
\\
\Omega_r = (2.4\cdot 10^{4} h^2)^{-1} \;\;\;,\;\;\; h=0.67.\nonumber\end{aligned}$$
Results {#sec:Dis_and_Con}
=======
In some theoretical schemes (e.g., in the model of spin-flavor precession in a magnetic field) the Sun can emit a large flux of antineutrinos. The LSD experiment [@26] sets the upper limit on this flux, $\Phi_{\tilde \nu}/\Phi_{\nu}\le 1.7$ %. The corresponding constraints on diffuse neutrino background from primordial black holes were obtained in the work of authors [@11].
Recently, the KamLAND experiment obtained a new limit on electron antineutrino flux from the Sun, according to which this flux is less than $2.8\times 10^{-4}$ of the SSM ${}^8B$ $\nu_e$ flux. The neutrino detection was carried out in both experiments using the reaction $$\label{27.1}
\tilde \nu_e + p \to n + e^{+}.$$ No candidates were found in [@3] for an expected background of $1.1\pm 0.4$ events. We calculate the limit on the antineutrino flux using the formula $$\label{27.2}
\Phi_{\tilde \nu_e}=\frac{1.1}{\bar\sigma\cdot\Delta E\cdot T\cdot N_p\cdot 4\pi},$$ where $\sigma$ is the cross-section of the reaction (\[27.1\]) averaged over neutrino background spectrum from PBHs, $\Delta E$ is the neutrino energy range measured in the experiment ($8.3-14.8$ MeV), $T=1.6\cdot 10^{6}$ s is the livetime, and $N_p=4.6\cdot 10^{31} $ is the number of target protons in the fiducial volume.
Using the results of calculations of electron neutrino background spectra from PBHs (the examples of such spectra are given on Fig.1 for the case of Page-Hawking collapse) and normalization of the perturbation amplitude $\delta_H (k)$ on COBE data (for details see [@11]), we obtain the constraints on the spectral index $n$ shown on Fig.2.
K. Eguchi [*et al*]{} (KamLAND Collaboration) [*hep-ex/0310047*]{}. W. H. Press and P. Schechter, Astrophys. J. [**187**]{}, 425 (1974). A. R. Liddle and D. H. Lyth, Phys. Rept. [**231**]{} 1-105 (1993). D. Page and S. Hawking, Astrophys. J. [**206**]{}, 1 (1976). D. Blais [*et al*]{}, Phys. Rev. D [**67**]{}, 024024 (2003). E. V. Bugaev, K. V. Konishchev, Phys. Rev. D [**66**]{} 084004 (2002). E. V. Bugaev, K. V. Konishchev [*astro-ph/0202519*]{}. E. V. Bugaev and K. V. Konishchev, Phys. Rev. D [**65**]{}, 123005 (2002). H. I. Kim and C. H. Lee, Phys. Rev. D [**54**]{}, 6001 (1996). : 10 B. J. Carr and S. W. Hawking, Mon. Not. R. Astron. Soc. [**168**]{}, 399 (1974). J. C. Niemeyer and K. Jedamzik, Phys. Rev. Lett. [**80**]{}, 5481 (1998). J. C. Niemeyer and K. Jedamzik, Phys. Rev. D [**59**]{}, 124013 (1999). E. V. Bugaev and K. V. Konishchev, [*astro-ph/*]{}0005295. J. H. MacGibbon and B. R. Webber, Phys. Rev. D [**41**]{}, 3052 (1990); J. H. MacGibbon, Phys. Rev. D [**44**]{}, 376 (1991). F. Halzen, B. Keszthelyi, and E. Zas, Phys. Rev. D [**52**]{}, 3239 (1995). H. I. Kim, C. H. Lee, and J. H. MacGibbon, Phys. Rev D [**59**]{}, 063004 (1999). E. V. Bugaev and V. A. Naumov, Phys. Lett. B [**232**]{}, 391 (1989). M. Aglietta [*et al*]{}., Pis’ma Zh. Eksp. Teor. Fiz. [**63**]{}, 753 (1996).
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{
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---
abstract: 'We explore the escape dynamics in open Hamiltonian systems with multiple channels of escape continuing the work initiated in Part I. A thorough numerical investigation is conducted distinguishing between trapped (ordered and chaotic) and escaping orbits. The determination of the location of the basins of escape towards the different escape channels and their correlations with the corresponding escape periods of the orbits is undoubtedly an issue of paramount importance. We consider four different cases depending on the perturbation function which controls the number of escape channels on the configuration space. In every case, we computed extensive samples of orbits in both the configuration and the phase space by numerically integrating the equations of motion as well as the variational equations. It was found that in all examined cases regions of non-escaping motion coexist with several basins of escape. The larger escape periods have been measured for orbits with initial conditions in the vicinity of the fractal structure, while the lowest escape rates belong to orbits with initial conditions inside the basins of escape. In addition, we related the model potential with applications in the field of reactive multichannel scattering. We hope that our numerical analysis will be useful for a further understanding of the escape mechanism of orbits in open Hamiltonian systems with two degrees of freedom.'
author:
- 'Euaggelos E. Zotos'
date: 'Received: 9 January 2015 / Accepted: 12 May 2015 / Published online: 29 May 2015'
title: 'Escapes in Hamiltonian systems with multiple exit channels – Part II.'
---
=1
Introduction {#intro}
============
Over the last decades a huge amount of research work has been devoted on the subject of escaping particles from open dynamical systems. Especially the issue of escape in Hamiltonian systems is a classical problem in nonlinear dynamics (e.g., \[[@C90], [@CE04] – [@CHLG12], [@STN02]\]). It is well known that several types of Hamiltonian systems have a finite energy of escape. For values of energy lower than the escape energy the equipotential surfaces of the systems are close which means that orbits are bound and therefore escape is impossible. For energy levels above the escape energy on the other hand, the equipotential surfaces open and exit channels emerge through which the particles can escape to infinity. The literature is replete with studies of such “open" Hamiltonian systems (e.g., \[[@BBS09], [@KSCD99], [@NH01], [@STN02], [@SCK95] – [@SKCD96], [@Z13] – [@Z14b]\]). At this point we should emphasize that all the above-mentioned references on escapes in Hamiltonian system are exemplary rather than exhaustive, taking into account that a vast quantity of related literature exists.
Nevertheless, the issue of escaping orbits in Hamiltonian systems is by far less explored than the closely related problem of chaotic scattering. In this situation, a test particle coming from infinity approaches and then scatters off a complex potential. This phenomenon is well investigated as well interpreted from the viewpoint of chaos theory (e.g., \[[@BTS96] – [@BM92], [@CDO90], [@CPR75], [@DGO90] – [@EJ86], [@GR89], [@H88], [@JRS92] – [@JT91], [@LMG00] – [@ML02], [@OT93], [@PH86], [@RJ94], [@SASL06] – [@SS10]\]).
During the last half century, dynamical systems made up of perturbed harmonic oscillators have been extensively used in order to describe local motion (i.e., near an equilibrium point) (e.g., \[[@AEFR06], [@CZ12], [@FLP98a] – [@HH64], [@SI79], [@Z12b] – [@Z14b]\]). In an attempt to reveal and understand the nature of orbits in these systems, scientists have used either numerical (e.g., \[[@CZ12], [@KV08], [@ZC12]\]) or analytical methods (e.g., \[[@D91], [@DE91], [@E00], [@ED99]\]). Furthermore, potentials made up of harmonic oscillators are frequently used in galactic Astronomy, as a first step for distinguishing between ordered and chaotic local motion in galaxies, since it is widely accepted that the motion of stars near the central region of a galaxy can be approximated by harmonic oscillations (e.g., \[[@Z12a]\]). One of the most characteristic Hamiltonian systems of two degrees of freedom with three escape channels is undoubtedly the well-known Hénon-Heiles system \[[@HH64]\]. A huge load of research on the escape properties of this system has been conducted over the years (e.g., \[[@AVS01] – [@AVS03], [@BBS08], [@BBS09], [@FLP98a], [@FLP98b]\]).
In open Hamiltonian systems an issue of paramount importance is the determination of the basins of escape, similar to basins of attraction in dissipative systems or even the Newton-Raphson fractal structures. An escape basin is defined as a local set of initial conditions of orbits for which the test particles escape through a certain exit in the equipotential surface for energies above the escape value. Basins of escape have been studied in many earlier papers (e.g., \[[@BGOB88], [@C02], [@KY91], [@PCOG96], [@SO00]\]). The reader can find more details regarding basins of escape in \[[@C02]\], while the review \[[@Z14a]\] provides information about the escape properties of orbits in a multi-channel dynamical system of a two-dimensional perturbed harmonic oscillator. The boundaries of an escape basins may be fractal (e.g., \[[@AVS09], [@BGOB88], [@dML99]\]) or even respect the more restrictive Wada property (e.g., \[[@AVS01]\]), in the case where three or more escape channels coexist in the equipotential surface.
The layout of the present paper is as follows: a detailed presentation of the properties of the Hamiltonian system is given in Section \[modpot\]. In Section \[rescat\] we relate our model potential with applications in the field of reactive multichannel scattering. All the computational techniques used in order to determine the character (ordered vs. chaotic and trapped vs. escaping) of orbits are described in Section \[cometh\]. In the following Section \[numres\] a thorough numerical analysis of several cases regarding the total number of escape channels is conducted. Our paper ends with Section \[disc\], where the discussion and the main conclusions of this work are presented. The text structure of the paper as well as all the numerical methods are the same as in Part I.
Presentation of the Hamiltonian system {#modpot}
======================================
The potential of a two-dimensional perturbed harmonic oscillator is $$V(x,y) = \frac{1}{2}\left(\omega_1^2 x^2 + \omega_2^2 y^2 \right) + \varepsilon V_1(x,y),
\label{genform}$$ where $\omega_1$ and $\omega_2$ are the unperturbed frequencies of oscillations along the $x$ and $y$ axes respectively, $\varepsilon$ is the perturbation parameter, while $V_1$ is the function containing the perturbing terms.
As in Part I, we shall use a two-dimensional perturbed harmonic oscillator at the 1:1 resonance, that is when $\omega_1 = \omega_2 = \omega$. Therefore the corresponding potential is $$V(x,y) = \frac{\omega^2}{2}\left(x^2 + y^2 \right) + \varepsilon V_1(x,y),
\label{pot}$$ being $\omega$ the common frequency of oscillations along the two axes. Without the loss of generality, we may set $\omega = 1$ and $\varepsilon = 1$ for more convenient numerical computations.
The basic equations of motion for a test particle with a unit mass $(m = 1)$ are $$\ddot{x} = - \frac{\partial V}{\partial x}, \ \ \
\ddot{y} = - \frac{\partial V}{\partial y},
\label{eqmot}$$ where, as usual, the dot indicates derivative with respect to the time. Furthermore, the variational equations governing the evolution of a deviation vector $\vec{w} = (\delta x, \delta y, \delta \dot{x}, \delta \dot{y})$, which joins the corresponding phase space points of two initially nearby orbits, needed for the calculation of standard chaos indicators (the SALI in our case) are given by $$\begin{aligned}
\dot{(\delta x)} &=& \delta \dot{x}, \ \ \ \dot{(\delta y)} = \delta \dot{y}, \nonumber \\
(\dot{\delta \dot{x}}) &=& -\frac{\partial^2 V}{\partial x^2}\delta x - \frac{\partial^2 V}{\partial x \partial y}\delta y, \nonumber \\
(\dot{\delta \dot{y}}) &=& -\frac{\partial^2 V}{\partial y \partial x}\delta x - \frac{\partial^2 V}{\partial y^2}\delta y.
\label{variac}\end{aligned}$$
Consequently, the Hamiltonian to potential (\[pot\]) (with $\omega = \varepsilon = 1$) reads $$H = \frac{1}{2}\left(\dot{x}^2 + \dot{y}^2 + x^2 + y^2\right) + V_1(x,y) = h,
\label{ham}$$ where $\dot{x}$ and $\dot{y}$ are the momenta per unit mass conjugate to $x$ and $y$ respectively, while $h > 0$ is the numerical value of the Hamiltonian, which is conserved. Thus, an orbit with a given value for it’s energy integral is restricted in its motion to regions in which $h \leq V(x,y)$, while all other regions are forbidden to the test particle. The Hamiltonian can also be written in the form $$H = H_0 + H_1,
\label{ham2}$$ with $H_0$ being the integrable term and $H_1$ the non-integrable correction.
The function with the perturbation term $V_1(x,y)$ plays a key role as it determines the location as well as the total number of the escape channels in the configuration $(x,y)$ space. In Part I, we considered perturbing terms that create between two and four escape channels, while now we will investigate the escape dynamics of orbits in the cases where five, six, seven and eight exits are present in the configuration space. At this point we should emphasize that this is the first time that the escape properties of test particles in Hamiltonian systems with more than four escape channels are systematically explored. In order to obtain the appropriate perturbing terms for the required number of escape channels we need a generating function. In polar $(r,\theta)$ coordinates we can easily define functions of the form $r^n \ \sin(n \ \theta)$, where $n$ is the desired number of exits (see e.g., \[[@HM09]\]). Then we can convert them to rectangular cartesian $(x,y)$ coordinated by following a simple three steps procedure: (i) first split up sums and integer multiples that appear in arguments of trigonometric functions, (ii) expand out products of trigonometric functions into sums of powers, using trigonometric identities when possible and (iii) replace everywhere with $\cos(\theta) = y/r$, $\sin(\theta) = x/r$ and $r = \sqrt{x^2 + y^2}$. In our study we want to work on the $(x,\dot{x})$ phase plane and this type of plane is constructible only if the $V_1(x,y)$ function has terms with even powers regarding the $y$ coordinate. The above-mentioned generating function however, gives terms with even powers of $y$ only for odd values of $n$. Therefore, we need two types of generating functions $$V_1(r,\theta) = \left\{
\begin{array}{rl}
r^n \ \sin(n \ \theta), &\mbox{ when $n$ is odd and $n \geq 3,$} \\
r^n \ \cos(n \ \theta), &\mbox{ when $n$ is even and $n \geq 4,$}
\end{array} \right.
\label{gens}$$ regarding how many channels we want the configuration $(x,y)$ space to have. In Appendix A we provide a list of the perturbation functions for the first nine cases, that is for $n \in [2, 10]$[^1].
Applications to reactive scattering {#rescat}
===================================
Dynamical models with many exits of the form (\[pot\]) have the nice interpretation as scattering models for rearrangement scattering. We may interpret each exit as a different arrangement in nuclear scattering or molecular scattering. Each channel means a different grouping of particles or atoms into the various fragments. By trajectories entering through one channel and leaving through another channel one can describes nuclear reactions or chemical reactions. As J.R. Taylor writes on page 318 in his well known book \[[@T76]\], we can imagine multichannel scattering as an irrigation system where water comes in through one channel and goes out through various other channels.
Let us try to explain the basic idea of potential models for rearrangement scattering for the simplest possible case. It is the case of collinear scattering of three particles (e.g., \[[@BS90], [@E88], [@J91]\]). So we have a one dimensional position space and the particles called $A$, $B$, $C$ moving in this position space with coordinates $q_A$, $q_B$, $q_C$. Now we change to relative coordinates $x = q_A - q_B$ and $y = q_C - q_B$. Then the configuration $(x,y)$ plane is the relevant configuration space for all the reactions. The motion of the center of mass of the whole system is irrelevant. Now we imagine a potential with a deep well around the origin and channels along the lines $x = 0$, $y = 0$ and $x - y = 0$. The potential goes to zero rapidly for all other directions. Far away from the origin the channels are straight and of constant depth. First let us assume a negative total energy $E$. Then the trajectory always stays in the central well and the channels. The motion in the central well describes motion where all 3 particles are close and interact. The motion in the channels describes motion of the various asymptotic arrangements. For example think of the channel along $x = 0$ which we call the arrangement channel $C$. In this channel the particles $A$ and $B$ are close enough and interact, while the particle $C$ is far away and moves freely. The general trajectory in channel $C$ moves along the channel with constant velocity in longitudinal direction and at the same time oscillates in transverse direction in the channel potential (i.e., the relative coordinate $q_A - q_B$ oscillates).
We have some kinetic energy $E_k$ for the longitudinal motion and some energy $E_v$ of transversal motion relative to the minimum of the potential channel. The interpretation is like this: A bound state (molecule) of particles (atoms) $A$ and $B$ moves far away from the free particle (atom) $C$. And the molecule $AB$ is in an vibrational state with energy $E_v$. $E_k$ on the other hand, is the kinetic energy of the motion of atom $C$ relative to the bound fragment $AB$. Similar for the other channels. We call the channel along the line $y = 0$ the arrangement channel $A$ and the potential channel along the line $x = y$ the arrangement channel $B$. Now let us assume we find a trajectory which comes in along channel $C$ with transverse energy $E_{v1}$ enters the central potential well, performs complicated motion in the central well for a finite time and then leaves along channel $B$ with transverse energy $E_{v2}$. In position space this event looks like this: Atom $C$ moves in direction of the molecule $AB$ which is in the vibrational state $E_{v1}$. Then they collide, the atoms perform complicated motion for a while and finally the atom $B$ flies away and leaves behind the molecule $AC$ in vibrational state $E_{v2}$. This is the microscopic description of a chemical reaction $AB + C \rightarrow B + AC$.
For total energy larger than 0 there is the additional possibility that all atoms fly away separately. This is the breakup channel 0. The trajectory in the configuration space then leaves the channels and goes away in a direction where the potential is 0. For more particles and for particles in a higher dimensional position space the basic idea remains the same. Take as configuration space a high dimensional space of appropriate relative coordinates and construct a potential with some deep potential well in the middle and some kind of tubes or layers going out into various directions along which certain relative coordinates between particles remain small. Of course, in a 3 dimensional (3D) position space we also include the various rotational states of the bound molecules.
In the potential (\[pot\]) we could imagine that far away from the origin the total potential converges to some form having the correct asymptotic properties of scattering theory (i.e., the potential goes to zero in most directions and otherwise has a finite number of straight channels of constant depth); such a modification should be doable somehow, while the inside well remain how it is.
Computational methods {#cometh}
=====================
In Hamiltonian systems the configuration as well as the phase space is divided into the escaping and non-escaping (trapped) regions. Usually, the vast majority of the trapped space is occupied by initial conditions of regular orbits forming stability islands where a third adelphic integral of motion is present. In many systems however, as we also seen in Part I, trapped chaotic orbits have also been observed. Therefore, we decided to distinguish between regular and chaotic trapped orbits. Over the years, several chaos indicators have been developed in order to determine the character of orbits. In our case, we chose to use the Smaller ALingment Index (SALI) method. The SALI \[[@S01]\] has been proved a very fast, reliable and effective tool, which is defined as $$\rm SALI(t) \equiv min(d_-, d_+),
\label{sali}$$ where $d_- \equiv \| {\bf{w_1}}(t) - {\bf{w_2}}(t) \|$ and $d_+ \equiv \| {\bf{w_1}}(t) + {\bf{w_2}}(t) \|$ are the alignments indices, while ${\bf{w_1}}(t)$ and ${\bf{w_2}}(t)$, are two deviations vectors which initially point in two random directions. For distinguishing between ordered and chaotic motion, all we have to do is to compute the SALI along time interval $t_{max}$ of numerical integration. In particular, we track simultaneously the time-evolution of the main orbit itself as well as the two deviation vectors ${\bf{w_1}}(t)$ and ${\bf{w_2}}(t)$ in order to compute the SALI. The variational equations (\[variac\]), as usual, are used for the evolution and computation of the deviation vectors. The time-evolution of SALI strongly depends on the nature of the computed orbit since when the orbit is regular the SALI exhibits small fluctuations around non zero values, while on the other hand, in the case of chaotic orbits the SALI after a small transient period it tends exponentially to zero approaching the limit of the accuracy of the computer $(10^{-16})$. Therefore, the particular time-evolution of the SALI allow us to distinguish fast and safely between regular and chaotic motion (e.g., \[[@ZC13]\]). Nevertheless, we have to define a specific numerical threshold value for determining the transition from regularity to chaos. After conducting extensive numerical experiments, integrating many sets of orbits, we conclude that a safe threshold value for the SALI is the value $10^{-7}$. In order to decide whether an orbit is regular or chaotic, one may use the usual method according to which we check after a certain and predefined time interval of numerical integration, if the value of SALI has become less than the established threshold value. Therefore, if SALI $\leq 10^{-7}$ the orbit is chaotic, while if SALI $ > 10^{-7}$ the orbit is regular. For the computation of SALI we used the `LP-VI` code \[[@CMD14]\], a fully operational routine which efficiently computes a suite of many chaos indicators for dynamical systems in any number of dimensions.
For investigating the escape escape process in our Hamiltonian system, we need to define samples of orbits whose nature (escaping or trapped) will be identified. For this purpose we define for each value of the energy (all tested energy levels are always above the escape energy), dense, uniform grids of initial conditions regularly distributed in the area allowed by the value of the energy. Our investigation takes place both in the configuration $(x,y)$ and the phase $(x,\dot{x})$ space for a better understanding of the escape mechanism. In both cases, the step separation of the initial conditions along the axes (or in other words the density of the grids) was controlled in such a way that always there are about 50000 orbits (a maximum grid of 225 $\times$ 225 equally spaced initial conditions of orbits). For each initial condition, we integrated the equations of motion (\[eqmot\]) as well as the variational equations (\[variac\]) using a double precision Bulirsch-Stoer `FORTRAN 77` algorithm (e.g., \[[@PTVF92]\]) with a small time step of order of $10^{-2}$, which is sufficient enough for the desired accuracy of our computations (i.e., our results practically do not change by halving the time step). Our previous experience suggests that the Bulirsch-Stoer integrator is both faster and more accurate than a double precision Runge-Kutta-Fehlberg algorithm of order 7 with Cash-Karp coefficients. In all cases, the energy integral (Eq. (\[ham\])) was conserved better than one part in $10^{-11}$, although for most orbits it was better than one part in $10^{-12}$.
In Hamiltonian systems with escapes an issue of paramount importance is the determination of the position as well as the time at which an orbit escapes. When the value of the energy $h$ is smaller than the escape energy, the Zero Velocity Curves (ZVCs) are closed. On the other hand, when $h > h_{esc}$ the equipotential curves are open and extend to infinity. An open ZVC consists of several branches forming channels through which an orbit can escape to infinity. At every opening there is a highly unstable periodic orbit close to the line of maximum potential \[[@C79]\] which is called a Lyapunov orbit. Such an orbit reaches the ZVC, on both sides of the opening and returns along the same path thus, connecting two opposite branches of the ZVC. Lyapunov orbits are very important for the escapes from the system, since if an orbit intersects any one of these orbits with velocity pointing outwards moves always outwards and eventually escapes from the system without any further intersections with the surface of section (see e.g., \[[@C90]\]). Additional details regarding the escape criteria are given in the Appendix B. The passage of orbits through Lyapunov orbits and their subsequent escape to infinity is the most conspicuous aspect of the transport, but crucial features of the bulk flow, especially at late times, appear to be controlled by diffusion through cantori, which can trap orbits far vary long time periods.
For the numerical integration we set a maximum time equal to $10^5$ time units. Our previous experience in this subject indicates that usually orbits need considerable less time to find one of the exits in the limiting surface and eventually escape from the system (obviously, the numerical integration is effectively ended when an orbit passes through one of the escape channels and intersects one of the unstable Lyapunov orbits). Nevertheless, we decided to use such a vast integration time just to be sure that all orbits have enough time in order to escape. Remember that there are the so called “sticky orbits" which behave as regular ones and their true chaotic character is revealed only after long time intervals of numerical integration. Here we should clarify that orbits which do not escape after a numerical integration of $10^5$ time units are considered as non-escaping or trapped.
Numerical results {#numres}
=================
The main target is to distinguish between trapped and escaping orbits for values of energy larger than the escape energy where the Zero Velocity Curves are open and several channels of escape are present. Furthermore, two important properties of the orbits will be investigated: (i) the directions or channels through which the particles escape and (ii) the time-scale of the escapes (we shall also use the term escape period). In particular, we examine these aspects for various values of the energy $h$, as well as for four different types of perturbation function $V_1(x,y)$. The grids of initial conditions of orbits whose properties will be determined are defined as follows: for the configuration $(x,y)$ space we consider orbits with initial conditions $(x_0, y_0)$ with $\dot{x_0} = 0$, while the initial value of $\dot{y_0}$ is always obtained from the energy integral of motion (\[ham\]) as $\dot{y_0} = \dot{y}(x_0,\dot{x_0},h) > 0$. Similarly, for the phase $(x,\dot{x})$ space we consider orbits with initial conditions $(x_0, \dot{x_0})$ with $y_0 = 0$, while again the initial value of $\dot{y_0}$ is obtained from the Hamiltonian (\[ham\]).
Our numerical calculations indicate that in almost all cases, apart from the escaping orbits there is an amount of non-escaping orbits. In general terms, the majority of non-escaping regions corresponds to initial conditions of regular orbits, where a third integral of motion is present, restricting their accessible phase space and therefore hinders their escape. However, there are also chaotic orbits which do not escape within the predefined time interval and remain trapped for vast periods until they eventually escape to infinity. At this point, it should be emphasized and clarified that these trapped chaotic orbits cannot be considered, by no means, neither as sticky orbits nor as super sticky orbits with sticky periods larger than $10^5$ time units. Sticky orbits are those who behave regularly for long time periods before their true chaotic nature is fully revealed. In our case on the other hand, this type of orbits exhibit chaoticity very quickly as it takes no more than about 100 time units for the SALI to cross the threshold value (SALI $\ll 10^{-7}$), thus identifying beyond any doubt their chaotic character. Therefore, we decided to classify the initial conditions of orbits in both the configuration and phase space into three main categories: (i) orbits that escape through one of the escape channels, (ii) non-escaping regular orbits and (iii) trapped chaotic orbits.
Here we would like to point out that all the following subsections containing the results of the four cases are formed having in mind flexibility. According the current text structure the reader can read any of the four subsections and have a clear view of the properties of the corresponding Hamiltonian system because each subsection is practical text-autonomous.
Case I: Five channels of escape {#case1}
-------------------------------
In this case $(n = 5)$, the non-integrable part of the Hamiltonian according to the first generating function (\[gens\]) is $$H_1 = V_1(x,y) = - \frac{1}{5}\left(x^5 - 10 x^3 y^2 + 5 x y^4 \right),
\label{ham1}$$ and the corresponding escape energy equals to 3/10. The total Hamiltonian of the system $H = H_0 + H_1$ has a special symmetry, that is $H$ is symmetric with respect to $y \rightarrow - y$. The equipotential curves of the total potential (\[pot\]) for various values of the energy $h$ are shown in Fig. \[pot5\]a. The equipotential corresponding to the energy of escape $h_{esc}$ is plotted with red color in the same plot. The open ZVC at the configuration $(x,y)$ plane when $h = 0.35 > h_{esc}$ is presented with green color in Fig. \[pot5\]b and the five channels of escape are shown. In the same plot, we denote the five unstable Lyapunov orbits by $L_i$, $i = 1,...,5$ using red color.
![Evolution of the average escape time of orbits $< t_{\rm esc} >$ on the configuration $(x,y)$ space as a function of the total orbital energy $h$.[]{data-label="tesc"}](Tesc.pdf){width="\hsize"}
We will investigate the trapped or escape dynamics of test particles for values of energy in the set $h$ = {0.31, 0.35, 0.40, 0.45, 0.50, 0.55, 0.60, 0.65, 0.70}. Our exploration begins in the configuration $(x,y)$ space and in Fig. \[xy5\] we present the structure of the $(x,y)$ plane for several values of the energy. Each initial condition is colored according to the escape channel through which the particular orbit escapes. The gray regions on the other hand, denote initial conditions where the test particles move in regular orbits and do not escape, while trapped chaotic orbits are indicated with black. The outermost solid line is the Zero Velocity Curve (limiting curve) which is defined as $V(x,y) = h$. It is seen that for values of energy larger but yet very close to the escape energy $(h < 0.40)$ a large portion of the $(x,y)$ plane is covered by stability islands which correspond to initial conditions of non-escaping regular orbits which are surrounded by a very rich fractal structure. Moreover, looking carefully the grids we observe that there is a highly sensitive dependence of the escape process on the initial conditions, that is, a slight change in the initial conditions makes the test particle escape through another channel, which is of course a classical indication of chaos. As the value of the energy increases the stability islands and the amount of non-escaping and trapped orbits is reduced and basins of escape emerge. Indeed, when $h = 0.70$ almost all the computed orbits of the grid escape and there is no indication of bounded motion or whatsoever. By the term basin of escape, we refer to a local set of initial conditions that corresponds to a certain escape channel. The escape basins become smoother and more well-defined as the energy increases and the degree of fractality decreases[^2]. The fractality is strongly related with the unpredictability in the evolution of a dynamical system. In our case, it can be interpreted that for high enough energy levels, the test particles escape very fast from the scattering region and therefore, the system’s predictability increases. It is seen in channel 5 that outside the Lyapunov orbit there is a stream of initial conditions of orbits which even though they are launched outside the unstable the Lyapunov orbits they escape from another exit, meaning that before escape they first enter the interior region.
The distribution of the escape times $t_{\rm esc}$ of orbits on the $(x,y)$ plane is given in the Fig. \[xyt5\], where light reddish colors correspond to fast escaping orbits, dark blue/purpe colors indicate large escape periods, while gray color denote both trapped chaotic and non-escaping regular orbits. It is observed that for $h = 0.31$, that is a value of energy just above the escape energy, the escape periods of the majority of orbits are huge corresponding to tens of thousands of time units. This however, is anticipated because in this case the width of the escape channels is very small and therefore, the orbits should spend much time inside the equipotential surface until they find one of the five openings and eventually escape to infinity. As the value of the energy increases however, the escape channels become more and more wide leading to faster escaping orbits, which means that the escape period decreases rapidly. We found that the longest escape rates correspond to initial conditions near the vicinity of the fractal regions. On the other hand, the shortest escape periods have been measured for the regions without sensitive dependence on the initial conditions (basins of escape), that is, those far away from the fractal basin boundaries. We would like to emphasize that by definition, the fractal basin boundaries contain initial conditions of orbits that will never escape from the system, as it coincides with the stable manifold of the non-attracting chaotic set, also known as chaotic saddle or strange saddle, that is formed by a set of Lebesgue measure zero of orbits that will never escape from the scattering region for both $t \rightarrow \infty$ or $t \rightarrow - \infty$. It is known that at the critical energy the escape time is infinity and it decreases if one moves away from the critical value. The evolution of the average value of the escape time $< t_{\rm esc} >$ of orbits on the configuration $(x,y)$ space as a function of the total orbital energy $h$ is given in Fig. \[tesc\]. It is seen, that for low values of energy, just above the escape value, the average escape time of orbits is about 750 time units, however it reduces rapidly tending asymptotically to zero which refers to orbits that escape almost immediately from the system.
The structure of the phase $(x,\dot{x})$ plane for the same set of values of the energy is shown in Fig. \[xpx5\]. A similar behavior to that discussed for the configuration $(x,y)$ plane can be seen. The outermost black solid line is the limiting curve which is defined as $$f(x,\dot{x}) = \frac{1}{2}\dot{x}^2 + V(x, y = 0) = h.
\label{zvc}$$ Here we must clarify that this $(x,\dot{x})$ phase plane is not a classical Poincaré Surface of Section (PSS), simply because escaping orbits in general, do not intersect the $y = 0$ axis after a certain time, thus preventing us from defying a recurrent time. A classical Poincaré surface of section exists only if orbits intersect an axis, like $y = 0$, at least once within a certain time interval. Nevertheless, in the case of escaping orbits we can still define local surfaces of section which help us to understand the orbital behavior of the dynamical system. It is interesting to note that the limiting curve is open at the right part due to the $x^5$ term entering the perturbation function. In the phase planes of Fig. \[xpx5\] one can distinguish fractal regions where it is impossible to predict the particular escape channel and regions occupied by escape basins. These basins are either broad well-defined regions, or elongated bands of complicated structure spiralling around the center. Once more we observe that for values of energy close to the escape energy there is a substantial amount of non-escaping regular orbits and the degree of fractalization of the rest phase plane is high. As we proceed to higher energy levels however, the rate of non-escaping regular orbits heavily reduces, the phase plane becomes less and less fractal and is occupied by well-defined basins of escape. We would like to note that at the right open part of the $(x,\dot{x})$ planes there is flow of initial conditions which extends asymptotically to infinity. The distribution of the escape times $t_{\rm esc}$ of orbits on the $(x,\dot{x})$ plane is shown in Fig. \[xpxt5\]. It is evident that orbits with initial conditions inside the exit basins escape from the system very quickly, or in other words, they possess extremely low escape periods. On the contrary, orbits with initial conditions located in the fractal parts of the phase plane need considerable amount of time in order to escape.
The following Fig. \[percs5\]a shows the evolution of the percentages of trapped and escaping orbits on the configuration $(x,y)$ plane when the value of the energy $h$ varies. Here we would like to point out that we decided to merge the percentages of non-escaping regular and trapped chaotic orbits together because our computations indicate that always the rate of trapped chaotic orbits is extremely small (less than 1%) and therefore, it does not contribute to the overall orbital structure of the dynamical system. We observe that when $h = 0.31$, that is just above the escape energy, trapped motion is the most populated family occupying about 22% of the configuration plane, while escaping orbits through exits 2 and 3 have the same rates with escaping orbits through channels 5 and 4, respectively. As the value of the energy increases however, the rate of trapped orbits drops rapidly and for $h > 0.60$ it practically vanishes. At the same time, the percentage of orbits escaping through exit channel 2 increases steadily and at the highest energy level studied it corresponds to about 40% of the configuration $(x,y)$ plane. The rates of escaping orbits through channels 4 and 5 exhibit a similar slow reduction for $h > 0.4$, while the percentages of escaping orbits through exits 1 and 3 seem less unaffected by the shifting of the energy being almost unperturbed around 12% and 23%, respectively. Therefore, one may conclude that for high energy levels $(h > 0.60)$, all orbits in the configuration $(x,y)$ plane escape and about 40% of them choose channel 2. In the same vein we present in Fig. \[percs5\]b the evolution of the percentages of trapped and escaping orbits on the phase plane as a function of the energy $h$. It is observed that the pattern and the evolution of the percentages is completely different with respect to that discussed in Fig. \[percs5\]a regarding the configuration plane. We see that escaping orbits through exit channel 1 dominate throughout, even though their rate reduces with increasing energy. Moreover, the percentages of escaping orbits through exits 2 and 3 display an identical increase from 5% to 25%, while the rates of exits 4 and 5 are much smaller (less than 10%). The only similarity with the configuration plane is the evolution pattern of trapped orbits. Taking all into account we can deduce that in the configuration space an orbit is more likely to escape form channel 1, while for sufficiently enough values of energy $(h > 1)$, we have numerical evidence that the rates of exits 1, 2 and 3 seem to converge thus sharing about 90% of the phase space.
Case II: Six channels of escape {#case2}
-------------------------------
We continue our exploration of escapes in a Hamiltonian system with six exit channels with escape energy equal to 1/3. In order to obtain this number of exits $(n = 6)$ in the limiting curve in the configuration $(x,y)$ plane, the perturbation term should be $$V_1(x,y) = - \frac{1}{6}\left(x^6 + 15 x^4 y^2 - 15 x^2 y^4 + y^6 \right),
\label{ham2}$$ according to the second generating function of Eqs. (\[gens\]). The corresponding Hamiltonian $H = H_0 + H_1$ is invariant under $x \rightarrow - x$ and/or $y \rightarrow - y$. In Fig. \[pot6\]a we see the equipotential curves of the total potential (\[pot\]) for various values of the energy $h$, while the equipotential corresponding to the energy of escape $h_{esc}$ is plotted with red color in the same plot. Furthermore, the open ZVC at the configuration $(x,y)$ plane when $h = 0.4 > h_{esc}$ is presented with green color in Fig. \[pot6\]b and the six channels of escape are shown. In the same figure, the six unstable Lyapunov orbits $L_i$, $i = {1,...,6}$ are denoted using red color.
In this case, we shall investigate the escape dynamics of unbounded motion of test particles for values of energy in the set $h$ = {0.34, 0.36, 0.38, 0.40, 0.42, 0.45, 0.50, 0.55, 0.60}. We begin with initial conditions of orbits in the configuration $(x,y)$ plane. The orbital structure of the configuration plane for different values of the energy $h$ is show in Fig. \[xy6\]. Again, following the approach of the previous case, each initial condition is colored according to the escape channel through which the particular orbit escapes. Stability islands on the other hand, filled with initial conditions of ordered orbits which do not escape are indicated as gray regions, while trapped chaotic orbits are shown in black. We observe that things are quite similar to that discussed previously in Fig. \[xy5\]. In fact, for energy levels very close to the escape energy, the central region of the $(x,y)$ plane is highly fractal and it is also occupied by several stability islands mainly situated at the outer parts of the plane. However, as we increase the value of the energy the regions of regular non-escaping orbits are reduced, the configuration plane becomes less and less fractal and well-defined basins of escape emerge. Additionally, we see that the area on the $(x,y)$ plane occupied by initial conditions of orbits that escape through exit channel 2 grows rapidly with increasing energy and at high energy levels $(h > 0.50)$ it dominates. Once more, as we discussed earlier in Fig. \[xy5\], we observe in channel 5 a vertical flow of initial conditions of orbits that escape through exit 2. The following Fig. \[xyt6\] shows how the escape times $t_{\rm esc}$ of orbits are distributed on the $(x,y)$ plane. Light reddish colors correspond to fast escaping orbits, dark blue/purpe colors indicate large escape periods, while gray color denote trapped orbits. This grid representation of the configuration plane gives us a much more clearer view of the orbital structure and especially about the trapped and non-escaping orbits. In particular, we see that even for the highest energy level studied, that is when $h = 0.60$, two tiny stability islands are still present in the configuration space.
The structure of the $(x,\dot{x})$ phase plane for the same set of values of the energy is shown in Fig. \[xpx6\]. It is worth noticing that in the phase plane the limiting curve is closed but this does not mean that there is no escape. Remember that we decided to choose such perturbation terms that create the escape channels on the configuration $(x,y)$ plane which is a subspace of the entire four-dimensional $(x,y,\dot{x},\dot{y})$ phase space of the system. We observe a similar behavior to that discussed earlier for the configuration $(x,y)$ plane in Fig. \[xy6\]. Again, we can distinguish in the phase plane fractal regions where the prediction of the particular escape channel is impossible and regions occupied by escape basins. For low values of the energy $(h < 0.38)$ we can identify initial conditions of trapped chaotic orbits at the boundaries of the two stability islands on the $\dot{x}$ axis. As we proceed to higher energy levels however, the extent of these stability islands is reduced and at relatively high values of the energy $(h > 0.60)$ they completely disappear. Furthermore, it is also seen that the extent of the escape basins of exits 1, 2 and 3 is significantly grows in size with increasing energy. In this case the limiting curves are close and therefore, there is no flow of initial conditions outwards. Fig. \[xpxt6\] shows the distribution of the escape times $t_{\rm esc}$ of orbits on the $(x,\dot{x})$ plane. It is evident that orbits with initial conditions inside the exit basins escape from the system after short time intervals, or in other words, they possess extremely small escape periods. On the contrary, orbits with initial conditions located in the fractal domains of the phase plane need considerable amount of time in order to find one of the exits and escape.
The evolution of the percentages of trapped and escaping orbits on the configuration $(x,y)$ plane when the value of the energy $h$ varies is presented in Fig. \[percs6\]a. It is seen that for $h = 0.34$, that is the first investigated energy level above the escape energy, escaping orbits through channels 1, 3, 4, and 6 share the same percentage (around 15%), escapers through channel 2 have a slightly elevated percentage (around 18%), while trapped orbits possess a low rate corresponding only to about 10% of the configuration plane. Once more, as we increase the value of the energy the rate of trapped orbits decreases and eventually vanishes for $h > 0.8$. Furthermore, we observe that the percentage of escaping orbits through channel 2 grows with increasing energy and remains always the most populated escape channel. The percentages of escaping orbits through channels 1, 3, 4 and 6 on the other hand, are almost unperturbed by the shifting on the orbital energy and the seem to saturate around 15%, while the rate of escaping orbits through exit 5 displays a gradual decrease. In general terms, we may conclude that throughout the energy range studied, the majority of orbits in the configuration $(x,y)$ plane choose to escape through exit channel 2, while exit 5 seems to be the least favorable among the escape channels. It is evident from Fig. \[percs6\]b where the evolution of the percentages of trapped and escaping orbits on the phase plane as a function of the value of the energy $h$ is presented that the pattern has many differences comparing to that discussed previously in Fig. \[percs6\]a. To begin with, we observe that for $h = 0.34$ more than 35% of the phase plane corresponds to initial conditions of orbits that do not escape, while all the escape channels are equiprobable taking into account that all channels have the same rate thus sharing about 60% of the phase space. As the value of the energy increases and we move away from the escape energy it is seen that the rate of trapped orbits is heavily reduced, while the percentages of the escape channels start to diverge following two different patterns. Being more specific, one may observe that the rates of escaping orbits through exits 1, 2 and 3 start to grow, while on the other hand the percentages of escapers through exits 4, 5 and 6 exhibit a gradual decrease. At the highest energy level studied $(h = 1.0)$, about 30% of the total orbits escape through channel 2, exit channels 1 and 3 share about half of the phase plane, while exit channels 4 and 6 share about 10% of the same plane. Thus, one may reasonably conclude that throughout the energy range studied, the vast majority of orbits in the phase $(x,\dot{x})$ plane choose to escape through channels 1, 2 and 3, while channels 4, 5 and 6 are much less likely to be chosen.
Case III: Seven channels of escape {#case3}
----------------------------------
Our escape quest continues considering a Hamiltonian system with seven exit channels where the escape energy is equal to 5/14. In order to obtain this number of exits $(n = 7)$ in the limiting curve in the configuration $(x,y)$ plane, the perturbation term should be $$V_1(x,y) = - \frac{1}{7}\left(x^7 - 21 x^5 y^2 + 35 x^3 y^4 - 7 x y^6 \right),
\label{ham3}$$ according to the first generating function of Eqs. (\[gens\]). We observe that the corresponding Hamiltonian $H = H_0 + H_1$ is symmetric with respect to $y \rightarrow - y$. In Fig. \[pot7\]a we see the equipotential curves of the total potential (\[pot\]) for various values of the energy $h$, while the equipotential corresponding to the energy of escape $h_{esc}$ is plotted with red color in the same plot. Furthermore, the open ZVC at the configuration $(x,y)$ plane when $h = 0.42 > h_{esc}$ is presented with green color in Fig. \[pot6\]b and the seven channels of escape are shown. In the same figure, the seven unstable Lyapunov orbits $L_i$, $i = {1,...,7}$ are denoted using red color.
In this case, the set of values of the total orbital energy of the test particles is $h$ = {0.36, 0.40, 0.44, 0.48, 0.52, 0.60, 0.72, 0.84, 0.96}. First, we consider initial conditions of orbits in the configuration $(x,y)$ plane and in Fig. \[xy7\] the orbital structure of the configuration plane for different values of the energy $h$ is presented. As in all previous cases, each initial condition is colored according to the escape channel through which the test particle escapes. For $h = 0.36$, that is an energy level just above the critical escape energy, the vast majority of the interior region of the configuration plane is covered by initial conditions of non-escaping regular orbits forming two stability islands which are separated by a highly fractal layer. As we increase the value of the energy the stability islands are reduced, while at the same time the fractality of the configuration plane is considerably reduced and well-formed basis of escape emerge. Furthermore, we see that the area on the $(x,y)$ plane occupied by initial conditions of orbits that escape through exit channel 3 grows rapidly with increasing energy and at high energy levels $(h > 0.60)$ they dominate. The outwards flow of initial conditions is once more present in channel 6. The distribution of the escape times $t_{\rm esc}$ of orbits on the configuration plane is given in Fig. \[xyt7\], where light reddish colors correspond to fast escaping orbits, dark blue/purpe colors indicate large escape periods, while gray color denote trapped and non-escaping orbits. It is evident that the two main stability islands even though they reduce in size with increasing energy they do not completely disappear since for $h = 0.96$ we still observe the presence of two tine stability islands inside the interior region of the configuration plane.
The following Fig. \[xpx7\] shows the orbital structure of the $(x,\dot{x})$ phase plane for the same set of values of the total energy $h$. It is seen that the phase space is divided into three types of regions: (i) regions of regular motion where the corresponding orbits do not escape; (ii) fractal regions where we cannot predict the particular escape channel for a given orbit and (iii) regions where the initial conditions of orbits define broad basins of escape. The first and the second type of regions occupy large portion of the phase space for low values of the energy $(h < 0.50)$, while for larger values their extent is considerably confined. The third type on the other hand, exhibits the complete opposite behavior. In particular, we observe that the basins of escape corresponding to exits 2, 3 and 4 grow significantly in size with increasing energy. Similarly as in Fig. \[xpx5\], a weak stream of initial conditions of orbits is identified in the phase planes. Fig. \[xpxt7\] depicts the distribution of the escape times $t_{\rm esc}$ of orbits on the phase $(x,\dot{x})$ plane. Once more we see that orbits with initial conditions inside the basins of escape have very small escape periods and therefore, they escape to infinity quite early. On the contrary, orbits with situated in the fractal regions of the phase plane require long time intervals in order to find one of the exits and escape.
In Fig. \[percs7\]a we see the evolution of the percentages of trapped and escaping orbits on the configuration $(x,y)$ plane when the value of the energy $h$ varies. For $h = 0.36$, that is an energy level just above the escape energy $h_{esc}$, trapped orbits occupy about one fourth of the entire configuration plane. In addition, the percentages of escaping orbits through exits 2 and 7 share about 30% of the $(x,y)$ plane, while all the other rates apart form that of exit 1 have about the same value around 10%. The portion of trapped orbits reduces as the value of the energy increases and for $h > 1$ they vanish. The percentage of escaping orbits through exit channel 3 on the other hand increases as we proceed to higher energy levels and for $h > 0.4$ it is the most populated type of orbits. The rate of escaping orbits through exit 2 also increases and for high values of energy it seems to saturate around 22%, while that of exit 6 decreases reaching 5% at the highest energy level studied. The percentages of all the remaining exit channels seem to be almost unperturbed by the change on the value of the energy holding values around 10% throughout. Therefore, one may conclude that in the configuration $(x,y)$ plane the majority of orbits choose to escape either through exit channel 2, or exit 3, while all the other exits are significantly less probable to be chosen by the test particles. In the same vein, we present in Fig. \[percs7\]b the evolution of the percentages of trapped and escaping orbits on the phase plane as a function of the value of the energy $h$. Here it is evident that things are quite different. At the lowest examined energy level $(h = 0.36)$ it is found that half of the phase space is occupied by initial conditions of orbits that escape through channel 1, about 35% of the integrated initial conditions correspond to trapped regular orbits, while the remaining 15% of the phase plane is shared by escaping orbits through channels 2 to 7. As the value of the energy increases the percentages of both trapped and escaping through exit 1 orbits are reduced however, the latter type of orbits remains throughout the most populated one. At the same time, the percentages of escaping orbits through channels 2 to 7 start to diverge and produce two distinct branches. The first branch contains the evolution of the rates of escaping orbits through exits 2, 3 and 4 which all of them exhibit a common increase and at the highest energy level studied $(h = 1.56)$ they share about 60% of the phase space. The second branch includes the percentages of escaping orbits through channels 5, 6 and 7 and we see that all of them are almost unperturbed by the energy shifting evolving at low values less than 5%. Taking into account the above-mentioned results regarding the phase $(x,\dot{x})$ space we may say that throughout the energy range studied, the vast majority of orbits choose to escape through one of the first four channels (exits 2, 3 and 3 are practically equiprobable), while the remaining channels (5 to 7) are significantly less likely to be chosen.
Case IV: Eight channels of escape {#case4}
---------------------------------
The last case under investigation is the case where the Hamiltonian system has eight channels of escape $(n = 8)$. The corresponding perturbation function is obtained from the second generating function (\[gens\]) and it reads $$V_1(x,y) = - \frac{1}{8}\left(x^8 - 28 x^6 y^2 + 70 x^4 y^4 - 28 x^2 y^6 + y^8 \right),
\label{ham1}$$ while the corresponding escape energy is equal to 3/8. The total Hamiltonian $H = H_0 + H_1$ is invariant under $x \rightarrow - x$ and/or $y \rightarrow - y$. The equipotential curves of the total potential (\[pot\]) for various values of the energy $h$ are shown in Fig. \[pot8\]a. The equipotential corresponding to the energy of escape is plotted with red color in the same plot. The open ZVC at the configuration $(x,y)$ plane when $h = 0.44 > h_{esc}$ is presented with green color in Fig. \[pot5\]b and the eight channels of escape are shown. In the same plot, we denote the eight unstable Lyapunov orbits by $L_i$, $i = 1,...,8$ using red color.
The escape properties and mechanism of unbounded motion of test particles for values of energy in the set $h$ = {0.38, 0.40, 0.45, 0.50, 0.60, 0.70, 0.80, 0.90, 1.00} will be examined. We begin, as usual, with initial conditions of orbits in the configuration $(x,y)$ plane. Fig. \[xy8\] shows the orbital structure of the configuration plane for different values of the energy $h$. Again, following the same approach as in all previous cases, each initial condition is colored according to the escape channel through which the particular orbit escapes. Areas corresponding to non-escaping regular orbits on the other hand, are indicated as gray regions, while trapped chaotic orbits are shown in black. We see that for values of energy very close to the escape energy $(h < 0.50)$ the majority of the central region of the configuration plane is fractal, some small stability islands are present situated mainly at the outer parts of the interior region, while we observe a week stream of initial conditions of orbits that escape from channel 3 which crosses vertically the $(x,y)$ plane and flows outwards from channel 7 (a similar phenomenon was also observed in all previous cases). As the value of the energy increases this stream becomes more and more strong evolving to a wide basin of escape which eventually takes over most of the interior region of the configuration plane. Moreover, additional smaller basins of escape emerge mainly around the unstable Lyapunov orbits, while the stability islands containing the initial conditions of non-escaping regular orbits are reduced in size. Here we should like to note that in general terms, throughout the energy range the structure of the configuration $(x,y)$ plane is somehow symmetrical with respect to the $x = 0$ axis. The distribution of the escape times $t_{\rm esc}$ of orbits on the configuration plane is given in Fig. \[xyt8\]. Light reddish colors correspond to fast escaping orbits, dark blue/purpe colors indicate large escape periods, while gray color denote trapped and non-escaping orbits orbits. Here, we have a better view regarding the amount of trapped orbits. We see that for $h = 1.0$ all orbits escape from the system. Moreover we observe that orbits with initial conditions close to the area occupied by trapped orbits have significantly high escape periods, while on the other hand, orbits located near the exit channels escape very quickly having escaping rates of about two orders smaller.
We proceed with the phase $(x,\dot{x})$ plane, the structure of which for the same set of values of the energy is presented in Fig. \[xpx8\]. It is seen that this time the limiting curve is open at both sides. One may observe that for $h < 0.4$ a large part of the phase plane is covered by several sets of stability islands corresponding to non-escaping orbits, the remaining has a highly fractal structure, while only three basins of escape are shown; one in the central region of the phase plane and two other one above and one below it. However, as the value of the energy increases and we move far away for the escape energy, the extent of these three basins of escape grows and for $h > 0.70$ they dominate. At the same time, small elongated spiral basins of escape emerge inside the fractal region which surrounds the central escape basin. Furthermore, at very high energy levels $(h > 1.0)$ we see that non-escaping regular orbits disappear completely from the grid and the three main basins of escape take over the vast majority of the phase plane, while the elongated escape basins remain confined to the central region. As we noticed previously when discussing the configuration $(x,y)$ plane, there is also a symmetry in the phase plane. In particular, throughout the energy range the structure of the phase plane $(x,\dot{x})$ is somehow symmetrical (though not with the strick sense) with respect to the $\dot{x} = 0$ axis. In this case, the limiting curves in the phase plane are open in both sides thus, we observe the existence of two streams of initial conditions leaking out and extend to infinity. The following Fig. \[xpxt8\] shows the distribution of the escape times $t_{\rm esc}$ of orbits on the phase $(x,\dot{x})$ plane. It is clear that orbits with initial conditions inside the exit basins escape to infinity after short time intervals, or in other words, they possess extremely small escape periods. On the contrary, orbits with initial conditions located in the fractal parts of the phase plane need considerable amount of time in order to find one of the four exits and escape. It is seen that at the highest energy level studied $(h = 1.0)$ there is no indication of bounded motion and all orbits escape to infinity sooner or later.
It is of particular interest to monitor the evolution of the percentages of trapped and escaping orbits on the configuration $(x,y)$ plane when the value of the energy $h$ varies. A diagram depicting this evolution is presented in Fig. \[percs8\]a. We see that for $h = 0.38$, that is an energy level just above the escape energy, about 17% of the configuration plane is covered by initial conditions of trapped orbits. As the value of the energy increases however, the rate of trapped orbits drops rapidly and eventually at $h > 1.0$ it vanishes. We also observe that the evolution of the percentages of orbits escaping through channels 1, 2 and 6 coincide with the evolution of the percentages escaping through channels 5, 4 and 8, respectively. We anticipated this behaviour of the escape percentages, which is a natural result of the symmetrical structure of the configuration $(x,y)$ plane. We also anticipated the domination of escaping orbits through exit 3 due to the strong escape stream. It is seen that initially $(h = 0.38)$ the rates of escaping orbits through exits 2, 4, 6, 7 and 8 coincide at about 12%. Then, with increasing energy the rates of these types of escaping orbits start to diverge but only escapers through exit 7 decrease; all the others remain almost unperturbed. At the highest energy studied, escaping orbits through channels 2 and 4 share about 30% of the configuration plane, escaping orbits through channels 6 and 8 share about 20% of the same plane, while escaping orbits trough channels 1 and 5 occupy only about 12% of the grid. Therefore, one may reasonably conclude that in general terms, throughout the range of the values of the energy studied, the majority of orbits in the configuration $(x,y)$ plane choose to escape through channels 2, 3 and 4. The evolution of the percentages of trapped and escaping orbits on the phase plane as a function of the value of the energy $h$ is given in Fig. \[percs8\]b. For $h = 0.38$, we see that trapped orbits is the most populated type of orbits as they occupy about 27% of the phase space. However as usual, with increasing energy the dominance of trapped orbits deteriorates rapidly due to the increase of the rates of escaping orbits forming basins of escape. We observe that the evolution of several rates of escaping orbits coincide due to the symmetry of the phase space. In particular, there are three different branches, given with decreasing strength (rate): (i) containing the rates of exits 1 and 5; (ii) containing the rates of exits 2, 3 and 4; (iii) containing the rates of exits 6, 7 and 8. The first branch seems to be unaffected by the change on the value of the energy and exits 1 and 5 share about half of the phase space throughout. The rates of escaping orbits that belong to the second branch exhibit a small increase and an the highest energy level studied they share about 40% of the grid. The percentages of escaping orbits that belong to the third branch on the other hand, are reduced and when $h = 1.5$ they share the remaining 10% of the phase plane. Thus, we may conclude that the vast majority of orbits in the phase $(x,\dot{x})$ plane displays clear sings of preference through exits 1 and 5, while channels 6, 7 and 8 have considerable less probability to be chosen.
![Evolution of the fractal uncertainty dimension $D_0$ of the $(x,h)$-planes of Figs. 25(a-d) as a function of the total energy $h$. $D_0 = 1$ means total fractality, while $D_0 = 0$ implies zero fractality.[]{data-label="frac"}](fract.pdf){width="\hsize"}
An overview analysis {#geno}
--------------------
The color-coded grids in configuration $(x,y)$ as well as the phase $(x,\dot{x})$ plane provide information on the phase space mixing however, for only a fixed value of energy. Hénon back in the late 60s \[[@H69]\], introduced a new type of plane which can provide information not only about stability and chaotic regions but also about areas of trapped and escaping orbits using the section $y = \dot{x} = 0$, $\dot{y} > 0$ (see also \[[@BBS08]\]). In other words, all the orbits of the test particles are launched from the $x$-axis with $x = x_0$, parallel to the $y$-axis $(y = 0)$. Consequently, in contrast to the previously discussed types of planes, only orbits with pericenters on the $x$-axis are included and therefore, the value of the energy $h$ can be used as an ordinate. In this way, we can monitor how the energy influences the overall orbital structure of our Hamiltonian system using a continuous spectrum of energy values rather than few discrete energy levels. Figs. \[xh\](a-d) shows the structure of the $(x,h)$-plane for the four types of Hamiltonians presented in the previous subsections. It is seen that in all four plots the boundaries between bounded and unbounded motion are now seen to be more jagged than in the previous types of grids. In addition, we found in the blow-ups of the diagrams many tiny islands of stability[^3]. We observe that for low values of the energy close to the escape energy, there is a considerable amount of trapped orbits inside stability regions surrounded by a highly fractal structure. This pattern however changes for larger energy levels, where there are no trapped orbits and the vast majority of the grids is covered by well-formed basins of escape, while fractal structure is confined only near the boundaries of the escape basins. It would be of particular interest to monitor how the total orbital energy $h$ influences the percentages of all types of orbits. The following Fig. \[percs\](a-d) shows the evolution of the percentages of all types of orbits identified in the $(x,h)$ planes of Figs. \[xh\](a-d), respectively as a function of the total orbital energy.
In all previous subsections we discussed fractality of the configuration and phase space in a qualitative way. In particular, rich and highly fractal domains are those in which we cannot predict through which exit channel the particle will escape since the particle chooses randomly an exit. On the other hand, inside the escape basins where the degree of fractality is zero the escape process of the particles is well known and predictable. At this point, we shall provide a quantitative analysis of the degree of fractality for the grids shown in Figs. \[xh\](a-d). In order to measure the fractality we have computed the uncertainty dimension \[[@O93]\] for different values of the total energy. Obviously, this quantity is independent of the initial conditions used to compute it. We follow the numerical way according to \[[@AVS01]\]. We calculate the exit for certain initial condition $(x,h)$. Then, we compute the exit for the initial conditions $(x - \epsilon, h)$ and $(x + \epsilon, h)$ for a small $\epsilon$ and if all of them coincide, then this point is labeled as “certain”. If on the other hand they do not, it will be labeled as “uncertain”. We repeat this procedure for different values of $\epsilon$. Then we calculate the fraction of initial conditions that lead to uncertain final states $f(\epsilon)$. There exists a power law between $f(\epsilon)$ and $\epsilon$, $f(\epsilon) \propto \epsilon^{\alpha}$, where $\alpha$ is the uncertainty exponent. The uncertainty dimension $D_0$ of the fractal set embedded in the initial conditions is obtained from the relation $D_0 = D - \alpha$, where $D$ is the dimension of the phase space. It is typical to use a fine grid of values of $x$ and $h$ to calculate the uncertainty dimension. The evolution of the uncertainty dimension $D_0$ when the energy is increased is shown in Fig. \[frac\](a-d) for the corresponding $(x,h)$ grids of Fig. \[xh\](a-d), respectively. As it has just been explained, the computation of the uncertainty dimension is done for only a “1D slice” of initial conditions of Figs. \[xh\](a-d) and for that reason $D_0 \in (0,1)$. It is remarkable that the uncertainty dimension tends to one when the energy tends to its minimum value $(E_{esc})$. This means that for that critical value, there is a total fractalization of the grid, and the chaotic set becomes “dense” in the limit. Consequently, in this limit there are no smooth sets of initial conditions and the only defined structures that can be recognized are the Kolmogorov-Arnold-Moser (KAM)-tori of quasi-periodic orbits. When the energy is increased however, the different smooth sets appear and tend to grow, while the fractal structures that coincide with the boundary between basins decrease. Finally for values of energy much greater than the escape energy the uncertainty dimension tends to zero (no fractality). Furthermore, it is seen that there is a hierarchy in four curves shown in Fig. \[frac\]. In particular, the order of the curves follow the number of exits (channels); the more the exits the higher the corresponding curve with more fractality. This makes sense, because if there are more basins it seems to be more probable that your closest point in the exit basin belongs to a different basin.
The rich fractal structure of the $(x,h)$ planes shown in Figs. \[xh\](a-d) implies that all four Hamiltonians have also a strong topological property, which is known as the Wada property \[[@AVS01]\]. The Wada property is a general feature of two-dimensional (2D) Hamiltonians with three or more escape channels. A basin of escape verifies the property of Wada if any initial condition that is on the boundary of one basin is also simultaneously on the boundary of three or even more escape basins (e.g., \[[@BSBS12], [@KY91]\]). In other words, every open neighborhood of a point $x$ belonging to a Wada basin boundary has a nonempty intersection with at least three different basins. Hence, if the initial conditions of a particle are in the vicinity of the Wada basin boundary, we will not be able to be sure by which one of the three exits the orbit will escape to infinity. Therefore, if a Hamiltonian system has this property the unpredictability is even stronger than if it only had fractal basin boundaries. If an orbit starts close to any point in the boundary, it will not be possible to predict its future behavior, as its initial conditions could belong to any of the other escape basins. In Fig. \[wada\](a-d) we present zoom plots of characteristic exit channels in the configuration $(x,y)$ space for the system with five, six, seven and eight escape channels, respectively, while the corresponding Lyapunov orbits are shown in dashed white. We see than no matter the scale, all colors are fully mixed and therefore we have an indication that our Hamiltonian system verify this special property. However, it should be pointed out that the only mathematically precise method to verify the Wada property in a Hamiltonian system is to paint the unstable manifold of the Lyapunov orbit and show that it crosses all basins (see e.g., \[[@NY96]\]). This special topological property has been identified and studied in several dynamical systems (e.g., \[[@AVS09], [@KY91], [@PCOG96]\]) and it is a typical property in open Hamiltonian systems with three or more escape channels.
Figure $x_0$ $y_0$ $t_{esc}$ $h$ Outside exit
----------- ------------- ------------- ----------- ------ --------- ------
\[orbs\]a 0.51600000 -1.40000000 14.52 0.35 $L_5$ 3
\[orbs\]b 0.08555443 -1.37457465 40.66 0.40 $L_5$ 1
\[orbs\]c -0.37030768 -1.15722900 12.15 0.44 $L_6$ 3
\[orbs\]d 0.06095617 -1.45418326 64.62 0.45 $L_7$ 5
: Initial conditions, escape period and value of the energy of the orbits shown in Fig. \[orbs\](a-d).[]{data-label="table1"}
It is evident from the results presented in Figs. \[txh\](a-d) that the escape times of the orbits are strongly correlated to the escape basins. In addition, one may conclude that the smallest escape periods correspond to orbits with initial conditions inside the escape basins, while orbits initiated in the fractal regions of the planes have the highest escape rates. In all four cases the escape times of orbits are significantly reduced with increasing energy. Thus, combining all the numerical outcomes presented in Figs. \[xh\] and \[txh\] we may say that the key factor that determines and controls the escape times of the orbits is the value of the orbital energy (the higher the energy level the shorter the escape rates), while the fractality of the basin boundaries varies strongly both as a function of the energy and of the spatial variable. Another interesting way of measuring the escape rate of an orbit in the phase $(x,\dot{x})$ space is by counting how many intersections the orbit has with the axis $y = 0$ before it escapes. The regions in Figs. \[iters\](a-b) are colored according to the number of intersections with the axis $y = 0$ upwards $(\dot{y} > 0)$ and this is another type of grid representation showing a characteristic example of each Hamiltonian system. We observe that orbits with initial conditions inside the green basins escape directly without any intersection with the $y = 0$ axis. We should also note here that orbits with initial conditions located at the vicinity of the stability islands or at the boundaries of the escape basins perform numerous intersections with the $y = 0$ axis before they eventually escape to infinity. On the other hand, orbits with initial conditions inside the elongated spiral bands need only a couple of intersection until they escape.
Before closing this section, we would like to emphasize that orbits with initial conditions outside the unstable Lyapunov orbits do not necessarily escape immediately from the dynamical system. In Figs. \[orbs\](a-d) we present one characteristic example for each Hamiltonian and in Table \[table1\] we provide the exact initial conditions, the escape period and the value of the energy for all the depicted orbits. We observe that even though all orbit are initiated outside but relatively close to one of the unstable Lyapunov orbits that bridge the escape channels they do not escape right away from the system. On the other hand, they enter the interior region and only after some non-zero time units of chaotic motion they eventually escape from one of the exit channels. Moreover, another interesting fact is that all four orbits escape from channels which do not coincide with the original at which they have been initiated. Thus it is evident that the initial position itself does not furnish a sufficient condition for escape, since the escape criterion is in fact a combination of the coordinates and the velocity of the test particles. More computational details regarding the escape criteria can be found in Appendix B.
Conclusions and discussion {#disc}
==========================
The aim of this work was to numerically investigate the escape dynamics in open Hamiltonian systems with multiple exit channels of escape. This type of dynamical systems has the key feature of having a finite energy of escape. In particular, for energies smaller than the escape value, the equipotential surfaces are closed and therefore escape is impossible. For energy levels larger than the escape energy however, the equipotential surfaces open and several channels of escape appear through which the test particles are free to escape to infinity. Here we should emphasize that if a test particle has energy larger than the escape value, this does not necessarily mean that the test particle will certainly escape from the system and even if escape does occur, the time required for an orbit to cross an unstable Lyapunov orbit and hence escape to infinity may be very long compared with the natural crossing time. The non-integrable part of the Hamiltonian containing the perturbation terms affects significantly the structure of the equipotential surface and determines the exact number of the escape channels in the configuration space. In Part I, we chose such perturbing terms creating between two and four escape channels, while here in Part II the escape channels in the $(x,y)$ plane vary between five and eight. Here we would like to emphasize that in this paper we introduce and explore for the first time potential functions that correspond to Hamiltonian systems with more than four escape channels and this is the main novelty of our work.
We defined for several values of the total orbital energy dense, uniform grids of initial conditions regularly distributed in the area allowed by the corresponding value of the energy in both the configuration $(x,y)$ and the phase $(x,\dot{x})$ space. In both cases, the density of the grids was controlled in such a way that always there were about 50000 orbits to be examined. For the numerical integration of the orbits in each grid, we needed roughly between 1 minute and 6 days of CPU time on a Pentium Dual-Core 2.2 GHz PC, depending both on the amount of trapped orbits and on the escape rates of orbits in each case. For each initial condition, the maximum time of the numerical integration was set to be equal to $10^5$ time units however, when a test particle escapes the numerical integration is effectively ended and proceeds to the next initial condition.
By conducting a thorough and systematical numerical investigation we successfully revealed the structure of both the configuration and the phase space. In particular, we managed to distinguish between trapped (non-escaping) and escaping orbits and we located the basins of escape leading to different exit channels, also finding correlations with the corresponding escape times of the orbits. Among the escaping orbits, we separated between those escaping fast or late from the system. Our extensive numerical calculations strongly suggest that the overall escape process is very dependent on the value of the total orbital energy. The main numerical results of our investigation can be summarized as follows:
1. In all four Hamiltonian systems studied, areas of non-escaping orbits and regions of initial conditions leading to escape in a given direction (basins of escape), were found to exist in both the configuration and the phase space. The several escape basins are very intricately interwoven and they appear either as well-defined broad regions or thin elongated spiral bands. Regions of trapped orbits first and foremost correspond to stability islands of regular orbits where a third adelphic integral of motion is present.
2. We observed that in several exit regions the escape process is highly sensitive dependent on the initial conditions, which means that a minor change in the initial conditions of an orbit leads the test particle to escape through another exit channel. These regions are the opposite of the escape basins, are completely intertwined with respect to each other (fractal structure) and are mainly located in the vicinity of stability islands. This sensitivity towards slight changes in the initial conditions in the fractal regions implies that it is impossible to predict through which exit the particle will escape.
3. A strong correlation between the extent of the basins of escape and the value of the total orbital energy $h$ was found to exists. Indeed, for low values of $h$ the structure of both the configuration and the phase space exhibits a large degree of fractalization and therefore the majority of orbits escape choosing randomly escape channels. As the value of $h$ increases however, the structure becomes less and less fractal and several basins of escape emerge. The extent of these basins of escape is more prominent at relatively high energy levels, where they occupy about nine tenths of the entire area on the girds.
4. Our numerical computations revealed that the escape times of orbits are directly linked to the basins of escape. In particular, inside the basins of escape as well as relatively away from the fractal domains, the shortest escape rates of the orbits had been measured. On the other hand, the longest escape periods correspond to initial conditions of orbits in the vicinity of stability islands or inside the fractal structures. It was also found that as we proceed to high energy levels far above the escape energy the proportion of fast escaping orbits increases significantly. This phenomenon can be justified, if we take into account that with increasing energy the exit channels on the equipotential surfaces become more and more wide thus the test particles can find easily and faster one of the exits and escape to infinity.
5. We provided numerical evidence that our open Hamiltonian systems have a strong topological property, known as the Wada property. This means that any initial condition that is on the boundary of an escape basin, is also simultaneously on the boundary of at leats other two basins of escape. We also concluded that if a dynamical system verifies the property of Wada, the unpredictability is even stronger than if it only had fractal basin boundaries.
6. In all four examined cases, we identified a small portion of chaotic orbits with initial conditions close enough to the outermost KAM islands which remain trapped in the neighbourhood of these islands for vast time intervals having sticky periods which correspond to hundreds of thousands time units. It should be pointed out however, that the amount of these trapped chaotic orbits is significantly smaller that that reported in the case of four exit channels of Part I.
7. In both the configuration as well the phase space we reported the existence of streams of initial conditions which correspond to orbits that start outside the unstable Lyapunov orbits then they enter the interior region and finally escape from some escape channel which however do not coincide with the original one in which they have been initiated. These streams flow from the inside to the outside of the equipotential surfaces and extend asymptotically to infinity.
We hope that the present numerical analysis to be useful in the active field of open Hamiltonian systems which may have implications in different aspects of chaotic scattering with applications in several areas of physics. For example, we related the current model potential with applications in the field of reactive multichannel scattering. Moreover, it is in our future plans to expand our investigation in other more complicated potentials, focusing our interest in reveling the escape mechanism of stars in galactic systems such as star clusters, binary stellar systems, or barred spiral galaxies.
Acknowledgments {#acknowledgments .unnumbered}
===============
I would like to express my warmest thanks to Prof. James D. Meiss and Jacobo Aguirre for all the illuminating and inspiring discussions during this research and also to Prof. Christof Jung for pointing out the interesting subject of multichannel chaotic scattering. My thanks also go to the two anonymous referees for the careful reading of the manuscript and for all the apt suggestions and comments which allowed us to improve both the quality and the clarity of the paper.
Compliance with Ethical Standards {#compliance-with-ethical-standards .unnumbered}
=================================
- Funding: The author states that he has not received any research grants.
- Conflict of interest: The author declares that he has no conflict of interest.
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APPENDIX A: LIST OF PERTURBATION FUNCTIONS {#apex1 .unnumbered}
==========================================
In the following Table \[table2\] we provide the equations containing the perturbing terms derived by the generating functions (\[gens\]), for the first nine cases, that is when the Hamiltonian system has between two and ten channels of escape in the configuration $(x,y)$ space. Note that in Part I for the case of four exits we adopted the perturbation function $V_1(x,y) = - x^2 y^2$, simply because it was also used in many earlier works, while in Table (\[table2\]) we give the general function according to the corresponding generating function.
Channels Perturbation function $V_1(x,y)$
---------- ------------------------------------------------------------------------------------------------
$n = 2$ $V_1 = - x y^2$
$n = 3$ $V_1 = - \frac{1}{3} (x^3 - 3 x y^2)$
$n = 4$ $V_1 = - \frac{1}{4} (x^4 - 6 x^2 y^2 + y^4)$
$n = 5$ $V_1 = - \frac{1}{5} (x^5 - 10 x^3 y^2 + 5 x y^4)$
$n = 6$ $V_1 = - \frac{1}{6} (x^6 + 15 x^4 y^2 - 15 x^2 y^4 + y^6)$
$n = 7$ $V_1 = - \frac{1}{7} (x^7 - 21 x^5 y^2 + 35 x^3 y^4 - 7 x y^4)$
$n = 8$ $V_1 = - \frac{1}{8} (x^8 - 28 x^6 y^2 + 70 x^4 y^4 - 28 x^2 y^6 + y^8)$
$n = 9$ $V_1 = - \frac{1}{9} (x^9 - 36 x^7 y^2 + 126 x^5 y^4 - 84 x^3 y^6 + 9 x y^8)$
$n = 10$ $V_1 = - \frac{1}{10} (x^{10} - 45 x^8 y^2 + 210 x^6 y^4 - 210 x^4 y^6 + 45 x^2 y^8 - y^{10})$
: Equations of perturbing terms when $n \in [2, 10]$.[]{data-label="table2"}
APPENDIX B: ESCAPE PROCEDURE & CRITERIA {#apex2 .unnumbered}
=======================================
Here we would like to present a step by step explanation of the escape procedure of orbits and analyze all the corresponding computational aspects. We consider the case of the Hamiltonian system with eight channels of escape (obviously in all other cases with less escape channels things are much simpler) and we choose the energy level $h = 0.45 > h_{esc}$. In Fig. \[ang\] the corresponding equipotential curve is shown in black, while the eight unstable Lyapunov orbits are denoted using red color. The initial conditions $(x_0,y_0)$ of orbits in the configuration space are divided into two main categories: (i) orbits with initial conditions in the interior region (green), that is inside the Lyapunov orbits and (ii) orbits initiated at the exterior region (yellow), that is outside the Lyapunov orbits. The gray regions on the other hand, correspond to the forbidden area where motion is impossible.
![The equipotential curve for the Hamiltonian with eight channels of escape when $h = 0.45$ is shown in black color, while the unstable Lyapunov orbits are indicated with red color. The configuration plane is divided into three domains: (i) the interior region (green), the exterior region (yellow) and (iii) the forbidden regions (gray). The blue straight lines define the angular sectors for each channel of escape, while the dashed, magenta line corresponds to the limiting circle.[]{data-label="ang"}](angles.pdf){width="\hsize"}
Let us first deal with the orbits initiated in the interior region. It is evident from Fig. \[ang\] that the escape channels are very close to one another and this behavior becomes stronger in Hamiltonians with more exits $(n > 8)$. However, in any case, it is possible to define appropriate angles that embrace each channel as it is seen in Fig. \[ang\]. Due to the overall symmetry of the dynamical system it is $\theta_1 = \theta_2 = \theta_3 = \theta_4 = \theta_5 = \theta_6 = \theta_7 = \theta_8 = 30^{\circ}$. Now we need to determine where each angle starts and where it ends so as to divide the configuration space into eight angular sectors. For this purpose, we define a polar angle which starts counting from the $x$-axis $(y = 0)$. Then we have for each sector
sector 1: $\theta_1 < 15$ or $\theta_1 > 345$,
sector 2: $30 < \theta_2 < 60$,
sector 3: $75 < \theta_3 < 105$,
sector 4: $120 < \theta_4 < 150$,
sector 5: $165 < \theta_5 < 195$,
sector 6: $210 < \theta_6 < 240$,
sector 7: $255 < \theta_7 < 285$,
sector 8: $300 < \theta_8 < 330$.
Along each time step of the numerical integration we monitor the position of the test particle given by the coordinates $(x,y)$ as well its velocity vector. When a test particle crosses one of the Lyapunov orbits with velocity pointing outwards then the escape takes place. In order to determine through which exit channel, or in other words through which sector the orbit has escaped we need to calculate the corresponding angle through the Cartesian coordinates. Therefore we define $z = y/x$ and the polar angle reads $$\theta = \left\{
\begin{array}{lr}
\tan^{-1}(z), &\mbox{ if $x > 0$ and $y \geq 0$,} \\
\tan^{-1}(z) + \pi, &\mbox{ if $x < 0$,} \\
\tan^{-1}(z) + 2\pi, &\mbox{ if $x > 0$ and $y < 0$,}
\end{array} \right.
\label{angz}$$ where the output is given in radians. We can easily transform the result into degrees by multiplying with $180^{\circ} \theta/\pi$. Thus following this procedure we can determine the exit channels of orbits initiated in the interior region.
Orbits with initial conditions outside the unstable Lyapunov orbits exhibit a different behavior. In Fig. \[orbs\](a-d) we saw that orbits with initial conditions in the exterior region do not escape directly to infinity but on the other hand they enter the interior region and after some countable (non-zero) time they escape. For this type of orbits we use the above-mentioned technique for determining the exact channel of escape. However, the vast majority of orbits with initial conditions in the exterior region escape directly to infinity without entering the interior region and therefore crossing any Lyapunov orbit. In this case, we consider an orbit to escape when $x^2 + y^2 > q$, where $q$ is a real number depending in the particular dynamical system (for $n = 8$ we have $q = 7$). We may say that the equality $x^2 + y^2 = q$ defines a limiting circle that determines the escape of orbits initiated in the exterior region.
![Magnification of channel 7 when $h = 0.45$. The stream of blue initial conditions corresponding to exit 3, flows outside the unstable Lyapunov orbit (red) and extends vertically to infinity.[]{data-label="flow"}](flow.pdf){width="\hsize"}
When studying the escape dynamics of the configuration space we found the existence of streams of initial conditions which correspond to orbits that start outside the Lyapunov orbits, then they enter the interior region and finally escape from an exit which however do not coincide with the original one in which they have been initiated. Fig. \[flow\] shows a magnification of channel 7 when $h = 0.45$. We observe the stream of blue initial conditions corresponding to exit 3, that flows outside the unstable Lyapunov orbit and extends vertically to infinity. In the same figure we plotted the limiting circle for $q = 7$. It is evident that the value of $q$ strongly depends on the size of the grid. In our calculations we considered in all four cases initial conditions of orbits inside the square area $-2 \leq x \leq 2$ and $-2 \leq y \leq 2$. For creating Fig. \[flow\] where $y_{max} = -3$, we increased the radius of the limiting circle to $q = 12$, in order to correctly determine the escape process of orbits in the outflow stream.
[^1]: The perturbation function for the case $n = 2$ is not derived by the generating function.
[^2]: The fat-fractal exponent increases, approaching the value 1 which means no fractal geometry, when the energy of the system is high enough (see \[[@BBS08]\]).
[^3]: From chaos theory we expect an infinite number of islands of (stable) quasi-periodic (or small scale chaotic) motion.
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- Mohammed Bachir
title: 'On representations of isometric isomorphisms between some monoid of functions.'
---
[*Laboratoire SAMM 4543, Université Paris 1 Panthéon-Sorbonne, Centre P.M.F. 90 rue Tolbiac 75634 Paris cedex 13*]{}
[*Email : [email protected]*]{}
**Abstract.** We prove that each isometric isomorphism, between the monoids of all nonegative $1$-Lipschitz maps defined on invariant metric groups and equiped with the inf-convolution law, is given canonically from an isometric isomorphism between their groups of units.
[**Keyword, phrase:**]{} Inf-convolution; $1$-Lipschitz; isometries; group and monoid structure; Banach-Stone theorem.\
[**2010 Mathematics Subject:**]{} 46T99; 26E99; 20M32; 47B33.\
Introduction. {#introduction. .unnumbered}
=============
Given a metric space $(X,d)$, we denote by $Lip^1_+(X)$ the set of all nonnegative $1$-Lipschitz maps on $X$ equipped with the metric $$\rho(f,g):= \sup_{x\in X} \frac{|f(x)-g(x)|}{1+|f(x)-g(x)|}, \hspace{2mm} \forall f,g\in Lip^1_+(X).$$
If $X$ is a group and $f, g: X\longrightarrow \R$ are two functions, the inf-convolution of $f$ and $g$ is defined by the following formula $$\begin{aligned}
\label{Al}
(f\oplus g) (x) &:=& \inf_{y,z\in X/ yz=x}\left\{f(y)+g(z)\right\}; \hspace{3mm}\forall x\in X.\end{aligned}$$
We recall the following definition.
\[invgroup\] Let $(X,d)$ be a metric group. We say that $(X,d)$ is invariant if and only if, $$d(xy,xz)=d(yx,zx)=d(y,z), \hspace{2mm} \forall x, y, z \in X.$$ If moreover $X$ is complete for the metric $d$, then we say that $(X,d)$ is an invariant complete metric group.
Examples of invariant metric groups are given in [@Ba1]. In all the paper $(X,d)$ and $(Y,d')$ will be assumed to be invariant metric groups having respectively $e$ and $e'$ as identity element and $(\overline{X},\overline{d})$ (resp. $(\overline{Y},\overline{d'})$) denotes the group completion of $(X,d)$ (resp. of $(Y,d')$).
Recently, we established in [@Ba1] that the set $(Lip^1_+(X),\oplus)$ enjoys a monoid structure, having the map $\delta_e: x\mapsto d(x,e)$ as identity element and that the group completion $(\overline{X},\overline{d})$ of $(X,d)$ is completely determined by the metric monoid structure of $(Lip^1_+(X),\oplus, \rho)$. In other words, $(Lip^1_+(X),\oplus, \rho)$ and $(Lip^1_+(Y),\oplus, \rho)$ are isometrically isomorphic as monoids if and only if, $(\overline{X},\overline{d})$ and $(\overline{Y},\overline{d'})$ are isometrically isomorphic as groups. Also, we proved that the group of all invertible elements of $Lip^1_+(X)$ coincides, up to isometric isomorphism, with the group completion $\overline{X}$ (See \[Theorem 1., [@Ba1]\] and \[Theorem 2., [@Ba1]\]). The main result of [@Ba1], gives a Banach-Stone type theorem. The representations of isometries between Banach spaces of Lipschitz maps defined on metric spaces and equipped with their natural norms, was considered by several authors [@GJ], [@W], [@JV]. In general, such isometries are given, under some conditions, canonically as a composition operators. Other Banach-Stone type theorems are also given for unital vector lattices structure [@GJ1]. In this note, we provide the following result which gives complete representations of isometric isomorphisms for the monoid structure between $Lip^1_+(X)$ and $Lip^1_+(Y)$. Our result complement those given in [@Ba1] and [@Ba].
\[Thm0\] Let $(X,d)$ and $(Y,d')$ be two invariant metric groups. Let $\Phi$ be a map from $(Lip^1_+(X),\oplus, \rho)$ into $(Lip^1_+(Y),\oplus, \rho)$. Then the following assertions are equivalent.
$(1)$ $\Phi$ is an isometric isomorphism of monoids
$(2)$ there exists an isometric isomorphism of groups $T: (\overline{X},\overline{d})\longrightarrow (\overline{Y},\overline{d'})$ such that $\Phi (f)=(\overline{f}\circ T^{-1})_{|Y}$ for all $f\in Lip^1_+(X)$, where $\overline{f}$ denotes the unique $1$-Lipschitz extenstion of $f$ to $\overline{X}$ and $(\overline{f}\circ T^{-1})_{|Y}$ denotes the restriction of $\overline{f}\circ T^{-1}$ to $Y$.
If $A$ (resp. X) is a metric monoid (resp. a metric group), by $Is_m(A)$ (resp. $Is_g(X)$) we denote the group of all isometric automorphism of the monoid $A$ (resp. of the group $X$). The symbol ”$\simeq$” means “[*isomorphic as groups*]{}”. An immediate consequence of Theorem \[Thm0\] is given in the following corollary.
\[Corintro\] Let $(X,d)$ be an invariant metric group. Then, $$Is_m(Lip^1_+(X)) \simeq Is_g(\overline{X}).$$
As application of the results of this note, we discover new semigroups law on $\R^n$ (different from the usual operation $+$) having some nice properties. We treat this question in Example \[Exemp1\] at Section \[S4\], where it is shown that each finite group structure $(G, \cdot)$, extend canonically to a semigroup structure on $\R^{n}$ (where $n$ is the cardinal of $G$). In other words, there always exists a semigroup law $\star_G$ on $\R^{n}$ and an injective group morphism $i$ from $(G, \cdot)$ into $(\R^{n},\star_G)$ such that the maximal subgroup of $(\R^{n},\star_G)$ having $e:=(0,1,1,...,1)$ as identity element is isomorphic to the group $G\times \R$. The idea is simply based on the use of the results of this paper and the identification between $(\R^{n},\star_G)$ and $(Lip(G),\oplus)$ where $G$ is equiped with the discrete metric, and $Lip(G)$ denotes the space of all Lipschitz map on $G$.
This note is organized as follows. Section \[S0\] concern the proof of Theorem \[Thm0\] and is divided on two subsections: in Subsection \[S1\] we prove some useful lemmas and in Subsection \[S2\], we give the proof of the main result Theorem \[Thm0\]. In Section \[S3\], we give some properties of the group of invertible elements for the inf-convolution law. In section \[S4\], we review the results of this paper in the algebraic case.
Proof of the main result. {#S0}
=========================
Preliminary results {#S1}
-------------------
We follow the notation of [@Ba1]. For each fixed point $x\in X$, the map $\delta_{x}$ is defined from $X$ into $\R$ as follows $$\begin{aligned}
\delta_{x}: X &\rightarrow& \R \nonumber\\
z &\mapsto& d(z,x)=d(zx^{-1},e).\nonumber\end{aligned}$$
We define the subset $\mathcal{G}(X)$ of $Lip^1_+(X)$ as follows $$\mathcal{G}(X):=\left\{\delta_{x}: x\in X\right\}\subset Lip^1_+(X).$$
We consider the operator $\gamma_X$ defined as follows $$\begin{aligned}
\gamma_X : X & \rightarrow & \mathcal{G}(X) \nonumber \\
x & \mapsto & \delta_{x} \nonumber \end{aligned}$$
We are going to prove some lemmas.
\[lem3\] Let $(X,d)$ and $(Y,d')$ be two invariant complete metric groups having respectively $e$ and $e'$ as identity elements. Let $\Phi$ be a map from $(Lip^1_+(X),\oplus, \rho)$ onto $(Lip^1_+(Y),\oplus, \rho)$ which is an isometric isomorphism of monoids. Then, the following asserions holds.
$(1)$ for all $f\in Lip^1_+(X)$, $\inf_{Y} \Phi(f)=\inf_{X} f$ and for all $r\in \R^+$, $\Phi(r)=r$.
$(2)$ there exists an isometric isomorphism of groups $T : (X,d)\longrightarrow (Y,d')$ such that $\Phi(r+\delta_x)=r+\delta_{T(x)}=r+ \delta_x \circ T^{-1}$, for all $r\in \R^+$ and for all $x\in X$.
$(3)$ $\Phi(f+r)=\Phi(f)+ r$, for all $f\in Lip^1_+(X)$ and for all $r\in \R^+$.
Since an isomorphism of monoids, sends the group of unit onto the group of unit, then using \[Theorem 1., [@Ba1]\], the restriction $T_1:=\Phi_{|\mathcal{G}(X)}$ is an isometric group isomorphism from $\mathcal{G}(X)$ onto $\mathcal{G}(Y)$. On the other hand, the map $\gamma_X : X\longrightarrow \mathcal{G}(X)$ gives an isometric group isomorphism by \[Lemma 2., [@Ba1]\]. Thus, the map $T:=\gamma^{-1}_Y \circ T_1 \circ \gamma_X$, gives an isometric group isomorphism from $X$ onto $Y$ and we have that for all $x\in X$, $\Phi(\delta_x):=T_1(\delta_x)=T_1 \circ \gamma_X(x)=\gamma_Y\circ T(x)=\delta_{T(x)}=\delta_x \circ T^{-1}$.
We prove the part $(1)$. Note that $f\oplus 0=0\oplus f=\inf_{x\in X} f$ for all $f\in Lip^1_+(X)$. First, we prove that $\Phi(0)=0$. Indeed, for all $x\in X$, we have that $0\oplus \delta_x=0$. Thus, $\Phi(0)=\Phi(0)\oplus \Phi(\delta_x)=\Phi(0)\oplus\delta_{Tx}$. Using the surjectivity of $T$, we obtain that for all $y\in Y$, we have that $\Phi(0)=\Phi(0)\oplus \delta_y$. So, using the definition of the if-convolution, we get $\Phi(0)(z)=\inf_{ts=z}\lbrace \Phi(0)(t)+\delta_{y}(s) \rbrace \leq \Phi(0)(zy^{-1})$ for all $y,z\in Y$. By taking the infinimum over $y\in Y$, we obtain that $\Phi(0)(z)\leq \inf_Y \Phi(0)$, for all $z\in Y$. It follows that $\Phi(0)=\inf_Y \Phi(0)$ is a constant function. Now, since $\Phi(0)$ is a constant function, we have $2\Phi(0)= \Phi(0)\oplus \Phi(0)=\Phi(0\oplus 0)=\Phi(0)$, it follows that $\Phi(0)=0$. Finaly, we prove that $\Phi(r)=r$ for all $r\in \R^+$. Indeed, since $r\oplus 0=r$ and $\Phi(0)=0$, it follows that $\Phi(r)=\Phi(r)\oplus 0=\inf_Y \Phi(r)$, which implies that $\Phi(r)$ is a constant function. Using the fat that $\Phi$ is an isometry, we get that $\rho(\Phi(r),0)=\rho(\Phi(r),\Phi(0))=\rho(r,0)$. In other word, $\frac{\Phi(r)}{1+\Phi(r)} =\frac{r}{1+r}$, which implies that $\Phi(r)=r$. Now, we have $\inf_{y\in Y} \Phi(f)=\Phi(f)\oplus 0=\Phi(f)\oplus \Phi(0)=\Phi(f\oplus 0)=\Phi(\inf_{x\in X} f)=\inf_{x\in X} f$.
We prove the part $(2)$. Let $r\in \R^+$ and set $g=\Phi(r+\delta_e)\in Lip^1_+(Y)$. We first prove that $g=r+\delta_{e'}$. Using the part $(1)$, we have that $r=\Phi(r)=\Phi(\inf_{x\in X}(r+ \delta_e))=\inf_{y\in Y} \Phi(r+\delta_e)\leq \Phi(r+\delta_e)=g$. Thus $g-r\geq 0$ and so $g-r \in Lip^1_+(Y)$. On the other hand, since $Lip^1_+(Y)$ is a monoid having $\delta_{e'}$ as identity element, we have that $g=(g-r)\oplus(r+\delta_{e'})=(r+\delta_{e'})\oplus (g-r)$. Now, since $\Phi^{-1}$ is a monoid morphism, we get that $$\begin{aligned}
r+\delta_e &=& \Phi^{-1}(g)\\
&=&\Phi^{-1}(g-r)\oplus \Phi^{-1}(r+\delta_{e'})=\Phi^{-1}(r+\delta_{e'})\oplus \Phi^{-1}(g-r).\end{aligned}$$ As above we prove that $\Phi^{-1}(r+\delta_{e'})-r\geq 0$. Thus, $\Phi^{-1}(r+\delta_{e'})-r\in Lip^1_+(X)$. Since $r$ is a constant function, the above equality is equivalent to the following one $$\begin{aligned}
\delta_e &=&\Phi^{-1}(g-r)\oplus (\Phi^{-1}(r+\delta_{e'})-r)=(\Phi^{-1}(r+\delta_{e'})-r)\oplus \Phi^{-1}(g-r).\end{aligned}$$ Since from \[Theorem 1, [@Ba1]\], the invertible element in $Lip^1_+(X)$ are exactely the element of $\mathcal{G}(X)$ and since $\mathcal{G}(X)$ is a group by \[Lemma 2, [@Ba1]\], we deduce from the above equality that $\Phi^{-1}(r+\delta_{e'})-r \in \mathcal{G}(X)$ and $\Phi^{-1}(g-r) \in \mathcal{G}(X)$ and there exists $\alpha(r), \beta(r) \in X$ such that $$\left\{
\begin{array}{cl}
&e = \alpha(r)\beta(r)\\
&\Phi^{-1}(r+\delta_{e'})-r = \delta_{\alpha(r)}\\
&\Phi^{-1}(g-r) = \delta_{\beta(r)}
\end{array}
\right.$$ This implies that $$\label{eq0}
\left\{
\begin{array}{cl}
&e = \alpha(r)\beta(r)\\
&\Phi(r+\delta_{\alpha(r)})=r+ \delta_{e'}\\
& g= r+\Phi(\delta_{\beta(r)}) =r+\delta_{T(\beta(r))}
\end{array}
\right.$$ We need to prove that $\alpha(r)=\beta(r)=e$ for all $r\in \R^+$. Indeed, since $\Phi$ is an isometry, we have that $$\begin{aligned}
\rho(\Phi(r+\delta_{\alpha(r)}),\Phi(\delta_e))=\rho(r+\delta_{\alpha(r)},\delta_e).\end{aligned}$$ Using the above formula, the second equations in (\[eq0\]) and the definition of the metric $\rho$ with the fact that $\Phi(\delta_e)=\delta_{e'}$, we get
$$\begin{aligned}
\frac{r}{1+r}&=&\rho(r+ \delta_{e'},\delta_{e'})\\
&=&\rho(\Phi(r+\delta_{\alpha(r)}),\Phi(\delta_e))\\
&=&\rho(r+\delta_{\alpha(r)},\delta_e)\\
&=&\sup_{t\in X}\frac{|r+\delta_{\alpha(r)}(t)-\delta_e(t)|}{1+|r+\delta_{\alpha(r)}(t)-\delta_e(t)|}.\\
&\geq& \frac{r+\delta_{\alpha(r)}(e)}{1+r+\delta_{\alpha(r)}(e)}\end{aligned}$$
A simple computation of the above inequality, gives that $\delta_{\alpha(r)}(e)\leq 0$ i.e. $d(\alpha(r),e)\leq 0$. In other word, we have that $\alpha(r)=e$ for all $r\in \R^{+}$. On the other hand, using the first equation of (\[eq0\]), we get that $\beta(r)=e$ for all $r\in \R^{+}$. It follows from the equation (\[eq0\]) that $\Phi(r+\delta_e)=r+\delta_{e'}$ for all $r\in \R^+$. Now, it is easy to see that for all $r\in R^+$ and all $x\in X$ we have $$\begin{aligned}
r+\delta_x=(r+\delta_e)\oplus \delta_x.\end{aligned}$$ It follows that $$\begin{aligned}
\Phi(r+\delta_x)&=&\Phi(r+\delta_e)\oplus \Phi(\delta_x)\\
&=& (r+\delta_{e'})\oplus \delta_{T(x)}\\
&=& r+\delta_{T(x)}\end{aligned}$$ Since $T$ is isometric, we obtain that $\Phi(r+\delta_x)=r+\delta_{T(x)}=r+\delta_{x}\circ T^{-1}$.
Now, we prove the part $(3)$. Let $f \in Lip^1_+(X)$ and $r\in \R^+$. It is easy to see that $f+r=f\oplus (r+\delta_e)$. So, using the part $(2)$, we obtain that $\Phi(f+r)=\Phi(f)\oplus \Phi(r+\delta_e)=\Phi(f)\oplus (r+\delta_{e'})=\Phi(f)+r$.
\[lem1.2\] Let $(X,d)$ be an invariant metric group. Let $f\in Lip^1_+(X)$. Then, for all $x \in X$ and all positive real number $a$ such that $a\geq f(x)$, we have that $$f(x)= (\inf(\delta_e,a)\oplus f)(x).$$
Let $x\in X$ and $a \geq 0$ such that $f(x)\leq a$. We have that $$\begin{aligned}
(\inf(\delta_e,a) \oplus f)(x) = \inf_{t \in X} \lbrace \inf (d(xt^{-1},e),a) + f(t) \rbrace\\
= \inf_{t \in X} \lbrace f(t) + \inf (d(t,x),a)\rbrace\\
= \min \lbrace \inf_{t\in X/ d(t,x)\leq a} \lbrace f(t) + \inf (d(t,x),a)\rbrace ; \\
\inf_{t\in X/ d(t,x)\geq a} \lbrace f(t) + \inf (d(t,x), a)\rbrace \rbrace\\
= \min \lbrace \inf_{t/ d(t,x)\leq a} \lbrace f(t) + d(t,x)\rbrace, \inf_{t/ d(t,x)\geq a} \lbrace f(t) + a\rbrace \rbrace.\end{aligned}$$
Since $f$ is $1$-Lipschitz we have that $ f(x)=\inf_{t/ d(t,x)\leq a} \lbrace f(t) + d(t,x)\rbrace$. It follows that
$$\begin{aligned}
(\inf(\delta_e,a) \oplus f)(x) &=& \min \lbrace f(x), \inf_{t/ d(t,x)\geq a} \lbrace f(t)\rbrace +a \rbrace\\
&=& f(x).\end{aligned}$$
\[lem1.3\] Let $(X,d)$ be an invariant metric group. Then, the following assertions hold.
$(1)$ for each $f\in Lip^1_+(X)$ and for each bounded function $h\in Lip^1_+(X)$, the function $f\oplus h \in Lip^1_+(X)$ is bounded.
$(2)$ Let $f, g \in Lip^1_+(X)$, then the following assertions are equivalent.
$(a)$ $f\leq g$
$(b)$ $h\oplus f \leq h \oplus g$, for all function $h \in Lip^1_+(X)$ which is bounded.
$(1)$ Since $0 \leq f\oplus h (x)\leq f(e)+h(x)$ for all $x\in X$ and since $h$ is bounded, it follows that $f\oplus h$ is bounded. On the other hand, $f\oplus h \in Lip^1_+(X)$ since $Lip^1_+(X)$ is a monoide.
$(2)$ The part $(a) \Longrightarrow (b)$ is easy. Let us prove the part $(b) \Longrightarrow (a)$. Indeed, let $x\in X$ and chose a positive real number $a\geq \max(f(x), g(x))$. Set $h:= \inf(\delta_e,a)$. It is clear that $h \in Lip^1_+(X)$ and is bounded. So, from the hypothesis $(b)$ we have that $(\inf(\delta_e,a)\oplus f) \leq (\inf(\delta_e,a)\oplus g)$. Using Lemme \[lem1.2\], we obtain that $f(x) \leq g (x)$.
\[lem1\] Let $A$ be a nonempty set and $f,g :A\longrightarrow \R$ be two functions. Then, the following assertions are equivalent.
$(1)$ $\sup_{x\in A}|f(x)-g(x)| <+\infty$.
$(2)$ $\sup_{x\in A} \frac{|f(x)-g(x)|}{1+|f(x)-g(x)|}< 1$.
Suppose that $(1)$ hold. Using \[Lemma 1., [@Ba1]\], we have that $\sup_{x\in A} \frac{|f(x)-g(x)|}{1+|f(x)-g(x)|}=\frac{\sup_{x\in A}|f(x)-g(x)|}{1+\sup_{x\in A}|f(x)-g(x)|} < 1$. Now, suppose that $(2)$ holds. Set $\alpha=\sup_{x\in A} \frac{|f(x)-g(x)|}{1+|f(x)-g(x)|}<1$. Then, we obtain that $|f(x)-g(x)| \leq\frac{\alpha}{1-\alpha}$, for all $x\in A$. This implies that $\sup_{x\in A}|f(x)-g(x)| <+\infty$.
\[lem2\] Let $(X,d)$ and $(Y,d')$ be two invariant complete metric groups. Let $$\Phi: (Lip^1_+(X),\rho)\longrightarrow (Lip^1_+(Y), \rho)$$ be an isometric isomorphism of monoids. Then, for all $f, g \in Lip^1_+(X)$, we have $$f\leq g \Longleftrightarrow \Phi(f) \leq \Phi(g).$$
The proof is divided on two cases.
[*Case1: (The case where $f$ and $g$ are bounded.)*]{} Let $f,g \in Lip^1_+(X)$ be bounded functions. In this case we have $\sup_{x\in X}|f(x)-g(x)|< +\infty$, so using Lemma \[lem1\] and the fact that $\Phi$ is isometric, we get also that $\sup_{y\in Y}|\Phi(f)(y)-\Phi(g)(y)|< +\infty$. Using \[Lemma 1. [@Ba1]\] and the fact that $\Phi$ is isometric, we obtain that $$\frac{\sup_{y\in Y}|\Phi(f)(y)-\Phi(g)(y)|}{1+\sup_{y\in Y}|\Phi(f)(y)-\Phi(g)(y)|}=\frac{\sup_{x\in X}|f(x)-g(x)|}{1+\sup_{x\in X}|f(x)-g(x)|}.$$ This implies that $$\sup_{y\in Y}|\Phi(f)(y)-\Phi(g)(y)|=\sup_{x\in X}|f(x)-g(x)|.$$ Set $r:=\sup_{y\in Y}|\Phi(f)(y)-\Phi(g)(y)|=\sup_{x\in X}|f(x)-g(x)|<+\infty$. By applying the above arguments to $f+r$ and $g$ which are bounded, we also get that $$\sup_{y\in Y}|\Phi(f+r)(y)-\Phi(g)(y)|=\sup_{x\in X}|(f+r)(x)-g(x)|.$$ Using the fact that $\Phi(f+r)=\Phi(f)+r$ (by Lemma \[lem3\]) and the choice of the number $r$, we get that $$\sup_{x\in X}\lbrace \Phi(f)(x)-\Phi(g)(x) +r \rbrace=\sup_{x\in X}\lbrace f(x)-g(x)+r \rbrace$$ which implies that $$\sup_{y\in Y}\lbrace \Phi(f)(y)-\Phi(g)(y) \rbrace=\sup_{x\in X}\lbrace f(x)-g(x)\rbrace.$$ It follows that $f\leq g \Longleftrightarrow \Phi(f) \leq \Phi(g).$ Replacing $\Phi$ by $\Phi^{-1}$ we also have $k\leq l \Longleftrightarrow \Phi^{-1}(k) \leq \Phi^{-1}(l),$ for all bounded functions $k, l \in Lip^1_+(Y)$.
[*Case2: (The general case.)*]{} First, note that for each bounded function $k\in Lip^1_+(Y)$, we have that $\Phi^{-1}(k) \in Lip^1_+(X)$ is bounded. Indeed, there exists $r\in \R^+$ such that $0\leq k \leq r$. Using the above case, we get that $\Phi^{-1}(0) \leq \Phi^{-1}(k)\leq \Phi^{-1}(r)$. This shows that $\Phi^{-1}(k)$ is bounded, since $\Phi^{-1}(0)=0$ and $\Phi^{-1}(r)=r$ by Lemma \[lem3\].
Now, let $f,g \in Lip^1_+(X)$ be two functions such that $f\leq g$. Let $k\in Lip^1_+(Y)$ be any bounded function. It follows that $f \oplus \Phi^{-1}(k) \leq g \oplus \Phi^{-1}(k)$. From the part $(1)$ of Lemma \[lem1.3\], we have that $f \oplus \Phi^{-1}(k), g \oplus \Phi^{-1}(k) \in Lip^1_+(X)$ are bounded. Using [*Case1.*]{}, we get that $\Phi(f \oplus \Phi^{-1}(k))\leq \Phi(g \oplus \Phi^{-1}(k))$. Since $\Phi$ is a morphism, we have that $\Phi(f) \oplus k \leq \Phi(g) \oplus k$, which implies that $\Phi(f) \leq \Phi(g)$ by using the part $(2)$ of Lemma \[lem1.3\]. The converse is true by changing $\Phi$ by $\Phi^{-1}$.
\[lem4\] Let $(X,d)$ and $(Y,d')$ be two invariant metric groups and let $\Phi$ be a monoid isomorphism $\Phi: (Lip^1_+(X),\oplus, \rho)\longrightarrow (Lip^1_+(Y),\oplus, \rho)$. Then, the following assertions are equivalent.
$(1)$ for all $f, g \in Lip^1_+(X)$, we have that $(f\leq g \Longleftrightarrow \Phi(f)\leq \Phi(g)).$
$(2)$ for all $f', g' \in Lip^1_+(Y)$, we have that $(f'\leq g' \Longleftrightarrow \Phi^{-1}(f')\leq \Phi^{-1}(g')).$
$(3)$ for all familly $(f_i)_{i\in I}\subset Lip^1_+(X)$, where $I$ is any nonempty set, we have $\Phi(\inf_{i\in I} f_i)=\inf_{i\in I} \Phi(f_i)$.
The part $(1)\Longleftrightarrow (2)$ is clear. Let us prove $(1)\Longrightarrow (3)$. Let $(f_i)_{i\in I}\subset Lip^1_+(X)$, where $I$ is any nonempty set. First, it is easy to see that the infinimum of a nonempty familly of nonnegative and $1$-Lipschitz functions is also nonnegative and $1$-Lipschitz function. So, $\inf_{i\in I} f_i \in Lip^1_+(X)$. For all $i\in I$, we have that $\inf_{i\in I} f_i \leq f_i$, which implies by hypothesis that $\Phi(\inf_{i\in I} f_i)\leq \Phi(f_i)$ for all $i\in I$. Consequently we have that $\Phi(\inf_{i\in I} f_i)\leq \inf_{i\in I}\Phi(f_i)$. On the other hand, since $\inf_{i\in I}\Phi(f_i)\leq \Phi(f_i)$ for all $i\in I$, using $(2)$, we have that $\Phi^{-1}(\inf_{i\in I}\Phi(f_i))\leq f_i$, for all $i\in I$. It follows that, $\Phi^{-1}(\inf_{i\in I}\Phi(f_i))\leq \inf_{i\in I} f_i$. Using $(1)$, we obtain that $\inf_{i\in I}\Phi(f_i)\leq \Phi(\inf_{i\in I} f_i)$. Hence, $\inf_{i\in I}\Phi(f_i)= \Phi(\inf_{i\in I} f_i)$. Now, let us prove that $(3)\Longrightarrow (1)$. First, let us show that from $(3)$ we also have that $\Phi^{-1}(\inf_{i\in I} g_i)=\inf_{i\in I} \Phi^{-1}(g_i)$, where $I$ is a nonempty set and $g_i\in Lip^1_+(Y)$ for all $i\in I$. Indeed, since $\Phi$ is bijective, there exists $(f_i)_{i\in I}\subset Lip^1_+(X)$ such that $g_i=\Phi(f_i)$ for all $i\in I$. Thus, $\inf_{i\in I} g_i=\inf_{i\in I} \Phi(f_i)=\Phi(\inf_{i\in I} f_i)=\Phi(\inf_{i\in I} \Phi^{-1}(g_i))$, which implies that $\Phi^{-1}(\inf_{i\in I} g_i)=\inf_{i\in I} \Phi^{-1}(g_i)$. Now, let $f, g \in Lip^1_+(X)$. We have that $f\leq g \Longleftrightarrow f=\inf(f,g)$, so if $f\leq g$ then $\Phi(f)=\Phi(\inf(f,g))=\inf(\Phi(f),\Phi(g))$. This implies that $\Phi(f)\leq \Phi(g)$. Conversely, if $\Phi(f)\leq \Phi(g)$ then $\Phi(f)=\inf(\Phi(f), \Phi(g))$ and so $f=\Phi^{-1}(\Phi(f))=\Phi^{-1}(\inf(\Phi(f), \Phi(g)))=\inf(\Phi^{-1}(\Phi(f)),\Phi^{-1}(\Phi(g)))=\inf(f,g)$. This implies that $f\leq g$.
Proof of the main result. {#S2}
-------------------------
Now, we give the proof of the main result.
We know from \[Lemma 3. , [@Ba1]\] that the map $$\begin{aligned}
\chi_X : (Lip^1_+(X),\oplus,\rho) &\rightarrow& (Lip^1_+(\overline{X}),\oplus,\rho)\nonumber\\
f &\mapsto& \overline{f}\nonumber
\end{aligned}$$ is an isometric isomorphism of monoids, where $\overline{f}$ denotes the unique $1$-Lipschitz extention of $f$ to $\overline{X}$. Let us define the map $\overline{\Phi} : (Lip^1_+(\overline{X}),\oplus,\rho) \longrightarrow (Lip^1_+(\overline{Y}),\oplus,\rho)$ by $\overline{\Phi}:= \chi_Y \circ \Phi \circ \chi^{-1}_X$. Then, $\overline{\Phi}$ is an isometric isomorphism of monoids.
$(1) \Longrightarrow (2)$. Since $Lip^{1}_+(\overline{X})$ is a monoid having $\delta_e: \overline{X}\ni x\mapsto \overline{d}(x,e)$ as identity element, we have that $\overline{f}=\delta_e\oplus \overline{f}$ for all $\overline{f}\in Lip^{1}_+(\overline{X})$. Thus, $\overline{f}=\inf_{t\in \overline{X}}\lbrace \overline{f}(t)+\delta_t\rbrace$ for all $\overline{f}\in Lip^{1}_+(\overline{X})$. Using Lemma \[lem4\] together with Lemma \[lem2\], we have that for all $\overline{f}\in Lip^1_+(\overline{X})$, $\overline{\Phi}(\overline{f})=\overline{\Phi}(\inf_{t\in \overline{X}}\lbrace \overline{f}(t)+\delta_t\rbrace)=\inf_{t\in \overline{X}} \overline{\Phi}(\overline{f}(t)+\delta_t)$. Using Lemma \[lem3\], there exists an isometric isomorphism of groups $T:(\overline{X},\overline{d}) \longrightarrow (\overline{Y},\overline{d'})$ such that $\overline{\Phi}(\overline{f}(t)+\delta_t)=\overline{f}(t)+ \delta_{T(t)}$, for all $t\in \overline{X}$. Thus, we get that $\overline{\Phi}(\overline{f})=\inf_{t\in \overline{X}}\lbrace \overline{f}(t)+ \delta_{T(t)}\rbrace$. Equivalently, for all $y\in \overline{Y}$, we have $$\begin{aligned}
\overline{\Phi}(\overline{f})(y)&=& \inf_{t\in \overline{X}}\lbrace \overline{f}(t)+ \delta_{T(t)}(y)\rbrace\\
&=& \inf_{t\in \overline{X}}\lbrace \overline{f}(t)+ \overline{d'}(y,T(t))\rbrace\\
&=& \inf_{t\in \overline{X}}\lbrace \overline{f}(t)+ \overline{d}(T^{-1}(y),t)\rbrace\\
&=& (\delta_e\oplus \overline{f})(T^{-1}(y))\\
&=& \overline{f}(T^{-1}(y))\\
&=& \overline{f}\circ T^{-1} (y).\end{aligned}$$ From the formulas $\Phi=\chi^{-1}_Y\circ \overline{\Phi}\circ \chi_X$, we get that $\Phi (f)=(\overline{f}\circ T^{-1})_{|Y}$ for all $f\in Lip^1_+(X)$.
$(2) \Longrightarrow (1)$. If $T:(\overline{X},\overline{d}) \longrightarrow (\overline{Y},\overline{d'})$ is an isometric isomorphism of groups, then clearly the map $\overline{\Phi}$ defined by $\overline{\Phi}(\overline{f}):=\overline{f}\circ T^{-1}$ for all $\overline{f}\in Lip^1_+(\overline{X})$, gives an isometric isomorphism from $(Lip^1_+(\overline{X}),\oplus,\rho)$ onto $(Lip^1_+(\overline{Y}),\oplus,\rho)$. Thus, the map $\Phi:=\chi^{-1}_Y\circ \overline{\Phi}\circ \chi_X$ gives an isometric isomorphism from $(Lip^1_+(X),\oplus,\rho)$ onto $(Lip^1_+(Y),\oplus,\rho)$. Now, it clear that $\Phi(f)=(\overline{f}\circ T^{-1})_{|Y}$ for all $f\in Lip^1_+(X)$.
\[rem1\] $(1)$ The description of all isomorphisms seems to be more complicated than the representations of the isometric isomorphisms. Here is two examples of isomorphisms which are not isometric.
$(a)$ The map $\Phi : Lip^1_+(X) \longrightarrow Lip^1_+(X)$ defined by $\Phi(f)=f+\inf_X(f)$ for all $f \in Lip^1_+(X)$, is an isomorphism of monoids which respect the order but is not isometric for $\rho$ (the proof is similar to the proof of \[Theorem 7., [@Ba]\]. Note that we always have $\inf_{X}(f\oplus g)=\inf_X(f)+\inf_Y(g)$).
$(b)$ The map $\Phi : Lip^1_+(\R) \longrightarrow Lip^1_+(\R)$ defined by $\Phi(f)(x)=f(x+\inf_X(f))$ for all $f \in Lip^1_+(\R)$ and all $x\in \R$, is an isomorphism but not isometric for $\rho$.
$(2)$ Following the proof of Theorem \[Thm0\] and changing “$1$-Lipschitz function” by “$1$-Lipschitz and convex function”, we get a positive answer to the problem 2. in [@Ba].
The group of units. {#S3}
===================
In order that the inf-convolution of two functions $f$ and $g$ takes finit values i.e $f\oplus g >-\infty$, we need to assume that $f$ and $g$ are bound from below. Since, we work with Lipschitz maps, for simplicity, we assume in this section, that $(X,d)$ is a bounded invariant metric group. By $Lip^1_0(X)$ we denote the set of all $1$-Lipschitz map $f$ from $X$ into $\R$ such that $\inf_X(f)=0$. By $Lip^1(X)$ (resp. $Lip(X)$, ) we denote the set of all $1$-Lipschitz map (resp. the set of all Lipchitz map) defined from $X$ to $\R$. We have that $$Lip^1_0(X) \subset Lip^1_+(X) \subset Lip^1(X) \subset Lip(X).$$
\[MS\] Let $(X,d)$ be a bounded invariant metric (abelian) group. Then, the sets $Lip^1_0(X)$, $Lip^1_+(X)$ and $Lip^1(X)$ are (abelian) monoids having $\delta_e$ as identity element and $Lip(X)$ is a (abelian) semigroup.
The proof is similar to \[Proposition 1., [@Ba1]\].
Note that since $(X,d)$ is bounded, each function $f \in Lip^1(X)$ (resp. $f \in Lip(X)$) is dounded and so $d_{\infty}(f,g):=\sup_{x\in X}|f(x)-g(x)|< +\infty$ for all $f, g \in Lip^1(X)$ (resp. $f,g \in Lip(X)$). In this case, from \[Lemma 1., [@Ba1]\], we have that $$\rho=\frac{d_{\infty}}{1+d_{\infty}}$$ on $Lip(X)$. We also consider the following metric: $$\theta_{\infty}(f, g):= d_{\infty}(f-\inf_X(f),g-\inf_X(g))+|\inf_X(f)-\inf_X(g)|, \hspace{3mm} \forall f, g \in Lip(X).$$
\[Ptau\] Let $(X,d)$ be a bounded invariant metric group. Then, the following map $$\begin{aligned}
\tau : (Lip^1(X),\theta_{\infty}) &\longrightarrow& (Lip^1_0(X)\times \R,d_{\infty}+|.|)\\
f &\mapsto & (f-\inf_X(f),\inf_X(f)).\end{aligned}$$ is an isomeric isomorphism of monoids, where $Lip^1_+(X)\times \R$ is equiped with the operation $\overline{\oplus}$ defined by $(f,c)\overline{\oplus} (f',c'):=(f\oplus f',c+c')$..
Clearly, $(Lip^1_+(X)\times \R,\overline{\oplus})$ is a monoid having $(\delta_e,0)$ as identity element, since $(Lip^1_+(X),\oplus)$ is a monoid having $\delta_e$ as identity element. It is also clear that $\tau$ is a monoid isomorphism. Now, $\tau$ is isometric by the defintion of $\theta_{\infty}$. It follows that $\tau$ is an isometric isomorphism,
The following proposition gives an alternative way to consider the group completion of invariant metric groups. Recall that if $(M,\cdot)$ is a monoid having $e_M$ as identity element, the group of units of $M$ is the set $$\mathcal{U}(M):=\lbrace m\in M/\hspace{1mm} \exists m'\in M: m\cdot m'=m'\cdot m=e_M \rbrace.$$ The symbol $\cong$ means isometrically isomorphic as groups. We give below an analogue to \[Corollary 1., [@Ba1]\], for each of the spaces $Lip^1_0(X), Lip^1(X)$ and $Lip(X).$ Note that in the part $(1)$ of the following proposition as in \[Corollary 1.,[@Ba1]\], we do not need to assume that $X$ is bounded.
\[Punit\] Let $(X,d)$ be a bounded invariant metric group. Then, we have that
$(1)$ $(\mathcal{U}(Lip^1_0(X)),d_{\infty})=(\mathcal{U}(Lip^1_+(X)),d_{\infty})\cong (\overline{X}, d),$
$(2)$ $(\mathcal{U}(Lip^1(X)),\theta_{\infty})\cong (\overline{X}\times \R, d+ |.|).$
$(3)$ The group $\mathcal{U}(Lip^1(X))$ is the maximal subgroup of the semigroup $Lip(X)$, having $\delta_e$ as identity element.
$(1)$ The fact that $(\mathcal{U}(Lip^1_+(X)),d_{\infty})\cong (\overline{X}, d)$, is given in \[Corollary 1., [@Ba1]\]. On the other hand, since, $\mathcal{G}(X)\subset \mathcal{U}(Lip^1_0(X))\subset \mathcal{U}(Lip^1_+(X))$ and since $\overline{\mathcal{G}(X)}\cong \overline{X}$ (see \[Lemma 2., [@Ba1]\]) we get that $\mathcal{U}(Lip^1_0(X))=\mathcal{U}(Lip^1_+(X))$. Let us prove the part $(2)$. Indeed, since $\tau$ (Proposition \[Ptau\]) is an isometric isomorphism, it sends isometrically the group of units onto the group of units. Hence, from Proposition \[Ptau\] we have $$(\mathcal{U}(Lip^1(X)),\oplus,\theta_{\infty})\cong (\mathcal{U}(Lip^1_0(X)\times \R),\overline{\oplus},d_{\infty}+|.|).$$ Since $\mathcal{U}(Lip^1_0(X)\times \R)=\mathcal{U}(Lip^1_0(X))\times \R$, the conclusion follows from the first part. For the part $(3)$, let $f$ be an element of the maximal group having $\delta_e$ as identity element. Then, $f\oplus \delta_e=f$ and so it follows that $f$ is $1$-lipschitz map i.e $f\in Lip^1(X)$. Thus, $f\in \mathcal{U}(Lip^1(X))$.
The algebraic case. {#S4}
===================
Let $G$ be an algebraic group having $e$ as identity element and let $f : G\longrightarrow \R^+$ be a function, we denote $Osc(f):=\sup_{t,t'\in G}|f(t)-f(t')|$ and by $G^*$ we denote the following set : $$G^*:=\lbrace f : G \longrightarrow \R^+/ Osc(f)\leq 1\rbrace.$$ Note that the set $G^*$ is juste the set $Lip^1_+(G)$ where $(G,disc)$ is equipped with the discrete metric $"disc"$, which is an invariant complete metric. So, $(G^*, \oplus)$ is a monoid having $\delta_e$ as identity element, where $\delta_e(\cdot):=disc(\cdot,e)$ i.e. $\delta_e(e)=0$ and $\delta_e(t)=1$ for all $t\neq e$. Observe also that two algebraic groups $G$ and $G'$ are isomorphic if and only they are isometrically isomorphic when equipped respectively with the discrete metric. Thus, we obtain that the algebraic group structure of any group $G$ is completely determined by the algebraic monoid structure of $(G^*, \oplus)$.
\[cor1\] Let $G$ and $G'$ be two groups. Then the following assertions are equivalent.
$(1)$ the groups $G$ and $G'$ are isomorphic
$(2)$ the monoids $(G^*, \oplus, \rho)$ and $({G'}^*, \oplus, \rho)$ are isometrically isomorphic
$(3)$ the monoids $(G^*, \oplus, d_{\infty})$ and $({G'}^*, \oplus, d_{\infty})$ are isometrically isomorphic (where $d_{\infty}(f,g):=\sup_{t\in G}|f(t)-g(t)|< +\infty$, for all $f, g \in G^*$)
$(4)$ the monoids $(G^*, \oplus)$ and $({G'}^*, \oplus)$ are isomorphic.
Moreover, $\Phi : (G^*, \oplus, \rho) \longrightarrow ({G'}^*, \oplus, \rho)$ (resp. $\Phi : (G^*, \oplus, d_{\infty}) \longrightarrow ({G'}^*, \oplus, d_{\infty})$) is an isometric isomorphism of monoids, if and only if there exists an isomorphism of groups $T: G\longrightarrow G'$ such that $\Phi(f)=f\circ T^{-1}$ for all $f\in G^*$.
Since $G^*=Lip^1_+(G)$, where $G$ is equipped with the discrete metric and since $G$ and $G'$ are isomorphic if and only if $(G,disc)$ and $(G',disc)$ are isometrically isomorphic, then the part $(1)\Longleftrightarrow (2)$ is a direct consequence of Theorem \[Thm0\]. The part $(2)\Longrightarrow (3)$, follows from the fact that $\rho =\frac{d_{\infty}}{1+d_{\infty}}$ by using \[Lemma 1, [@Ba1]\]. The part $(3)\Longrightarrow (4)$ is trivial. Let us prove $(4)\Longrightarrow (1)$. Since an isomorphism of monoids sends the group of unit onto the group of unit, and since the group of unit of $G^*$ (resp. of ${G'}^*)$ is isomorphic to $G$ (resp. to $G'$) by Proposition \[Punit\], we get that $G$ and $G'$ are isomorphic. The last assertion is given by Theorem \[Thm0\].
As mentioned in Remark \[rem1\], if $T: G\longrightarrow G'$ is an isomorphism, then $\Phi(f):=f\circ T^{-1} + \inf_G(f)$ for all $f\in G^*$ gives an isomorphism of monoids between $G^*$ and ${G'}^*$ which is not isometric. In the following exemple, we treat the case where $G$ is a finite group.
\[Exemp1\] Let $n\geq 1$ and $(\R^n,d_{\infty})$ the usual $n$-dimentional space equiped with the max-distance. The subsets $M^n_+$ and $M^n$ of $\R^n$ are defined as follows $$M^n_+:=\lbrace (x_k)_{1\leq k \leq n} \in \R_+^n/ |x_i-x_j|\leq 1, \hspace{2mm} \hspace{2mm} 1\leq i, j \leq n\rbrace.$$ $$M^n:=\lbrace (x_k)_{1\leq k \leq n} \in \R^n/ |x_i-x_j|\leq 1, \hspace{2mm} \hspace{2mm} 1\leq i, j \leq n\rbrace.$$
Let $G:=\lbrace g_1, g_2,... g_n\rbrace$, be a group of cardinal $n$, where $g_1$ is the identity of $G$. We define the law $\star_G$ on $\R^n$ depending on $G$ as follows: for all $x=(x_k)_k, y=(y_k)_k\in \R^n$, $$x\star_G y=(z_k)_{1\leq k \leq n},$$ where for each $1\leq k \leq n$, $$z_k:=\min \lbrace x_i+y_j/ g_i\cdot g_j=g_k, 1\leq i, j \leq n\rbrace.$$ Then,
$(1)$ The set $(\R^n,\star_G)$ has a semigroup structure (and is abelian if $G$ is abelian).
$(2)$ The sets $(M^n_+,\star_G)$ and $(M^n,\star_G)$ are monoids having $e=(0,1,1,1,...,1)$ as identity element.
$(3)$ Let $G$ and $G'$ be two groups of cardinal $n$. The monoids $(M^n_+,\star_G)$ and $(M^n_+,\star_{G'})$ are isomorphic if and only if the groups $G$ and $G'$ are isomorphic.
$(4)$ We have that $$\mathcal{U}(M^n_+)\simeq G,$$ $$\mathcal{U}(M^n) \simeq G\times \R.$$ Moreover, the maximal subgroup of $(\R^n,\star_G)$ having $e$ as identity element is isomorphic to the group $G\times \R.$
$(5)$ We have that $$Is_m(M^n_+) \simeq Aut(G).$$
The properties $(1)-(5)$ follows easily from the results of this note. It sufficies to see that the space $\R^n$ can be identified to the space $Lip(G)$ of all real-valued Lipschitz map on $(G,disc)$. Indeed, the map
$$\begin{aligned}
i: Lip(G)&\longrightarrow& \R^n\\
f &\mapsto& (f(g_1),...,f(g_n))\end{aligned}$$
is a bijective map. Then, we observe that the operation $\star_G$ on $\R^n$ is just the operation $\oplus$ on $Lip(G)$. On the other hand, the subset $M^n_+$ is identified to $Lip^1_+(G)$ and $M^n$ is identified to $Lip^1(G)$.
[999]{} M. Bachir, *A Banach-Stone type theorem for invariant metric groups*, Topology Appl. 209 (2016) 189-197. M. Bachir, *The inf-convolution as a law of monoid. An analogue to the Banach-Stone theorem*, J. Math. Anal. Appl. 420, (2014) No. 1, 145-166. M.I. Garrido, J.A. Jaramillo, *Variations on the Banach-Stone Theorem*, Extracta Math. 17, (2002) N. 3, 351-383. M.I. Garrido, J.A. Jaramillo, *Lipschitz-type functions on metric spaces*, J. Math. Anal. Appl. 340 (2008) 282-290. A. Jiménez-Vargas, M. Villegas-Vallecillos, *2-Local Isometries on Spaces of Lipschitz Functions*, Canad. Math. Bull. 54,(2011), 680-692. N. Weaver, Isometries of noncompact Lipschitz spaces, Canad. Math. Bull., 38 (1995), 242-249.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Sympathetic cooling of an atomic Fermi gas by a Bose gas is studied by solution of the coupled quantum Boltzmann equations for the confined gas mixture. Results for equilibrium temperatures and relaxation dynamics are presented, and some simple models developed. Our study illustrate that a combination of sympathetic and forced evaporative cooling enables the Fermi gas to be cooled to the degenerate regime where quantum statistics, and mean field effects are important. The influence of mean field effects on the equilibrium spatial distributions is discussed qualitatively.'
address: |
School of Physics, Georgia Institute of Technology\
Atlanta, Georgia 30332-0430
author:
- 'W. Geist, L. You, and T.A.B. Kennedy'
title: 'Sympathetic cooling of an atomic Bose-Fermi gas mixture'
---
epsf
Introduction \[sec:intro\]
===========================
The magnetic trapping and forced evaporative cooling of alkali metal vapors has lead to the Bose-Einstein condensation of a number of different atomic species [@bec1]. As a consequence, the many-body physics of confined, weakly interacting Bose gases are now amenable to experimental investigation. To date most experimental and theoretical research on degenerate atomic gases has focused on bosons [@bec2]. Recently, some exotic systems have been investigated experimentally, these include the Bose condensation of spin mixtures of $^{87}$Rb using sympathetic cooling [@mixtures], and a report of spin domains of $^{23}$Na in an optical trap [@ketterle]. There is now a growing interest in the properties of degenerate atomic Fermi gases [@cornell; @stoof; @collisions; @classic; @baranov; @marco; @rb] and boson-fermion mixtures [@moelmer], although to date a degenerate atomic Fermi gas has not been achieved.
In this paper we investigate the process of sympathetic cooling of an initially non-degenerate Fermi gas to the quantum degenerate regime using a Bose gas as coolant. Sympathetic cooling of a Fermi gas to degeneracy is necessary as a result of the suppression of s-wave scattering between identical fermions in spin symmetric states. Evaporative cooling of a pure fermion gas trapped in a single hyperfine state, which depends on rethermalization through atomic collisions, is thus ineffective at temperatures sufficiently low that only the lowest few partial waves contribute. The Fermi gas may instead be cooled by thermal contact with a cold Bose gas, which may be either already condensed before thermal contact with the Fermi gas, or evaporatively cooled in its presence. We investigate the dynamics using the quantum Boltzmann equation (QBE) for the Bose-Fermi gas mixture. The QBE has been applied to describe the population dynamics for a Bose gas by a number of authors who solved it by direct integration [@snoke; @wally], trajectory simulation [@holland], and Bird’s simulation method [@wu]. The first two sets of authors make the assumption that the distribution is ergodic at all times [@surkov] to simplify the numerical computation, whereas in Bird’s method collision dynamics are constructed from simulated particle trajectories. In this paper we solve the QBE in the ergodic approximation by direct integration of the coupled differential equations which describe the Fermi and Bose gas distribution functions.
We consider an harmonically trapped gas of $N_f$ fermions with Fermi temperature given by $E_F\equiv k_B T_F =
\hbar \omega (6 N_f)^{1/3}$, which is in thermal contact with a similarly confined gas of $N_b$ bosons with condensation temperature $k_B T_C = \hbar \omega (N_b/1.202)^{1/3}$. By evaporatively cooling the Bose gas through $T_C$ down to a temperature $\eta T_C$ ($0< \eta <1$), the Fermi gas will equilibrate to the Fermi temperature provided $N_f < \eta^3
N_C/7.212$, where $N_C < N_b$ is the number of condensed atoms remaining. Solution of the QBE gives both the energy and spatial distributions of the Bose and Fermi gases. An important limitation of our present treatment is the neglect of mean field effects on the dynamics. These are potentially important when the Bose and Fermi gas are degenerate and the mean field energy exceeds the trap frequency. In a recent paper M[ø]{}lmer [@moelmer] has used a simple mean field model to study the spatial distributions of a Bose-Fermi gas mixture at $T=0\;K$. The distributions depend strongly on the relative sign and magnitude of the boson-boson and boson-fermion scattering lengths. A complete and numerically tractable approach to quantum kinetics which incorporates such mean field effects in addition to the population dynamics, is still under investigation. A qualitative discussion of the influence of mean field effects on our results is given in section \[sec3\].
The remainder of this paper is organized as follows. In section \[sec1\] we discuss the equilibrium temperatures for an initially hot Fermi gas placed in thermal contact with a cold Bose gas. We also derive a simple dynamical model for the thermalization using the assumption that both gases are described by a Maxwell Boltzmann distribution at all times. In section \[sec2\] we describe a theoretical model based on the QBE for the Fermi-Bose mixture and the treatment of forced evaporative cooling. In section \[sec3\] we present results of solutions of the QBE which provide information on the kinetic temperature and spatial distributions of the mixture. Using recent atomic data we also present some results for the sympathetic cooling of the $^{40}$K/$^{39}$K Fermi/Bose potassium isotopes. Finally, in section \[sec4\], we summarize our conclusions.
Equilibrium properties and cooling without forced evaporation {#sec1}
==============================================================
Equilibrium temperature
-----------------------
If two gases at different initial temperatures are brought into contact they will rethermalize to a common equilibrium temperature which can be obtained using conservation of particle number and energy (over bar indicates average energy) $$\begin{aligned}
\bar\epsilon_{{\rm tot}}&=&
\bar\epsilon_f(0)+\bar\epsilon_b(0)
\nonumber\\
&=&\bar\epsilon_f(\infty)+\bar\epsilon_b(\infty)\end{aligned}$$ assuming that there are no losses during relaxation. For a Bose-Fermi gas mixture the equilibrium temperature $T_{\infty}$ can be found numerically using the Bose-Einstein and Fermi-Dirac distribution functions as follows. Given the number of bosons $N_b$, and fermions $N_f$, and the initial temperatures $T_b(0)$, $T_f(0)$ we set $T_f=T_b$ at each iteration step and compute the fugacity $z_f$ for the Fermi gas by solving $$\label{it1}
N_f=\sum_{\epsilon_i}
g(\epsilon_i)(z_f^{-1}e^{\epsilon_i/k_BT_f}+1)^{-1}$$ with $g(\epsilon_i)$ the degeneracy of states with energy $\epsilon_i$. We then compute the mean energy $\bar\epsilon_f(z_f,T_f)$ for the Fermi gas and determine $T_b$ and $z_b$ for the Bose gas by the following iteration: set $\bar\epsilon_b=\bar\epsilon_{{\rm tot}}-\bar\epsilon_f(z_f,T_f)$ and compute $T_b$ from $$\label{it2}
\bar\epsilon_b=\sum_{\epsilon_i}\epsilon_i
g(\epsilon_i)(z_b^{-1}e^{\epsilon_i/k_B T_b}-1)^{-1}.$$ Then compute $z_b$ by solving $$\label{it3}
N_b=\sum_{\epsilon_i}
g(\epsilon_i)(z_{b}^{-1}e^{\epsilon_i/k_BT_b}-1)^{-1}$$ and repeat this iteration until $[N_b(z_b,T_b)-N_b]/N_b<10^{-7}$ and $[\bar\epsilon_b(z_b,T_b)-\bar\epsilon_b]/\bar\epsilon_b<10^{-7}$. We then set $T_f=T_b$ and repeat the procedure until $(T_f-T_b)/T_f<10^{-4}$.
For temperatures $T_f \gtrsim 0.5 T_F$ and $T_{\infty} < T_C$ an approximate equation can be obtained using a classical Maxwell Boltzmann distribution for the Fermi gas and a Bose distribution with $z_b=1$ for the Bose gas. This yields $$\begin{aligned}
\label{approxt1}
g_4(1) \frac{T_{\infty}^4}{T_F^4} +\frac{T_{\infty}}{6T_F}=
g_4(1)\frac{T_b^4(0)}{T_F^4} +\frac{T_f(0)}{6T_F}.\end{aligned}$$ with the Bose-Einstein function $g_4(1)\approx 1.082$. If $T_{\infty}>T_C$ we can approximate the Bose-Einstein distribution with a Maxwell-Boltzmann distribution to obtain an explicit expression for the temperature $$\label{approxt2}
T_{\infty}=\frac{ N_f T_{f}(0)+ N_b T_{b}(0)} { N_f+N_b}.$$ In Fig. \[fig0\] we show the equilibrium temperature as a function of the initial Fermi gas temperature scaled in units of $T_F$. The Bose gas consists of $10^6$ atoms at initial temperature $T_b(0)= 0.1 T_F(0)$, where $T_F$ depends on the number of fermions which is varied between $10^4$ and $10^6$. In terms of the BEC temperature the initial temperature of the Bose gas lies in the range $0.02T_C < T_b(0) < 0.2T_C$. The full numerical results agree well with approximation Eq. (\[approxt1\]) for $N_f=10^3,10^4$ and $10^5$ in the temperature range considered. This is shown by the curves at the bottom of the figure which almost overlap the approximate solution. The equilibrium temperature $T_{\infty}$ stays below the critical temperature $T_C$ and in agreement with Eq. (\[approxt1\]) $T_{\infty}/T_F$ does not depend on the number of fermions. If we increase the number of fermions or alternatively the initial Fermi gas temperature, the Bose gas will be heated above $T_C$ and the equilibrium temperature shows the linear dependence on $T_f(0)$ as derived in (\[approxt2\]). From our numerical studies we find that Fermi statistics become significant only for $T_f \lesssim 0.5 T_F$ (see also [@Butts]), which in general can be reached with additional evaporative cooling of the gas mixture.
Thermalization of a two-component mixture
-----------------------------------------
We consider Bose and Fermi gases brought into thermal contact in a confining potential, at different initial temperatures $T_b(0)$ and $T_f(0)$, respectively. We are primarily interested here in a regime prior to a stage of forced evaporative cooling, in which the Bose gas as well as the Fermi gas is non-degenerate. The thermalization may result in a significant alteration in the energy and spatial distribution of fermionic atoms in the trap. A simple dynamical description of the thermalization can be derived using the classical Boltzmann equation in the ergodic approximation. The mean energy of component $k = (b,f)$ is given by $$\begin{aligned}
\bar\epsilon_{k}(t) = \int \epsilon \; {\mathcal F}_k(\epsilon,t)\;
\rho(\epsilon) d\epsilon,\end{aligned}$$ where ${\mathcal F}_k(\epsilon,t)$ is the ergodic distribution function for component $k$, and $\rho(\epsilon) =
\epsilon^{2}/2(\hbar \omega)^3$ is the density of states for an harmonic trap of mean frequency $\omega=(\omega_x\omega_y\omega_z)^{1/3}$. For simplicity we asssume that both components see the same trapping potential.
The time dependence of the energy of the Fermi gas is then obtained by integrating the Boltzmann equation [@snoke] over all energies $$\begin{aligned}
\label{energy}
\frac{d\bar E_f}{dt}&=&\frac{1}{\tau_0}
\frac{1}{(\hbar\omega)^5}
\int dE_1\int dE_2
\int dE_3\int dE_4 E_{{\rm min}}^2/2 \; E_1
\delta(E_1+E_2-E_3-E_4)
\nonumber\\
&&
\left[
{\mathcal F}_f(E_4,t){\mathcal F}_b(E_3,t)- {\mathcal F}_f(E_2,t)
{\mathcal F}_b(E_1,t)\right],\end{aligned}$$ where all energies written with upper case E are dimensionless, i.e, $E \equiv \epsilon / \hbar \omega$, $E_{{\rm min}}
\equiv {\rm min}\{E_1,E_2,E_3,E_4\}$, and the natural timescale $\tau_0$ is given in terms of the Bose-Fermi [*s*]{}-wave collision cross section $\sigma_{bf}$ by $ 1 / \tau_0
\equiv (\hbar\omega)^2 m\sigma_{bf} / \pi^2\hbar^3$. We have dropped the term which represents collisions between fermionic atoms assuming that the [*p*]{}-wave contribution is neglibible at the low energies under consideration.
To get some qualitative insight into the thermalization dynamics we need to further simplify the model. We assume that the component distribution functions are Boltzmann-like at all times, and parametrized by time dependent fugacity $z_{k}(t)$ and dimensionless temperature $\bar T_k(t) \equiv k_B T_k(t)/
\hbar\omega$ as follows, $$\begin{aligned}
{\mathcal F}[E,z_k(t),\bar T_k(t)] = z_k(t) e^{-E/\bar T_k(t)}.\end{aligned}$$ The average energy and particle number for a trapped Maxwell-Boltzmann distribution are given by $$\begin{aligned}
\bar E_k &=&\frac{z_k}{2}
\bar T_k^4\int dx x^3\;e^{-x}, \\
N_k &=& \frac{z_k}{2}\bar T_k^3\int dx x^2\;e^{-x}\end{aligned}$$ from which it follows that $z_k = N_k/\bar T_k^3$ , and $\bar E_k
= 3 N_k \bar T_k $. The fugacity of component $k$ can therefore be eliminated in terms of the particle number and mean energy. Another simplification which results from the approximation is that the scattering of two bosons does not alter the average energy of the Bose gas.
It is convenient to write the average energy in terms of a dimensionless temperature, thus $$\begin{aligned}
\label{tf}
\frac{d\bar T_f}{dt}&=&\frac{N_b r^3}{3 \tau_0} \int dx_1\int dx_2
\int dx_3\int dx_4 \frac{x_{{\rm min}}^2}{2}x_1
\delta(x_1+x_2-x_3-x_4)
\left( e^{-x_4}e^{-r x_3}- e^{-x_2}e^{-r x_1}\right)
\nonumber\\
&=&\frac{ N_b }{3 \tau_0} P(T_f / T_b),
\\
\frac{d\bar T_b}{dt}&=&-\frac{ N_f}{3 \tau_0 }\;\; P(T_f / T_b),\end{aligned}$$ where $r \equiv \bar T_f /\bar T_b$, and the rational function $$P(r)=\frac{1+3r+2r^2-2r^3-3r^4-r^5}{(1+r)^5}.$$ The equilibrium temperature $\bar T_\infty$ is obtained from $d\bar
T_f/d\bar T_b=- N_b/ N_f$. Integrating from $t=0$ to $\infty$, we recover equation (\[approxt2\]). The analysis shows that the timescale for relaxation is approximately $
\bar T_b \tau_0/ N_f $. In Fig. \[fig1\] we compare the predictions of this simple dynamical model, with the full dynamics of the QBE as discussed in the following section. The solutions show that the time scale that describes equilibration is on the order of $\tau_r\equiv \bar T_b\tau_0/ N_f \approx 0.043\tau_0$, as predicted by Eqs. (\[tf\]) and (13).
If forced evaporative cooling is applied to the gas mixture, atoms from the hot energy tail are removed. The rate of evaporation must be chosen to be much smaller than the relaxation rate of the gas as discussed above. Another way to determine the relaxation time is to calculate the initial mean collision rate $\gamma$ using the Boltzmann collision integral. Consider the collisions of “test" atoms, with energies $E_1$, with atoms of energy, $E_2$, into final states with energies $E_3$ and $E_4$. One sums over all initial energies $E_1$, and divides by the number of particles to get the energy averaged rate per particle $$\begin{aligned}
\label{scatter}
\gamma(\bar T,N)&=& \frac{1}{\tau_0}\frac{N}{\bar T}
\int dx_1 dx_2 dx_3 dx_4
\delta(x_1+x_2-x_3-x_5)
\frac{x_{{\rm min}}^2}{2} e^{-x_1}e^{-x_2}\\
\nonumber
&=& \frac{N}{2\tau_0\bar T},\end{aligned}$$ in qualitative agreement with the simple model above. We note, however, that the relaxation rate we have discussed here should not be regarded as the characteristic time scale for condensation [@smerzi; @anglin; @li].
Quantum Boltzmann Equation and forced evaporation {#sec2}
=================================================
Quantum Boltzmann equation
--------------------------
The QBE for a harmonically confined Bose gas has been discussed elsewhere within the ergodic approximation [@holland; @gard]. The QBE for an interacting two component Bose-Fermi mixture in the harmonic trap can be written ($\tau = t/\tau_0$) $$\begin{aligned}
\label{bos1}
g(E_i)\frac{db_{E_i}}{d\tau}&=&
\alpha_b \sum_{E_j,E_k,E_l}
\delta_{E_i +E_j, E_k ,E_l}
g(E_i,E_j,E_k,E_l)
\nonumber\\
&&\left[b_{E_k}b_{E_l}(1+b_{E_j})(1+ b_{E_i})-b_{E_i} b_{E_j}(1+
b_{E_k})(1+b_{E_l})\right]+
\nonumber\\
&& \sum_{E_j,E_k,E_l}
\delta_{E_i +E_j, E_k+E_l}
g(E_i,E_j,E_k,E_l)
\nonumber\\
&&\left[ b_{E_k}f_{E_l}(1-f_{E_j})(1+b_{E_i})-
b_{E_i}f_{E_j}(1-f_{E_l})(1+ b_{E_l})\right],\end{aligned}$$ $$\begin{aligned}
\label{ferm1}
g(E_i) \frac{df_{E_i}}{d\tau}&=&
\alpha_f\sum_{E_j,E_k,E_l}
\delta_{E_i +E_j, E_k
+E_l}\;g(E_i,E_j,E_k,E_l)
\nonumber\\
&&\left[f_{E_k}f_{E_l}(1-f_{E_j})(1- f_{E_i})-f_{E_i} f_{E_j}(1-
f_{E_k})(1-f_{E_l})\right]+
\nonumber\\
&& \sum_{E_j,E_k,E_l}
\delta_{E_i +E_j, E_k+E_l}\;
g(E_i,E_j,E_k,E_l)
\nonumber\\
&&\left[ b_{E_l}f_{E_k}(1-f_{E_i})(1+b_{E_j})-
b_{E_j}f_{E_i}(1-f_{E_k})(1+ b_{E_l})\right],\end{aligned}$$ where $b_{n}$ and $f_{n}$ are the number of Bose and Fermi atoms in state $E_n$. The collision matrix elements are approximately given by $$\begin{aligned}
g(E_i,E_j,E_k,E_l)=g(E_{{\min}}),\end{aligned}$$ with $g(E_n)$ the degeneracy of energy level $E_n$, and $E_{{\rm min}}$ is the minimum energy of all four energies involved in the scattering process, as defined earlier. Although this approximation is not quantitatively accurate for the lowest several states of the trap, it is sufficient to illustrate the main qualitative features of sympathetic cooling. In an isotropic trap the degeneracy of the energy state $\epsilon_n=\hbar\omega (n-1),\;n=1,2\cdots$, is $g(E_n)=n(n+1)/2$. The coefficients $\alpha_b=\sigma_{bb}/ \sigma_{bf}$ and $\alpha_f=\sigma_{ff}/\sigma_{bf}$, give the ratios of the cross sections for boson-boson and fermion-fermion scattering, respectively, to the boson-fermion cross section. Exchange symmetry leads to $\alpha_f = 0$ since the [*s*]{}-wave cross section vanishes for identical fermions. Of course, this is the reason we must employ sympathetic cooling with a Bose gas refrigerant.
Forced evaporative cooling
--------------------------
Evaporative cooling in magnetic traps is performed by inducing transitions to untrapped states with a radio-frequency field. This is modeled here by the following procedure. Particles that are scattered into states with energy larger than the time varying cut-off energy $E_{{\rm cut}}(\tau)$ are lost. The latter is a given decreasing function of time in the case of forced evaporative cooling. A particle may be scattered into a state with energy $E_i
< E_{{\rm cut}}(\tau)$, by two-body scattering of atoms in states with energies $E_k$ and $E_l$, i.e., $E_k,\;E_l \rightarrow E_i,\;E_j$, in which $E_j > E_{{\rm
cut}}(\tau)$, so that one particle is lost from the trap. Similarly a particle in energy level $E_i$ can be scattered out of this energy level, $E_i,\;E_j
\rightarrow E_k,\;E_l$, resulting in one particle loss from the trap when either $E_k > E_{{\rm cut}}(\tau)$, or $E_l > E_{{\rm cut}}(\tau)$. Explicitly,\
(a) The gain process for energy level $E_i$\
$$\begin{aligned}
g(E_i)\frac{dn_{E_i}}{d\tau}&=&
\alpha\; \sum_{E_j > E_{{\rm cut}}(\tau)>E_k,E_l}
^{E_{j} < 2 E_{{\rm cut}}(\tau)}
\delta_{E_i +E_j, E_k ,E_l}
g(E_i,E_j,E_k,E_l)
n_{E_k}n_{E_l}(1 \pm
n_{E_i}) .
\end{aligned}$$ (b) The loss process for energy level $E_i$\
$$\begin{aligned}
g(E_i)\frac{dn_{E_i}}{d\tau}&=&
-2\alpha \;\sum_{E_k > E_{{\rm cut}}(\tau)>E_j,E_l}
^{E_{k} < 2 E_{{\rm cut}}(\tau)}
\delta_{E_i +E_j, E_k ,E_l}
g(E_i,E_j,E_k,E_l)
n_{E_i} n_{E_j}(1 \pm n_{E_l}),
\end{aligned}$$ where $n_{E_i}$ denotes the distribution function for Fermi or Bose atoms with energy $E_i$, as appropriate.
The kinetics of forced evaporative cooling is modeled by adding these terms to the QBE, Eqs. (\[bos1\]) and (\[ferm1\]), which include all two-body collision processes between initial and final states $i,j,k$, and $l$ with energies $E_i ,E_j, E_k ,E_l < E_{{\rm
cut}}(\tau)$ conserving the total number of particles in the trap.
Results and discussion {#sec3}
======================
In this section we illustrate the dynamics of sympathetic cooling of Bose-Fermi gas mixtures, through their energy, state and spatial distribution functions. The spatial distributions of confined degenerate Bose and Fermi atomic gases are quite different. An ideal Bose condensate has a size determined by the quantum width of the trap ground state $l = \sqrt{\hbar/2M \omega}$, whereas the size of a Fermi gas is governed by the Fermi width $R_F = (E_F/2M
\omega^2)^{1/2}$, which scales as $R_F \sim N_f^{1/6} \; l$, as a result of the Pauli exclusion principle. For an interacting Bose gas, with positive scattering length, the condensate is larger than $l$, and for strongly condensed gases its size can be estimated using mean field theory in the Thomas-Fermi approximation [@eddi]. Mean field effects, which can be significant well below the condensation temperature, are not included in our model. These may be important in the final stages of cooling if the Bose gas is already strongly condensed at this stage. Our illustrations of the spatial distributions of both Fermi and Bose gas employ the universal scaling described by Butts and Rokhsar [@Butts], who showed that for a harmonically trapped ideal Fermi gas at $T = 0$ $$\begin{aligned}
\label{nrf}
n_f(r) = \frac{N_f}{R_F^3} \frac{8}{\pi^2}
\left[ 1 - \left(\frac{r}{R_F}\right)^2 \right]^{3/2}.\end{aligned}$$ It should be remembered that with evaporative cooling the number of fermionic and bosonic atoms is a time dependent variable, and therefore so is $R_F$. In the figures we always scale with respect to the instantaneous value of $R_F$.
In Figs. \[fig1\] and \[fig2\] we present the rethermalization of a non-degenerate Fermi gas immediately after it is placed in thermal contact with a Bose gas which is initially at the Bose condensation temperature $T_C$. The calculation is performed by numerical integration of the QBE without any forced evaporative cooling. In Fig. \[fig2\] the temperature of the Fermi gas alters considerably over a timescale $
\tau_0\bar{T}_f/N_b$ \[Eq. (\[tf\])\]. Initially a hot tail of atoms extends to the trap extremities and during the early stages of equilibration the Fermi gas distribution deviates significantly from a Fermi-Dirac distribution which is fit to the average energy and particle number. The gas then equilibrates to a non-degenerate state as can be seen from inspection of the peak of the spatial distribution function, $n_f(r=0) \ll 1$, [@Butts]. The Bose gas has one hundred times more particles than the Fermi gas, and completely envelopes the Fermi gas at all times. Fig. \[fig2\] compares the simple model of thermalization discussed in the last section with the QBE. The model is very good in the early stages, but the agreement deteriorates in the intermediate regime before steady state is achieved.
In Figs. \[fig3\] and \[fig4\] we consider the forced evaporative cooling of both gases. In contrast to Figs. \[fig1\] and \[fig2\] there are $10^5$ bosons and $10^4$ fermions, initially. The forced cooling begins after the initial equilibration stage during which the boson temperature increases (Fig. \[fig4\]), following thermal contact of the two gases. The evaporative cooling time scale is chosen to be $\tau_0$, which is much longer than both the relaxation time scale of the one component Bose gas \[Eq. (\[scatter\])\] and the relaxation time scale of the Fermi gas with the Bose gas \[Eq. (\[tf\])\]. The Bose gas energy distribution shows the formation of the condensate and the corresponding spatial distribution contracts to that of the condensate with a small thermal component. The degenerate Fermi gas is then exposed and its spatial distribution is close to the zero temperature limit \[Eq. (\[nrf\])\] which has a maximum density at the trap center $n_f(0)R_F^3/N_f=8/\pi^2\approx 0.81$. The inset shows the state occupancy for the Fermi distribution with the characteristic smearing of the Fermi surface at finite temperature, and near unit occupancy for low lying levels. It is interesting to note that evaporative cooling of the Fermi gas still proceeds at later times when the spatial overlap between fermions and bosons is mainly in a small region at the center of the trap where the condensate is located. The collisions which result in cooling involve orbits of hot fermions through the trap center where they collide with cold condensed bosons. At this stage evaporation mainly results in depletion of fermions as can be seen in Fig. \[fig4\]. We also simulate the case when the evaporative cooling involves only the loss of Bose atoms from the trap. In Figs. \[fig5\] and \[fig6\] we consider the same initial conditions as for Figs. \[fig3\] and \[fig4\], but only ramp down the cut-off energy for the bosons. The results are qualitatively the same as in the former case where both Bose and Fermi gas particles evaporate, except that we end up with more particles left in the Fermi gas. As mentioned earlier we have not included the effect of the bosonic mean field which can alter the spatial distributions of each component depending on the mean field strength, the ratio of the scattering lengths, and the particle numbers of both components [@moelmer].
A possible experimental scenario for sympathetic cooling of a Bose-Fermi mixture involves two isotopes of potassium. Recent calculations predict the [*s*]{}-wave scattering length for the bosonic isotope $^{39}$K to be $a(^{39}{\rm K}) \equiv a_b =4.3$ (nm) with corresponding cross section $\sigma_{bb}=8\pi^2a_b^2$ and the [*s*]{}-wave scattering length between $^{39}$K and the fermionic isotope $^{40}$K to be $a(^{40}{\rm K}-^{39}{\rm
K})\equiv a_{bf} = 2.5$ (nm) with cross section $\sigma_{bf}=4\pi^2a_{bf}^2$ [@potassium]. Using $M(^{40}{\rm
K})=6.6\times 10^{-26}kg$ and $\omega=400\; $(Hz) sets the time scale $\tau_0 = 1254 \;({\rm s})
\approx 20$(min). Mean field effects become important when the mean field strength $E_{{\rm mean}}=4\pi\hbar^2a_b/ (M(^{40}{\rm K})\bar
n_0)$ is of the order of the level spacing $\hbar\omega$ [@singh]. At the onset of BEC the density profile of an ideal Bose gas is almost Gaussian and the mean density of the ground state becomes $\bar n_0
= N_0/\pi^{3/2}l^3$. The ratio of the mean field strength to the trap energy is then $$\gamma=\frac{E_{{\rm mean}}}{\hbar\omega}=
\sqrt{\frac{1}{\pi}}
\frac{a_b}{l}N_0=1.72\times10^{-3}N_0 .$$ Further discussions of the influence of mean-field effects on our results is given below. In Figs. \[fig7\] and \[fig8\] we present an example of evaporative cooling for a spin polarized $^{40}{\rm K}-^{39}$K mixture of $10^6$ bosons and $10^5$ fermions at initial temperatures $T_b=T_C\equiv 0.3$ ($\mu$K) and $T_f=7.2T_F\equiv1.8$ ($\mu$K) and cut-off energy $E_{{\rm
cut}}(0)= 1000$. We estimate the boson scattering rate using Eq. (\[scatter\]) which yields $\approx 5\times 10^3/\tau_0$. From $\tau=0$ until $\tau=0.04$ the cut-off energy remains at $E_{{\rm
cut}}=1000$. During the thermalization the Bose distribution completely envelopes the Fermi distribution which deviates significantly from its equilibrium distribution. As one can see from Fig. \[fig8\] the number of fermions almost remains the same whereas a large number of bosons are evaporated. After $\tau=0.04$ the cut-off energy is ramped down exponentially with rate $\gamma_{{\rm evap}}=100/\tau_0$ until $\tau=0.064 \cong 80.25$ (s). At the early stages of the forced evaporation the number of bosons decreases until most of the bosons are in the condensate and evaporation mainly leads to depletion of fermions. The simulation shows that the Fermi gas can be cooled to a temperature $T_f\approx
0.1 T_F$ with more than $2\times 10^4$ fermions left in the trap.
In practice there are some additional issues that must be considered. The isotopes have different mass and magnetic moment, this means that in general the clouds will be displaced with respect to one another due to the combined effects of gravity and the magnetic trapping force [@mixtures]. Sympathetic cooling can only proceed efficiently if good overlap between the gases is maintained [@ketterle2]. Even if we assume this has been achieved, the difference in magnetic moments will cause the trap frequencies to be different for the Bose and Fermi gas. For example if the bosonic $^{39}$K ( $I=7/2$, where I is the nuclear spin) is polarized in the state $|F=2, M_F=2 \rangle$, and the fermion isotope $^{40}$K ($I=4$) is polarized in the state $|F=7/2,
M_F=-7/2\rangle$, the trap frequencies would be in the ratio of $7:9$. In our calculation we assume the trap frequencies and masses to be identical.
For degenerate Bose and Fermi gases mean field effects can strongly influence the spatial distributions. Here we discuss how these effects qualitatively change the stationary distributions presented above, using the zero temperature model discussed by M[ø]{}lmer [@moelmer]. If the number of bosons is much larger than the number of fermions then as $a_{bf}$ increases relative to $a_b$ ($a_{bf},\;a_b > 0$), the fermion distribution is displaced further and further outside of the central core of the trap occupied by the bosons. When the particle numbers are similar the bosons are displaced outside the fermion region in the same limit. On the other hand, for large negative fermion-boson scattering length the stationary distributions may be unstable and dynamically change as a result. In the latter case mean field effects qualitatively change the nature of the problem, and thus our results will not apply.
In our simulations the number of bosons is always much larger than the number of fermions. In this case we can to first order neglect the influence of the fermions on the the boson spatial distribution. The boson density is then given by the Thomas-Fermi approximation $$\label{tommy}
n_b(\vec{r})=[\mu -V_{{\rm ext}}(\vec{r})]/a_{b}$$ where $V_{{\rm ext}}(\vec{r})= M\omega^2r^2 / 2$ denotes the harmonic trapping potential, and the chemical potential $\mu$ is fixed through the condition $\int d\vec r n_b(\vec r)=N_b$. Explicitly, this yields $$\mu=[(\frac{m\omega^2}{2})^{3/2}\frac{15}{8\pi}N_b a_b]^{2/5}.$$ The bosonic mean-field produces the well known broadening of the boson density distribution relative to the ground state length of the trap. The corresponding spatial distribution of fermions can be found from the equation [@moelmer] $$\label{molly}
\frac{\hbar^2}{2M}[6\pi^2n_f(\vec{r})]^{2/3} +
(1-\frac{a_{bf}}{a_{b}}) V_{{\rm
ext}}(\vec{r})+\frac{a_{bf}}{a_{b}}\mu=E_F$$ where $E_F$ is determined by $\int d\vec r n_f(\vec r)=N_f$. Explicitly the density is given by $$\begin{aligned}
\label{nrf2}
n_f(r) = \frac{N_f}{R_F^3} \frac{8}{\pi^2}
\left[ 1 - \frac{a_{bf}}{a_b}\frac{\mu}{E_F}-(1-\frac{a_{bf}}{a_b})
\left(\frac{r}{R_F}\right)^2 \right]^{3/2}\end{aligned}$$ which may be compared with equation (\[nrf\]) for an ideal gas. Clearly the mean field effects cause a broadening in the spatial distribution of the fermion cloud for $a_{bf} < a_b$. If $a_{bf}>a_{b}$ the fermions experience an inverted harmonic oscillator potential near the origin which repells them from this region. In our results presented above, and in particular for the example of the potassium isotopes, we always have $a_{bf}<a_{b}/2$. As a result mean field effects will result in a broadening of both Bose and Fermi gas spatial distributions, but not a relative displacement of the clouds.
Conclusion {#sec4}
==========
We have discussed the cooling of a confined non-degenerate Fermi gas to quantum degeneracy using an ultracold Bose gas coolant and evaporative cooling. Results for the stationary distributions and dynamics based on solutions of coupled QBE equations for the Bose-Fermi mixture were presented. These include investigations of the use of forced evaporative cooling to enhance the degeneracy of the Fermi gas. While the QBE does not include mean field effects, which are potentially important in the quantum degenerate regime, we have discussed their qualitative effects on the results presented, when $a_b > a_{bf} > 0$. In this instance mean fields lead to a broadening of both the Bose and Fermi spatial distributions, but not a relative displacement of the clouds.
We acknowledge support from the NSF.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We provide some new necessary and sufficient conditions which guarantee arbitrary pole placement of a particular linear system over the complex numbers. We exhibit a non-trivial real linear system which is not controllable by real static output feedback and discuss a conjecture from algebraic geometry concerning the existence of real linear systems for which all static feedback laws are real.'
address:
- |
-Joachim Rosenthal\
Department of Mathematics\
University of Notre Dame\
Notre Dame, IN 46556, USA
- |
-Frank Sottile\
Department of Mathematics\
University of Toronto\
100 St. George Street\
Toronto, Ontario M5S 3G3\
CANADA
author:
- Joachim Rosenthal
- Frank Sottile
date: 'January 28, 1997'
title: |
Some Remarks on\
Real and Complex Output Feedback
---
Preliminaries
=============
Let ${{\mathbb F}}$ be an arbitrary field and let $m,p,n$ be fixed positive integers. Let $A,B,C$ be matrices with entries in ${{\mathbb F}}$ of sizes $n\times n$, $n\times m$, and $p\times n$ respectively. Identify the space of monic polynomials having degree $n$, $$s^n+a_{n-1}s^{n-1}+\cdots+a_1s+a_0\in {{\mathbb F}}[s],$$ with the vector space ${{\mathbb F}}^n$. In its simplest form, the static output pole placement problem asks for conditions on the matrices $A,B,C$ which guarantee that the pole placement map $$\label{pole-map}
\chi_{(A,B,C)}:\, {{\mathbb F}}^{mp}\longrightarrow{{\mathbb F}}^n,\hspace{3mm}
K\longmapsto \det(sI-A-BKC)$$ is surjective. A dimension argument shows the necessity of $mp\geq n$. This question has been studied intensively and we refer to the survey articles [@by89; @ro97] and the recent papers [@ki95p; @le95; @ro95; @ro96; @wa92; @wa96] for details. We summarize some of the most important results.
A matrix pair $(A,B)$ defined over a field ${{\mathbb F}}$ is [ *controllable*]{} if the matrix pencil $\left[ sI-A \mid B\right]$ is left coprime. Equivalently, if the full size minors of the pencil $\left[ sI-A \mid B\right]$ have no common non-trivial polynomial factor. Similarly, a matrix pair $(A,C)$ is [*observable*]{} if the matrix pencil ${\left[
\begin{array}{c} sI-A \\ C
\end{array} \right]}$ is right coprime. Then we have:
$\chi_{(A,B,C)}$ is surjective only if $(A,B)$ is a controllable pair and $(A,C)$ is an observable pair.
The following identity immediately establishes the claim: $$\det(sI-A-BKC)=\det{\left[
\begin{array}{ccc} sI-A &\;& -B \\ -KC &\;& I \end{array}
\right]}=
\det{\left[
\begin{array}{ccc} sI-A &\;& -BK \\ -C &\;& I \end{array}
\right]}$$
The necessary conditions $mp\geq n$, controllability, and observability are not sufficient to guarantee arbitrary pole assignability. When $p=1$, the following straightforward lemma provides exact conditions for arbitrary pole assignability over any field ${{\mathbb F}}$.
\[p=1suff\] Let $p=1$ and let $d^{-1}(s)(n_1(s),\ldots,n_m(s))$ be a left coprime factorization of the transfer function $C(sI-A)^{-1}B$. Then the pole placement map (\[pole-map\]) is surjective if and only if $n_1(s),\ldots,n_m(s)$ span the vector space of polynomials of degree at most $n-1$.
One readily establishes a similar result when $m=1$. Lemma \[p=1suff\] gives algebraic conditions on the set of systems parameters. To make this precise, identify the set of matrices $(A,B,C)$ having fixed sizes $n\times n$, $n\times m$, and $p\times n$ with the vector space $V:={{\mathbb F}}^{n(n+m+p)}$. Recall that a subset $G\subset V$ is [*generic*]{} if a non-trivial polynomial vanishes on its complement $V\setminus G$. Thus Lemma \[p=1suff\] implies that if $p=1$ and $m\geq n$, then the set of systems which can be arbitrarily pole assigned forms a generic set.
Since non-controllable systems $(A,B,C)$ cannot be arbitrarily pole assigned, pole placement results are often restricted to a generic class of systems. If the base field ${{\mathbb F}}$ is the real numbers ${{\mathbb R}}$ or the complex numbers ${{\mathbb C}}$, then a generic set $G\subset V$ is open and dense with respect to the usual Euclidean topology, and its complement $V\setminus G$ has measure zero.
If the pole placement map $\chi$ is surjective for a generic set of systems and some fixed base field ${{\mathbb F}}$ we will say in short that $\chi$ is [*generically surjective*]{}.
The major results are as follows:
\[BrBy\] If the base field ${{\mathbb F}}$ is algebraically closed and if $mp\geq
n$ then $\chi$ is generically surjective. Moreover if $mp=n$ then for a generic set of systems the cardinality of $\chi^{-1}(\phi)$ (when counted with multiplicity) is independent of the closed loop polynomial $\phi\in {{\mathbb F}}^n$ and is equal to $$\label{1}
d(m,p)= \frac{1!2!\cdots (p-1)!(mp)!}{m!(m+1)!\cdots(m+p-1)!}$$
Since $mp\geq n$ is necessary for $\chi$ to be surjective, Theorem \[BrBy\] gives the best possible bound when the base field ${{\mathbb F}}$ is algebraically closed.
The number $d(m,p)$ is the degree of the Grassmann variety, which was computed in the last century by Schubert [@sh91]. Although the real numbers ${{\mathbb R}}$ are not algebraically closed and Theorem \[BrBy\] therefore does not apply one still has the following Corollary:
\[oddR\] If ${{\mathbb F}}={{\mathbb R}}$, $mp=n$, and $d(m,p)$ is odd, then $\chi$ is generically surjective.
If $(A,B,C)$ are real matrices then the set $\chi^{-1}(\phi)$ is closed under complex conjugation for every closed loop polynomial $\phi\in {{\mathbb R}}^n$. Therefore, for a generic set of systems, $\chi^{-1}(\phi)$ contains a real point for each $\phi$.
As an example, consider the case ${{\mathbb F}}={{\mathbb R}}$, $m=2$, $p=3$ and $n=6$. Here, $d(2,3)=5$. At least one of the 5 points $\chi^{-1}(\phi)$ is real, so $\chi$ is generically surjective even over the reals.
Berstein determined when $d(m,p)$ is odd.
The number $d(m,p)$ is odd if and only if $\min (m,p)=1$ or $\min (m,p)=2$ and $\max (m,p)=2^t-1,$ where $t$ is a positive integer.
When $d(m,p)$ is even, the best known sufficiency result over the reals is due to Wang:
If ${{\mathbb F}}={{\mathbb R}}$ and $mp > n$, then $\chi$ is generically surjective.
For an elementary direct proof of this important sufficiency result we refer to [@ro95].
For generic surjectivity over the reals, there is a difference of one degree of freedom between sufficiency ($mp>n$) and necessity ($mp\geq n$). As we already noted, $mp\geq n$ is sufficient if $d(m,p)$ is an odd number. One may ask if $mp\geq n$ might be always sufficient?
If ${{\mathbb F}}={{\mathbb R}}$ and if $m=p=2$ and $n=4$ then there is an open Euclidean neighborhood $U\subset V={{\mathbb R}}^{32}$ having the property that $\chi_{(A,B,C)}$ is not surjective if $(A,B,C)\in U$. In particular $\chi$ is not generically surjective.
It has been conjectured by S.-W. Kim that $m=p=2$, $n=4$ is the only case where $mp=n$ is not a sufficient condition for $\chi$ to be generically surjective over the reals. In the next section we exhibit a counterexample.
Main Results
============
The result by Brockett and Byrnes provides a sufficiency result for a generic set of systems. We provide exact conditions which guarantee that a particular plant $(\bar{A},\bar{B},\bar{C})$ is arbitrarily pole assignable. Our approach is geometric, utilizing the central projection of the Grassmann variety induced by the pole placement map [@br81; @wa96].
Let $D^{-1}(s)N(s)=C(sI-A)^{-1}B$ be a left coprime factorization of the transfer function having the property that $\det
(sI-A)=\det D(s)$. Then the closed loop characteristic polynomial can be written as: $$\label{polypol}
\det(sI-A-BKC)=\det{\left[
\begin{array}{ccc} D(s) &\;& N(s) \\ -K &\;& I \end{array}
\right]}=
\sum_\alpha g_\alpha(s)k_\alpha,$$ where the numbers $k_\alpha$ are the Plücker coordinates (full size minors) of the compensator $[-K\ I]$ inside $\wedge^m{{\mathbb F}}^{m+p}$ and the polynomials $g_\alpha(s)$ are (up to sign) the corresponding Plücker coordinates of $[D(s)\ N(s)]$. Let ${\mathbb P}^N$ be the projective space ${\mathbb
P}(\wedge^m{{\mathbb F}}^{m+p})$ and let $$E_{(A,B,C)}:=\left\{ k\in {\mathbb P}^N \mid \sum_\alpha
g_\alpha(s)k_\alpha =0 \right\} .$$ Since each $g_\alpha(s)$ has degree at most $n$, $E_{(A,B,C)}$ has dimension at least $N-n-1$, and its dimension equals $N-n-1$ precisely when the $g_\alpha(s)$ span the vector space of polynomials of degree at most $n$. In this case, the central projection induced by $\chi$ (see [@wa96]) $$\label{central}
L_{(A,B,C)} \; :\;
{\mathbb P}^N-E_{(A,B,C)} \ \longrightarrow \
{\mathbb P}^n,\hspace{9mm}
k\ \longmapsto \ \sum_\alpha g_\alpha(s)k_\alpha$$ is surjective.
By, there is a unique Plücker coordinate $\bar{\alpha}$ with $g_{\bar{\alpha}}(s)$ of degree $n$, namely that corresponding to the minor $\det D(s)$ of $[D(s)\ N(s)]$. Moreover, $k_{\bar{\alpha}}=1$ and all other $g_\alpha(s)$ have degree at most $n-1$. Identify ${{\mathbb F}}^N\subset{\mathbb P}^N$ with those points whose $\bar{\alpha}$th coordinate is 1. Then the central projection $L_{(A,B,C)}$ maps ${{\mathbb F}}^N$ to the set of monic polynomials of degree $n$, and its complement ${\mathbb
P}^N-{{\mathbb F}}^N$ to polynomials of degree at most $n-1$.
Every $m\times p$ compensator $K$ defines a $m$-dimensional linear subspace of ${{\mathbb F}}^{m+p}$, the row space of $[-K\ I]$ and therefore a point of the Grassmann variety Grass$(m,{{\mathbb F}}^{m+p})\subset{\mathbb P}^N.$ The previous paragraph shows this point is in ${{\mathbb F}}^N$. Conversely, all points in Grass$(m,{{\mathbb F}}^{m+p})\cap{{\mathbb F}}^N$ are of the form rowspace$[-K\ I]$ ([*cf.*]{} [@br81]).
The main theorem we have is:
\[main\] Let ${{\mathbb F}}$ be algebraically closed and $n\leq mp$. Then the pole placement map $\chi_{(\bar{A},\bar{B},\bar{C})}$ is surjective for a particular system $(\bar{A},\bar{B},\bar{C})$ if and only if $\dim E_{(\bar{A},\bar{B},\bar{C})}=N-n-1$ and, for any $y\in {{\mathbb F}}^N-E_{(\bar{A},\bar{B},\bar{C})}\cap{{\mathbb F}}^N$, $$\label{iff}
{\rm span}\left( E_{(\bar{A},\bar{B},\bar{C})},y\right)\cap
{\rm Grass}(m,{{\mathbb F}}^{m+p})\ \neq\ E_{(\bar{A},\bar{B},\bar{C})}\cap
{\rm Grass}(m,{{\mathbb F}}^{m+p}).$$
Suppose $\chi_{(\bar{A},\bar{B},\bar{C})}$ is surjective. Then the central projection $L_{(\bar{A},\bar{B},\bar{C})}$ is surjective and so $\dim E_{(\bar{A},\bar{B},\bar{C})}=N-n-1$. If for some $\hat{y}\in
{{\mathbb F}}^N-E_{(\bar{A},\bar{B},\bar{C})}\cap{{\mathbb F}}^N,$ $${\rm span}\left( E_{(\bar{A},\bar{B},\bar{C})},\hat{y}\right)\cap
{\rm Grass}(m,{{\mathbb F}}^{m+p})\ = \ E_{(\bar{A},\bar{B},\bar{C})}\cap
{\rm Grass}(m,{{\mathbb F}}^{m+p}),$$ then there is also equality in for all $y\in {\rm
span}\left( E_{(\bar{A},\bar{B},\bar{C})},\hat{y}\right).$ In particular, we see that the set $\chi_{(\bar{A},\bar{B},\bar{C})}^{-1}(L_{(\bar{A},\bar{B},
\bar{C})}(\hat{y}))$ is empty, a contradiction.
Conversely, if $\dim E_{(\bar{A},\bar{B},\bar{C})}=N-n-1$, then $L_{(\bar{A},\bar{B},\bar{C})}$ is surjective. Let $\phi\in{\mathbb P}^n$ be any closed loop polynomial and $y\in{\mathbb P}^N$ satisfy $L_{(\bar{A},\bar{B},\bar{C})}(y)=\phi$. Then necessarily $y\in
{{\mathbb F}}^N$, and condition guarantees that there exists $P\in {\rm Grass}(m,{{\mathbb F}}^{m+p})$ with $L_{(\bar{A},\bar{B},\bar{C})}(P)=\phi$. But then $P$ is the row space of $[-K\ I]$, for some compensator $K$. Hence $\chi_{(\bar{A},\bar{B},\bar{C})}(K)=\phi$.
A system $(\bar{A},\bar{B},\bar{C})$ is [*nondegenerate*]{} if $E_{(\bar{A},\bar{B},\bar{C})}\cap {\rm
Grass}(m,{{\mathbb F}}^{m+p})=\emptyset$. In [@br81] it was shown that nondegenerate systems can be arbitrarily pole assigned and that the set of nondegenerate systems forms a generic set if and only if $mp\leq n$.
The remainder of the paper is concerned with the question of when the condition $mp=n$ is also sufficient for the pole placement map $\chi$ to be generically surjective over the reals. If $(A,B,C)$ are real matrices and if $\chi_{(A,B,C)}:
{{\mathbb R}}^{mp}\longrightarrow{{\mathbb R}}^n$ is the real pole placement map, we let $\tilde{\chi}_{(A,B,C)}: {{\mathbb C}}^{mp}\longrightarrow{{\mathbb C}}^n$ denote the corresponding complexified map.
\[criter\] Let ${{\mathbb F}}={{\mathbb R}}$ and assume that $mp=n$ and $d(m,p)$ is even. Then $\chi$ is not generically surjective if and only if there exists a system $(\bar{A},\bar{B},\bar{C})$ and a polynomial $\bar{\phi}\in{{\mathbb R}}[s]$ such that $\tilde{\chi}_{(\bar{A},\bar{B},\bar{C})}^{-1}(\bar{\phi})\subset{{\mathbb C}}^{mp}$ consists of $d(m,p)$ different complex points, none of them real.
Assume $\chi$ is not generically surjective. Then there exists a Euclidean open neighborhood $U\subset{{\mathbb R}}^{n(n+m+p)}$ for which $\chi_{(A,B,C)}$ is not surjective if $(A,B,C)\in U$. Since $U$ is open, there exists a nondegenerate plant $(\bar{A},\bar{B},\bar{C})\in U$ having the property that $\tilde{\chi}_{(\bar{A},\bar{B},\bar{C})}^{-1}(\phi)$ consists of $d(m,p)$ points independent of $\phi$. Choosing a polynomial $\bar{\phi}$ which is not in the image of $\chi$ establishes one direction of the proof.
On the other hand, if $\tilde{\chi}_{(\bar{A},\bar{B},\bar{C})}^{-1}(\bar{\phi})\subset{{\mathbb C}}^{mp}$ consists of $d(m,p)$ different complex points, then necessarily $(\bar{A},\bar{B},\bar{C})$ is a nondegenerate plant. It follows that there exists an open Euclidean neighborhood $U$ of $(\bar{A},\bar{B},\bar{C})$ consisting solely of nondegenerate systems, none of which can be assigned the closed loop characteristic polynomial $\bar{\phi}$.
Theorem \[criter\] is interesting since it seeks a geometric configuration where all discrete solutions are purely complex. We use it to show that besides the case of $m=p=2$ and $n=4$, there are other situations where $mp=n$ is not sufficient to guarantee that $\chi$ is generically surjective over the reals. This disproves the conjecture by S.-W. Kim mentioned in §1.
If ${{\mathbb F}}={{\mathbb R}}$, $p=2$, $m=4$, and $n=8$ then $\chi$ is not generically surjective.
By Lemma 2.5, it suffices to exhibit a real system $(\bar{A},\bar{B},\bar{C})$ and a polynomial $\bar{\phi}$ of degree $8$ with $8$ real roots such that $\tilde{\chi}_{(\bar{A},\bar{B},\bar{C})}^{-1}(\bar{\phi})\subset{{\mathbb C}}^8$ consists of exactly $d(4,2)=14$ purely complex solutions. Here is such a system:
Let $(\bar{A},\bar{B},\bar{C})$ be a minimal realization of the system represented through a coprime factorization $D^{-1}(s)N(s)$, where
$$D(s)= \left[\begin{array}{cccccc} {x}^{4}\!-\! 16
{x}^{3}\!+\! 3 {x}^{2}\!+\! 11 x&
\!-\! 26 {x}^{3}\!+\! 10 {x}^{2}\!+\! 7 x\!+\! 16\\
6 {x}^{3}\!-\! 4 {x}^{2}\!-\! 9x\!-\! 5& {x}^{4}\!+\! 3
{x}^{3}\!-\! {x}^{2}\!-\! 16 x\!-\! 13
\end{array}\right]$$
[$$N(s)= \left[
\begin{array}{cccccc} \!\! 9{x}^{3}\!-\! 12
{x}^{2}\!+\! 13 x\!-\! 17& \!-\! 31 {x}^{3}\!-\! 16
{x}^{2}\!+\! 43 x\!-\! 23& {x}^{3}\!-\! 36 {x}^{2}\!+\! 8
x\!-\! 13&
23 {x}^{3}\!-\! {x}^{2}\!+\! 2 x\!-\! 21\!\!\\
8 {x}^{3}\!-\! 6{x}^{2}\!+\! 5 x\!+\! 15& 26 {x}^{3}\!-\! 14
{x}^{2}\!-\! 11 x\!+\! 12& 11 {x}^{3}\!+\! 5 {x}^{2}\!+\! 11
x\!+\! 33& \!-\! 7 {x}^{2}\!+\! 11x\!+\! 5
\end{array}\right]\!.$$]{}
Let $$\bar{\phi}(s):=(s+8)(s+6)(s+4)(s+2)(s-1)(s -2)(s- 3)(s- 4).$$
We claim that $\tilde{\chi}_{(\bar{A},\bar{B},\bar{C})}^{-1}(\bar{\phi})$ consists of 14 purely complex solutions (displayed below). First, we discuss how we compute $\tilde{\chi}_{(\bar{A},\bar{B},\bar{C})}^{-1}(\bar{\phi})$ for such a system with $n=mp$. Identify ${{\mathbb C}}^{mp}$ with the set of compensators $K$. Then the $mp$ polynomial equations $$\label{dets}
\det{\left[
\begin{array}{ccc} D(s) &\;& N(s) \\ -K &\;& I \end{array}
\right]}\ =\ 0$$ as $s$ ranges over the roots of $\bar{\phi}$ generate the ideal of $\tilde{\chi}_{(\bar{A},\bar{B},\bar{C})}^{-1}(\bar{\phi})$ in ${\mathbb C}^{mp}$. We used the software package SINGULAR [@SINGULAR] to compute an elimination Gröbner basis [@clo92] of this ideal and verify that $\tilde{\chi}_{(\bar{A},\bar{B},\bar{C})}^{-1}(\bar{\phi})$ is zero-dimensional with degree 14. This calculation on the system requires 59 seconds of CPU on a HP 9000 D250, 800 Series computer.
This Gröbner basis contains a univariate polynomial, the eliminant, whose roots are the values of that variable for the solutions. We used the [realroot]{} routine of Maple to determine the number of real roots of the eliminant and [ fsolve]{} to compute its roots numerically. Since we only obtain one coordinate of each solution, we repeated this procedure to find the others and to match the coordinates with the solutions.
Here are numerical solutions of the system.
$$\left[
\begin{array}{cc}
- 548.1543631072859\pm 539.02172783574002\sqrt{-1}&
- 2966.220011381735449\pm 1301.806890926492508\sqrt{-1}\\
227.99002317474104\mp 195.29675914098226\sqrt{-1}&
1189.40572416765385\mp 428.190112835481936\sqrt{-1}\\
253.619670619102274\mp 128.997418066861599\sqrt{-1}&
1192.66093663708038\mp 127.782426659628597\sqrt{-1}\\
-373.4608141108503\mp 376.1870941851628\sqrt{-1}&
-907.2715490825303837\mp 2040.657619029875556\sqrt{-1}
\end{array}\right]$$
$$\left[
\begin{array}{cc}
182.1974051162797\pm 1524.2891350121054\sqrt {-1}&-
3910.9491667600289\pm 3319.9425134666556\sqrt {-1}\\ -
92.76689536072804\mp 494.390627883840\sqrt {-1}&
1206.13014159582817\mp 1171.58923461208352\sqrt {-1} \\
202.71121387564936\mp 458.78014215695346\sqrt {-1}&
1652.30669576900037\mp 280.820983264097575\sqrt {-1}\\ -
999.496765955436554\mp 938.918292576740638\sqrt {-1}&
771.9810394973421\mp 4516.958140814761213\sqrt {-1}
\end{array}\right]$$
$$\left[
\begin{array}{cc}
2792.9110057318105\mp 969.00549705135278\sqrt {-1}&
3350.9339523791667\mp 832.762320679797284\sqrt {-1}\\ -
338.608141548768\mp 31.1420684422097\sqrt {-1}&-
390.733153481711\mp 71.9581835765450\sqrt {-1} \\ -
858.10666480772375\pm 463.34803831698071\sqrt {-1}&-
1047.08493981311276\pm 448.52274247532122\sqrt {-1}\\ -
1736.0182637110866\pm 473.54602107116131\sqrt {-1}&-
2069.7786151738302\pm 367.88390311074763\sqrt {-1}
\end{array}\right]$$
$$\left[
\begin{array}{cc}
566.14047176252718\mp 390.1690631954798\sqrt {-1}&
894.7573009772359\mp 213.7664118348474\sqrt {-1}\\ -
28.9144418101747\mp 8.82325220859399\sqrt {-1}&-
31.9889032754154\mp 25.1286025912621\sqrt {-1} \\ -
101.611268377237\pm 166.198294126534\sqrt {-1}&-
207.075559094765\pm 158.433905818864\sqrt {-1}\\ -
433.109410705026\pm 160.543671922194\sqrt {-1}&-
618.358581551134\mp 8.42746099774335\sqrt {-1}
\end{array}\right]$$
$$\left[
\begin{array}{cc}
- 1328.31492831596508\pm 780.43146580510958\sqrt {-1}&
2115.8811996413627\mp 363.25099106004349\sqrt {-1}\\
277.0599315399026\mp 134.0101686258348\sqrt {-1}&-
426.505631447159\pm 38.4080785894925\sqrt {-1} \\
242.753288068855\mp 128.748683783964 \sqrt {-1}&-
380.517275415650\pm 48.1897454846160\sqrt {- 1}\\
809.814164981704\mp 420.527784827832\sqrt {-1}&-
1263.86094232894868\pm 149.27131835292291\sqrt {-1}
\end{array}\right ]$$
$$\left[
\begin{array}{cc}
- 74.07812921055438\mp 1186.0867962658997\sqrt {-1}&
481.83814937211068\mp 659.46539248077808\sqrt {-1}\\
131.85311577768057\pm 223.6599712395458\sqrt {-1}&-
28.4575338243835\pm 176.018708417247\sqrt {-1} \\
50.0398731323218\pm 311.162560564792 \sqrt {-1}&-
110.484321267527\pm 186.531966999705\sqrt {- 1}\\
120.94035205524575\pm 693.23751296762126\sqrt {-1}&-
241.138619140528\pm 419.709352592197\sqrt {-1}
\end{array}\right]$$
$$\left[
\begin{array}{cc}
- 466.3420096818032\pm 2560.3776496553293\sqrt {-1}&-
477.06216348936717\pm 1505.4573962873226\sqrt {-1}\\
206.16217936754085\mp 504.1659905544772\sqrt {-1}&
162.819554092696\mp 287.715806475160\sqrt {-1} \\
198.483315335125\mp 690.317301079547 \sqrt {-1}&
172.179197573658\mp 400.2773514799496 \sqrt {-1}\\
350.2539156691074\mp 1658.3575908118343\sqrt {-1}&
337.47012412920796\mp 971.424525500586678\sqrt {-1}
\end{array}\right]$$
After discovering this example, we did a systematic search for others. In all, we generated 70 pairs $D(s),N(s)$ with random integral polynomial entries, and, for each of the 70, considered 25 degree 8 polynomials $\bar{\phi}(s)$ with distinct integral roots in $[-12,12]$. Of the 1750 instances of $\tilde{\chi}_{(\bar{A},\bar{B},\bar{C})}^{-1}(\bar{\phi})$ we tested, [*none*]{} had 14 purely complex solutions, and only 3 had the ‘opposite’ situation of 14 purely real solutions. This suggests that these extreme situations of real systems with real data giving only purely complex (or purely real) solutions are quite rare. Despite this, we believe that it is always possible to find such examples. Specifically:
If $d(m,p)$ is even and $n=mp$, then $\chi$ is not generically surjective over ${{\mathbb R}}$.
Consider now the ‘opposite’ situation. Namely, for which $m,p,n$ with $n=mp$ does there exist a real system $(\bar{A},\bar{B},\bar{C})$ and a polynomial $\bar{\phi}$ all of whose $(n)$ roots are real such that $\tilde{\chi}_{(\bar{A},\bar{B},\bar{C})}^{-1}(\bar{\phi})$ consists of exactly $d(m,p)$ real solutions? Similar questions have recently been of interest in algebraic geometry (see [@rtv95; @so97dmj; @so97mega] or the survey [@so97sc]). In fact, there is a precise conjecture of Shapiro and Shapiro which is relevant to systems theory:
\[conjSS\] Let $(\bar{A},\bar{B},\bar{C})$ be a minimal realization of the system represented through a coprime factorization $D^{-1}(s)N(s)$, where the matrix $\left[D(s)\,\mid\,N(s)\right]$ has the following form: The first row is $$s^{m+p-1}, s^{m+p-2}, \ldots, s^2, s, 1$$ and, for $1\leq j<p$, the $(j+1)$st row consists of the derivative of the $j$th row divided by $j$.
Then the system is nondegenerate, and for any polynomial $\bar{\phi}$ of degree $mp$ with distinct real roots, $\tilde{\chi}_{(\bar{A},\bar{B},\bar{C})}^{-1}(\bar{\phi})$ consists of exactly $d(m,p)$ real solutions.
For example, if $m=p=3$, then we have $$\left[D(s)\,\mid\,N(s)\right]\ =\ \left[\begin{array}{rrr|rrr}
s^5 & s^4 & s^3 & s^2 & s& 1\\
5s^4&4s^3 &3s^2 & 2s& 1& 0\\
10s^3 &6s^2 &3s& 1&0&0\end{array}\right].$$
For such a system, $\chi(\underline{0}) = s^{mp}$ and $\chi^{-1}(s^{mp})=\underline{0}$, a real point with multiplicity $d(m,p)$. Here, $\underline{0}$ is the null compensator, the matrix of all 0’s. Prior to learning of this conjecture, one of us (Rosenthal) had suggested that it might be possible to perturb $s^{mp}$ and obtain a polynomial $\bar{\phi}$ all of whose roots are real so that $\chi^{-1}(\bar{\phi})$ consists of $d(m,p)$ real solutions.
When $p$ or $m$ is 1, this conjecture follows from Corollary \[oddR\], and when $m=p=2$, it can be verified by hand. All other cases remain open. There is strong computational evidence in support of this conjecture: In every instance we have checked, $\tilde{\chi}_{(\bar{A},\bar{B},\bar{C})}^{-1}(\bar{\phi})$ consists of exactly $d(m,p)$ real solutions. When $m=4, p=2$ (so that $d(4,2)=14$), we checked about 50 polynomials $\bar{\phi}$. In light of the search described above, we feel this gives overwhelming evidence for this conjecture. In addition, we have considered numerous instances when $m=3,p=2$, and a handful of instances for each of $m=5, p=2$ and $m=3, p=3$. For each of these last two cases, $d(m,p)$ is $42$. Unfortunately, the task of computing an elimination Gröbner basis for larger $m,p$ overwhelms the HP 9000 computer we use for these calculations.
There are other methods for solving systems of polynomials which we have not tried, but which should work for larger $m,p$. When $(m,p)=(5,2), (6,2)$, or $(4,3)$, we can compute a Gröbner basis, and there are linear algebraic methods for solving a polynomial system, given a Gröbner basis [@clo97 §2.4]. Also, homotopy continuation [@ag90] algorithms which are optimized for these systems have been developed [@hss97], and are presently being implemented.
The row space of the matrix $\left[D(s)\,\mid\,N(s)\right]$ of Conjecture \[conjSS\] is a $p$-plane $H(s)$ which osculates the moment, or rational normal curve in ${{\mathbb R}}^{m+p}$. The rational normal curve is the image of the map $$s\ \longmapsto\ (s^{m+p-1}, s^{m+p-2}, \ldots, s^2, s, 1).$$
This observation, together with the fact that all non-degenerate rational curves of degree $m+p-1$ in ${\mathbb
P}^{m+p-1}$ are projectively equivalent, show that the conditions of Conjecture \[conjSS\] may be relaxed somewhat to the following:
The row span of the matrix $\left[D(s)\,\mid\,N(s)\right]$ equals the row span of a matrix $P(s)$ of real polynomials, where
1. The first row of $P(s)$ is a basis for all polynomials of degree at most $m+p-1$ and therefore defines a non-degenerate rational curve of degree $m+p-1$.
2. For $1\leq j<p$, the $(j+1)$st row of $P(s)$ is the derivative of the $j$th row of $P(s)$.
Thus the Conjecture of Shapiro and Shapiro proposes a family of real systems $(\bar{A},\bar{B},\bar{C})$ for which $\tilde{\chi}_{(\bar{A},\bar{B},\bar{C})}^{-1}(\bar{\phi})$ consists of exactly $d(m,p)$ real solutions, whenever $\bar{\phi}$ has all real roots.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We demonstrate a canted magnetization of biatomic zigzag Co chains grown on the ($5 \times 1$) reconstructed Ir(001) surface using density functional theory calculations and spin-polarized scanning tunneling microscopy (SP-STM) experiments. Biatomic Co chains grow in three different structural configurations and are all in a ferromagnetic state. Two chain types possess high symmetry due to two equivalent atomic strands and an easy magnetization direction which is along one of the principal crystallographic axes. The easy magnetization axis of the zigzag Co chains is canted away from the surface normal by an angle of $33^\circ$. This giant effect is caused by the broken chain symmetry on the substrate in combination with the strong spin-orbit coupling of Ir. SP-STM measurements confirm stable ferromagnetic order of the zigzag chains with a canted magnetization.'
author:
- Bertrand Dupé
- 'Jessica E. Bickel'
- Yuriy Mokrousov
- Fabian Otte
- Kirsten von Bergmann
- André Kubetzka
- Stefan Heinze
- Roland Wiesendanger
title: 'Giant magnetization canting due to symmetry breaking in zigzag Co chains on Ir(001)'
---
Low dimensional magnetic nanostructures at surfaces such as single atoms, clusters, and atomic chains constitute model systems to explore spintronic concepts at the ultimate scale. In view of their enhanced magnetocrystalline anisotropy energy (MAE) such nanomagnets are attractive for further miniaturization of data storage as the MAE acts as a barrier and stabilizes the magnetization against thermal fluctuations. One of the most striking examples is the giant anisotropy reported for single atom Co chains grown at the step edges of a Pt(111) surface allowing the observation of ferromagnetic order [@gambardella2002]. Due to the step edge the easy magnetization axis is not oriented along one of the high symmetry axis but it is canted from the surface normal by about 43$^{\circ}$ towards the upper Pt terrace [@gambardella2002] which is caused by the competition of contributions to the magnetocrystalline anisotropy from the Co and Pt atoms [@Ujf2004; @PhysRevB.69.212410; @Baud_PRB_2006]. Addition of more atomic Co strands at the step edge reorients the magnetization direction away from the upper terrace and the direction oscillates until it is nearly perpendicular to the vicinal surface for a coverage of a monolayer [@gambardella2004; @baud2006ss]. Since these pioneering experiments were performed, quasi one-dimensional chains at surfaces have been the subject of intense research, in particular in theoretical studies (e.g. [@Ujf2004; @PhysRevB.69.212410; @Baud_PRB_2006; @baud2006ss; @SpisakSS2003; @SpisakPRB2003; @Mok2007; @Lounis2008; @MazPRB2009; @MokrousovPRB2009; @Has2010]). However, very few other systems have been characterized experimentally concerning their magnetic state [@hirjibehedin2006spi; @mmmPrl2012; @Loth13].
One promising surface on which to grow quasi-one-dimensional chains is the (5 $\times$ 1) surface reconstruction of Ir(001) which exhibits a trench structure that allows self-assembly of different types of biatomic chains [@Gilarowski2000290; @PhysRevB.67.12542]. Two of the chain configurations possess a high symmetry with two equivalent atomic strands that adsorb either on the inner or on the outer hollow site of the trench (see Figs. \[fig:structure\](a,b)). For such biatomic Fe chains it has been reported previously that due to the hybridization between Fe and Ir the magnetic coupling and the easy magnetization axis depend significantly on the adsorption site of the atoms [@MazPRB2009; @MokrousovPRB2009; @mmmPrl2012]. If one of the two strands of the biatomic chain adsorbs on the inner hollow site while the other adsorbs on the outer hollow site a zigzag chain forms which lacks a mirror plane along its axis (see Figs. \[fig:structure\](c,d)). This type of chain has not been observed for Fe, however, as we show here it can be formed by depositing Co.
In this paper, we demonstrate that the symmetry breaking of zigzag chains on a surface can lead to a giant canting of their easy magnetization direction. This is remarkable, since both the chain and the substrate separately possess high symmetry and exhibit a relatively small buckling when brought in contact with each other. It is in clear contrast to the case of atomic Co chains at a Pt(111) step edge where the surface structure already has broken symmetry and a canting of the magnetization is expected.
Our first-principles calculations based on density functional theory (DFT) show that biatomic zigzag Co chains on the $(5\times1)$ reconstructed Ir(001) surface are ferromagnetic and that their easy axis is canted from the surface normal by an angle of $33^\circ$. We explain that this very large effect is due to the local symmetry breaking of the bridge chain of Ir atoms in between the Co strands, which provides the dominant contribution to the MAE and favors large canting of the magnetization. Experiments performed using spin-polarized scanning tunneling microscopy (SP-STM) confirm ferromagnetic order at 8 K with a canted magnetization direction of the Co zigzag chains.
![(color online) (a-d) Top and (e-h) side view of the three different chain configurations and the (5 $\times$ 1) reconstructed Ir(001) surface. In (c) the inner hollow (IH) and the outer hollow (OH) sites and in (g) the bridge Ir atom are marked. For the C1 (a,e) and C4 (b,f) chains the two strands are in the IH and OH sites, respectively, while for the C3 chain (d,h) one is in the IH and the other in the OH site. []{data-label="fig:structure"}](Fig1.png){width="8.5cm"}
We study the magnetic properties of biatomic Co chains on the (5 $\times$ 1) reconstructed Ir(001) surface by first-principles calculations based on DFT applying the film version of the full potential linearized augmented plane wave (FLAPW) method as implemented in the FLEUR code [@FLEUR]. We used a symmetric slab consisting of 37 Ir atoms (7 substrate layers) and 4 Co atoms. The resulting ($5 \times 1$) supercell has inversion symmetry with Co chains on both surfaces and adjacent Co chains are separated by $13.51$ Å. The structural relaxation was carried out using a mixed LDA/GGA functional introduced by De Santis *et al* [@PhysRevB.75.205432] to treat systems of 3$d$- and 5$d$-transition metals. Further computational details can be found in [@suppl].
We denote the biatomic chains with both atomic strands adsorbed in the inner (IH) or outer hollow (OH) sites as C1 and C4, respectively [@SpisakSS2003], while the zigzag chain is referred to as C3 chain (cf. Fig. \[fig:structure\]). The separation between the two Co strands increases as we move from the C1 to the C3 and to the C4 chain from 2.26 Å to 3.18 [Å]{} up to 4.10 [Å]{}. For the zigzag (C3) chain, there is a small buckling between the two inequivalent Co strands amounting to only 0.14 [Å]{}. The magnetic moments of the Co atoms depend both on the hybridization between the two strands and of the strands with the Ir substrate. The resulting values for the Co atoms in C1 are $1.9~\mu_B$ and decrease to $1.8~\mu_B$ for the Co in the C4 configuration. For the C3 chains the moments are comparable with the high symmetry configurations: $1.9~\mu_B$ for the IH site atom and $1.8~\mu_B$ for the OH site atom. For all chain configurations strong ferromagnetic exchange interaction is found along the chain. Due to the larger spacing between the two strands, the ferromagnetic coupling between the two strands is reduced by one order of magnitude for the C3 and C4 chain (values can be found in [@suppl]).
We calculated the MAE of the Co chains using the magnetic force theorem [@doi-10.1139/p80-159] and checked several configurations using self-consistent calculations [@suppl]. For the biatomic chains possessing mirror symmetries along and perpendicular to their axis, the easy axis of the magnetization can only lie along one of the high symmetry directions: out-of-plane (OP) to the surface, in-plane along the chain axis ($[110]$ direction) and in-plane perpendicular to the chain axis ($[\bar{1}10]$ direction) (see Fig. \[fig:structure\]). The easy magnetization axis of the C1 chain, where the two strands are closest, is out-of-plane with energy differences of 0.17 meV/Co-atom and 0.40 meV/Co-atom with respect to the $[110]$ direction and $[\bar{1}10]$ direction, respectively. For the C4 chain, the easy axis is oriented along the chain axis and the hard axis is along the surface normal (1.00 meV/Co-atom) and the $[\bar{1}10]$ direction is the intermediate state (0.59 meV/Co-atom). Qualitatively, the same results were found for biatomic Fe chains [@MokrousovPRB2009].
![(color online) (a) Magnetocrystalline anisotropy energy (MAE) calculated for the zigzag (C3) Co chain. The SQA is rotated in a plane perpendicular to the chain axis as shown in the inset. Beside the total energy, calculations are shown in which SOC has been turned off in one of the two Co strands (${\rm Co}_{\rm IH}$: SOC off in the outer hollow strand and vice versa for ${\rm Co}_{\rm OH}$). The dotted lines represent a fit to the MAE according to uniaxial anisotropy expression. (b) Orbital moments of the two inequivalent Co atoms, all Ir atoms, and the bridge Ir atom.[]{data-label="fig:MAE"}](Fig2.pdf "fig:"){width="8.cm"}\
Due to symmetry breaking, the situation becomes more complicated for the deposited zigzag (C3) chains and the easy magnetization direction does not need to align with a high symmetry direction, although it has to do so for the free-standing zigzag chain. The total energy was therefore calculated by rotating the spin quantization axis (SQA) also in a plane perpendicular to the chain axis. As can be seen in Fig. \[fig:MAE\](a), a minimum of about 1 meV/Co-atom is obtained for an angle of $\theta_0=33^{\circ}$ (see red curve).
In the spirit of the Bruno formula which links the energy minimum to the maximum of the orbital moment [@PhysRevB.39.86], we can interpret this result based on the orbital contributions of individual atoms displayed in Fig. \[fig:MAE\](b). Since the zigzag chain is composed of two non-equivalent strands, we observe different sizes and angular dependencies for the inner and outer hollow Co atoms. While the maximum is at a positive angle of $15^\circ$ for the inner hollow Co strand it is at $-23^\circ$ for the outer hollow strand. This leads to an opposite preference of the favorable magnetization direction for the two Co atoms. In agreement, the calculated energy displays a minimum of $\theta \approx 0^\circ$ if we turn off the SOC contribution from the entire substrate (not shown) [@note_noco].
The driving force behind the giant canting stems from interplay of the broken chain symmetry and the large Ir substrate contribution to the anisotropy of total energy and orbital moments. When SOC is switched on in the substrate and in only one of the Co strands, we still acquire a total energy minimum at positives angles (Fig. \[fig:MAE\](a)) irrespective of which Co chain is “switched on”. This shows that although one of the Co strands favors canting with a negative angle, its contribution is overwhelmed by that of the substrate. In agreement with the total energy, the maximum orbital moment of all Ir atoms in the first surface layer is found at a large positive angle of $44^\circ$. The main contribution to this orbital moment comes from the chain of Ir atoms at the bridge site between the two Co strands (cf. Fig. \[fig:structure\]). For each of the bridge Ir atoms the nearest neighbor Co atoms, which are different in their electronic structure due to different coordination, form a triangle with identical orientation. This causes fundamental breaking of local symmetry which results in all Ir bridge atoms favoring strong canting of magnetization towards a giant positive angle.
![(color online) (a) STM topography image and (b) simultaneously obtained d$I$/d$U$ image of Co chains on the $(5 \times 1)$ Ir(001) surface. (c) Line profiles of a C4 chain in image (a) and two profiles along a chain which changes structure along its axis from C1 to C3 (magnified in inset). Measurement parameters: W-tip, *T* = 7.8 K, $U_{bias}$ = $-$60 mV, $I$ = 0.2 nA.[]{data-label="fig:topo"}](Fig3.png){width="8.cm"}
To confirm the predicted ferromagnetic order and the canted magnetization of the zigzag Co chains we have performed spin-polarized STM experiments [@wiesendanger_RevModPhys_2009]. Bulk Cr tips are used with an arbitrary magnetization direction that is not changed by the application of an external magnetic field. Details on the sample and tip preparation can be found in [@suppl].
Figure \[fig:topo\](a),(b) show STM topography and d$I$/d$U$ images of Co deposited on the (5 $\times$ 1) reconstructed Ir(001) surface. As expected wires form along the trench structure of the surface. Several contrast levels are present in the d$I$/d$U$ image, indicating differences in the electronic structure, which demonstrates that multiple chain structures are present on the surface [@note_defect]. The chain configurations can be determined from line profiles shown in Fig. \[fig:topo\](c) taken perpendicular to the chains as marked in the inset and Fig. \[fig:topo\](a). Three different profiles can be distinguished: the dashed red curve shows a single peak with a narrow chain profile, the dot-dashed green curve shows a double peak with a wide chain profile, and the solid blue line a double peak with a profile that matches the narrow chain on the right and the wider chain on the left. When these three profiles are compared to the possible atomic arrangements in the Ir(001)-($5 \times 1$) trench, the chains can be identified as C1, C4, and C3, respectively. The line profiles obtained from our DFT calculations based on the Tersoff-Hamann model [@TH1983] confirm this interpretation and demonstrate a good agreement of the relaxed geometries with the experimental result [@suppl].
The anticipated SP-STM experiment is sketched in Fig. \[fig:tipcant-v2\]: due to the symmetry of the sample two mirror-symmetric Co zigzag chains are expected on the reconstructed Ir(001) surface, each with two possible magnetization directions along the easy axis. Within the spin-polarized Tersoff-Hamann model [@Wortmann_prl_2001] the tunneling current can be written as $I=I_0 + I_{\rm SP} {\mathbf m}_{\rm T} {\mathbf m}_{\rm S}$ where the first and second term are the non-spin-polarized and spin-polarized contribution, respectively, and ${\mathbf m}_{\rm T}$ and ${\mathbf m}_{\rm S}$ are the unit vectors of tip and sample magnetization. Therefore, with a perfectly out-of-plane ($\theta = 0°$) or in-plane magnetized tip ($\theta = 90°$) only two contrast levels are measured. However, a suitable canted tip magnetization can in principle discern all four possible magnetization directions of the chains, as demonstrated for the tip sketched in Fig. \[fig:tipcant-v2\]. As the magnetization of the tip is close to the easy axis of one type of zigzag chains (left, C3A) there is a large positive (C3A,$\uparrow$) or negative (C3A,$\downarrow$) contribution from the spin-polarized current for the two magnetization directions leading to a high or low d$I$/d$U$ signal, i.e. providing a high magnetic contrast. For the other type of zigzag chains (right, C3B) the projection of the tip magnetization onto the chain magnetization is much smaller leading to d$I$/d$U$ signals which are closer in value. Depending on the exact tip angle and the noise in the experiment the variation of the d$I$/d$U$ signal, i.e. magnetization direction, of the latter chain type will not be resolved and thus instead of a four-level contrast only a three-level contrast is obtained.
![(color online) Sketch of a magnetic tip and the two types of zigzag chains with the OH site strand either on the right (C3A) or left (C3B). The easy axis is canted by 33$^\circ$ from the surface normal in the direction of the OH site atom, i.e. $\theta$=$\pm$33$^\circ$ for C3A and C3B chains, respectively. The tip is chosen at $\theta$=50$^\circ$.[]{data-label="fig:tipcant-v2"}](Fig4.pdf){width="7.5cm"}
Spin-polarized STM measurements on six zigzag Co chains are shown in Fig. \[fig:mag\]. The two mirror-symmetric chain configurations can be identified in the line profiles of the topography (a); while most chains exhibit a single configuration one chain in the image area changes from C3A to C3B at a defect. The virgin state d$I$/d$U$ image in Fig. \[fig:mag\](b) exhibits either a very high (yellow) or comparably low (blue) signal for the C3A-chains, while the C3B-chains show a uniform intermediate (gray) signal. This uniform contrast level on each chain is indicative of ferromagnetic order. The observation of a three level contrast which is also visible in the line sections of Fig. \[fig:mag\](b) is in agreement with the considerations related to the sketch in Fig. \[fig:tipcant-v2\] confirming a canted chain magnetization.
The application of an external magnetic field induces a magnetization reversal for the chains with antiparallel magnetization components: Fig. \[fig:mag\](c) shows that the three upper C3A-chains have turned into the same magnetization state as the C3A-chain on the lower left and they appear red in the difference image (d). Still the magnetic contrast for the C3B-chain is in the intermediate state, leading to a two-level contrast in applied magnetic field as seen in the right panel of Fig. \[fig:mag\](c). While the exact magnetization angle cannot be determined experimentally, we can conclude that the Co-chains are ferromagnetic with a considerable canting of the magnetization away from the high-symmetry crystallographic axes, in agreement with the theoretical findings.
![(color online) (a) SP-STM image of zigzag Co chains which have the strand in the OH site either on the upper (red squares, C3A) or on the lower side (green circles, C3B) of the chain (left). Line profiles as marked in the image showing the difference between C3A and C3B structures. (b,c) Left panels show differential conductance images taken at $B$ = 0 and +2 T respectively. Right panels display line profiles of the d$I$/d$U$ signal taken at positions marked in (a) showing the contrast of the C3A and C3B chains with different magnetization directions. The dashed vertical line indicates the center of the chain. (d) difference image of the two d$I$/d$U$ maps (b) and (c). Measurement parameters: Cr-tip, *T* = 7.5 K, $U_{\mathrm{bias}}$ = 300 mV, $I$ = 3 nA.[]{data-label="fig:mag"}](Fig5.pdf){width="8.5cm"}
In conclusion, we have demonstrated that the local breaking of symmetry of the substrate due to proximity of an atomic chain can have a gigantic effect on the direction of the chain’s magnetization which is not anticipated from intuitive symmetry arguments. Our results provide a direction for further advances in the area of control of complex low-dimensional magnets based on employing the reduced symmetry of nano-magnets at surfaces.
We thank Matthias Menzel, Gustav Bihlmayer, and Stefan Blügel for many fruitful discussions. J.E.B. thanks the Alexander von Humboldt Foundation. Y.M. gratefully acknowledges funding under the HGF-YIG programme VH-NG-513. S.H. thanks the Peter-Grünberg Institute of the Forschungszentrum Jülich for its hospitality during his stay. B.D., Y.M., F.O., and S.H. thank the HLRN for providing computational resources and the DFG for financial support under project HE3292/8-1. J.E.B., A.K., K.v.B. and R.W. thank Matthias Menzel for experimental help and the DFG via SFB668-A8 and the ERC via the Advanced Grant FURORE for financial support.
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{
"pile_set_name": "ArXiv"
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---
abstract: 'The spherical QRPA method is used for the calculations of the $\beta$-decay properties of the neutron-rich nuclei in the region near the neutron magic numbers N=82 and N=126 which are important for determination of the r-process path. Our calculations differ from previous works by the use of realistic forces for the proton-neutron interaction. Both the allowed and first-forbidden $\beta$-decays are included. Detailed comparisons with the experimental measurements and the previous shell-model calculations are performed. The results for half-lives and beta-delayed neutron emission probabilities will serve as input for the r-process nucleosynthesis simulations.'
author:
- 'Dong-Liang Fang$^{a,b}$, B. Alex Brown$^{a,b,c}$ and Toshio Suzuki$^{d,e}$'
title: 'Investigating $\beta$-decay properties of spherical nuclei along the possible r-process path'
---
Introduction
============
The synthesis of the heavy elements in the universe is one of the important and interesting topics in modern physics. Different processes and astrophysical events are involved in understanding the measured isotopic abundances. One kind of process is the so-called rapid-neutron capture process (r-process), that gives rise to heavy elements beyond iron in our solar system [@WHW02; @CTT91]. The site of this process is still not clear. One of the popular ideas is the high-entropy wind of core collapse Type II supernovae [@CTT91; @FKMPJTT09; @FKPRTT10]. The obstacles to the simulation of the r-process are two-fold. (1) The unclear site gives uncertainties to the astrophysical environmental parameters which are crucial for the initial conditions of the simulations of the r-process evolution, such as the initial neutron richness, the temperature [*etc*]{} [@FKPRTT10]. (2) Most of the nuclei involved in the evolution are exotic neutron-rich ones which are currently out of the reach of experiments, so their reaction and decay rates are still uncertain. For the r-process the nuclei are very neutron-rich and some are near the neutron dripline. The general pattern of solar element abundance shows peaked distributions around $A\sim130$ and $A\sim 190$. Surveys [@CTT91] show that in order to produce these two heavy element peaks, the properties of nuclei around $A\sim 130$ and $190$ are important especially those near the proton or neutron magic numbers. The nuclear chart can be divided into regions near the magic numbers where the low-lying states are best described in a spherical basis, and regions in between the magic numbers where they are best described in a deformed basis. The QRPA method has been developed for each of these. There is a transitional region that, at present, must be interpolated in terms of the spherical and deformed limits. In a previous paper we focused on the deformed region of nuclei centered on Z=46 and N=66 [@FBS12]. In this work we will focus on the heavy spherical nuclei near the magic numbers. In these regions, some shell-model calculations can be performed with a truncated model space. Early calculations only included Gamow-Teller (GT) decay [@ML99; @LM03], but recent calculations [@SYKO12; @CMLNB07; @ZCCLMS13] have also included First-Forbidden (FF) decays which turn out to be important for the $N=126$ isotones [@SYKO12]. QRPA methods have previously been used, such as the QRPA method with separable forces in [@MR90; @MPK03; @HBHMKO96], the self-consistent QRPA from DFT [@EBDBS99] and the continuum QRPA [@BG00; @Bor03] [*etc*]{}. As an improved alternative, we propose the pn-QRPA method with realistic forces as introduced in [@STF88; @CFT87]. The advantage of this method is that with the G-matrix obtained from the Bethe equations with realistic potentials fitted from the nucleon-nucleon scattering data, one can obtain the full spectrum for the ground as well as excitation states of the odd-odd daughter nuclei. The full spectrum provides an exact treatment for the excitation energies which are missing in most other QRPA methods. The inclusion of states with more spin-parities also means that we can deal with the negative parity FF transitions which are missing in some calculations [@MR90; @MPK03].
This article is arranged as follows. In next section the QRPA method and its application to allowed and forbidden beta decay is outlined. The choice of model parameters is discussed in Sec. III. The results are presented in Sec. IV in comparison with experiment and to previous calculations. Sec. V presents the conclusions.
Formalism and Method
====================
In this work, we will calculate both the allowed Gamow-Teller (GT) and first forbidden (FF) $\beta$-decays for proton-neutron even-even and odd-even nuclear isotones near $N=82$ and $N=126$. The half-lives for these decays can be expressed generally as: $$\begin{aligned}
T_{1/2}=\frac{\ln 2}{\Gamma}.\end{aligned}$$ Here $\Gamma=\sum_i\Gamma_i$ is the total decay width and is a sum over all the possible decay widths from the ground state of initial parent nucleus to different states of final daughter nucleus with the specific selection rules. For GT decay, the width can be expressed as: $$\begin{aligned}
\Gamma_i^{{\rm GT}}=({\it f_0}/K_0)g_A^2 B_i({\rm GT}^-).
\label{GT}\end{aligned}$$ The term ${\it f_0}$ is the dimensionless phase-space factor depending on $\beta$-decay Q value [@BB82], while $K_0$ is a combinations of constants defined as $K_0=\frac{2 \pi^3 \hbar^7}{m_e^5 c^4}$. The GT strength $B({\rm GT}^-)$ can be expressed in terms of the reduced matrix element for spherical nuclei $B({\rm GT}^-)=|M_i({\rm GT}^-)|^2/(2 J+1)$, here $J$ is the spin of the parent nucleus. A frequently used quantities here is the log$ft$ value, which is defined here as log$ft$=log$[\frac{C}{g_A^2 B({\rm GT}^-)}]$ with $C=\ln 2 K_0=
6170$.
For FF decay, the expression is more complicated, as in [@BB82]: $$\begin{aligned}
\Gamma_i^{{\rm FF}}=\frac{f_i}{8896} (s^{-1}),\end{aligned}$$ with $$\begin{aligned}
f_i=\int_{1}^{\omega} \mathcal{C}(\omega)F(Z,\omega)p \omega (\omega_0-\omega)^2
d\omega,\end{aligned}$$ and with $\mathcal{C}(\omega)$ defined as $$\begin{aligned}
\mathcal{C}(\omega)=k+ka\omega+kb/\omega+kc \omega^2.
\label{FF}\end{aligned}$$ $\omega$ is the electron energy in the unit of electron mass $\omega=E_e/m_ec^2$, $F(Z,\omega)$ is the Fermi function as expressed in [@BB82] and $k$, $ka$, $kb$ and $kc$ are the nuclear matrix elements depending on the nuclear structure. The detailed expression of these matrix elements are given in eq.(8) in Ref. [@ZCCLMS13] and eq.(3,4,5) in Ref. [@SYKO12]. The log$ft$ values in this case are defined as log$ft$=log($f_0C/\Gamma_i^{{\rm FF}}$), here $f_0$ is the phase space factor for GT decay. In this work, for the matrix element calculations, we adopt the pn-QRPA (proton-neutron Quasi-particle Random Phase Approximation) method. The concept of quasi-particle starts with the nuclear pairing. The most common description of pairing in nuclear physics is the BCS formalism. Under the BCS formalism, one can define the quasi-particle operator $\alpha_\tau=u_\tau c_\tau+v_\tau
\tilde{c}_\tau^\dagger$, where $u_\tau$, $v_\tau$ are the BCS coefficients solved from BCS equations, and $c^\dagger_{\tau}$ is the single particle creation operator. With the quasi-particle operators, one can define the QRPA phonons as [@STF88]: $$Q^{m\dagger}_{J^{\pi}M}=\sum_{pn}
{X^{m}_{pn}A^\dagger_{pn,J^{\pi}M}-Y^{m}_{pn}\tilde{A}_{pn,J^{\pi}M}}.$$ Here the two quasi-particle operators are defined as $A^\dagger_{pn,{J^{\pi}M}}\equiv [\alpha_{p}^\dagger\alpha_{n}^\dagger]_{J^{\pi}M}$, with $p$, $n$ being proton and neutron respectively. The coefficients $X$’s and $Y$’s here are the forward and backward amplitudes respectively, they can be derived from the solutions of QRPA-equations [@CFT87; @STF88]: $$\begin{aligned}
\left( \begin{array}{cc}
A & B \\
B & A \end{array} \right)
\left(\begin{array}{c}X \\ Y\end{array}\right)
=\omega\left(\begin{array}{cc}1 & 0 \\ 0 & -1 \end{array}\right)
\left(\begin{array}{c}X \\ Y\end{array}\right).\end{aligned}$$ Here, $A_{pn,p'n'}=[A_{pn},[H,A^\dagger_{p'n'}]]$ and $B_{pn,p'n'}=[{A}_{pn}^\dagger,[H,\tilde{A}_{p'n'}^\dagger]]$. The Hamiltonian and the detailed expressions of $A$ and $B$ with realistic interactions were presented in Ref. [@STF88]. In this scenario, the different excited states are defined as: $|J^{\pi}M;m>\equiv Q^{m\dagger}_{J^{\pi}M}|0>$ with the QRPA phonon $Q^{m}_{J^{\pi}M}$ acted on even-even vacuum $|0>$ .
For decays of even-even nuclei we choose the BCS vacuum as the ground state: $$\begin{aligned}
|0\rangle_i=|QRPA\rangle\approx|BCS\rangle.\end{aligned}$$ While the final states in the odd-odd nuclei are the pn-QRPA excited states as defined by: $$\begin{aligned}
|m\rangle_f=Q^{m}_{J^\pi}|0\rangle.\end{aligned}$$ We choose the state with the smallest eigenvalue from QRPA equations to be the ground state of odd-odd final nucleus. The energies of all the excited states are then: $E_m=\omega_{m,J^\pi}-\omega_{g.s.}$, here the $\omega_m$ refer to the eigenvalues of the $m$th state from the QRPA solutions in the spherical systems while $\omega_{g.s.}$ is the smallest eigenvalues of the solutions. The effective Q values for each state are $Q_m=Q-E_m$, where Q is the mass difference between the two ground states of parent and daughter nuclei.
To obtain the lifetime, besides the Q values, one also needs the matrix elements in equations \[GT\]-\[FF\], for even-even to odd-odd decay. These can be expressed in our formalism as: $$\begin{aligned}
\langle J^\pi m||\tau^+ O_{I}^{K^\pi}||0\rangle_i=\delta_{J,K}\sum_{pn}\langle
p||\tau^+ O_{I}^{K\pi}||n\rangle \\ \nonumber
\times (X^{J^\pi,m}u_{p}v_{n}+Y^{J^\pi,m}v_{p}u_{n}).\end{aligned}$$ Here the $O_{I}^{K^\pi}$ is the nuclear transition operator for $\beta$-decay. The reduced matrix elements from the Wigner-Eckart theorem are independent of M (the projection of total angular momentum on z axis). For the allowed decay, $O_{GT}^{1^+}={\bf \sigma}$ with the selection rule $\Delta J=K=1$ and $\Delta \pi=1$, while for first-forbidden decay, the expressions of the operators are more complicated (the detailed forms given in [@SYKO12]) with the selection rules $\Delta J=K=0,~1,~2$ and $\Delta \pi=-1$.
For decays of odd-even nuclei, one has an unpaired nucleon, the simplest scenario is that given in Ref. [@MR90]. One quasi-particle or one quasi-particle plus one QRPA-phonon acts on the BCS vacuum (the spectator mode in [@MR90]). In Ref. [@MR90], for the single quasi-particle excitations, the correction from the coupling between the single quasi-particle and the QRPA-phonon was taken into consideration with the assumption of the weak coupling limit. In our calculation we find that this correction gives only small changes. Since it takes a much longer time for the calculation, we neglect this effect in the present calculation.
We have two kinds of states for odd-even nuclei. First, the single particle state as given by: $$\begin{aligned}
|\tau\rangle_i=\alpha_{\tau_i}^\dagger|0\rangle. \nonumber\end{aligned}$$ Here $\tau$ can be either proton or neutron. The ground states of the parent nuclei are chosen to be the one with the lowest quasi-particle energies, and this also holds for the even-odd daughter nuclei. For all the other single quasi-particle excitations, the relative excitation energies to the ground state are $E_i=E_{\tau,i}-E_0$.
Another kind of states for the daughter nuclei is the quasi-particle plus phonon state mentioned above without the possible mixing between then: $$\begin{aligned}
|\omega_{K^\pi M',m},\tau';J^\pi M
\rangle=C_{K,M';j_{\tau'}m_{\tau'}}^{J^\pi,M}Q_{K^\pi,m}^\dagger\alpha_{\tau'}^\dagger|0\rangle.\end{aligned}$$ The energies of such states are determined as follows. If we compare such states with the odd-odd nuclei, the only difference is the spectator single quasi particle, the difference of excitation energies in the two systems is simply equivalent to the difference of the Q values in these two systems (corresponding the difference of the ground states), this gives $E_{m,0}=Q_{oe}-Q_{ee}+\omega_{m}-\omega_{g.s.}$.
The matrix elements for $\beta$-decay of odd-even nuclei to these two different kinds of final states can be written as: $$\begin{aligned}
\langle n_i||\tau^+O^{K^\pi}_{I}||p_0\rangle=u_{p_0} u_{n_i} \langle
p_0||\sigma||n_i\rangle \nonumber\end{aligned}$$ and $$\begin{aligned}
\langle \omega_{K^\pi,m},p';J^\pi
M||\tau^+O^{K^\pi}_{I}||p_0\rangle=-\sqrt{(2J+1)(2j_{p_0}+1)}&& \\ \nonumber
\times \delta_{p_0,p'} \left\{
\begin{array}{ccc}
K & J & j_{p_0} \\
j_{p_0} & 0 & K
\end{array}
\right\} \langle \omega_{K^\pi,m}||\tau^+ O^{K^\pi}_{I}||0\rangle&&\end{aligned}$$ With $O_I^{K\pi}$ the same form as for the even-even case.
With these derived excitation energies and matrix elements, we can calculate the beta decay properties with the results presented in the next sections.
Choice of Parameters
====================
------------ ---- ----------- --------- -------------- ----------- --------- -------------- -- --
Calc.
N $E_{1^+}$ log$ft$ $t_{1/2}(s)$ $E_{1^+}$ log$ft$ $t_{1/2}(s)$
$^{118}$Cd 70 0 3.91 3018(12) 0 3.84 2484
$^{120}$Cd 72 0 4.10 50.80(21) 0 3.90 32.2
$^{122}$Cd 74 0 3.95 5.24(3) 0.04 3.97 4.76
$^{124}$Cd 76 1.25(2) 0.40 4.02 1.79
$^{126}$Cd 78 0.515(17) 0.70 4.09 0.67
$^{128}$Cd 80 1.17 4.17 0.28(4) 1.02 4.11 0.25
$^{130}$Cd 82 2.12 4.10 0.162(7) 1.18 4.14 0.12
$^{132}$Cd 84 0.097(10) 4.62 4.27 0.14
------------ ---- ----------- --------- -------------- ----------- --------- -------------- -- --
: The experimental $1^+$ energies and corresponding log$ft$ values of the largest decay branch are listed here for different even-even Cd isotopes, also the corresponding half-lives are shown. A comparison between the experimental measurements and theoretical calculations has been made with quenching factors $Q\equiv g_A/g_{A0}=0.4$. Here $g_{A0}=1.26$ is the axial-vector-coupling constant for free neutrons.
\[cd\]
In this work, we are interested in nuclei near or on the neutron magic numbers 82 and 126. We chose the isotonic chains with $N$=80, 82 and 84 and $N$=124, 126 and 128. Many of these nuclei are on the r-process path and their decay properties are important for r-process path that determines the final element productions near the peaks.
For single-particle (SP) energies, we adopt here those derived from solution of Hartree-Fock equations with the SkX Skyrme interaction [@SKX]. For the unbound positive-energy states, we make an approximate extrapolation for their discrete energies. We choose the QRPA model space as follows. We include all the SP levels with energies up to 5 MeV for neutrons and protons. For the pairing part, the BCS equations are solved with constant pairing gaps obtained from the symmetric five-term formula [@AW03], where the pairing gaps are derived from the odd and even mass differences. The obtained BCS coefficients and quasi-particle energies are then used as inputs for the QRPA equations as described above. For odd-even nuclei, the BCS solutions are especially important for nuclei with small Q values, since they determined which single-particle transitions are important for the lowest energies.
------------ ------------- ------------------- ----------------- -------------- ----------- ----------- ------------- ----------
Exp.[@nucldata] Theo.
$J^{\pi}_i$ $J^{\pi}_f$ $E_{ex}$ log$ft$ log$ft_q$ log$ft_u$ $J^{\pi}_f$ $E_{ex}$
$0^+$ $1^-,0^-$ 0.080 6.14 6.15 5.55 $0^-$ 0
$^{140}$Xe $(0,1^-)$ 0.515 6.82 6.58 5.98 $1^-$ 0.127
$0^{(-)},1^{(-)}$ 0.653 5.98 5.90 5.29 $1^-$ 0.586
$(1,2^-)$ 0.800 $\approx$7.1 7.36 6.22 $2^-$ 0.370
$1^{(-)}$ 0.966 6.77 6.56 5.95 $1^-$ 1.350
$0^+$ $2^-$ 0.011 $>$8.5 7.62 7.01 $2^-$ 0.041
$^{138}$Xe $(1)^-$ 0.016 7.2 7.01 6.41 $1^-$ 0.111
$1^-,2^-$ 0.258 7.32 7.42 6.82 $2^-$ 0.349
$1^-$ 0.412 6.79 6.29 5.68 $1^-$ 0.563
$0^-,1^-$ 0.450 6.59 6.52 5.92 $0^-$ 0
$0^+$ $(1^-)$ 0 $>$6.7 6.72 6.16 $1^-$ 0.169
$^{136}$Te $(0^-,1,2^-)$ 0.222 7.23 6.79 6.19 $2^-$ 0.540
$(0^-,1)$ 0.334 6.27 6.00 5.39 $1^-$ 0.749
$(0^-,1)$ 0.631 6.28 6.37 5.77 $1^-$ 0
$(0^-,1,2^-)$ 0.738 7.57 7.70 7.09 $2^-$ 0.194
------------ ------------- ------------------- ----------------- -------------- ----------- ----------- ------------- ----------
: Decay schemes for three even-even isotopes near $^{132}$Sn. For understanding the FF decays, we show the spin-parties, excitation energies and log$ft$ values for several important FF decay branches. For the two theoretical log$ft$ values, the one without subscript uses the queching factors $g_A=0.5g_{A0}$ and $g_V=0.5g_{V0}$ while the subscript $u$ means no quenching has been taken into account. For both cases, an enhancement factor $\epsilon=2$ for the tensor part of $0^-$ transitions is adopted [@War91].
\[ff82ee\]
---------------- ------------- --------------- ----------------- --------- ------------- ---------- ---------
Exp.[@nucldata] Theo.
$J^{\pi}_i$ $J^{\pi}_f$ $E_{ex}$ log$ft$ $J^{\pi}_f$ $E_{ex}$ log$ft$
$7/2^+$ $7/2^-$ 0.051 6.88 $7/2^-$ 0 6.63
$^{139}$Cs $9/2^-$ 1.283 7.4 $9/2^-$ 0.824 6.77
$5/2^-,7/2^-$ 2.349 7.3 $7/2^-$ 3.308 6.66
$3/2^+$ $3/2^-$ 0.051 5.63 $3/2^-$ 0.333 5.18
$^{203}$Au $3/2^-$ 0.225 6.61 $1/2^-$ 0 7.16
$5/2^-$ 0 5.19 $5/2^-$ 0.418 5.11
$3/2^+$ $1/2^-$ 0 5.79 $1/2^-$ 0 7.06
$^{205}$Au $3/2^-$ 0.468 6.43 $3/2^-$ 0.336 5.15
$5/2^-$ 0.379 6.37 $3/2^-$ 0.761 5.07
$1/2^+$ $1/2^-$ 0.379 5.11 $1/2^-$ 0 5.02
$^{207}$Tl[^1] $3/2^-$ 0.898 6.16 $3/2^-$ 0.624 6.56
---------------- ------------- --------------- ----------------- --------- ------------- ---------- ---------
: The decay schemes for several odd-even isotopes have been illustrated for both the experimental measurements and the theoretical calculations. We show the spin-parities, excitation energies and log$ft$ values for several decay branches which have the largest branch ratios. The parameters adopted here are $g_{pp}=1$ and $g_A(g_V)=0.5 g_{A0}(g_{V0})$.
\[ff128oe\]
For the residual interactions, we adopt the Brückner G-matrix derived from the CD-Bonn potential [@Mac89]. Two different parts of the interaction are included, namely the particle-hole channel and particle-particle channel interactions. The two body matrix elements are calculated with Harmonic-Oscillator radial wavefunctions. Renormalization of the matrix elements has been introduced to empirically take into consideration the effect of the truncations of the model space over the infinite Hilbert space into those orbitals in model space. For simplicity, two overall factors are introduced, namely $g_{ph}$ (for the particle-hole channel) and $g_{pp}$ (for the particle-particle channel). The parameter $g_{ph}$ mainly determines the position of the Giant Gamow-Teller resonance (GTR). Since there is no coupling between the GTR and the low-lying states the GT beta decay to low-lying states is not affected by the value of $g_{ph}$. Although for large $Q$ values some of the decay could go to the lower part of the GTR. This contributes little to the total decay width due to their higher energies, but it could be important for the beta delayed neutron decay branches. A value of $g_{ph}$=1 which reproduces the experimental energy is adopted here.
The other parameter, $g_{pp}$, affects the energies of low-lying excited states and the GT matrix elements to these states. The $1^+$ excitation energies are sensitive to the value of $g_{pp}$. For nuclei near N=82, $g_{pp}$ values around $0.8$ best reproduce details energy levels (see Table \[cd\]). For nuclei near N=126 there is limited data on the final state energies (especially for $1^+$), as we shall show later, with the quenching values we chose seems to agree with the experimental results, we find $g_{pp}=1.0$ better reproduces the half-lives. These parameters are obtained for the even-even nuclei, and for consistency the same values are used in odd-mass nuclei.
In this work, we also introduce a quenching effect since our calculations show an overall underestimation of half-lives compared with the experimental measurements for cases where there is good agreement for excitation energies. This quenching of the QRPA calculations for the spherical nuclei may have two origins. The GT strengths obtained with shell-model calculations in the $sd$ shell [@sdgt] and $pf$ shell [@pfgt] are systematically larger than experiment by about a factor of two. The “quenching" of experiment relative to theory by about a factor of 0.5 is consistent with results obtained by second-order perturbation theory corrections that take into account the excitation of nucleons outside of these model spaces [@arima; @towner].
Secondly, in spherical QRPA calculation, only one-phonon excitations have been taken into account, while shell-model calculations show that there exists the mixing between the single- and multi-phonon states. Two kinds of mixing outside of our model could be present: the low-lying part mixes with the GTR which shifts strength to higher energy; and the charge conserving 1p-1h excitations coupled with charge exchange GT excitations which spreads the GT strength and shifts some of the strength to higher energies. We determine the quenching empirically by comparing the calculated log$ft$ values and half-lives with the experimental ones for the Cd isotopes. The results are shown in table \[cd\]. The log$ft$ values are for individual final states, while the half-lives take into account the decay to all final states (GT is the dominate decay mode). A choice of $g_A/g_{A0}\approx0.4$ generally reproduces the half-lives. The largest difference comes from $^{130}$Cd due to the fact that we predict a smaller excitation energy compared to experiment. From this comparison, we adopt the value of $g_{A}/g_{A0}=0.4$ for the GT calculation as an optimal choice in our calculation for both $N=82$ and $N=126$ regions.
Compared to the simple operator $\sigma$ operator for GT decays, the FF operators are more complicated with several different spin-parity combinations. As was shown in [@War91], the tensor part of the $0^+$ to $0^-$ FF operator is enhanced due to mesonic-exchange currents. As in Ref. [@ZCCLMS13] we use an enhancement factor$\epsilon=2$. In general, we could use different quenchings for the other types of FF operators as was done for the shell-model calculations of Ref. [@ZCCLMS13]. However, this is quite complicated and even with least-square fits in Table. I of Ref. [@ZCCLMS13], we still see some large deviations, We cannot simply use the same quenching adopted in the shell-model approach since the origins of quenching for the shell-model and QRPA are different. For the shell model it comes from the model-space truncation (with a rather small model space compared with QRPA calculation), while for QRPA, it is from the configuration-space truncation (only one-phonon excitations have been taken into account in QRPA). In this sense, different queching schemes should be used. For simplicity we use an overall quenching factor for all the FF operator and we vary this value to find the best agreement to a rather limited set of data.
The quenching factors $g_A/g_{A0}=0.5$ and $g_V/g_{V0}=0.5$ for FF transitions are obtained this way by comparing the theory to experiment for some relatively strong FF transitions to low-lying states. In Table \[ff82ee\], the comparison between experiment and theory with and without the chosen quenching factors for the decay of Te and Xe isotopes is shown. Without quenching, there is a general underestimation about $0.6$ in the log$ft$ values. When we use above queching, for most decay branches, this deviation reduces to less than $0.3$. The same quenching factors are used in the N=126 region.
For the odd-mass nuclei, the same parameter sets ($g_{pp}$ and $g_A$) are adopted. The results for some FF branch ratios are shown in Table \[ff128oe\]. The deviation of the log$ft$ values are generally larger than the even-even case. We see reasonable agreement for $^{139}$Cs and $^{203}$Au. The $^{207}$Tl to $^{207}$Pb decay is particularly simple. When experimental single-particle energies are used the transition is dominated by the $3s_{1/2}$ to $2p_{1/2}$ contribution and we obtain log$ft$=5.11 (with quenching) compared to the experimental value of 5.02. In general, our QRPA approach is not very good for cases with one or two nucleons removed from the double-magic nuclei where the pairing is weak and the BCS solution sometimes overestimates the pairing effect. Also transitions to a few low-lying states are sensitive to the precise value of the single-particle energies. (The SkX single-particle energies differ from experiment by up to 0.5 MeV, see Fig. 4 in [@SKX].)
For the global parameters such as Q values and neutron separation energies, we use the experimental values if they are available, otherwise we use the masses predicted by some phenomenological mass models. For comparisons, we used two mass model here, the FRDM model [@MPK03] and HFB21 [@GCP09].
Results and Discussion
======================
 
With the determined parameters, we proceed to the calculations of isotonic chains on the r-process path in the vicinity of $N=82$ and $N=126$. We first show the results with neutron magic number $N=82$ and $N=126$ where the shell model results for these isotones are available. In Figs. \[tcmp\]-\[ffcmp\], we show the comparison of our results with shell-model calculations from Ref. [@SYKO12; @ZCCLMS13] (SMII results[@SYKO12] are updated by making a correction of a calculations error) and other QRPA calculations such as the QRPA with separable force from Ref. [@MPK03] and the continuum QRPA from Ref. [@Bor03; @Bor11]. For the first-forbidden (FF) part of the decay, Ref. [@MPK03] uses the gross theory instead of microscopic calculations, while for all the other calculations, the FF parts are calculated explicitly.
 
We first compare the most important observables – the half-lives predicted by different methods in Fig. \[tcmp\]. For $N=82$ isotones, there is good agreement among different methods except those from separable-force model of Ref. [@MPK03] for which there are systematic overestimation of half-lives on the even-even isotones compared with other methods for the high Z (low Q) isotones. This is due to their over-estimation of the excitation energies and underestimation of the matrix elements due to the lack of particle-particle interactions for QRPA. This results in an odd-even staggering behaviors for the half-lives in separable-force calculations. However, when the Q becomes large, this effect is reduced as we will see later. Except for Ref. [@MPK03], the general discrepancy among different methods for most nuclei is within a factor of two. This shows a good convergence of the predicted half-lives in microscopic calculations. While comparing with the experiments, our method underestimates the half-lives for $^{131}$In and $^{130}$Cd, especially for the former. The reason for this is that the BCS method does not work well near the doubly-magic nuclei where the pairing is weak and the BCS solution sometimes overestimates the pairing effect as we mentioned previously. For $^{130}$Cd the reason for the disagreement is due to the energy of $1^{+}$ state being too low compared to experiment as seen in Table \[cd\]. However, we find better agreement for $^{129}$Ag, compared with other methods. The results with FRDM mass model seem agree well with the shell-model calculations while those of the HFB21 mass give a smoother behavior over the isotonic chains. Compared with the shell-model calculation, we find the trend that when Q value becomes larger, our results give some overestimation on the half-lives. For $N=126$ isotones shown in Fig. \[tcmp\] the discrepancy between different methods becomes larger. Again, there is a systematic overestimation for results from the separable-force model Ref. [@MPK03] for even-even nuclei for larger Z. For the continuum-QRPA method, two sets of results are shown; the results from [@Bor03] systematically underestimate the half-lives, while the results from Ref. [@Bor11] are closer to other calculations. The two Shell Model calculations give similar results for the half-lives. The difference for the two shell-model calculations comes from the different quenching factors and model space they adopted that produces a nearly small constant difference within a factor of about two. In our calculations, there is difference of less than a factor of two with the two different mass models. The results with the FRDM masses shows a good agreement with the shell-model calculation while those with HFB21 masses give longer half-lives. For most of the nuclei here, with the FRDM mass model, our results differs with the shell model by a factor less than two. Overall, we find good agreement for the half-lives for the $N=126$ isotones among the different methods except those from Ref. [@MPK03] at high Z and Ref. [@Bor03] at low Z.
 
Next we compare the results for the $\beta$-delayed neutron emission probability $P_n$ in Fig. \[pncmp\]. For the $N=82$ isotones, unlike the half-life results, there are now larger differences among different methods. There is a general trend that the odd-even nuclei have larger $P_n$ values than their neighboring even-even nuclei except results from Ref. [@Bor03]. We predict lower $P_n$ values than the other calculations, especially for the even-even nuclei. For $^{130}$Cd, our result is smaller than the experimental value of 3.5%. In this case the neutrons come from beta decay to the region of excitation in $^{130}$In below the beta-decay Q value of 8.9 MeV and above the neutron decay separation energy of 5.1 MeV. Our low $P_n$ value indicates that the QRPA GT distribution in the region of 8 MeV is not spread out enough due to the lack of coupling with particle-hole states. The FRDM+QRPA(Sep.) [@MPK03] seems to have better predictions for the $P_n$ values, but this comes from the fact that the energy of the lowest $1^+$ state is too high in this case as they predict a much longer half-life for this nucleus, that is, about four times larger than the experimental half-life. The same situation happens for the $N=126$ isotones (right panel of Fig. \[pncmp\]). The differences among the calculations are quite large for even-even isotones, while there is better agreement for the odd-even cases. In the region $Z<70$, the $P_n$ values are close to $1$. This can be important for the r-process evolution. The shell-model calculation [@ZCCLMS13] predicts larger $P_n$ values than most other calculations for both $N=82$ and $N=126$ cases, the reason for this is that in shell-model calculations, due to their large configuration space, the strength is fragmented compared with QRPA calculations and more strength has been distributed above the neutron separation energy threshold. In order to see the importance of inclusion of the first-forbidden part, we compared the ratios of FF part to the overall decay width in Fig. \[ffcmp\]. For $N=82$, except Ref. [@MPK03] (as they haven’t calculated the FF parts explicitly), the three methods (QRPA, cQRPA and shell model) predict generally similar ratios. We have lower FF ratios which are close to those from cQRPA for high Z and higher FF ratios which are close to shell-model calculations for low Z nuclei. On the other hand, this ratio is about the same for the two mass models in our calculation There is a systematic difference between the cQRPA and shell-model calculations and our results are in between. In general, for the $N=82$ isotonic chain, all the calculations show that the first-forbidden part plays a less important role in $\beta$-decay. However, this is not the case for the $N=126$ (right panel of Fig. \[ffcmp\]) isotonic chain, where there are large discrepancies among different methods. We obtain FF ratios that decrease with Z in contrast to the other methods where they increase. Comparing our results carefully with those of Fig. 16 in Ref.[@ZCCLMS13], we found the difference comes from the fact that the two methods have different GT strength distributions. Although the excitation energies for $1^+$ states are similar (around 2 MeV for both methods for three nuclei $^{194}$Er,$^{196}$Yb and $^{198}$Hf), the structures of these states are different. In QRPA calculation, we see an small increase of the energies for the first $1^+$ states as the proton number increase, but the energies of the $1^+$ states which give dominant contribution to the GT decay however decrease sharply with the increasing proton number. This is not seen in shell-model calculations. This means that there may be too much strength presented in the low-lying states for N=126 isotones with higher Z due to the configuration space truncations. But on the other hand, shell-model calculations have too limited model space and could result in this different behaviors of GT strength distributions as well. Although Ref. [@Bor03] has also a very large FF ratio, compared with the same method in Ref. [@Bor11], this would be just due to a wrongly accounting of parameters in their calculations and their FF ratios for low Z isotones may need some updates. On the other hand we see that in the two shell-model calculations, the different quenchings and model spaces also give some different ratios as they have similar quenching for the GT part. So as explained above, the difference between our calculation and the shell-model calculations may came from two aspects, the different configuration space and the different model space, and further investigation is needed to explain the discrepancy for the ratios of FF parts.
 
 
The advantage of QRPA methods is that we can calculate with a larger model space more nuclei than those on the magic number line as in the shell model calculations. So we present more results around the magic number vicinity for even-end nuclei. First, we present the Q value sets we use in our calculation in Fig. \[Q82\]. As mentioned above, we adopt the Q values if they are available from experimental measurements as those in [@AWT02]. Otherwise, we use two mass models for the sake of comparison, the FRDM [@MPK03] mass model come from the macroscopic droplet models and the HFB21 [@GCP09] model from self-consistent microscopic Hartree-Fock-Boglyubov calculations. From Fig. \[Q82\], we see that around the neutron number $N=80$, the Q-values obtained from the two mass tables are basically the same for most even-even nuclei while there are $\sim 1MeV$ differences for most odd-even ones, generally FRDM predicts smaller Q-Values except for those nuclei with lower Z. However, for $N=82$ and $N=84$, the contrary happens and FRDM generally predicts larger Q-values. As for $N=124$, the same situation happens as for $N=82$, large difference occurs at low Z. As For $N=126$ and $128$, the discrepancy from the two mass models becomes irregular.
 
We calculated the half-lives of the isotones with the two mass models and the results are presented in Fig. \[t82v\]. For $N=$80, 82, 84 isotones, we make comparison with those results in Ref. [@MPK03] as well as some experimental values. When compared with the experimental results, we find very good agreement. In general, the agreement gets better when the nuclei are far away from the proton magic number $Z=50$. The results with $N=80$ and 84 has the same trends as N=82 we analyzed previously. Compared with results obtained in Ref. [@MPK03], we would see the same behaviour of overestimation of even-even nuclei in their calculation with moderate Q values. When the Q value increases the results in both calculations get closer and for much larger Q values, Ref.[@MPK03] predicts shorter half-lives than ours. The reason for this is due to the different treatments of the excitation energies. In Ref. [@MPK03], QRPA energies have been used directly as the excitation energies, which are in fact about several MeV higher than the actual excitation energies. Although the strength spreading has been introduced, even without the quenching effect, the deviation is still large. If Q values are large, then the effect from this difference between QRPA and excitation energies is weakened. Meanwhile the quenching reduces our decay strength and make our predicted half-lives longer than theirs. While in the case of odd nuclei, the single particle transition may play dominant role for low-lying states, we would not see this large deviation of half-lives for isotones with large proton numbers (Z). Generally, for low Q values, deviation occurs for even-even nuclei and agreement is achieved for odd-even nuclei, and for large Q values, we have longer half-lives due to the inclusion of quenching. If we compare the results obtained for the two mass models, we see that for the same nuclei, large Q value shortens the half-life. However, exceptions do exist as the mass model also affect the empirical gap parameters and hence change the structure indirectly. Such as the case of $^{132}$Cd, with the same Q value we have a small difference with two mass sets due to the different gap parameters we obtained. Different mass sets produce different half-lives and their impact on r-process is to be investigated. Also, the smoothing behavior of the half-lives on the proton number Z depends on the mass set one chooses. A choice of mass sets may produce the unwanted even-odd staggering behavior. It is somehow hard to tell, whether the staggering comes from the microscopic approaches or the Q value sets, due to the uncertainty of the theory. The same conclusion can be drawn on $N=124$, 126, 128. One has only limited experimental data in this region, and the agreement for these limited data with the calculations are very poor, as the Q values for these nuclei are very small. For $N=$124 and 126, with the increasing Q values, our calculation tends to agree with calculations of Ref. [@MPK03] for odd mass nuclei, but for N=128, the deviations always exist. In general, the different mass sets produce a deviation less than a factor of two close to the deviations produced from different methods we discussed for N=126 isotonic chain.
As for another measurable $P_n$, we have even less experimental data for comparison. Our calculation has smaller $P_n$ values compared with experiments, but the deviation is not large especially when $P_n$ is close to $1$. For $N=80$ and 82, the $P_n$ values increase as the decrease of proton numbers. For very neutron-rich nuclei, the beta-decay is followed immediately with neutron emissions. We see an odd-even staggering behavior as previously found in shell-model calculations. The deviation between our results and those from Ref [@MPK03] is large in magnitude but keeps the same trends (the reason of this has been explained above as the strength has been shifted to high-energies region systematically in their calculations). However, the two sets of results seem to agree with each other when $P_n\sim 1$. This is because the neutron separation energies here are pretty smaller, nearly all the strength lies in the window between $E_n$ and Q. When one crosses the magic number line $N=82$, there appear increased $P_n$ values as the nuclei here become less stable against the neutron emission, which agrees with results in Ref. [@MPK03]. For the heavier nuclei region with N around 126, the neutron emission probability hasn’t been measured for any nuclei. The general trends here for the two calculations again keeps the same while they differ in numbers as above. The nuclei beyond the neutron magic number $N=126$ show the large $P_n$ values as for the region of $N\sim
82$.
Due to the drawbacks of the QRPA methods, limited accuracy has been obtained especially for $P_n$ values. But one could see some improvements of the QRPA calculations, the half-lives are closer to the experimental values as well as those by shell model calculations compared with results in[@MPK03]. For regions where shell-model calculations are absent, the decent results have been obtained. To further improve the predicted $P_n$ values, the strength spreading from multi-phonon effect could be introduced such as those from the particle-vibration couplings [@LR06; @CSB10].
conclusion
==========
In this work, we calculated the weak decay properties of even proton number isotones near the neutron magic numbers 82 and 126 with the spherical QRPA method with realistic forces. Our results agree well with other calculations on the neutron magic number chains 82 and 126. We give the predictions for more nuclei in these region and compared with results in Ref. [@MPK03]. Different mass models have been used for the sake of comparison, and they produce a moderate difference for the final results. In general, we make some improvements on the accuracy of decay rates compared with Ref. [@MPK03] and its impact on the r-process simulations are to be investigated.
This work was supported by the US NSF \[PHY-0822648 and PHY-1068217\].
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[^1]: For this nucleus, we used the BCS coefficients extract from shell model to replace the solved bcs coefficients which are poorly reproduced near double magic nuclei.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
Software Transactional Memory systems (STMs) have garnered significant interest as an elegant alternative for addressing synchronization and concurrency issues with multi-threaded programming in multi-core systems. For STMs to be efficient, they must guarantee some progress properties. This work explores the notion of one of the progress property, i.e., **, in STMs. An STM system is said to be if every thread invoking a transaction gets the opportunity to take a step (due to the presence of a fair scheduler) such that the transaction eventually commits.
A few ** algorithms have been proposed in the literature in context of single-version STMs. These algorithms are priority based i.e. if two transactions are in conflict, then the transaction with lower priority will abort. A transaction running for a long time will eventually have the highest priority and hence commit. But the drawback with this approach is that if a set of high-priority transactions become slow, then they can cause several other transactions to abort. So, we propose multi-version ** STM system which addresses this issue.
Multi-version STMs maintain multiple-versions for each transactional object. By storing multiple versions, these systems can achieve greater concurrency. In this paper, we propose multi-version ** STM, , which as the name suggests achieves while storing $K$-$versions$ of each . Here $K$ is an input parameter fixed by the application programmer depending on the requirement. Our algorithm is dynamic which can support different values of $K$ ranging from one to infinity. If $K$ is infinite, then there is no limit on the number of versions. But a separate garbage-collection mechanism is required to collect unwanted versions. On the other hand, when $K$ is one, it becomes the same as a single-version ** STM system. We prove the correctness and ** property of the algorithm.
To the best of our knowledge, this is the first multi-version STM system that satisfies **. We implement and compare its performance with single-version ** STM system () which works on the priority principle. Our experiments show that gives an average speedup on the worst-case time to commit of a transaction by a factor of 1.22, 1.89, 23.26 and 13.12 times over , , NOrec STM and ESTM respectively for counter application. performs 1.5 and 1.44 times better than and but 1.09 times worse than NOrec for low contention KMEANS application of STAMP benchmark whereas performs 1.14, 1.4 and 2.63 times better than , and NOrec for LABYRINTH application of STAMP benchmark which has high contention with long-running transactions.
author:
- 'Ved Prakash Chaudhary [^1]'
- Chirag Juyal
- Sandeep Kulkarni
- Sweta Kumari
- 'Sathya Peri[^2]'
bibliography:
- 'citations.bib'
title: 'Achieving Starvation-Freedom in Multi-Version Transactional Memory Systems [^3]'
---
[^1]: A part of this work was submitted towards the fulfillment of M.Tech thesis requirement by the author.
[^2]: Author sequence follows a lexical order of last names.
[^3]: A preliminary version of this work was accepted in AADDA 2017 as **work in progress**.
|
{
"pile_set_name": "ArXiv"
}
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---
abstract: 'We present $BV$ photometry centered on the globular cluster M54 (NGC 6715). The color–magnitude diagram clearly shows a blue horizontal branch extending anomalously beyond the zero age horizontal branch theoretical models. These kinds of horizontal branch stars (also called “blue hook” stars), which go beyond the lower limit of the envelope mass of canonical horizontal branch hot stars, have so far been known to exist in only a few globular clusters: NGC 2808, $\omega$ Centauri (NGC 5139), NGC 6273, and NGC 6388. Those clusters, like M54, are among the most luminous in our Galaxy, indicating a possible correlation between the existence of these types of horizontal branch stars and the total mass of the cluster. A gap in the observed horizontal branch of M54 around $T_{\rm eff} = 27\,000$ K could be interpreted within the late helium flash theoretical scenario, a possible explanation for the origin of those stars.'
author:
- Alfred Rosenberg
- 'Alejandra Recio-Blanco'
- 'Macarena García-Marín'
title: Discovery of Blue Hook Stars in the Massive Globular Cluster M54
---
Introduction
============
The horizontal branch (HB) hosts stars with a helium-burning core of about 0.5 $M_{\odot}$, and a hydrogen-burning shell. The masses of the hydrogen envelopes vary from more than 0.2 $M_{\odot}$ to less than 0.02 $M_{\odot}$. Furthermore, the less massive the hydrogen envelope is, the hotter is the corresponding HB star. In the case of a star cluster, we find a color spread of the HB stars which is called the HB morphology. To a first approximation, the different color extensions of observed cluster HBs are described in terms of the variation of metal abundance, the [*first parameter*]{} (metal-rich clusters tend to have short red HBs, while metal-poor ones exhibit predominantly blue HBs). However, some other parameter (or set of parameters) has also to be at work, as clusters with nearly identical metallicities can show very different HB color distributions [@vandenbergh67; @sandage67], leading to the so called [*second parameter*]{} debate.
Horizontal branch stars with very low envelope masses ($\leq$ 0.02 $M_{\odot}$, $T_{\rm eff} > 20\,000$ K), known as extended or “extreme HB” (EHB) stars, are probably the most extreme expression of the second parameter problem. They have lost up to twice the mass during the red giant branch (RGB) ascent than other HB stars in the same cluster [@dcruz96]. As a result, in contrast to the more massive blue HB stars, EHB stars do not ascend the asymptotic giant branch (AGB) but evolve directly onto the white dwarf domain [@sweigart74]. Recently, @whitney98 and @dcruz00 revealed the existence of a particular kind of EHB star: a population of hot subluminous HB stars, lying up to 0.7 mag below the ZAHB and forming a hook-like feature in the far-UV color–magnitude diagram (CMD) of $\omega$ Centauri. These “blue hook” stars have effective temperatures up to $40\,000$ K and cannot be produced by canonical HB evolution [@brown01]. In the optical, for effective temperatures higher than 10000 K, ultraviolet radiation constitutes the main part of the energy flux coming from the stellar surface, making the HB, in practice, vertical in the classical $V$ [*vs.*]{} ($B-V$) plane become of bolometric correction. Hence, in optical CMDs blue hook stars are located at the faintest extreme of the HB.
In this letter, we present $BV$ photometry centered on the globular cluster (GC) M54. The CMD clearly shows a blue HB anomalously extending beyond ZAHB models. Previous photometric studies of this cluster, in ($V-I$), were not suitable for properly revealing this extremely hot stellar population. Initially, blue hook stars were detected in the clusters NGC 2808 and $\omega$ Centauri. More recently, @busso03 have reported their presence also in the blue HB tail of NGC 6388. In addition, as noted by @brown01, the CMD of NGC 6273 shown by @piotto99 shows a blue HB extending to $M_V$ $>$ 5, and therefore beyond theoretical ZAHB models. All these clusters are among the most massive GCs in the Galaxy, as well as M54, which is the second most massive GC known. In Section 2, we describe the observations and the photometric reduction techniques. In Section 3, we analyze the extended HB of M54 and its interpretation inside the late helium flashers scenario. Finally, in Section 4, we summarize the results and consider their wider implications.
Observations and Reduction
==========================
The observational data base consist of four images, two in $B$ and two in $V$, of 30 and 900 s each, centered on M54. Images were observed in service mode on 2002 July $8$ at the ESO 3.5 m New Technology Telescope (NTT), with a seeing of $\sim0.5$ arcsec. The detector, the Superb Seeing Imager (SUSI-2), is a mosaic of two 2k $\times$ 4k EEV CCDs, with a size of $0.08$ arcsec per pixel and a total field of view of $5.5$ $\times$ $5.5$ arcmin that were binned $2\times2$.
The images were corrected for bias and spatial sensitivity variations using the respective master flats, computed as the median of all available sky flats of the specific run. Afterward, photometry was performed using the DAOPHOT/ALLSTAR/ALLFRAME software [@stetson87; @stetson94].
The absolute calibration of the observations, which include the $BVRI$ filters, will be published in a forthcoming paper. It is based on six fields of standard stars from the catalog of [@landolt92], and the absolute zero-point uncertainties of our calibration are $\leq0.02$ mag for each of the four bands.
The extended HB of M54 and its interpretation
=============================================
Figure 1 shows the CMD of M54 in an area away from the crowded region with $R$ $<$ 90 arcsec, where $R$ is the projected distance from the cluster center. We can identify at least three stellar systems: the GC M54, the Sagittarius Dwarf Galaxy, and the Milky Way bulge [@layden00]. All stars plotted were selected using the sharp parameter ($-$0.25 $<$ sh $<$ 0.25) and the $V$ error ($\leq0.1$). In this paper we are interested in M54, for which the main sequence (MS), RGB, and HB are clearly visible. The whole CMD has been shifted in color and magnitude by $E(B-V)=0.16$ and $(m-M)_0=17.25$ in order to fit the ZAHB model. The CMD shows an extended HB, which spans almost 4.75 mag in $V$ and extends down to $M_V\simeq$ 5.25, i.e., $\sim$ 1.5 mag below the turn-off. The thick line shows the ZAHB model by @cassisi97 for a metallicity of \[Fe/H\] $=$ $-1.31$.
Crowding and completeness experiments where also performed along the HB sequence. The completeness was found to be higher than $50\%$ for the entire CMD shown ($60\%$ and $70\%$ zones are labeled in Fig. 1.)
From the crowding experiments, we have obtained the photometric dispersion for synthetic stars located on the HB theoretical model line, to which we have add three times the typical dispersion in color (0.025 mag) measured from the region between $1.0\leq{M_V}\leq4.5$ of the HB. These limits are represented by the dashed lines and all 78 stars within these two lines are considered as the HB stars in the following discussion.
Although, because of bolometric correction, the ($B-V$) color is not a good temperature indicator for the hot HB stars, the CMD clearly shows a horizontal branch $\sim$ 0.5 mag deeper than the ZAHB model. At least nine stars in Fig. 1 seem to have effective temperatures higher than $33\,600$ K, which is the end of the ZAHB model by @cassisi97. As mentioned in the introduction, a different evolutionary path for populating the very hot end of the HB is therefore needed. One possible explanation is that these stars experience a delayed helium core flash. @castellani93 and @dcruz96 [@dcruz00], showed that even if a star suffers a very high mass loss during its RGB phase, it can still ignite He burning after evolving off the RGB.
If the helium flash occurs somewhere between the tip of the RGB and the top of the helium dwarf cooling curve ([*early*]{} hot flashers), the star will settle onto the EHB without any mixing between its helium core and hydrogen envelope, following a canonical evolutionary path to the EHB. The reason for this can be found in the entropy barrier caused by the hydrogen-burning shell, which prevents the convection zone produced by the helium flash from penetrating into the hydrogen envelope [@iben76]. Early hot flasher models by @brown01 predict that these stars reach a maximum temperature of $31\,500$ K on the EHB, and that their luminosities are almost indistinguishable from the luminosities of canonical EHB stars. @brown01 claimed that if the helium flash occurs while the star is descending the white dwarf cooling curve ([*late*]{} hot flashers), the entropy barrier carried by the hydrogen-burning shell becomes a too weak to prevent mixing between the core and the star’s envelope. The resulting star will have a temperature of about $37\,000$ K and will lie as much as 0.7 mag below the ZAHB in the [*UV*]{} CMD, further supporting the argument that blue hook stars are the progeny of late hot flashers. The fact that blue hook stars are both hotter and more helium-rich than classical EHB stars has been observationally confirmed by the spectroscopic analysis of @moehler02 in $\omega$ Centauri. In addition, full evolutionary computation of helium flash induced-mixing in Population II stars has recently been developed by @cassisi03. They modeled the incursion of the He flash-driven convective zone into the H-rich external layers, with a subsequent surface enrichment in He and C. In agreement with the observations, models experiencing this dredge-up event are significantly hotter than their counterparts with H-rich envelopes. They also compare their abundance predictions of He with measurements by @moehler02.
On the other hand, the sharp transition between the early and the hot flashers, corresponding to a difference in mass loss of only 10$^{-4}$ $M_{\odot}$ along the RGB, would produce a gap in the observed stellar distribution. The CMDs of $\omega$ Centauri by @kaluzny97 and @lee99, and NGC 6273 by @piotto99 do not seem to show an EHB gap. @brown01 suggest that the gap in $\omega$ Centauri is perhaps blurred by the metallicity distribution of the cluster. The differential reddening and magnitude limit could be the reason in the case of NGC 6273. They test their blue hook models by trying to reproduce the luminosity function of NGC 2808 by @bedin00. They first assume that the EHB and the blue hook zone are uniformly populated and then include the HB evolution of the models to brighter $V$ magnitudes. They find that the gap in the observed ZAHB distribution corresponds to the gap between the canonical ZAHB and their blue hook models with mixed envelopes, further supporting the scenario of the late hot flashers.
In Figure 2, we compare the stellar distribution in $M_V$ of the M54 horizontal branch from our photometry (shown as filled triangles) with that of NGC 2808 by @bedin00 (open circles). The total number of stars in the NGC 2808 distribution (160) was normalized to that of M54 (78) for a better comparison. For the latter cluster, a distance modulus of ($M-m$)$_V=$ 15.74 [@bedin00] was used. Error bars for M54 correspond to the square root of the number of the stars per bin, and the bin size is 0.30 mag. Interestingly, M54 presents a gap in the stellar distribution at the same absolute $V$ magnitude ($M_V$ $\sim$ 4.3) as NGC 2808 (marked in Figure 2 with a vertical arrow). As pointed out before, in the late helium flasher scenario, this gap would separate the canonical EHB from the blue hook stars and would be only apparent because of the mixed envelopes of blue hook stars [@brown01]. If this is true, a quite abundant blue hook population is clearly seen in M54 at the same absolute magnitudes (effective temperatures) as in NGC 2808. From this histogram, we found that the number of star in the blue hook region, compared to the whole HB star sample (for $1.0 \leq M_V \leq 5.5$) is $35\%$ in the case of M54 and less than $20\%$ in the case of NGC 2808. We do not know if this difference is caused by the metallicities, which are very similar for both clusters (\[Fe/H\] $= -1.25$ and \[Fe/H\] $= -1.11$ for M54 and NGC 2808, respectively, in the Carretta & Gratton 1997 scale [@rutledge97]), which helps the direct comparison of their HBs. On the other hand, it would again be necessary to take into account the HB evolution of the models to brighter V luminosities in order to explain the location of the gap at $T_{\rm eff}\sim 27\,000$ K, as already claimed by @brown01 for the case of NGC 2808. On the other hand, we note that the first gap in the HB of NGC 2808 pointed out by @bedin00 at $V\sim18.5$ mag ($M_V\sim2.8$ mag) is not visible in M54. On the other hand, there is an underpopulated zone at lower magnitudes around $M_V\sim3.2$ mag.
Finally, the peculiar bimodality of the NGC 2808 HB morphology, which includes both a blue HB and a red HB clump, may also be present in M54. This would increase the similarities in the peculiar features observed in the HB of these two clusters. However, although there is evidence for a red HB both in our diagram and already published $VI$ photometry (see, for example, @sarajedini95), this feature could belong to the Sagittarius (Sgr) dwarf galaxy field. @layden00, in their extensive $VI$ photometry of M54 and the Sgr galaxy, conclude that their subtractions of the Sgr field from the M54 CMD cannot ascertain beyond doubt whether the anonymous red HB belongs to Sgr or to M54.
On the other hand, in the recent work by @monaco03, it is clear that the Sgr dwarf galaxy also possesses an old and relatively metal-poor stellar component that populates the blue HB tail. Interestingly, @monaco03 find that the the population of faintest ($V>18.6$) blue HB stars is relatively lower in the Sgr field than in M54. Moreover, the spread of the stars in the hottest part of the Sgr field HB is rather high. In our Figure 1 we can see that the stars that we present as blue hook stars are clearly grouped in a relatively small region of the CMD. They are clearly separate from the MS, the rest of our CMD being extremely clean. For all these reasons we believe that the stars in our CMD, located under the GAP, are really cluster members.
Discussion
==========
As described in the previous section, we can conclude that the M54 horizontal branch hosts a blue hook stellar population that extends the HB to fainter magnitudes than ZAHB models. Understanding the origin of hot HB stars is important not only for our fine-tuning of stellar evolution theory, but has wider applications in astrophysics. Indeed, hot HB are now considered to be the prime contributors to the ultraviolet emission in elliptical galaxies [@greggio90; @brown01]. Blue hook stars are not, however, a common feature of all GCs with an extended blue HB morphology (see for instance [@moehler00], for the cluster NGC 6752).
At this point, it might be useful to examine the similarities in the physical properties of this cluster with the other GCs with already confirmed or suspected blue hook stars in their HBs—that is NGC 2808, NGC 6388, NGC 6273, and $\omega$ Centauri—in order to shed more light on the origin of this peculiar kind of star. The most striking analogy between these five clusters is that they are among the most massive clusters in our Galaxy. Such characteristics would explain the presence of a larger EHB population, but would not, in principle, be directly considered as a justification for the bluer HB morphology of these clusters. The total absolute $V$ magnitude ($M_V$) can give a good estimate of these clusters’ total luminosities (as all GCs have similar color indices and hence similar bolometric corrections) and therefore a good measurement of the baryonic mass of these old stellar systems. From (@harris96, updated to the new catalog version of 2003 February) we find $M_V$ $=$ $-9.18$ for NGC 6273, $-9.39$ for NGC 2808, $-9.42$ for NGC 6388, $-10.01$ for M54, and $-10.29$ for $\omega$ Centauri.
Nevertheless, we could think that if there is a distribution in mass loss along the RGB, then the high mass loss tail of this distribution would be more likely to be occupied in a more massive cluster. If this is true, a correlation between the number of hot HB stars per stellar mass in a cluster and its HB extension and/or total mass at constant metallicity should also exist. The first results of a multivariate analysis (Recio-Blanco et al., in preparation), based on a photometric database of 74 GCs [@piotto02] seem to exclude such correlation.
On the other hand, both M54 and $\omega$ Centauri are suspected of being the nuclei of a current and former dwarf galaxy, respectively. In fact, M54 must play an important role in the star formation history of the Sagittarius galaxy, as it lies in the high density region of Sgr [@ibata97], and, as pointed out by @layden00, it marks one of the earliest epochs of star formation in Sgr. M54 may be similar to the nuclear star clusters seen in nucleated dwarf elliptical galaxies [@sarajedini95]. On the other hand, because of the unusual properties of $\omega$ Centauri (mainly abundance variations and metallicity spread), the scenario that this cluster may be also the core of a disrupted dwarf galaxy (e.g., @freeman93) had a continuous infall of gas to its center, leading to a variable star formation history, is becoming popular. Moreover, the NGC$~$2808 HB bimodality could be interpreted within a similar scenario of cluster self enrichment if we consider the @dantona02 suggestion of the influence of a possible second generation of He-rich stars in the final cluster HB morphology. In this sense, the correlation between the high mass of these clusters and the existence of blue hook stars (and so of their progenitors as the proposed late hot helium flashers) could also be linked to the second parameter debate regarding the more general problem of GC HB morphology.
It is interesting to point out, in addition, that age differences from cluster to cluster would not be enough to explain the second parameter problem. This is the case for NGC$~$2808, coeval with other clusters of similar metallicity but much shorter HBs such as NGC 362, NGC 1261, or NGC 1851, all of them still $20\%$ younger than NGC 288, NGC 5904, or NGC 6218 [@rosenberg99; @rosenberg00a; @rosenberg00b]. In fact, many other massive GCs of different metallicities show particularly extended HB morphologies: NGC$~$6266 ($M_V = -9.19$, \[Fe/H\] $\sim
-1.3$), NGC$~$2419 ($M_V = -9.58$, \[Fe/H\] $\sim -2.1$=) and NGC$~$6441 ($M_V = -9.64$, \[Fe/H\] $\sim -0.5$). Deeper photometry in the blue or in the ultraviolet could reveal the presence of a blue hook population for those clusters as the one already detected in NGC$~$2808, $\omega$ Centauri, NGC 6388, NGC 6273, and now in M54.
We would like to thank our anonymous referee for many valuable comments and suggestions. This report is based on observations with the ESO [NTT + SUSI2]{}, located at the La Silla Observatory, Chile (ESO proposal 69.D-0655(A)). We thank L. R. (“Rolly”) Bedin for providing the NGC 2808 HB photometry and Santino Cassisi for his ZAHB models. We are grateful to Antonio Aparicio, Carme Gallart, Giampaolo Piotto and Ivo Saviane for allowing us to used these data, from a larger project entitled “Relative Ages of Outer Halo Globular Clusters”, in advance of publication. ARB recognizes the sustenance from [MIUR]{} and from [ASI]{}.
Bedin, L.R., Piotto, G., Zoccali, M., Stetson, P.B., Saviane, I., Cassisi, S., & Bono, G. 2000, , 363, 159 Brown, T.M., Sweigart, A.V., Lanz, T., Landsman, W.B. & Hubeny, I. 2001, , 562, 368 Busso, G., Piotto, G. & Cassisi, S. 2003, MemSAI, astro-ph/0308341 Carreta, E., & Gratton, R.G. 1997, , 121, 95 Cassisi, S., Schlattl, H., Salaris, M. & Weiss A., 2003,, 582, L43 Cassisi, S., & Salaris M. 1997, , 285, 593 Castellani, M., & Castellani, V. 1993, , 407, 649 D’Antona, F., Caloi, V., Montalbán, J., Ventura, P. & Gratton, R. 2002, , 395, 69 D’Cruz, N.L., Dorman, B., Rood, R.T. & O’Connell, R.W. 1996, , 466, 359 D’Cruz, N.L., et al. 2000, , 530, 352 Freeman, K.C. 1993, ASP Conf. Ser., 48, 27 Greggio, L., & Renzini, A. 1990, , 364, 35 Harris, W. 1996, , 112, 1487 Ibata, R.A., Wyse, R.F., Gilmore, G., Irwin, M.J. & Suntzeff, N.B. 1997, , 113, 634 Iben, I. 1976, , 208, 165 Kaluzny, J., Kubiak, M., Szymański, M., Udalski, A., Krsemiński, W. & Mateo, M. 1997, , 125, 343 Layden, A.C., & Sarajedini, A. 2000, , 119, 1760 Landolt, A.U. 1992, , 104, 340 Lee, Y.W., Joo, J.M., Sohn, Y.J., Rey, S.C., Lee, H.C. & Walker, A.R. 1999, , 402, 55 Moehler, S., Sweigart, A.V., Landsman, W.B. & Heber, U. 2000, , 360, 120 Moehler, S., Sweigart, A.V., Landsman, W.B. & Driezler, S. 2002, , 395, 37 Monaco, L., Bellazzini, M., Ferraro, F.R., & Pancino, E. 2003, , 597, L25 Piotto, G., Zoccali, M., King, I.R., Djorgovski, S.G., Sosin, C., Rich, R.M. & Meylan, G. 1999, , 118, 1727 Piotto, G., et al. 2002, , 391, 945 Rosenberg, A., Saviane, I., Piotto, G. & Aparicio, A. 1999, , 118, 2306 Rosenberg, A., Piotto, G., Saviane, I. & Aparicio, A. 2000a, , 144, 5 Rosenberg, A., Aparicio, A., Saviane, I. & Piotto, G. 2000b, , 145, 451 Rutledge, G.A., Hesser, J.E., & Stetson, P.B. 1997, , 109, 907 Sandage, A., & Wildey, R. 1967, , 150, 469 Sarajedini, A., & Layden, A.C. 1995, , 109, 1086 Stetson, P.B. 1987, , 99, 191 Stetson, P.B. 1994, , 106, 250 Sweigart, A.V., Mengel, J.G., & Demarque, P. 1974, , 30, 13 Van den Bergh, S. 1967, , 72, 70 Walker, A.R. 1999, , 118, 432 Whitney, J.H., et al. 1998, , 495, 284
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{
"pile_set_name": "ArXiv"
}
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---
author:
- |
Xuemei Li$^a$ and Zaijiu Shang$^b$\
$^{a}$ Key Laboratory of High Performance Computing and Stochastic Information Processing,\
Department of Mathematics, Hunan Normal University, Changsha, Hunan 410081, P. R. China\
$^{b}$ 1. HLM, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China\
2. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China\
title: 'A KAM-Theorem for Persistence of Quasi-periodic Invariant Tori in Bifurcation Theory of Equilibrium Points [^1] '
---
0.3in
[**Abstract.**]{} In this paper, we establish a KAM-theorem for ordinary differential equations with finitely differentiable vector fields and multiple degeneracies. The theorem can be used to deal with the persistence of quasi-periodic invariant tori in multiple Hopf and zero-multiple Hopf bifurcations, as well as their subordinate bifurcations, of equilibrium points of continuous dynamical systems.
0.1in [*Keywords:*]{} Quasi-periodic invariant torus, Small frequency, Degeneracy, multiple Hopf bifurcation. 0.2in
1. Introduction {#introduction .unnumbered}
================
To study the bifurcations of equilibria of a system of differential equations (ODEs, PDEs and functional differential equations), one usually reduces such a system to a lower-dimensional one on the center manifold by the Center Manifold Theorem. Possibly the reduced system is finitely differentiable even if the original system is analytic. When the equilibrium is partially elliptic and the normal form of the reduced subsystem has a normal form of Birkhoff type on the center manifold, then the truncated normal form may possess quasi-periodic invariant tori. In this case, a question arises naturally: does the original system (equivalently, the reduced system on the center manifold) have quasi-periodic invariant tori with the same dimension ? This problem can be discussed by KAM theory and a careful study leads us to consider the existence of quasi-periodic tori of the following system $$\label{1.1}
\left\{\begin{array}{rl}
\dot I_1 & = \varepsilon^{q_1} [A_1(\xi,\varepsilon)I_1+\varepsilon^{q_2}g_1(I,\varphi; \xi,\varepsilon)]\\
\dot I_2 & = \varepsilon^{q_3} [A_2(\xi,\varepsilon)I_2+\varepsilon^{q_4}g_2(I,\varphi; \xi,\varepsilon)]\\
\dot{\varphi}_1 & = \varepsilon^{q_5}[\omega_1(\xi,\varepsilon)+\varepsilon^{q_6}g_3(I,\varphi; \xi,\varepsilon)]\\
\dot{\varphi}_2 & =\omega_2(\xi,\varepsilon)+\varepsilon^{q_7}g_4(I,\varphi; \xi,\varepsilon),
\end{array}
\right.$$ where $I={\rm col}(I_1,I_2)\in \Omega\subset \mathbb{R}^{n_{11}}\times \mathbb{R}^{n_{12}}=\mathbb{R}^{n_1}, \varphi={\rm col}(\varphi_1,\varphi_2)\in \mathbb{T}^{n_{21}}\times \mathbb{T}^{n_{22}}=\mathbb{T}^{n_2}, q_j\geq 0(j=1,\cdots,7), \xi\in \Pi\subset \mathbb{R}^{n_3}$ is the bifurcation parameter, $\varepsilon$ is a small perturbation parameter.
When $g_j=0, j=1,\cdots,4$, $I=0$ represents the quasi-periodic torus of the integrable part of which corresponds to the invariant torus of truncated normal forms. The aim of the present paper is to examine the persistence of the quasi-periodic torus $I=0$ under small perturbations (i.e., $g_j\neq 0,j=1,\cdots,4$). We meet some difficulties: the perturbation terms $g_j,j=1,\cdots,4$ are only finitely differentiable, there exist small frequencies, small twist and higher-order degeneracy in and the number of parameter variables is possibly less than the dimension of tori. We need to tackle these difficulties in constructing a new KAM theorem for .
In the context of finitely differentiable perturbations, the study on the persistence of quasi-periodic invariant tori has originated from the work of Moser [@Mos62] on area-preserving mappings of an annulus, which was extended to dissipative vector fields in [@BMS76] based on smoothing operator technique. Another important method, which can relax the requirement for regularity of perturbations, is to approximate a differentiable function by real analytic ones [@Mos66; @Russ70; @Zehn75; @Pos82; @Sh00; @CQ04; @Alb07; @Wag10]. Rüssmann proved an optimal estimate result on approximating a differentiable function by analytic ones. Following this approach Zehnder [@Zehn75] established a generalized implicit function theorem and applied it to the existence of parameterized invariant tori of nearly integrable Hamiltonian systems in finitely differentiable case, Pöschel [@Pos82] showed that on a Cantor set, invariant tori of the perturbed Hamiltonian system form a differentiable family in the sense of Whitney. The results and ideas of Moser and Pöschel are extended to the case of symplectic mappings by Shang [@Sh00] and to the case of lower dimensional elliptic tori by Chierchia and Qian [@CQ04], respectively. Wagener [@Wag10] extended the modifying terms theorem of Moser [@Mos67] (i.e., introducing additional parameters) to finitely differentiable and Gevrey regular vector fields.
The results mentioned above, except for [@CQ04], were restricted to the case where the integrable part is analytic in coordinate variables as well as in parameters. The integrable part in [@CQ04] is assumed to be Lipschitz with respect to parameters and the frequency map to be a Lipschitz homeomorphism. Of course, if the unperturbed (integrable) part and the perturbation are both of class $C^l$ with $l>2n$ ($n$ is the number of degrees of freedom), it is reduced to the case where the integrable part is analytic and the perturbation is of class $C^l$ by regarding the initial values of action variables as parameters. The KAM theorems in [@BHS96; @Wag10] can be applied to quasi-periodic bifurcations (i.e., bifurcations of quasi-periodic invariant tori). In this paper, we shall extend the result and method of Pöschel [@Pos82] to the dissipative system with degeneracies, and provide a convenient tool to investigate the persistence of quasi-periodic invariant tori in bifurcation theory of equilibrium points.
The perturbation was assumed to be $C^{333}$ originally in the work of Moser [@Mos62] on area-preserving mappings of an annulus, and then was weakened to $C^5$ by Rüssmann [@Russ70] and to $C^l (l>3)$ (meaning that the perturbation is of class $C^3$ and the derivatives of order 3 are Hŏlder continuous) by Rüssmann [@Russ83] and Herman [@Her83], where a counterexample for $l<3$ was given. For improvements on weakening the regularity of perturbations in the Hamiltonian case we refer to [@Alb07] and references therein.
The above mentioned results were proved under the so-called non-degeneracy conditions. In the context of degenerate KAM theory, i.e., if Kolmogorov’s non-degeneracy or Arnold’s iso-energetic non-degeneracy condition is violated, Arnol’d [@Arn63] established a properly degenerate KAM theorem (refined by [@Fej04; @Chier10]) to deal with quasi-periodic motions in the planetary many body problem. In this case the integrable part does not depend on the full set of action variables, and the non-degeneracy conditions are imposed additionally on the averaged perturbation. The ideas of Arnol’d [@Arn63] were extended to the resonant torus case in [@CW99; @LY03] and the normal zero-frequency case in [@GG05; @HLY06; @Gen07] for Hamiltonian systems and in [@BG01; @LY12; @Li16] for dissipative systems. Another method is to search for weaker non-degeneracy conditions concerning frequency maps, which have been studied in a series of papers, for example, by Bruno [@Bru92], Cheng and Sun [@CS94], Rüssmann [@Russ89; @Russ01], Han, Li and Yi [@HLY10] for finite dimensional Hamiltonian systems, and Bambusi, Berti and Magistrelli [@BBM11] for infinite dimensional case. The weaker non-degeneracy condition in [@CS94] is that the image of the frequency map in an open set includes a curved $C^{n+2}$ one-dimensional submanifold. Rüssmann [@Russ89; @Russ01] pointed out that the weaker non-degeneracy condition means that the image of the frequency map does not lie in an $(n-1)$-dimensional linear subspace of $\mathbb{R}^n$ (this condition is also necessary in the analytic case). An interesting and real analytic Hamiltonian of the form $$H(x,y,\varepsilon)=h_0(y^{n_0})+\varepsilon^{m_1} h_1(y^{n_1})+\cdots + \varepsilon^{m_a}h_a(y^{n_a})+\varepsilon^{m_a+1}P(x,y,\varepsilon)$$ with the degeneracy involving several time scales was considered in [@HLY10]. The degeneracy in is somewhat similar to the one in [@HLY10].
0.2in
2. Statement of results {#statement-of-results .unnumbered}
=======================
Let $\Omega_1$ and $\Omega_2$ be convex open neighbourhoods of the origin in $\mathbb{R}^{n_{11}}$ and $\mathbb{R}^{n_{12}}$, respectively, $\Omega=\Omega_1\times \Omega_2$, the parameter set $\Pi$ be a convex bounded open set of positive Lebesgue measure in $\mathbb{R}^{n_{3}}$. Let $|x|$ denote the maximum norm and $|x|_p$ the $p$-norm. In the following, $l$ and $\alpha$ represent the differentiability orders of functions in the space variables $(I,\varphi)$ and the parameter variables $\xi$, respectively.
\[def1\] Let $\alpha$ be a positive integer and $l>0$, $C^{l,\alpha}(\Omega\times \mathbb{T}^{n_2}, \Pi)$ be the class of all functions $f$ on $\Omega\times \mathbb{T}^{n_2}\times \Pi$ whose partial derivatives $\partial_{\xi}^{\beta}f$ with respect to the parameter variable $\xi\in\Pi$ (which means the Whitney derivative if $\Pi$ is a closed set) for all $\beta, 0\leq |\beta|_1\leq \alpha$ are of class $C^l$ in the space variable $x=(I,\varphi)\in \Omega\times \mathbb{T}^{n_2}$, that is, there is some positive constant $M$ such that the partial derivatives $D^k\left(\partial_{\xi}^{\beta}f\right)$ of $\partial_{\xi}^{\beta}f $ with respect to the space variable $x=(I,\varphi)\in \Omega\times \mathbb{T}^{n_2}$ satisfy $$\label{1.2}
\left|D^k\left(\partial_{\xi}^{\beta}f(x,\xi)\right)\right|\leq M$$ and $$\label{1.3}
\left|D^k\left(\partial_{\xi}^{\beta}f(x,\xi)\right)-D^k\left(\partial_{\xi}^{\beta}f(y,\xi)\right)\right|\leq M|x-y|^{l-[l]}, \quad |k|_1=[l]$$ for all $x,y\in \Omega\times \mathbb{T}^{n_2}$ and all $\beta,k$ with $0\leq |\beta|_1\leq \alpha, 0\leq |k|_1\leq [l]$, where $[l]$ is the integer part of $l:\, l-[l]\in [0,1)$, for nonnegative integer vectors $k, \beta$, $D^k=D_1^{k_1}\circ D_2^{k_2}\circ \cdots \circ D_{n_1+n_2}^{k_{n_1+n_2}}$, $D_j^{k_j}=\frac{\partial^{k_j}}{\partial x_j^{k_j}}$, $\partial^\beta_\xi=\frac{\partial^{|\beta|_1}}{\partial \xi_1^{\beta_1}\cdots \partial \xi_{n_3}^{\beta_{n_3}}}$.
In addition, define a norm $$||f||_{l,\alpha;\Omega\times \mathbb{T}^{n_2},\Pi}=\inf M,$$ which is the smallest M for which the inequalities and hold. $C^{l,\alpha}(\Omega\times \mathbb{T}^{n_2}, \Pi)$ is a Banach space with respect to the norm $||\cdot||_{l,\alpha;\Omega\times \mathbb{T}^{n_2},\Pi}$, which is a generalization of the Hölder space to a parameter-depending case. The norms $||\cdot||_{l,\alpha;\mathbb{T}^{n_2},\Pi}$ and $||\cdot||_{\alpha;\Pi}$ are defined in a similar way, which means that the associated function only depends on $\varphi\in \mathbb{T}^{n_2}, \xi\in \Pi$ and $\xi\in \Pi$, respectively.
When $l$ is integer, we also introduce a generalization of the Zygmund space $\hat{C}^{l,\alpha}(\Omega\times \mathbb{T}^{n_2}, \Pi)$ of all functions satisfying $$\label{1.4}
\left|D^k\left(\partial_{\xi}^{\beta}f(x,\xi)\right)\right|\leq M, \qquad 0\leq |k|_1\leq l-1$$ and $$\label{1.5}
\left|D^k\left(\partial_{\xi}^{\beta}f(x,\xi)\right)+D^k\left(\partial_{\xi}^{\beta}f(y,\xi)\right)-2D^k\left(\partial_{\xi}^{\beta}f(\frac{1}{2}(x+y),\xi)\right)\right|\leq M|x-y|, \quad |k|_1=l-1,$$ instead of and , respectively, and the norm $||f||_{\hat{C}^l,\alpha;\Omega\times \mathbb{T}^{n_2},\Pi}$ is the smallest $M$ for which the inequalities and hold. For non-integer $l>0$, $\hat{C}^{l,\alpha}(\Omega\times \mathbb{T}^{n_2}, \Pi)=C^{l,\alpha}(\Omega\times \mathbb{T}^{n_2}, \Pi)$.
Sometimes we shall drop parameters from functions whenever there is no confusion.
0.2in
a\) Assume
[**(H1)**]{} these non-negative constants $q_1, \cdots, q_7$ satisfy $$q_1>q_3\geq q_5,\quad q_7\geq q_2+q_5,\quad 0<q_2\leq \min\{q_4,q_6\};$$
0.2in [**(H2)**]{} $\omega_i,A_i\in C^{\alpha}(\Pi)$ with some positive integer $\alpha$ and $A_i$ is a diagonalizable matrix, $A_i(\xi,\varepsilon) =B_i(\xi,\varepsilon)\Lambda_i(\xi,\varepsilon)B_i(\xi,\varepsilon)^{-1}$ for some diagonal matrix $\Lambda_i, i=1,2$. Denote $\omega(\xi,\varepsilon)= {\rm col}(\varepsilon^{q_5}\omega_1, \omega_2), \,\Lambda_1(\xi,\varepsilon)={\rm diag} (\lambda_1, \cdots, \lambda_{n_{11}})$, $\Lambda_2(\xi,\varepsilon)= {\rm diag} (\lambda_{n_{11}+1}, \cdots, \lambda_{n_{1}})$. Furthermore, assume that there are positive constants $c_0, c_1$ and $\varepsilon*$ such that for all $\varepsilon \in (0,\varepsilon*]$ $$\inf_{\xi\in \Pi}|\lambda_j(\xi,\varepsilon)|\geq c_0,\quad \inf_{\xi\in \Pi}|\lambda_j(\xi,\varepsilon)-\lambda_i(\xi,\varepsilon)|\geq c_0,\quad i\neq j,\, 1\leq i,j\leq n_{11},\,{\rm or} \, n_{11}+1\leq i,j\leq n_1,$$ $$||B_i||_{\alpha;\Pi},\, ||B_i^{-1}||_{\alpha;\Pi},\, ||\Lambda_i||_{\alpha;\Pi}\leq c_1,\quad ||\omega_i||_{0;\Pi}\coloneqq \sup_{\xi\in \Pi}|\omega_i| \leq c_1,\quad i=1,2,$$ $$\parallel \partial_{\xi}^{\beta}\omega \parallel_{\Pi}\coloneqq \sup_{\xi\in\Pi}| \partial_{\xi}^{\beta}\omega(\xi,\varepsilon) |\leq c_1 \varepsilon^{q_5}, \quad 1\leq |\beta|_1\leq \alpha;$$
0.2in [**(H3)**]{} $g_j\in C^{l,\alpha}(\Omega\times \mathbb{T}^{n_2}, \Pi)(j=1,\cdots,4)$ with $l>2(\alpha+1)(\iota+2)+\alpha \iota,\, \iota>\alpha n_2-1$.
0.1in [**Remark 2.1**]{} The requirement that $A_i$ does not have multiple eigenvalues is not necessary, only for the sake of simplification. The difficulty caused by multiple eigenvalues may be overcome by the technique of Rüssmann [@Russ01].
\[theorem1\] Suppose that the system satisfies Assumptions (H1)-(H3). Then for any given $0<\gamma\ll1$, there is a sufficiently small $0<\varepsilon_0^*=o(\gamma^{\frac{\alpha+1}{q_2}})$ such that for $0<\varepsilon\leq \varepsilon_0^*$, there exists a Cantor set $\Pi_{\gamma}\subset \Pi$ and for each $\xi \in \Pi_{\gamma}$, the system admits a quasi-periodic invariant torus of the form $I_1=\Phi_1(\varphi;\xi),\, I_2=\Phi_2(\varphi;\xi),\, \varphi={\rm col}(\varphi_1,\varphi_2) \in \mathbb{T}^{n_{21}}\times \mathbb{T}^{n_{22}}$ with frequencies $\omega^*(\xi)=(\varepsilon^{q_5}\omega_1^*(\xi), \omega_2^*(\xi))$, which is of class $C^\mu$ in $\xi\in\Pi_{\gamma}$ in the sense of whitney and of class $\hat{C}^{l_1}(l_1=l-(\alpha+\mu+2)(\iota+1)-\alpha-2)$ in $\varphi\in \mathbb{T}^{n_2}$ together with derivatives up to order $\mu-1$ with respect to $\xi$ for $0< \mu\leq \alpha$ (positive integer $\mu$), the frequency map $\omega^*(\xi)$ is of class $C^\alpha $ in $\xi\in\Pi_{\gamma}$ in the sense of Whitney and satisfies $$\label{torusest}
||(\Phi_1,\Phi_2)||_{\hat{C}^{l_1},\mu-1; \mathbb{T}^{n_2},\Pi_\gamma}\leq C\varepsilon^{q_2}\gamma^{-(\mu+1)},$$ $$\label{freqest}
||\omega_1^*-\omega_1||_{C_W^\alpha(\Pi_{\gamma})}\leq C \varepsilon^{q_6},\qquad ||\omega_2^*-\omega_2||_{C_W^\alpha(\Pi_{\gamma})}\leq C \varepsilon^{q_7}.$$ Moreover, there exist closed subsets $\Pi_\nu$ of $\Pi$, frequency vectors $\omega^\nu(\xi)={\rm col}(\varepsilon^{q_5}\omega_1^\nu(\xi), \omega_2^\nu(\xi))$ and diagonal matrices $\Lambda^\nu(\xi)={\rm diag}(\varepsilon^{q_1}\Lambda_1^\nu(\xi), \varepsilon^{q_3}\Lambda_2^\nu(\xi))$ for $\nu=1,2,\cdots$, satisfying $$\label{frees}
||\omega_1^\nu-\omega_1||_{\alpha;\Pi_{\nu}}\leq c\varepsilon^{q_6},\quad ||\omega_2^\nu-\omega_2||_{\alpha;\Pi_{\nu}}\leq c\varepsilon^{q_7},\quad
||\Lambda_1^\nu-\Lambda_1||_{\alpha;\Pi_{\nu}}\leq c\varepsilon^{q_2},\quad ||\Lambda_2^\nu-\Lambda_2||_{\alpha;\Pi_{\nu}}\leq c\varepsilon^{q_4}$$ and $$\Pi_\nu=\Pi_{\nu-1}\setminus\bigcup_{k,m}\mathfrak{R}_{km}^\nu(\gamma)$$ such that $\Pi_{\gamma}=\bigcap_{\nu=0}^{\infty}\Pi_\nu$, where $$\mathfrak{R}_{km}^\nu(\gamma)=\left\{\xi\in \Pi_{\nu-1}:\, |\sqrt{-1}\langle k,\omega^{\nu-1}\rangle+\langle m,\Lambda^{\nu-1}\rangle|<\gamma \varepsilon^{q_5}|k|_2^{-\iota}\right\}$$ for $m\in\mathfrak{m}, k\in \mathbb{Z}^{n_2}$, $K_{\nu-1}<|k|_2\leq K_\nu$, $\mathfrak{m}=\{m\in \mathbb{Z}^{n_1}:\,|m|_1\leq 2, \sum_{j=1}^{n_1}m_j=0\, {\rm or}\, -1\}$, $\omega^0=\omega, \Lambda^0=\Lambda\coloneqq {\rm diag}(\varepsilon^{q_1}\Lambda_1, \varepsilon^{q_3}\Lambda_2), K_0=0$ and $K_\nu=[K_\nu^\prime ]+1, K_\nu^\prime =r^{-1}3^{\nu+1}[(l+(n_2+1)(\nu+1)-\alpha)\ln 3+(n_2+1)|\ln r|+\ln \widetilde{C}]$, $\Pi_0$ is a closed subset of $\Pi$ whose distance to the boundary of $\Pi$ is at least equal to $\gamma$, where $C$ and $c$ are constants independent of $\nu$, $\varepsilon$ and $\gamma$, $r$ is the radius of the neighbourhood $\Omega$, $||\cdot ||_{C_W^\alpha(\Pi_{\gamma})}$ is the whitney norm (see Appendix A.1), $ \widetilde{C}=24(n_2!)n_2^{n_2}e^{-n_2}$, $[K_\nu^\prime ]$ represents the integer part of $K_\nu^\prime$.
Here, we drop $\varepsilon$ from functions, the continuous differentiability of functions $\omega_i^\nu$ and $\Lambda_i^\nu (i=1,2)$ on the closed set $\Pi_\nu$ means that they are continuously differentiable in some neighbourhood of $\Pi\nu$. Here and in the sequel, we also regard the $\Lambda$ as a column vector of its diagonal elements when $\Lambda$ is a diagonal matrix.
0.1in b) The Cantor set $\Pi_{\gamma}$ is not empty and indeed the measure ${\rm meas}(\Pi\setminus \Pi_{\gamma})\rightarrow 0$ as $\gamma \rightarrow 0$ as long as we impose proper non-degeneracy conditions on frequencies. Since in applications the non-degeneracy conditions on frequencies are different, Theorem 1 does not involve the measure estimate of $\Pi_{\gamma}$ so that it can be used more widely. In the following theorem, we give some conditions to ensure that the Cantor set $\Pi_{\gamma}$ is not empty.
By the assumption (H2), we can write $\omega_2$ as $$\omega_2(\xi,\varepsilon)=\omega_{20}+\varepsilon^{q_5} \omega_{21}(\xi)+o(\varepsilon^{q_5}),$$ where $\omega_{20}$ is independent of $\xi$, $o(\varepsilon^{q_5})$ represents infinitely small quantity of $\varepsilon^{q_5}$ up to $\alpha$-th derivatives. Denote $\widetilde{\Lambda}_2(\xi)=\Lambda_2(\xi, \varepsilon)|_{\varepsilon=0}$ and $\widetilde{\omega}(\xi)={\rm col}(\omega_1(\xi,0), \omega_{21}(\xi))$ in the case $n_{22}\neq 0$, $\widetilde{\omega}(\xi)=\omega_1(\xi,0)$ in the case $n_{22}=0$.
\[theorem2\] Suppose that the system satisfies the assumptions in Theorem 1, moreover assume that $n_3+\cdots+n_3^{\alpha}\geq n_2$ and\
[(i)]{} for all $\xi\in \Pi$ $$\label{I11}
{\rm rank}\left(\widetilde{\omega},\frac{\partial^{|\beta|_1}\widetilde{\omega}}{\partial\xi^{\beta}}:\, 1\leq |\beta|_1\leq \alpha\right)=n_2 \quad {\rm in\, Case }\,\,n_{22}=0,$$ $$\label{I12}
{\rm rank}\left(\frac{\partial^{|\beta|_1}\widetilde{\omega}}{\partial\xi^{\beta}}:\, 1\leq |\beta|_1\leq \alpha\right)=n_2 \quad {\rm in\, Case }\,\,n_{22}\neq 0,$$
[(ii)]{} for all integer vectors $0\neq k\in \mathbb{Z}^{n_2}, m=(m_1,\cdots,m_{n_{12}})\in \mathbb{Z}^{n_{12}}$ with $1\leq |m|_1\leq 2$ and $m_1+\cdots+m_{n_{12}}=0$ or $-1$ $$\label{I2}
{\rm meas}\left\{\xi\in\Pi:\, \sqrt{-1}\langle k,\widetilde{\omega}_0+\varepsilon^{q_5}\widetilde{\omega}(\xi)\rangle+ \varepsilon^{q_3} \langle m,\widetilde{\Lambda}_2(\xi)\rangle =0 \right\}=0,$$ where $\widetilde{\omega}_0={\rm col}(0, \omega_{20})$.
Then the Cantor set $\Pi_{\gamma}$ defined in Theorem 1 is of positive Lebesgue measure and ${\rm meas}(\Pi\backslash\Pi_{\gamma})\rightarrow 0$ as $\gamma \rightarrow 0$.
The conditions , , Lemma \[lemmaA.3\] and Remark A.2 imply that there is a constant $c_2>0$ such that $$\label{Dn11}
\max_{0\leq \mu \leq \alpha}\parallel D^{\mu}\langle b,\widetilde{\omega}(\xi)\rangle\parallel \geq c_2 \qquad {\rm in \, Case}\,\, n_{22}=0$$ and $$\label{Dn22}
\max_{1\leq \mu \leq \alpha}\parallel D^{\mu}\langle b,\widetilde{\omega}(\xi)\rangle\parallel \geq c_2 \qquad {\rm in \, Case}\,\, n_{22}\neq 0$$ for all $\xi\in \Pi, b\in \mathcal{S}_{n_2,1}\coloneqq \{b\in \mathbb{R}^{n_2}:\, |b|_2=1\}$.\
Let $$K^*=\frac{32c_1}{c_2}n_3^{\frac{\alpha}{2}},\qquad f_{km}(\xi)=\langle k,\widetilde{\omega}_0+\varepsilon^{q_5}\widetilde{\omega}(\xi)\rangle+ \varepsilon^{q_3} \langle m,{\rm Im}\widetilde{\Lambda}_2(\xi)\rangle,$$ where ${\rm Im}\widetilde{\Lambda}_2$ is the imaginary part of $\widetilde{\Lambda}_2$.
\[theorem3\] If in Theorem 2, the condition (ii) is replaced by
[(ii)’]{} there is a constant $c_3>0$ such that $$\max_{1\leq \mu \leq \alpha}\parallel D^{\mu}f_{km}(\xi)\parallel \geq c_3\varepsilon^{q_5} \qquad {\rm for \, all}\,\, \xi\in\Pi,0<|k|_2<K^*,$$ then $${\rm meas}\Pi_{\gamma}={\rm meas}\Pi-O(\gamma^{\frac{1}{\alpha}})$$ for sufficiently small $\gamma$.
[**Remark 2.2**]{} If $q_3>q_5\geq 0$, or there a constant $c_2^{\prime}>0$ such that $$\inf_{\xi\in\Pi}|\langle m,{\rm Re}\widetilde{\Lambda}_2(\xi)\rangle|\geq c_2^{\prime} \qquad {\rm for}\,1\leq |m|_1\leq 2,\,m_1+\cdots+m_{n_{12}}=0\,{\rm or}\, -1,$$ then the conditions [*(ii)*]{} and [*(ii)’*]{} in Theorems 2 and 3, respectively, may be removed, see the proof of Theorems 2 and 3, and Remark 5.1 in Section 5.
More results on measure estimates of $\Pi_{\gamma}$ will be given in the forthcoming second part concerning on the persistence of quasi-periodic invariant tori in bifurcation theory.
0.1in c) we consider a specific form of for the case $n_{11}=n_{22}=0, n_2=n_3, q_3=q_5=0$ and $q_4=q_6=1$, which means that the first and fourth equations in are absent and the number of parameter variables equals the dimension of tori, the equation reads $$\label{eq1}
\left\{\begin{array}{rl}
\dot I & = A(\xi)I+\varepsilon g_1(I,\varphi; \xi,\varepsilon)\\
\dot{\varphi} & =\omega(\xi)+\varepsilon g_2(I,\varphi; \xi,\varepsilon).
\end{array}
\right.$$ Denote $\Lambda={\rm diag} (\lambda_1, \cdots, \lambda_{n_{1}})$ and $\omega={\rm col}(\omega_1,\cdots, \omega_{n_2})$, where $\lambda_1, \cdots, \lambda_{n_{1}} $ are the eigenvalues of $A$, $A(\xi) =B(\xi)\Lambda(\xi)B(\xi)^{-1}$. Assume 0.2in [**(H2)$^{\prime}$**]{} $\omega,A\in C^1(\Pi)$, the map $\xi\rightarrow \omega(\xi)$ is a diffeomorphism between $\Pi$ and its image, and there exist positive constants $c_0, c_1$ and $c_4$ such that $||B||_{1;\Pi}, ||B^{-1}||_{1;\Pi}, ||\Lambda||_{1;\Pi}, ||\omega||_{1;\Pi}\leq c_1$, $$\label{normalfre}
|\langle m,\Lambda(\xi)\rangle| \geq c_0, \quad \left|\left| \left(\frac{\partial\omega}{\partial \xi}\right)^{-1}\right|\right|\leq c_4 \qquad {\rm on}\quad \Pi$$ and $$\label{mixfre}
{\rm meas}\left\{\xi\in\Pi:\, \sqrt{-1}\langle k,\omega(\xi)\rangle+ \langle m,\Lambda(\xi)\rangle =0 \right\}=0$$ for all integer vectors $0\neq k\in \mathbb{Z}^{n_2}, m\in \mathbb{Z}^{n_1}$ with $1\leq |m|_1\leq 2$ and $m_1+\cdots+m_{n_1}=0$ or $-1$; 0.2in [**(H3)$^{\prime}$**]{} $g_j\in C^{l,1}(\Omega\times \mathbb{T}^{n_2}, \Pi)(j=1,2)$ with $l>5\iota+8,\, \iota>n_2-1$.
0.2in [**Remark 2.3**]{} (i) When the real part ${\rm Re} \Lambda$ of $\Lambda$ satisfies $\langle m,{\rm Re}\Lambda(\xi)\rangle \neq 0$ on $\Pi$, the condition holds spontaneously. In particular, the condition is satisfied if $\Lambda$ is independent of $\xi$.
\(ii) The Assumption (H2)$^{\prime}$ implies that the condition is satisfied if $$\label{Dfre}
\left(\left(\frac{\partial\omega}{\partial \xi}\right)^{-1}\right)^T\frac{\partial}{\partial \xi}\langle m,\Lambda(\xi)\rangle\neq \sqrt{-1} k \quad {\rm for}\, 0\neq |k|_1\leq 2n_2 c_1 c_4.$$
Theorems 1-3 imply
\[corollary1\] Suppose that the system satisfies Assumptions (H2)$^{\prime}$ and (H3)$^{\prime}$. Then for any given $0<\gamma \ll 1$, there is a sufficiently small $\varepsilon^*>0$ such that for $0<\varepsilon\leq \varepsilon^*$, there exists a Cantor set $\Pi_{\gamma}\subset \Pi$ with positive Lebesgue measure (the measure satisfies the estimate ${\rm meas}\Pi_{\gamma}={\rm meas}\Pi-c\gamma$ if replaces ) and for each $\xi\in \Pi_{\gamma}$, the system possesses a quasi-periodic invariant torus $I=\Phi(\varphi;\xi),\, \varphi \in \mathbb{T}^{n_2}$ consisting of quasi-periodic motions, which is of $\hat{C}^{l_1}(l_1=l-4(\iota+1)-3)$ in $\varphi \in \mathbb{T}^{n_2}$ and Lipschitz in $\xi\in \Pi_{\gamma}$, where $c$ is a constant independent of $\gamma$ and $\varepsilon$.
Usually the normal form (integrable part) of related bifurcation problems of actual models is only finitely differentiable, not analytic in the parameter $\xi$, and the frequency map is possibly degenerate so that we need the higher-order derivatives of the frequency map to estimate the Lebesgue measure of $\Pi_{\gamma}$ and obtain $\Pi_{\gamma}$ is the most part of $\Pi$. Hence, we want to establish an approximation lemma and the corresponding inverse approximation lemma in which a finitely differentiable function is approximated by a sequence of functions being analytic in space variables, but finitely differentiable in parameter variables. These comprise Section 3. The proofs of Theorems 1-3 are given in Sections 4 and 5, respectively.
0.2in
3 Approximation Lemmas {#approximation-lemmas .unnumbered}
======================
Zehnder [@Zehn75] established the approximation and inverse approximation Lemmas on a finitely differentiable real function approximated by a sequence of real analytic functions, which was generalized to the anisotropic case by Pöschel [@Pos82], and was sharpened to covering the finitely differentiable and Gevrey regular cases by Wagener [@Wag10], respectively. Here, we give generalized versions of Zehnder’s approximation and inverse approximation lemmas finite- smoothly depending on parameters, and obtain estimates of higher-order regularity. 0.2in
a\) We first introduce some notations. Let $m,n$ and $\alpha$ be positive integers, $\mathcal{U}\subset \mathbb{C}^m$ and $\Pi\subset \mathbb{R}^n$ be open sets, $\mathfrak{A}^\alpha (\mathcal{U},\Pi)$ be the class of all functions of $(z,\xi)$ on $\mathcal{U}\times\Pi$ which are analytic in $z\in \mathcal{U}$ and $\alpha$-times continuously differentiable in $\xi\in \Pi$. For $g\in \mathfrak{A}^\alpha (\mathcal{U},\Pi)$, define $$|g|_{\mathcal{U},\alpha;\Pi}=\sup_{|\beta|_1\leq \alpha}\sup_{(z,\xi)\in \mathcal{U}\times\Pi}\left|\partial_\xi^\beta g(z,\xi)\right|.$$ In particular, for $\mathcal{U}=\{z\in \mathbb{C}^m:\, |{\rm Im} z|\coloneqq sup_{1\leq j\leq m}|{\rm Im} z_j|<r\}$, we denote $|g|_{\mathcal{U},\alpha;\Pi}$ by $|g|_{r,\alpha;\Pi}$.
Take an even function $u_0\in C_0^\infty(\mathbb{R})$, vanishing outside the interval $[-1,1]$ and identically equal to 1 in a neighbourhood of 0 (see [@Wag10] for the construction of such a function). For $x\in \mathbb{R}^m$, let $u(x)=u_0(|x|_2^2)$ and $\tilde{u}$ be the inverse Fourier transform of $u$ $$\tilde{u}(z)=(2\pi)^{-m}\int_{\mathbb{R}^m} u(x)e^{\sqrt{-1}\langle z,x\rangle}dx.$$ Let $f$ be a real-valued function of class $C^{l,\alpha}(\mathbb{R}^m, \Pi)$ (see Definition\[def1\]), $f_r\,(0<r\leq 1)$ be defined by the convolution $$\label{frexpr}
f_r(x,\xi)\coloneqq (S_r(f(\cdot,\xi))(x)=r^{-m}\int_{\mathbb{R}^m} \tilde u(r^{-1}(x-y))f(y,\xi)dy$$ for $x\in \mathbb{C}^m$. We list some properties of the analytic smoothing operator $S_r$ in Section A.3 of the appendix, which will be used in the proof of the next lemma.
\[lemma2.1\] Let $f(x,\xi)$ be a real-valued function of class $C^{l,\alpha}(\mathbb{R}^m,\Pi)$ for some real number $l> 0$ and $\alpha \in \mathbb{N}$, where $\Pi\subset \mathbb{R}^n$ is an open set. Then for every $r\in (0,1]$, the function $f_r(x,\xi)$ is $\alpha$-times continuously differentiable in $\xi\in \Pi$, entire real analytic in $x\in \mathbb{C}^m$ together with derivatives up to order $\alpha$ with respect to $\xi$, and satisfies
[(i)]{} $||f_r-f||_{p,\alpha;\mathbb{R}^m,\Pi}\leq C_1(l,p)r^{l-p}||f||_{l,\alpha;\mathbb{R}^m,\Pi}$ for all $0\leq p\leq l$,
[(ii)]{} $|f_r-f_{r^\prime}|_{r^\prime,\alpha;\Pi}\leq C_2(l,p)r^{p}||f||_{p,\alpha;\mathbb{R}^m,\Pi}$ for all $0\leq p\leq l$ and $0<r^\prime\leq r$,
[(iii)]{} $|f_r|_{r,\alpha;\Pi}\leq C_3(l)||f||_{0,\alpha;\mathbb{R}^m,\Pi}\leq C_3(l)||f||_{l,\alpha;\mathbb{R}^m,\Pi}$,\
where $C_j\,(j=1,2,3)$ are constants depending on $l,p$ and the dimension $m$. Moreover, $f_r$ is $\omega$-periodic in some variable if in which $f$ is $\omega$-periodic.
[**Proof**]{} From it is clear that $f_r(x,\xi)$ is analytic in $x\in \mathbb{C}^m$, and $\alpha$-times continuously differentiable in $\xi\in \Pi$, taking real values on real variables $x$, and if $f$ is periodic in some variable, then so is $f_r$. As differentiation may commute with integration in for functions with bounded derivatives, we obtain $\partial_\xi^\beta f_r=S_r(\partial_\xi^\beta f)$ for $|\beta|_1\leq \alpha$. Of course, we also have $S_r(D^kf)=D^k(S_r(f))$ for $|k|_1\leq l, k\in \mathbb{Z}^m$. Hence we only need to prove the estimates (i)-(iii) in the case without parameter-dependence. In the following, we will use $C$ to denote some constant depending $l, p$ and $m$.
\(i) The case where $p$ is a integer, is proved by Chierchia [@Chier03], see Lemma \[lemmaA4\](f) in Appendix. Hence we only give the proof for the case $p=q+\mu\leq l, \mu\in (0,1), q\in \mathbb{Z}_+$. Denote $g(x)=D^\beta f, |\beta|_1=q$. Then by (a) and (b) in Lemma \[lemmaA4\], we have for $x,y\in \mathbb{R}^m$, $$\begin{aligned}
& & \sup_{x\neq y}|x-y|^{-\mu}|(g-S_rg)(x)-(g-S_rg)(y)|\\
\hskip 0.2in & & =\sup_{x\neq y}|x-y|^{-\mu}\left|\int_{\mathbb{R}^m}\tilde{u}(z)[g(x)-g(x-rz)-g(y)+g(y-rz)]dz\right|\equiv (*).\end{aligned}$$
Case I: $q=[l]$, the integer part of $l$. For $|x-y|\geq r$, by $g\in C^{l-q}$ and Lemma \[lemmaA4\] (d), we have $$\begin{aligned}
(*) & \leq & \sup_{x\neq y}|x-y|^{-\mu}\int_{\mathbb{R}^m}|\tilde{u}(z)|(|g(x)-g(x-rz)|+|g(y)-g(y-rz)|)dz\\
& \leq & 2r^{l-p}||f||_{l;\mathbb{R}^m}\int_{\mathbb{R}^m}|\tilde{u}(z)||z|^{l-q}dz\leq C r^{l-p}||f||_{l;\mathbb{R}^m}.\end{aligned}$$
For $|x-y|< r$, we also have $$\begin{aligned}
(*) & \leq & \sup_{x\neq y}|x-y|^{-\mu}\left(||f||_{l;\mathbb{R}^m}|x-y|^{l-q} + \int_{\mathbb{R}^m}|\tilde{u}(z)||g(x-rz)-g(y-rz)|dz\right)\\
& \leq & \left(1+\int_{\mathbb{R}^m}|\tilde{u}(z)|dz\right)|x-y|^{l-p}||f||_{l;\mathbb{R}^m}\leq C r^{l-p}||f||_{l;\mathbb{R}^m}.\end{aligned}$$ Hence, $||g-S_rg||_{\mu;\mathbb{R}^m}\leq C r^{l-p}||f||_{l;\mathbb{R}^m}$, which, combining with Lemma \[lemmaA4\] (f) for the case of integers, implies (i) for the case $q=[l]$.
Case II: $q<[l]$. For $|x-y|\geq r$, using the Taylor’s formula of $h(rz)=g(x-rz)-g(y-rz)$ at $z=0$ and Lemma \[lemmaA4\] (c), we obtain $$\begin{aligned}
(*) &= & \sup_{x\neq y}\frac{|x-y|^{-\mu}}{([l]-q)!}
\left|\int_{\mathbb{R}^m}\tilde{u}(z)(-rz\cdot \nabla )^{[l]-q}(g(x-\theta rz)-g(y-\theta rz))dz\right| \nonumber\\
& \leq & \sup_{x\neq y}|x-y|^{-\mu}r^{[l]-q}\sum_{|k|_1=[l]-q}\frac{1}{k!}
\left|\int_{\mathbb{R}^m}\tilde{u}(z)z^k (D^kg(x-\theta rz)-D^kg(y-\theta rz))dz\right|,\label{cpest1}\end{aligned}$$ where $rz\cdot \nabla=\sum_{j=1}^{m}rz_jD_j$ and $\theta\in (0,1)$. Thus, equivalently, we need to estimate the following expression $$(**)\equiv \sup_{x\neq y}|x-y|^{-\mu}
\left|\int_{\mathbb{R}^m}\tilde{u}(z)z^k (D^{k+\beta}f(x-\theta rz)-D^{k+\beta}f(y-\theta rz))dz\right|,\, |k+\beta|_1=[l].$$ By Lemma \[lemmaA4\] (c), we get $$\begin{aligned}
(**) &= & \sup_{x\neq y}|x-y|^{-\mu}\left|\int_{\mathbb{R}^m}\tilde{u}(z)z^k (D^{k+\beta}f(x-\theta rz)-D^{k+\beta}f(x))dz\right. \nonumber\\
& &+ \left.\int_{\mathbb{R}^m}\tilde{u}(z)z^k (D^{k+\beta}f(y)-D^{k+\beta}f(y-\theta rz))dz\right|\leq C r^{l-[l]-\mu}||f||_{l;\mathbb{R}^m}.\label{cpest2}\end{aligned}$$
For $|x-y|< r$, if $[l]-q\geq 2$, then similarly we have $$\label{cpest3}
(*)\leq \sup_{x\neq y}|x-y|^{-\mu}r^{[l]-q-1}\sum_{|k|_1=[l]-q-1}\frac{1}{k!}
\left|\int_{\mathbb{R}^m}\tilde{u}(z)z^k (D^kg(x-\theta rz)-D^kg(y-\theta rz))dz\right|.$$ The mean value theorem and Lemma \[lemmaA4\] (c) deduce $$\begin{aligned}
& & \sup_{x\neq y}|x-y|^{-\mu}
\left|\int_{\mathbb{R}^m}\tilde{u}(z)z^k (D^kg(x-\theta rz)-D^kg(y-\theta rz))dz\right| \nonumber\\
& \leq & \sup_{x\neq y}|x-y|^{-\mu}\sum_{|k'|_1=1}
\left|\int_{\mathbb{R}^m}\tilde{u}(z)z^k (x-y)^{k'}D^{k+k'}g(y-\theta rz+\theta_{kk'}(x-y))dz\right| \nonumber\\
& =& \sup_{x\neq y}|x-y|^{-\mu}\sum_{|k'|_1=1}
\left|\int_{\mathbb{R}^m}\tilde{u}(z)z^k (x-y)^{k'}(D^{k+k'}g(y-\theta rz+\theta_{kk'}(x-y))-D^{k+k'}g(x))dz\right| \nonumber\\
&\leq & \sup_{x\neq y}|x-y|^{-\mu}\sum_{|k'|_1=1}
|x-y|||f||_{l;\mathbb{R}^m}\int_{\mathbb{R}^m}|\tilde{u}(z)z^k| |\theta rz+(1-\theta_{kk'})(x-y)|^{l-[l]}dz \nonumber\\
&\leq & C r^{l+1-[l]-\mu}||f||_{l;\mathbb{R}^m},\label{cpest4}\end{aligned}$$ where $\theta_{kk'}\in (0,1)$. If $[l]-q=1$, then by the mean value theorem, $$\begin{aligned}
\label{cpest5}
(*) & \leq & \sup_{x\neq y}|x-y|^{-\mu}\sum_{|k'|_1=1}
\left|\int_{\mathbb{R}^m}\tilde{u}(z) (x-y)^{k'}(D^{k'}g(y+\theta_{1k'}(x-y))-D^{k'}g(y-rz+\theta_{2k'}(x-y)))dz\right| \nonumber \\
&\leq & \sup_{x\neq y}|x-y|^{-\mu}\sum_{|k'|_1=1}
|x-y|||f||_{l;\mathbb{R}^m}\int_{\mathbb{R}^m}|\tilde{u}(z)| |rz+(\theta_{1k'}-\theta_{2k'})(x-y)|^{l-[l]}dz \nonumber \\
&\leq & C r^{l+1-[l]-\mu}||f||_{l;\mathbb{R}^m}=C r^{l-p}||f||_{l;\mathbb{R}^m},\end{aligned}$$ where $\theta_{1k'},\theta_{2k'}\in (0,1)$. Hence, - and Lemma \[lemmaA4\] (f) imply (i) for the case $q<[l]$.
Obviously, Lemma \[lemmaA4\] (f) implies (ii), and the definition of $f_r$ and Lemma \[lemmaA4\] (b) and (d) imply (iii). 0.1in $\blacksquare$ 0.2in
From Lemma \[lemma2.1\], it follows the approximation lemma.
\[apprlem\] (Approximation Lemma) Let $f(x,\xi)$ be a real-valued function of class $C^{l,\alpha}(\mathbb{R}^m,\Pi)$ for some real number $l> 0$ and $\alpha \in \mathbb{N}$, where $\Pi$ is an open set, and let $\{ r_j\}_{j=0}^{\infty}$ be a monotonically decreasing sequence of positive numbers with $r_0\leq 1$ and tend to zero. Then there exists a sequence of functions $\{f_j(z,\xi)\}_{j=0}^{\infty}$, being of class $C^{\alpha}$ in $\xi \in \Pi$, and entire, real analytic in $z\in \mathbb{C}^m$ together with derivatives up to order $\alpha$ with respect to $\xi$, starting with $f_0\equiv 0$, such that $$\lim_{j\rightarrow \infty}||f_j-f||_{p,\alpha;\mathbb{R}^m,\Pi}=0 \qquad {\rm for \,\, all} \,\, 0\leq p<l$$ and $$|f_j-f_{j-1}|_{r_j,\alpha;\Pi}\leq C_0 r_{j-1}^l||f||_{l,\alpha;\mathbb{R}^m,\Pi} \qquad {\rm for } \,\, j\geq 1,$$ where the constant $C_0$ depends on $l$ and the dimension $m$. Moreover, the $f_j$ is $\omega$-periodic in each variable in which $f$ is $\omega$-periodic.
0.2in
b\) Now, we want to apply the approximation lemma to the proof of Theorem 1 and obtain sequences of real analytic functions approximating $g_i(i=1,\cdots,4)$ in the equation (1.1).
Without loss of generality, we take $$\Omega=\{I={\rm col}(I_1,I_2)\in \mathbb{R}^{n_1}:\, |I|<3\tilde{r}\}$$ for some constant $0<\tilde{r}\leq 1$. Let $$\Omega^*=\{I\in \mathbb{R}^{n_1}:\, |I|\leq 2\tilde{r}\},\qquad r_j=\tilde{r}3^{-j},\, j=0,1,2,\cdots.$$ Define complex neighbourhoods $\mathcal{U}_j$ of $\Omega^*\times \mathbb{T}^{n_2}$ for $j=0,1,2,\cdots$ by $$\mathcal{U}_j=\{(I,\varphi)\in \mathbb{C}^{n_1}\times \mathbb{C}^{n_2}:\, {\rm dist}(I,\Omega^*)<3r_j, |{\rm Im} \varphi|<3r_j\}\coloneqq \Omega^*\times \mathbb{T}^{n_2}+(3r_j,3r_j).$$ We first expand the definition domain $\Omega\times \mathbb{T}^{n_2}\times \Pi$ of $g_i(i=1,\cdots,4)$ to $\mathbb{R}^{n_1}\times \mathbb{T}^{n_2}\times \Pi$ in the following manner: we multiply $g_i$ by a $C^\infty$-function on $\mathbb{R}^{n_1}$ which identical 1 on $\Omega^*$ and vanishes outside $\Omega$. The obtained function belongs to $C^{l,\alpha}(\mathbb{R}^{n_1}\times \mathbb{T}^{n_2}, \Pi)$ and is equal to $g_i$ on $\Omega^*\times \mathbb{T}^{n_2}\times \Pi$, its norm is bounded by $c_l||g_i||_{l,\alpha;\Omega\times \mathbb{T}^{n_2},\Pi}$, where $c_l$ is a constant depending $l,n_1$ and the chosen $C^\infty$-function. Then by the approximation lemma (Lemma \[apprlem\]) we have the following corollary.
\[corollary2\] If the system (1.1) satisfies Assumption (H3), then there exist sequences $\{g_i^j(I,\varphi,\xi)\}_{j=0}^\infty$ $(i=1,\cdots,4)$ of real analytic functions, being of class $C^{\alpha}$ in $\xi \in \Pi$, and entire, real analytic in $(I,\varphi)\in \mathcal{U}_0$, periodic in the variables $\varphi$ with periodic $2\pi$ together with derivatives up to order $\alpha$ with respect to $\xi$, starting with $g_i^0\equiv 0$, such that $$\lim_{j\rightarrow \infty}||g_i^j-g_i||_{p,\alpha;\Omega^*\times \mathbb{T}^{n_2},\Pi}=0 \qquad {\rm for \,\, all} \,\, 0\leq p<l,$$ $$|g_i^j-g_i^{j-1}|_{\mathcal{U}_{j-1},\alpha;\Pi}\leq C_0 r_{j-1}^l||g_i||_{l,\alpha;\Omega\times \mathbb{T}^{n_2},\Pi} \qquad {\rm for } \,\, j\geq 1, i=1,\cdots,4,$$ where $C_0$ is a constant depending only on $l,n_1,n_2,\tilde{r}$ and $c_l$.
0.2in
c\) Let $\Omega\subset \mathbb{R}^m$ be an open convex set, and $\Pi_0\subset \mathbb{R}^n$ be a closed set, $$\mathcal{W}_j=\Omega+r_j,\quad \Pi_j=\bigcup_{\xi\in \Pi_0}\{\zeta \in \mathbb{R}^n: \, |\zeta-\xi|<s_j\},\qquad j=0,1,2,\cdots,$$ where $r_j=r_0\theta^j,0<\theta<1$ and $\{s_j\}_{j=0}^{\infty}$ is a monotonically decreasing sequence of positive numbers with $s_0\leq 1$ and tend to zero.\
\[invapplem\] (Inverse Approximation Lemma) Let $\{f_j(x,\xi)\}_{j=0}^{\infty}$ be a sequence of functions such that $f_0\equiv 0$, $f_j(x,\xi)$ is of class $C^{\alpha}$ in $\xi \in \Pi_j$, real analytic in $x\in \mathcal{W}_j$ together with derivatives up to order $\alpha$ with respect to $\xi$, and $$\label{fjest}
|f_j-f_{j-1}|_{\mathcal{W}_j,\alpha;\Pi_j}\leq M r_j^l$$ for every $ j\geq 1$ and some constant $M$. If there exists a constant $c^\prime_0>0$ such that $r_j^l\leq c^\prime_0s_j^\alpha, j=1,2,\cdots$, then there is a unique function $f(x,\xi)$ being of class $C^{\alpha}$ in $\xi \in \Pi_0$ in the sense of Whitney (see Appendix A.1), and of class $\hat{C}^l$ in $x\in \Omega$ together with derivatives up to order $\alpha-1$ with respect to $\xi$ such that $$||f||_{\hat{C}^l,\alpha-1;\Omega,\Pi}\leq C^\prime_0 M \qquad {\rm and} \quad
\lim_{j\rightarrow \infty}||f-f_j||_{p,\alpha-1;\Omega,\Pi}=0 \qquad {\rm for \,\, all} \quad 0\leq p<l.$$
Moreover, let $l=q+\mu,q\in \mathbb{Z}_+, \mu>0$ and if $r_j^\mu\leq c^\prime_1s_j^\delta$ for some constant $c^\prime_1$ and $0<\delta\leq 1$, then we may require the $(\alpha-1)$-order derivatives $\partial_\xi^\beta f(x,\xi)$ with $|\beta|_1= \alpha -1$ to be uniformly $\delta$-Hölder continuous in $\xi\in \Pi_0$ in the space $C^q(\Omega)$, that is, $$\label{lipcon}
||\partial_\xi^\beta f(\cdot,\xi)-\partial_\xi^\beta f(\cdot,\zeta)||_{C^q(\Omega)}\leq C^\prime_1M|\xi-\zeta|^{\delta} \quad {\rm for}\,\xi,\zeta\in \Pi_0,\,|\beta|_1= \alpha -1,$$ where the constant $C^\prime_0$ and $C^\prime_1$ depend on $l,m,n,\theta,c^\prime_0$ and $c^\prime_1$, $\hat{C}^l(\Omega)$ is the Zygmund space.
[**Proof**]{} By a similar proof to that of Lemma 2.2 (ii) in [@Zehn75] (also see the proof of Lemma 4.3 in [@LL10], Theorem A.3 in [@Wag10]), we can obtain that there exist functions $f^{(\beta)}\in \hat{C}^l(\Omega), |\beta|_1\leq \alpha$ such that $$\sup_{\xi\in \Pi}||f^{(\beta)}(\cdot,\xi)||_{\hat{C}^l(\Omega)}\leq C^\prime_0 M \quad {\rm and} \quad
\lim_{j\rightarrow \infty}||\partial_\xi^\beta f_j(\cdot,\xi)-f^{(\beta)}(\cdot,\xi)||_{C^p(\Omega)}=0$$ uniformly on $\Pi_0$ for all $0\leq p<l$ and $|\beta|_1\leq \alpha$. Set $f(x,\xi)= f^{(\beta)}(x,\xi)$ with $\beta=0$. To prove the rest of the lemma we only need to verify and the compatibility conditions in the definition of Whitney derivatives (see Appendix A.1) $$\label{compc}
f^{(\beta)}(x,\xi)=\sum_{|\beta+k|_1\leq \alpha-1}\frac{1}{k!}f^{(\beta+k)}(x,\zeta)(\xi-\zeta)^k+R^\beta(x,\xi,\zeta)$$ with $$\label{reterm}
\sup_{x\in\Omega}|R^\beta(x,\xi,\zeta)|\leq C M|\xi-\zeta|^{\alpha-|\beta|_1}$$ for all $\xi,\zeta\in \Pi_0, |\beta|_1\leq \alpha-1$ and some finite constant $C$.
Set $$h_j(x,\xi)=f_j(x,\xi)-f_{j-1}(x,\xi),\quad R_j^\beta(x,\xi,\zeta)=\partial_\xi^\beta h_j(x,\xi)-\sum_{|\beta+k|_1\leq \alpha-1}\frac{1}{k!}
\partial_\xi^{\beta+k} h_j(x,\zeta)(\xi-\zeta)^k$$ for $j\geq 1, |\beta|_1\leq \alpha-1$. Then $$\label{fdex}
f^{(\beta)}(x,\xi)=\sum_{j=1}^{\infty}\partial_\xi^\beta h_j(x,\xi),\quad
R^\beta(x,\xi,\zeta)=\sum_{j=1}^{\infty}R_j^\beta (x,\xi,\zeta),\quad |\beta|_1\leq \alpha-1.$$ If $s_{j_0+1}\leq |\xi-\zeta|<s_{j_0}$ for some positive integer $j_0$, then the line segment $L$ connecting $\xi$ to $\zeta$ is contained in $\Pi_j$ with $1\leq j\leq j_0$, and the Taylor expansion implies $$\sup_{x\in\Omega}|R_j^\beta(x,\xi,\zeta)|\leq C_1(\beta) Mr_j^l|\xi-\zeta|^{\alpha-|\beta|_1},\qquad 1\leq j\leq j_0.$$ And $$\sup_{x\in\Omega}|R_j^\beta(x,\xi,\zeta)|\leq C_2(\beta) Mr_j^ls_{j_0+1}^{-(\alpha-|\beta|_1)} |\xi-\zeta|^{\alpha-|\beta|_1},\qquad j\geq j_0+1.$$ Hence, $$\begin{aligned}
\sup_{x\in\Omega}|R^\beta(x,\xi,\zeta)| &\leq & M |\xi-\zeta|^{\alpha-|\beta|_1}\left(C_1\sum_{j=1}^{j_0} r_j^l+C_2\sum_{j=j_0+1}^{\infty}\left(\frac{r_j}{r_{j_0+1}}\right)^l\frac{r_{j_0+1}^l}{s_{j_0+1}^{\alpha-|\beta|_1}}\right)\\
& \leq & C M|\xi-\zeta|^{\alpha-|\beta|_1}\end{aligned}$$ If $|\xi-\zeta|\geq s_1$, then we also have $$\begin{aligned}
\sup_{x\in\Omega}|R^\beta(x,\xi,\zeta)| &\leq & C_2(\beta) M |\xi-\zeta|^{\alpha-|\beta|_1}\frac{r_1^l}{s_1^{\alpha-|\beta|_1}} \sum_{j=1}^{\infty}\left(\frac{r_j}{r_1}\right)^l\\
& \leq & C M|\xi-\zeta|^{\alpha-|\beta|_1}\end{aligned}$$ Thus, we prove the compatibility conditions and , and obtain $\partial_\xi^\beta f(x,\xi)=f^{(\beta)}(x,\xi)$ for $|\beta|_1\leq \alpha-1$.
Now, we prove . Let $$u_j(x,\xi)=\partial_\xi^\beta h_j(x,\xi) \quad {\rm and}\quad u(x,\xi)=\partial_\xi^\beta f(x,\xi),\qquad |\beta|_1=\alpha-1.$$ Then the implies $$\label{uexp}
u(x,\xi)=\sum_{j=1}^{\infty}u_j(x,\xi)\qquad {\rm for}\quad (x,\xi)\in \Omega\times \Pi_0.$$ By the Cauchy inequality and , we have $$\label{ujder}
|D^k u_j|_{\Omega,1;\Pi_j}\leq C(k) M r_j^{l-|k|_1} \qquad {\rm for} \quad |k|_1\leq q,$$ where $C(k)$ is a constant depending only on $k$. By a similar proof to one for the compatibility and replacing with , implies $$\sup_{x\in \Omega}|D^k u(x,\xi)-D^k u(x,\zeta)|\leq C_1^\prime M|\xi-\zeta|^{\delta} \quad {\rm for}\,\xi,\zeta\in \Pi_0,\,|k|_1\leq q.$$ The proof of the lemma is complete.0.4in $\blacksquare$
0.2in
4 Proof of Theorem 1 {#proof-of-theorem-1 .unnumbered}
====================
We first introduce some notation so that the system (1.1) is written in a compact form. Denote $$A^0={\rm diag}(\varepsilon^{q_1}A_1, \varepsilon^{q_3}A_2),\qquad B={\rm diag}(B_1,B_2),$$ $$\Lambda^0={\rm diag}(\varepsilon^{q_1}\Lambda_1, \varepsilon^{q_3}\Lambda_2),\qquad \omega^0={\rm col}(\varepsilon^{q_5}\omega_1,\omega_2),$$ $$P_1={\rm diag}(\varepsilon^{q_1}E_{n_{11}},\varepsilon^{q_3+q_4-q_2}E_{n_{12}}),\quad P_2={\rm diag}(\varepsilon^{q_5+q_6-q_2}E_{n_{21}},\varepsilon^{q_7-q_2}E_{n_{22}}),\quad P={\rm diag}(P_1,P_2),$$ where $E_n$ represents the $n\times n$ identity matrix. Then the system (1.1) reads $$\label{(4.1)}
\left(\begin{array}{c} \dot I\\ \dot\varphi\end{array}\right)=\left(\begin{array}{c} A^0(\xi,\varepsilon)I\\ \omega^0(\xi,\varepsilon)\end{array}\right)+PG(I,\varphi,\xi,\varepsilon)$$ with $G=\varepsilon^{q_2}{\rm col}(g_1,g_2,g_3,g_4)$. 0.2in
a\) [**Outline of the proof**]{} We are going to prove Theorem 1 by employing the KAM iteration process. By Corollary \[corollary2\] (see Section 3), we obtain a sequence of real analytic functions $G^0=0, G^j=\varepsilon^{q_2}{\rm col}(g_1^j,g_2^j,g_3^j,g_4^j)(j=1,2,\cdots)$ approximating $G$ and $$\label{gappr}
\lim_{j\rightarrow \infty}||G^j-G||_{p,\alpha;\Omega^*\times \mathbb{T}^{n_2},\Pi}=0 \qquad {\rm for \,\, all} \,\, 0\leq p<l,$$ $$\label{gjappr}
|G^j-G^{j-1}|_{\mathcal{U}_{j-1},\alpha;\Pi}\leq C_0 r_{j-1}^l||G||_{l,\alpha;\Omega\times \mathbb{T}^{n_2},\Pi} \qquad {\rm for }\,j\geq 1.$$ The definitions of $\Omega^*,\mathcal{U}_j$ etc are seen above Corollary \[corollary2\]. Denote $G_1^j=\varepsilon^{q_2}{\rm col}(g_1^j,g_2^j)$ and $G_2^j=\varepsilon^{q_2} {\rm col}(g_3^j,g_4^j)$. We truncate $G^1$ to its lower-degree terms $$\mathfrak{L}(G^1)\coloneqq \left(\begin{array}{c} G_1^1(0,\varphi)+\partial_IG_1^1(0,\varphi)I\\ G_2^1(0,\varphi)\end{array}\right)\coloneqq \left(\begin{array}{c} u_0^0(\varphi)+u_1^0(\varphi)I\\ w^0(\varphi)\end{array}\right)$$ and write (4.1) as $$\label{(4.2)}
\left(\begin{array}{c} \dot I\\ \dot\varphi\end{array}\right)=\left(\begin{array}{c} A^0I\\ \omega^0\end{array}\right)+P\left(\begin{array}{c} u_0^0(\varphi)+u_1^0(\varphi)I+H_1^0\\ w^0(\varphi)+H_2^0\end{array}\right)+P (G-G^1),$$ with $\partial_If(I,\varphi)$ represents the partial derivative (Jacobian matrix) of $f$ with respect to the variable $I$. Here, we drop parameters from functions and will do this also in the sequel whenever there is no confusion.
Moreover, the Cauchy inequality (see Lemma A.3 in [@Poschel96]) implies $$\label{u0est}
|u_0^0|_{r_0,\alpha;\Pi}\leq C_0 M \varepsilon^{q_2} r_0^l, \qquad |u_1^0|_{r_0,\alpha;\Pi}\leq C_0 M \varepsilon^{q_2} r_0^{l-1},$$ $$\label{w0est}
|w^0|_{r_0,\alpha;\Pi}\leq C_0 M \varepsilon^{q_2} r_0^l, \qquad H_1^0=O_{\mathcal{U}_1,\alpha;\Pi}(I^2),\quad H_2^0=O_{\mathcal{U}_1,\alpha;\Pi}(I)$$ and $$\label{H0est}
|H_1^0|_{\mathcal{U}_1,\alpha;\Pi}\leq 2C_0 M \varepsilon^{q_2} r_0^{l-2}, \qquad |H_2^0|_{\mathcal{U}_1,\alpha;\Pi}\leq C_0 M \varepsilon^{q_2} r_0^{l-1},$$ where $ M \varepsilon^{q_2}=||G||_{l,\alpha;\Omega\times \mathbb{T}^{n_2},\Pi}$.
We want to look for a transformation $T_1$ to eliminate the lower-degree terms of $PG^1$ such that in new coordinates the lower-degree terms of analytic part in are much smaller than the old ones. Assume that at the $\nu$-th step of the process, we have already found a coordinate transformation $T_{\nu}(\nu\geq 0$ with $T_0={\rm Id}$, the identity map) such that the system (4.1) is transformed into $$\left(\begin{array}{c} \dot I\\ \dot\varphi\end{array}\right)=\left(\begin{array}{c} A^\nu I\\ \omega^\nu\end{array}\right)+P\left(\begin{array}{c} \tilde{u}_0^\nu(\varphi)+\tilde{u}_1^\nu(\varphi)I+\tilde{H}_1^\nu\\ \tilde{w}^\nu(\varphi)+\tilde{H}_2^\nu\end{array}\right)+P\mathfrak{D}_\nu (G\circ T_\nu-G^\nu\circ T_\nu),$$ where $\tilde{H}_1^\nu=O(I^2),\tilde{H}_2^\nu=O(I), \mathfrak{D}_\nu=P^{-1}(DT_\nu)^{-1}P$, the circle “$\circ$” indicates composition of functions and $DT$ the Jacobian matrix of $T$ with respect to coordinate variables. Then we replace $G^\nu$ with $G^{\nu+1}$ which is closer to $G$, and the above equation is rewritten as $$\label{eqv}
\left(\begin{array}{c} \dot I\\ \dot\varphi\end{array}\right)=\left(\begin{array}{c} A^\nu I\\ \omega^\nu\end{array}\right)+P\left(\begin{array}{c} u_0^\nu(\varphi)+u_1^\nu(\varphi)I+H_1^\nu\\ w^\nu(\varphi)+H_2^\nu\end{array}\right)+P\mathfrak{D}_\nu (G\circ T_\nu-G^{\nu+1}\circ T_\nu),$$ where $$\left(\begin{array}{c} u_0^\nu(\varphi)+u_1^\nu(\varphi)I\\ w^\nu(\varphi)\end{array}\right)= \left(\begin{array}{c} \tilde{u}_0^\nu(\varphi)+\tilde{u}_1^\nu(\varphi)I\\ \tilde{w}^\nu(\varphi)\end{array}\right) + \mathfrak{L}(\mathfrak{D}_\nu (G^{\nu+1}\circ T_\nu-G^{\nu}\circ T_\nu)),$$ $$H_1^\nu=O(I^2),\qquad H_2^\nu=O(I).$$ We want to construct a coordinate change $T^{\nu+1}$ to eliminate the lower-degree terms in such that the lower-degree terms of the next step are much smaller. Repetition of this process leads to a sequence of transformation $T_\nu=T_{\nu-1}\circ T^\nu$ with $T_0={\rm Id}, \nu=1,2,\cdots$, the limit transformation of which , if converges, reduces (4.1) into a system without the lower-degree terms. Thus, we can obtain the quasi-periodic solution of (4.1). The proof of convergence is due to the following iteration lemma which describes quantitatively the KAM iteration process. 0.2in
b\) [**Iteration Lemma**]{} Before stating the iteration lemma we first introduce the iterative sequences and notations used at each iteration step. Set $$\varepsilon_0=\varepsilon^{q_2},\qquad ||G||_{l.\alpha;\Omega\times \mathbb{T}^{n_2},\Pi}=M \varepsilon_0,$$ $$\Omega=\{I\in \mathbb{R}^{n_1}:\, |I|<3\tilde{r}\},\quad \Omega^*=\{I\in \mathbb{R}^{n_1}:\, |I|\leq 2\tilde{r}\},\quad \Omega_0=\{I\in \mathbb{R}^{n_1}:\, |I|<\tilde{r}\}$$ with some constant $0<\tilde{r}\leq 1$. For $\nu\geq 1$, let
\(i) $ r_0=\tilde{r},\, r_\nu=\tilde{r}3^{-\nu}$, $$\mathcal{U}_\nu=\Omega^*\times \mathbb{T}^{n_2}+(3r_\nu,3r_\nu),\quad \mathcal{V}_\nu =\Omega_0\times \mathbb{T}^{n_2}+(r_\nu,r_\nu), \qquad \nu\geq 0,$$ $$\mathcal{V}_\nu^*=\Omega_0\times \mathbb{T}^{n_2}+(2r_\nu,2r_\nu)\subset \mathcal{V}_{\nu-1}\subset \mathcal{U}_\nu;$$
\(ii) $K_0=0,\quad K_\nu=[K_\nu^\prime ]+1, \quad K_\nu^\prime =3^\nu r_0^{-1}(\ln \widetilde{C}+(n_2+1)|\ln r_0|+(l+(n_2+1)\nu-\alpha)\ln3), \, \widetilde{C}=24(n_2!)n_2^{n_2}e^{-n_2}$, $[K_\nu^\prime ]$ is the integer part of $K_\nu^\prime$;
\(iii) $s_0=\gamma, \quad s_\nu=\gamma (16c_1n_3\sqrt{n_2}K_\nu^{\iota+1})^{-1},\quad \Pi_\nu^{s_\nu}=\{\xi \in \mathbb{R}^{n_3}:\, {\rm dist}(\xi, \Pi_\nu)<s_\nu\};$
\(iv) $\chi_\nu=r_\nu^{l-2(\alpha+1)(\iota+1)-\alpha-3},\quad X_\nu=\sum_{j=1}^{\nu}\chi_j,$\
the assumption $l>2(\alpha+1)(\iota+2)+\alpha\iota$ implies $X_\nu=\sum_{j=1}^{\infty}\chi_j<\frac{1}{2}$.
\(v) $\delta_{\nu\mu}=\gamma^{-\mu-1}r_\nu^{l-(\alpha+\mu+2)(\iota+1)-\alpha-3}C_0M\varepsilon_0,\quad 0\leq \mu\leq \alpha;$
\(vi) $f(I,\varphi,\xi)=O_{\mathcal{U},\alpha;\Pi}(I^k)$ denotes a map which is real analytic in coordinate variables $(I,\varphi)\in \mathcal{U}$, continuously differentiable up to order $\alpha$ in parameter $\xi \in \Pi$, and vanishes with $I$-derivatives up to order $k-1\geq 0$, and $f$ and its $\xi$-derivatives up to order $\alpha$ are bounded on $\mathcal{U}\times \Pi$.
\[iterationlemma\] (Iteration Lemma) Assume that for the equation with $\nu\geq 0$,
[(v.1)]{} (Frequency condition) let $A^\nu={\rm diag}(\varepsilon^{q_1}A_1^\nu, \varepsilon^{q_3}A_2^\nu),\quad \Lambda^\nu={\rm diag}(\varepsilon^{q_1}\Lambda_1^\nu, \varepsilon^{q_3}\Lambda_2^\nu), \quad \Lambda_1^\nu={\rm diag}(\lambda_1^\nu,\cdots,\lambda_{n_{11}}^\nu), \quad \Lambda_1^\nu={\rm diag} (\lambda_{n_{11}+1}^\nu,\cdots,\lambda_{n_{1}}^\nu)$, $A_i^\nu=B_i\Lambda_i^\nu B_i^{-1}\, (i=1,2)$ and $ \omega^\nu={\rm col}(\varepsilon^{q_5}\omega_1^\nu,\omega_2^\nu)$ satisfy, for $\varepsilon \in (0,\varepsilon^*]$, $$\inf_{\xi\in \Pi_\nu^{s_\nu}}|\lambda_j|\geq c_0(1-X_\nu)>\frac{c_0}{2},\quad \inf_{\xi\in \Pi_\nu^{s_\nu}}|\lambda_j-\lambda_i|\geq c_0(1-X_\nu)>\frac{c_0}{2}$$ for $i\neq j,\, 1\leq i,j\leq n_{11},\,{\rm or} \, n_{11}+1\leq i,j\leq n_1$, and $$||\Lambda_i^\nu||_{\alpha;\Pi_\nu^{s_\nu}}\leq c_1(1+X_\nu)<2c_1, \qquad \parallel \partial_{\xi}^{\beta}\omega^\nu \parallel_{\Pi_\nu^{s_\nu}}\leq c_1(1+X_\nu) \varepsilon^{q_5}<2c_1\varepsilon^{q_5}, \, 1\leq |\beta|_1\leq \alpha,$$ $$\label{freerror}
||\Lambda_i^\nu-\Lambda_i^{\nu-1}||_{\alpha;\Pi_\nu^{s_\nu}}\leq \tilde{c}_0 C_0M\varepsilon_0\varepsilon^{b_i}r_{\nu-1}^{l-(\alpha+1)(\iota+2)-1},\,
||\omega_i^\nu-\omega_i^{\nu-1}||_{\alpha;\Pi_\nu^{s_\nu}}\leq C_0M\varepsilon_0\varepsilon^{b_{i+2}}r_{\nu-1}^{l-(\alpha+1)(\iota+2)}, \,\nu\geq 1$$ for $i=1,2$, where $\tilde{c}_0$ is a positive constant, $c_0$ and $c_1$ are given in Assumption (H2), $b_1=0, b_2=q_4-q_2, b_3=q_6-q_2, b_4=q_7-q_2$;
[(v.2)]{} (Small condition) the terms $u_0^\nu, u_1^\nu$ and $w^\nu$ satisfy the following estimates $$|u_0^\nu|_{r_\nu,\alpha;\Pi_\nu^{s_\nu}}\leq 4 C_0M\varepsilon_0r_{\nu}^{l-\alpha},\quad |u_1^\nu|_{r_\nu,\alpha;\Pi_\nu^{s_\nu}}\leq C_0M\varepsilon_0r_{\nu}^{l-(\alpha+1)(\iota+2)-1},\quad |w^\nu|_{r_\nu,\alpha;\Pi_\nu^{s_\nu}}\leq C_0M\varepsilon_0r_{\nu}^{l-(\alpha+1)(\iota+2)},$$ $H_1^\nu(I,\varphi,\xi)$ and $H_2^\nu(I,\varphi,\xi)$ fulfill $$\label{hest}
H_1^\nu=O_{\mathcal{V}_\nu,\alpha;\Pi_\nu^{s_\nu}}(I^2),\quad H_2^\nu=O_{\mathcal{V}_\nu,\alpha;\Pi_\nu^{s_\nu}}(I),\quad |H_i^\nu-H_i^{\nu-1}|_{\mathcal{V}_\nu,\alpha;\Pi_\nu^{s_\nu}}\leq \chi_\nu C_0M\varepsilon_0$$ for $\nu\geq 1, \,i=1,2$;
[(v.3)]{} (Transformation) the transformation $T_\nu:\mathcal{V}_\nu\times\Pi_\nu^{s_\nu}\rightarrow \mathcal{U}_\nu$ is real analytic in coordinate variables $(I,\varphi)\in \mathcal{V}_\nu$ and continuously differentiable up to order $\alpha$ in the parameter $\xi \in \Pi_\nu^{s_\nu}$, satisfies $$\label{transest}
|T_\nu-T_{\nu-1}|_{\mathcal{V}_\nu,\mu;\Pi_\nu^{s_\nu}}\leq (1+X_\nu)C_1 C_0M\varepsilon_0\gamma^{-\mu-1}r_{\nu}^{l-(\alpha+\mu+2)(\iota+1)-\alpha-2}<r_\nu \chi_\nu,$$ $$\label{dTvest}
|P^{-1}(DT_\nu-DT_{\nu-1})P|_{\mathcal{V}_\nu,\mu;\Pi_\nu^{s_\nu}}\leq 2(1+X_\nu)C_1 C_0M\varepsilon_0\gamma^{-\mu-1}r_{\nu}^{l-(\alpha+\mu+2)(\iota+1)-\alpha-3}< \chi_\nu$$ with $T_0={\rm Id}$ and $0\leq \mu\leq \alpha$, where $C_1$ is a constant independent of $\nu$.
Then there exists a closed set $\Pi_{\nu+1}\subset \Pi_\nu$ $$\Pi_{\nu+1}=\left\{\xi \in \Pi_\nu:\, |\sqrt{-1}\langle k,\omega^{\nu}\rangle+\langle m,\Lambda^{\nu}\rangle|\geq \gamma \varepsilon^{q_5}|k|_2^{-\iota},m\in \mathfrak{m},k\in\mathbb{Z}^{n_2}, K_{\nu}<|k|_2\leq K_{\nu+1} \right\}$$ (see Theorem 1 and (H3) for definitions of $ \mathfrak{m}$ and $\iota$, respectively) and a coordinate transformation $$T^{\nu+1}:\mathcal{V}_{\nu+1}\times\Pi_{\nu+1}^{s_{\nu+1}}\rightarrow \mathcal{V}_{\nu+1}^*\subset \mathcal{V}_\nu \subset \mathcal{U}_{\nu+1}$$ in the form $$\label{vtran}
I=\rho +v_0^\nu(\phi,\xi)+v_1^\nu(\phi,\xi)\rho,\qquad \varphi=\phi+\Phi^\nu(\phi,\xi),$$ where $\rho$ and $\phi$ are new coordinate variables, and all terms in the transformation are real analytic in $\phi$ and continuously differentiable in $\xi$ up to order $\alpha$, satisfy the estimates $$\label{phiv}
|\Phi^\nu|_{2r_{\nu+1},\alpha;\Pi_{\nu+1}^{s_{\nu+1}}}\leq C_1 C_0M\varepsilon_0\gamma^{-\alpha-1}r_{\nu+1}^{l-(\alpha+1)(2\iota+3)},$$ $$\label{v0v}
|v_0^\nu|_{2r_{\nu+1},\alpha;\Pi_{\nu+1}^{s_{\nu+1}}}\leq C_1 C_0M\varepsilon_0\gamma^{-\alpha-1}r_{\nu+1}^{l-(\alpha+1)(\iota+1)-\alpha},$$ $$\label{v1v}
|v_1^\nu|_{2r_{\nu+1},\alpha;\Pi_{\nu+1}^{s_{\nu+1}}}\leq C_1 C_0M\varepsilon_0\gamma^{-\alpha-1}r_{\nu+1}^{l-(\alpha+1)(2\iota+3)-1}$$ and $$\label{dest}
|P^{-1}(DT^{\nu+1})^{-1}P|_{\mathcal{V}_{\nu+1},0;\Pi_{\nu+1}^{s_{\nu+1}}}<1+\chi_{\nu+1},\quad \left|\partial_\xi^\beta \left(P^{-1}(DT^{\nu+1})^{-1}P\right)\right|_{\mathcal{V}_{\nu+1},0;\Pi_{\nu+1}^{s_{\nu+1}}}<\chi_{\nu+1}$$ for $ 1\leq |\beta|_1\leq \alpha$, such that the equation is transformed into $$\left(\begin{array}{c} \dot \rho\\ \dot\phi\end{array}\right)=\left(\begin{array}{c} A^{\nu+1} \rho\\ \omega^{\nu+1}\end{array}\right)+P\left(\begin{array}{c} u_0^{\nu+1}(\phi)+u_1^{\nu+1}(\phi)\rho+H_1^{\nu+1}\\ w^{\nu+1}(\phi)+H_2^{\nu+1}\end{array}\right)+P\mathfrak{D}_{\nu+1} (G\circ T_{\nu+1}-G^{\nu+2}\circ T_{\nu+1})$$ and the conditions (v.1)-(v.3) are satisfied by replacing $\nu$ by $\nu+1$ and $(I,\varphi)$ by $(\rho,\phi)$, respectively, where $T_{\nu+1}=T_\nu\circ T^{\nu+1}, \mathfrak{D}_{\nu+1}=P^{-1}(DT_{\nu+1})^{-1}P$.
0.2in
c\) [**Proof of Theorem 1**]{} Theorem 1 is easy to be proven by the Iteration Lemma and Inverse Approximation Lemma.
First the system (1.1) has been written in the form just as satisfying the conditions (v.1)-(v.3) with $\nu=0$ in the Iteration Lemma by Assumptions (H2) and (H3), and . We use the Iteration Lemma inductively to obtain a sequence of transformations $T_\nu$ mapping $\mathcal{V}_\nu\times \Pi_\nu^{s_\nu}$ into $\mathcal{V}_0$ and satisfying the estimate . Noting that $\mathcal{V}_\nu$ and $\Pi_\nu^{s_\nu}$ are exactly regarded as those neighbourhoods of the open convex set $\Omega_0\times\mathbb{T}^{n_2}\subset \mathbb{R}^{n_1+n_1}$ and closed subset $\Pi_\gamma\subset \Pi$, respectively, and $r_\nu^{l_1}/s_\nu^{\mu}\rightarrow 0$ as $\nu\rightarrow \infty$ $(l_1=l-(\alpha+\mu+2)(\iota+1)-\alpha-2$ and the positive integer $\mu\leq\alpha)$ by the definition of $s_\nu$, the Inverse Approximation Lemma and Condition (v.3) imply that for every $\xi \in \Pi_\gamma$, the limit map $T=\lim_{\nu\rightarrow \infty}T_\nu$ exists in $C^{p,\mu-1}(\Omega_0\times\mathbb{T}^{n_2},\Pi_\gamma)$ for $0\leq p<l_1$ and $T:\Omega_0\times\mathbb{T}^{n_2}\times\Pi_\gamma\rightarrow \Omega^*\times\mathbb{T}^{n_2}$ for sufficiently small $\varepsilon$, and is of the form $$T:\qquad I=\rho +V_0(\phi,\xi)+V_1(\phi,\xi)\rho,\quad \varphi=\phi+\Phi(\phi,\xi)$$ by , which is of class $C^\mu$ in $\xi\in\Pi_{\gamma}$ in the sense of whitney and of class $\hat{C}^{l_1}$ in $\varphi\in \mathbb{T}^{n_2}$ together with derivatives up to order $\mu-1$ with respect to $\xi$ for $0< \mu\leq \alpha$. Moreover, by , we obtain $$\label{remest}
\lim_{\nu\rightarrow \infty}||G\circ T_\nu-G^{\nu+1}\circ T_\nu||_{p,\mu-1;\Omega_0\times \mathbb{T}^{n_2},\Pi_\gamma}=0 \qquad {\rm for \,\, all} \,\, 0\leq p<l_1,\,0<\mu\leq \alpha$$ and by Condition (v.3) and , $$\label{ddest}
|\mathfrak{D}_\nu-\mathfrak{D}_{\nu-1}|_{\mathcal{V}_\nu,\mu;\Pi_\nu^{s_\nu}}\leq C_2 C_0M\varepsilon_0\gamma^{-\mu-1}r_{\nu}^{l-(\alpha+\mu+2)(\iota+1)-\alpha-3},$$ where $C_2$ is a constant independent of $\nu,\gamma$ and $\varepsilon_0$. It follows from , and that System (4.1) is transformed by $T$ into the system $$\label{infeq}
\left\{\begin{array}{rl}
\dot{\rho} & =A^*(\xi)\rho+P_1O(\rho^2)\\
\dot{\phi} & = \omega^*(\xi)+P_2O(\rho)
\end{array}\right.$$ for $(\rho,\phi)\in \Omega_0\times\mathbb{T}^{n_2}, \xi\in\Pi_\gamma$, where $A^*(\xi)={\rm diag}(\varepsilon^{q_1}A_1^*(\xi),\varepsilon^{q_3}A_2^*(\xi)), \omega^*(\xi)={\rm col}(\varepsilon^{q_5}\omega_1^*(\xi),\omega_2^*(\xi))$, $A_i^*=\lim_{\nu\rightarrow \infty}A_i^\nu$ and $\omega_i^*=\lim_{\nu\rightarrow \infty}\omega_i^\nu (i=1,2)$ exist by Condition (v.1) and are of class $C^\alpha$ in $\xi\in\Pi_\gamma$ in the sense of Whitney by the Inverse Approximation Lemma since $r_\nu^{l-(\alpha+1)(\iota+2)-1}s_\nu^{-\alpha}\rightarrow 0$ as $\nu\rightarrow \infty$. Thus, we obtain the quasi-periodic invariant torus of (4.1) $$I=V_0(\phi,\xi),\quad \varphi=\phi+\Phi(\phi,\xi),\quad \phi=\omega^*(\xi) t+\phi_0$$ satisfying the estimates , and by Conditions (v.1) and (v.3). The rest of Theorem 1 can be derived immediately from the Iteration Lemma. 0.4in $\blacksquare$ 0.2in
d\) [**Proof of Iteration Lemma**]{} To simplify the notation, we denote quantities referring to $\nu+1$ with $+$ such as $u^{\nu+1}$ by $u^+$, $r_{\nu+1}$ by $r_+$, and those referring to $\nu$ without the $\nu$ such as $u^\nu$ by $u$, $r_\nu$ by $r$. Substituting the transformation $T^+$ into , the transformation $T^+$ will be obtained by solving the homological equations $$\begin{aligned}
& &\partial_{\phi}v_0\cdot \omega-A v_0 =P_1\Gamma_{K_+} u_0(\phi),\label{v0eq}\\
& &\partial_{\phi}v_1\cdot \omega+v_1A-Av_1 =P_1(\Gamma_{K_+}u_1(\phi)-B{\rm diag}(B^{-1}\widehat{u_1}(0)B)B^{-1}),\label{v1eq}\\
& &\partial_{\phi}\Phi \cdot \omega =P_2(\Gamma_{K_+} w(\phi)-\widehat{w}(0)),\label{phieq},\end{aligned}$$ where $B={\rm diag}(B_1,B_2), {\rm diag}(B^{-1}\widehat{u_1}(0)B)$ denotes a diagonal matrix whose elements are the diagonal elements of $B^{-1}\widehat{u_1}(0)B$, $\widehat{u_1}(0)$ and $\widehat{w}(0)$ denote the mean values (that is, the zero-order coefficients of the Fourier series expansions) of $u_1$ and $w$ over $ \mathbb{T}^{n_2}$, respectively, $\Gamma_{K_+}$ is the truncation operator of the Fourier series expansions defined in Lemma \[lemmaA5\] and the notation $\partial_\phi f\cdot \omega$ means $\partial_\phi f\cdot \omega=\sum_{j=1}^{n_2}t_j\frac{\partial f}{\partial \phi_j}$ for $\omega={\rm col}(t_1,\cdots,t_{n_2})$. Here, the homological equations are approximated by truncating the Fourier series expansions of $u_0,u_1$ and $w$ so that the solutions are defined on an open set of parameters. This idea is due to Arnol’d[@Arn63a] and Pöschel[@Pos82].
d1) Solutions of - and estimates. Set $$\Pi_+=\left\{\xi \in \Pi_\nu:\, |\sqrt{-1}\langle k,\omega\rangle+\langle m,\Lambda\rangle|\geq \gamma \varepsilon^{q_5}|k|_2^{-\iota},m\in \mathfrak{m}, K<|k|_2\leq K_{+} \right\}$$ and $$\Pi_+^{s_+}=\{\xi\in \mathbb{R}^{n_3}:\, {\rm dist}(\xi,\Pi_+)<s_+\}\subset \Pi_\nu^{s_\nu}.$$
\[lemma4.1\] For every $\xi\in\Pi_+^{s_+}$, we have $$\label{smalld1}
|\sqrt{-1}\langle k,\omega(\xi)\rangle+\langle m,\Lambda(\xi)\rangle|\geq \frac{1}{4}\gamma \varepsilon^{q_5}|k|_2^{-\iota},\quad 0<|k|_2\leq K_{+}, m\in \mathfrak{m}.$$
[**Proof**]{} We first prove $$\label{smalld2}
|\sqrt{-1}\langle k,\omega(\xi)\rangle+\langle m,\Lambda(\xi)\rangle|\geq \frac{1}{2}\gamma \varepsilon^{q_5}|k|_2^{-\iota},\quad 0<|k|_2\leq K_{+}, m\in \mathfrak{m}$$ for every $\xi\in\Pi_+$. Noting the fact that $K_j^{\iota+1}r_j^{l-(\alpha+1)(\iota+2)-2} \rightarrow 0$ as $j\rightarrow \infty$, the implies that for $0<|k|_2\leq K_j,1\leq j \leq \nu$, $$\begin{aligned}
|\sqrt{-1}\langle k,\omega^j(\xi)-\omega^{j-1}(\xi)\rangle+\langle m,\Lambda^j(\xi)-\Lambda^{j-1}(\xi)\rangle| & \leq & \varepsilon^{q_5}(\sqrt{n_2}K_j+2\tilde{c}_0) C_0M\varepsilon_0r_{j-1}^{l-(\alpha+1)(\iota+2)-1}\\
& < & r_j\gamma \varepsilon^{q_5}K_j^{-\iota}\end{aligned}$$ for sufficiently small $\varepsilon_0$. As for $K_{j-1}<|k|_2\leq K_j$, $|\sqrt{-1}\langle k,\omega^{j-1}(\xi)\rangle+\langle m,\Lambda^{j-1}(\xi)\rangle|\geq \gamma \varepsilon^{q_5}|k|_2^{-\iota}$, hence, $$\begin{aligned}
& & |\sqrt{-1}\langle k,\omega^\nu(\xi)\rangle+\langle m,\Lambda^\nu(\xi)\rangle|\\
& & \quad \geq |\sqrt{-1}\langle k,\omega^{j-1}(\xi)\rangle+\langle m,\Lambda^{j-1}(\xi)\rangle|-
\sum_{i=j}^{\nu}|\sqrt{-1}\langle k,\omega^i(\xi)-\omega^{i-1}(\xi)\rangle+\langle m,\Lambda^i(\xi)-\Lambda^{i-1}(\xi)\rangle| \\
& & \quad \geq \gamma \varepsilon^{q_5}(|k|_2^{-\iota}-\sum_{i=j}^{\nu}r_iK_i^{-\iota})\geq \frac{1}{2}\gamma \varepsilon^{q_5}|k|_2^{-\iota},\end{aligned}$$ which implies .
For every $\xi\in\Pi_+^{s_+}\subset \Pi_\nu^{s_\nu}$, there is $\xi_0\in \Pi_+$ such that $|\xi-\xi_0|<s_+$. The condition (v.1) and imply $$\begin{aligned}
& & |\sqrt{-1}\langle k,\omega(\xi)\rangle+\langle m,\Lambda(\xi)\rangle|\\
& & \quad \geq |\sqrt{-1}\langle k,\omega(\xi_0)\rangle+\langle m,\Lambda(\xi_0)\rangle|-
|\sqrt{-1}\langle k,\omega(\xi)-\omega(\xi_0)\rangle+\langle m,\Lambda(\xi)-\Lambda(\xi_0)\rangle| \\
& & \quad \geq \frac{1}{2}\gamma \varepsilon^{q_5}|k|_2^{-\iota}-2c_1n_3\varepsilon^{q_5}(\sqrt{n_2}|k|_2+2)s_+\\
& & \quad \geq \frac{1}{4}\gamma \varepsilon^{q_5}|k|_2^{-\iota}. \hskip 0.4in \blacksquare\end{aligned}$$ 0.2in
The procedure of solving - is standard in KAM theory. Expanding the functions into the Fourier series in $\phi$, and substituting in - and comparing coefficients of the term $e^{\sqrt{-1}\langle k,\phi\rangle}$, one obtain the solutions $$\begin{aligned}
& &v_0(\phi)=P_1\sum_{|k|_2\leq K_+} B (\sqrt{-1}\langle k,\omega\rangle-\Lambda)^{-1}B^{-1}\widehat{u_0}(k)e^{\sqrt{-1}\langle k,\phi\rangle},\label{v0}\\
& &v_1(\phi)=P_1\sum_{|k|_2\leq K_+} BV_1(k)B^{-1}e^{\sqrt{-1}\langle k,\phi\rangle},\label{v1}\\
& &\Phi(\phi)=P_2\sum_{0<|k|_2\leq K_+} (\sqrt{-1}\langle k,\omega\rangle)^{-1}\widehat{w}(k)e^{\sqrt{-1}\langle k,\phi\rangle},\label{phi}\end{aligned}$$ where $$(V_1(k))_{ij}=\left\{\begin{array}{ll} (\sqrt{-1}\langle k,\omega\rangle+\varepsilon^{a_j}\lambda_j-\varepsilon^{a_i}\lambda_i)^{-1}(\widehat{U_1}(k))_{ij},\,& |k|+|i-j|\neq 0\\
0, & |k|+|i-j|=0,
\end{array}\right.$$ $U_1(\phi)=B^{-1}u_1(\phi)B,\, a_i=q_1$ if $1\leq i\leq n_{11},\, =q_3$ if $n_{11}+1\leq i\leq n_1$, $(V_1(k))_{ij}$ and $(\widehat{U_1}(k))_{ij}$ represent elements of the matrices $V_1(k)$ and $\widehat{U_1}(k)$, respectively, and $\hat{u}(k)$ is the $k$-order coefficients of the Fourier series expansions of $u$. Hence, Lemma \[lemma4.1\], - and Conditions (v.1)-(v.2) imply that $v_0,v_1$ and $\Phi$ are real analytic in $\phi \in \mathcal{W}\coloneqq \mathbb{T}^{n_2}+2 r_+$, continuously differentiable up to order $\alpha$ in $\xi\in\Pi_+^{s_+}$. Meanwhile using Lemma \[lemmaA7\], one easily gets the estimates - (we denote $|\cdot|_{r,\mu; \Pi_+^{s_+}}$ by $|\cdot|_{r,\mu; s_+}$, $|\cdot|_{r,\mu; \Pi_\nu^{s_\nu}}$ by $|\cdot|_{r,\mu; s}$ for simplification) and $$\label{dv0est}
|v_0|_{2r_+,\mu;s_+}\leq C_1\delta_{+\mu}r_+^{(\alpha+1)(\iota+1)+3}, \quad |P_1^{-1}\partial_\phi v_0|_{2r_+,\mu;s_+}\leq C_1\varepsilon^{-q_5}\delta_{+\mu}r_+^{(\alpha+1)(\iota+1)+2},$$ $$\label{dv1est}
|v_1|_{2r_+,\mu; s_+} \leq C_1\delta_{+\mu}r_+, \quad |P_1^{-1}v_1P_1|_{2r_+,\mu;s_+}\leq C_1\delta_{+\mu}r_+,
\quad |P_1^{-1}\partial_\phi v_1|_{2r_+,\mu;s_+}\leq C_1\varepsilon^{-q_5}\delta_{+\mu}$$ and $$\label{dphiest}
|\Phi|_{2r_+,\mu;s_+}\leq C_1\delta_{+\mu}r_+^2, \quad |P_2^{-1}\partial_\phi \Phi|_{2r_+,\mu;s_+}\leq C_1\varepsilon^{-q_5}\delta_{+\mu}r_+$$ for $0\leq \mu\leq \alpha$ and an appropriate choice of the constant $C_1$ independent of $\nu$.
It is easy to see that when the $\varepsilon_0$ is sufficiently small, the transformation $T^+$ maps $\mathcal{V}_+$ into $\mathcal{V}_+^*\subset \mathcal{V}$ and $\mathcal{V}_+^*$ into $\mathcal{V}$, respectively, and $$\label{vTest}
|T^+-{\rm Id}|_{\mathcal{V}_+^*,0;s_+}\leq C_1 r_+\delta_{+0},\quad |\partial_\xi^\beta T^+|_{\mathcal{V}_+^*,0;s_+}\leq C_1 r_+\delta_{+\mu},\quad 1\leq |\beta|_1=\mu \leq \alpha.$$
d2) Proof of . Corresponding to the transformation $T^+$, we have its Jacobian matrix $$\label{dT}
DT^+=\left(\begin{array}{cc} E_{n_1}+v_1 & \partial_\phi v_0+\partial_\phi v_1\rho\\
0 & E_{n_2}+\partial_\phi \Phi\end{array}\right)$$ and the inverse $$\label{dTin}
\left(DT^+\right)^{-1}=\left(\begin{array}{cc} (E_{n_1}+v_1)^{-1} & -(E_{n_1}+v_1)^{-1}(\partial_\phi v_0+\partial_\phi v_1\rho)(E_{n_2}+\partial_\phi \Phi)^{-1}\\
0 & (E_{n_2}+\partial_\phi \Phi)^{-1}\end{array}\right).$$ Thus, - imply $$\label{vdTest}
|P^{-1}DT^+P|_{\mathcal{V}_+,0;s_+}\leq 1+\chi_+ r_+^{\alpha(\iota+1)},\quad |P^{-1}(DT^+-E)P|_{\mathcal{V}_+,\mu;s_+}\leq C_1\delta_{+\mu}\quad {\rm for}\,1\leq \mu\leq \alpha.$$ Noting that the derivatives of $DT^+$ with respect to the parameter $\xi$ is sufficiently small and that for a matrix $M(\xi)$ with a small norm, differentiating the left- and right-hand sides of $(E+M(\xi))^{-1}(E+M(\xi))=E$ and using the Leibniz formula, we find $$\partial_\xi^\beta (E+M(\xi))^{-1}=-\sum_{k<\beta}\left(\begin{array}{c} \beta\\k\end{array}\right)\partial_\xi^k (E+M(\xi))^{-1}\cdot \partial_\xi^{\beta-k} (E+M(\xi))\cdot (E+M(\xi))^{-1},$$ where $k,\beta\in \mathbb{Z}_+^{n_3}, \left(\begin{array}{c} \beta\\k\end{array}\right)=\left(\begin{array}{c} \beta_1\\k_1\end{array}\right)\cdots \left(\begin{array}{c} \beta_{n_3}\\k_{n_3}\end{array}\right)$, $E$ is the identity matrix, the estimates and imply $$\begin{aligned}
& & |P_1^{-1}(E_{n_1}+v_1)^{-1}P_1|_{2r_+,0;s_+}\leq 1+\chi_+ r_+^{\alpha(\iota+1)+1}, \label{p1v1in}\\
& & |P_2^{-1}(E_{n_2}+\partial_\phi \Phi)^{-1}P_2|_{2r_+,0;s_+}\leq 1+\chi_+ r_+^{\alpha(\iota+1)+1}, \label{p2phiin}\\
& & |\partial_\xi ^\beta (P_1^{-1}(E_{n_1}+v_1)^{-1}P_1)|_{2r_+,0;s_+}\leq 2(1+r_+\chi_+)^2 |P_1^{-1}v_1P_1|_{2r_+,|\beta|_1;s_+}, \label{dp1v1in}\\
& & |\partial_\xi ^\beta (P_2^{-1}(E_{n_2}+\partial_\phi \Phi)^{-1}P_2)|_{2r_+,0;s_+}\leq 2(1+r_+\chi_+)^2 |P_2^{-1}\partial_\phi \Phi P_2|_{2r_+,|\beta|_1;s_+} \label{dp2phiin}\end{aligned}$$ for $1\leq |\beta|_1\leq \alpha$ and sufficiently small $\varepsilon_0$. Hence, follows from , , and -. Moreover, we have $$\label{ddest1}
\left.\begin{array}{r}|P_1^{-1}(E_{n_1}+v_1)^{-1}P_1F|_{\mathcal{V}_+,\alpha;s_+} \\|P_2^{-1}(E_{n_2}+\partial_\phi \Phi)^{-1}P_2F|_{\mathcal{V}_+,\alpha;s_+} \\ |P^{-1}(DT^+)^{-1}PF|_{\mathcal{V}_+,\alpha;s_+} \end{array}\right\} <(1+r_+\chi_+)|F|_{\mathcal{V}_+,\alpha;s_+}$$ for suitable $F$ which is real analytic in $\mathcal{V}_+$ and continuously differentiable up to order $\alpha$ in $\xi \in \Pi_+^{s_+}$.
d3) We proceed to verify (v.3) with $\nu+1$ replacing $\nu$. As the transformation $T^+:\, \mathcal{V}_+\times \Pi_+^{s_+}\rightarrow \mathcal{V}_+^*$ (or $ \mathcal{V}_+^*\times \Pi_+^{s_+}\rightarrow \mathcal{V}$) is real analytic in coordinate variables and continuously differentiable up to order $\alpha$ in parameter $\xi$, so is $T_+=T\circ T^+$.
We first prove and inductively. For $\nu=1$, by -, it implies $$|T_1-{\rm Id}|_{\mathcal{V}_1,\mu;s_1}\leq C_1 r_1\delta_{1\mu}<r_1\chi_1,\quad |DT_1-E|_{\mathcal{V}_1,\mu;s_1}\leq C_1\delta_{1\mu}<\chi_1$$ and $$|P^{-1}(DT_1-E)P|_{\mathcal{V}_1,\mu;s_1}\leq C_1\delta_{1\mu}<\chi_1$$ for $0\leq \mu\leq\alpha$. Assume that at the $\nu$-th step we have $$|T-T_{\nu-1}|_{\mathcal{V},\mu;s}\leq (1+X)C_1 r\delta_{\nu\mu}<r\chi,\quad |DT-DT_{\nu-1}|_{\mathcal{V},\mu;s}\leq(1+X) C_1\delta_{\nu\mu}<\chi$$ and $$|P^{-1}(DT-DT_{\nu-1})P|_{\mathcal{V},\mu;s}\leq 2(1+X) C_1\delta_{\nu\mu}<\chi$$ for $0\leq \mu\leq\alpha$, here we have omitted the subscript $\nu$ from the quantities referring to $\nu$. Then in view of the induction assumptions we obtain $$\label{dTest}
|DT|_{\mathcal{V},0;s}\leq 1+X,\quad |\partial_\xi^\beta DT|_{\mathcal{V},0;s}\leq X \quad {\rm for}\,1\leq |\beta|_1\leq \alpha.$$ Combining , and Lemma \[lemmaA6\] (i) we get $$\label{T+est}
|T_+-T|_{\mathcal{V}_+^*,\mu;s_+} =|T\circ T^+-T|_{\mathcal{V}_+^*,\mu;s_+} \leq (1+X_+)C_1 r_+\delta_{+\mu}<r_+\chi_+ \quad {\rm for}\,0\leq \mu \leq \alpha,$$ which, together with the Cauchy inequality, implies $$\label{dT+est}
|DT_+-DT|_{\mathcal{V}_+,\mu;s_+} \leq (1+X_+)C_1 \delta_{+\mu}<\chi_+ \quad {\rm for}\,0\leq \mu \leq \alpha.$$ Similarly, we have $$\label{dpTest}
|P^{-1}DT P|_{\mathcal{V},0;s} \leq 1+X,\quad |\partial_\xi^\beta (P^{-1}DT P)|_{\mathcal{V},0;s} \leq X \quad {\rm for}\,1\leq |\beta|_1 \leq \alpha$$ and $$\label{ddpTest}
|(P^{-1}DT P)\circ T^+-P^{-1}DTP|_{\mathcal{V}_+,\mu;s_+} \leq (1+\chi_+)(1+X)C_1 \delta_{+\mu} \quad {\rm for}\,0\leq \mu \leq \alpha$$ by , and Lemma \[lemmaA6\] (i). Based on the observation $$P^{-1}(DT_+-DT)P=((P^{-1}DT P)\circ T^+-P^{-1}DTP)(P^{-1}DT^+ P)+P^{-1}DT P(P^{-1}(DT^+-E) P),$$ from , , and the Leibniz formula, it follows $$|P^{-1}(DT_+-DT)P|_{\mathcal{V}_+,\mu;s_+} \leq 2(1+X_+)C_1 \delta_{+\mu}<\chi_+$$ for $0\leq \mu \leq \alpha$ and sufficiently small $\varepsilon_0$. Thus, we have proved and with $\nu+1$.
Now, we show that $T_+$ maps $\mathcal{V}_+$ into $\mathcal{U}_+$. Noting the expression of $T_+$ in angle variable direction is independent of $\rho$, we set $T_+(\rho,\phi)={\rm col}(v_+(\rho,\phi),\Phi_+(\phi))$. In view of the induction hypotheses, implies $$|T_+-{\rm Id}|_{\mathcal{V}_+^*,0;s_+}\leq \sum_{j=1}^{\nu+1}|T_j-T_{j-1}|_{\mathcal{V}_+^*,0;s_+}\leq \sum_{j=1}^{\nu+1}r_j<r_0,$$ and implies $$\label{dT+est1}
|DT_+|_{\mathcal{V}_+,0;s_+} < 1+X_+<2.$$ Hence, the first component of $T_+$ is mapped into $\Omega^*+r_+$. For $\phi$ with $|{\rm Im} \phi|<r_+$, there exists a $\phi_0\in \mathbb{T}^{n_2}$ such that $|\phi-\phi_0|<r_+$. Therefore, the $\Phi_+$ being real analytic and imply $$|{\rm Im} \Phi_+(\phi)|=|{\rm Im} (\Phi_+(\phi)- \Phi_+(\phi_0))|\leq |DT_+|_{\mathcal{V}_+,0;s_+} |\phi-\phi_0|<2r_+.$$ Thus, for $\xi \in \Pi_+^{s_+}$, $T_+$ maps $\mathcal{V}_+$ into $\Omega^*\times \mathbb{T}^{n_2}+(2r_+,2r_+)\subset \mathcal{U}_+$, as claimed. Furthermore, let $\mathfrak{G}(\rho,\phi)=\mathfrak{D}_+(G^{\nu+2}\circ T_+-G^{\nu+1}\circ T_+)$, then , , , Lemma \[lemmaA6\] (ii) and the Cauchy inequality imply $$\label{G+est}
|\mathfrak{G}|_{\mathcal{V}_+,\alpha;s_+} \leq 2r_+^{l-\alpha}C_0M\varepsilon_0$$ and $$\label{dG+est}
|\partial_\rho\mathfrak{G}|_{\mathcal{V}_+,\alpha;s_+} \leq 2r_+^{l-\alpha-1}C_0M\varepsilon_0$$ for sufficiently small $\varepsilon_0$.
d4) Estimates of remainder terms. Denote $$W(I,\varphi)={\rm col}(u_0(\varphi)+u_1(\varphi)I,w(\varphi)), \quad A^+=A+P_1\widetilde{A}, \quad \Lambda^+=\Lambda+P_1\widetilde{\Lambda},\quad \omega^+=\omega+P_2\widetilde{\omega},$$ where $$\widetilde{A}=B({\rm diag}(B^{-1}\widehat{u_1}(0)B))B^{-1},\qquad \widetilde{\Lambda}={\rm diag}(B^{-1}\widehat{u_1}(0)B),\qquad \widetilde{\omega}=\widehat{w}(0).$$ Then the assumption (H2) implies that there is a constant $\tilde{c}_0\geq 1$ such that $$\label{A+est}
|{\rm diag}(B^{-1}\widehat{u_1}(0)B)|_{\alpha;s_+} \leq \tilde{c}_0 |u_1|_{r,\alpha;s},\qquad |\widetilde{A} |_{\alpha;s_+} \leq \tilde{c}_0 |u_1|_{r,\alpha;s}.$$ By Assumption (H1) and Condition (v.2), it is easy to see $A^+, \Lambda^+$ and $\omega^+$ satisfy (v.1) with $\nu$ replaced by $\nu+1$. We have found the transformation $T^+$ which transforms the equation , by using -, into the following one in the new variables $$\begin{aligned}
\left(\begin{array}{c} \dot \rho\\ \dot\phi\end{array}\right)& = & \left(\begin{array}{c} A^+ \rho\\ \omega^+\end{array}\right)+P \mathfrak{D}^+\left\{ P^{-1}(E-DT^+)P \left(\begin{array}{c} \tilde{A}\rho\\ \tilde{\omega}\end{array}\right)+({\rm Id}-\Gamma_{K_+})W(\rho,\phi)\right.\\
& & \left.+W\circ T^+(\rho,\phi)- W(\rho,\phi)+\left(\begin{array}{c} H_1\\ H_2\end{array}\right)\circ T^+ (\rho,\phi)\right\}+P\mathfrak{G}(\rho,\phi) +P\mathfrak{D}_+ (G-G^{\nu+2})\circ T_+(\rho,\phi),\end{aligned}$$ where $\mathfrak{D}^+=(P^{-1}DT^+P)^{-1}, \mathfrak{D}_+=(P^{-1}DT_+P)^{-1}$. We use the notation $Lf(\rho,\phi)$ to denote the linear part of a function $f$ in $\rho$, that is $$Lf(\rho,\phi)=f(0,\phi)+\partial_\rho f(0,\phi)\rho$$ and denote $\mathfrak{G}(\rho,\phi)={\rm col}(\mathfrak{G}_1(\rho,\phi),\mathfrak{G}_2(\rho,\phi))$, rewrite the above equation in the form of $$\label{eqv+}
\left(\begin{array}{c} \dot \rho\\ \dot\phi\end{array}\right)=\left(\begin{array}{c} A^+ \rho\\ \omega^+\end{array}\right)+P\left(\begin{array}{c} u_0^+(\phi)+u_1^+(\phi)\rho+H_1^+\\ w^+(\phi)+H_2^+\end{array}\right)+P\mathfrak{D}_+ (G-G^{\nu+2})\circ T_+(\rho,\phi),$$ where $$\begin{aligned}
& & w^+ = P_2^{-1}(E_{n_2}+\partial_\phi\Phi)^{-1}P_2\left[-P_2^{-1}\partial_\phi\Phi P_2 \tilde{\omega}+({\rm Id}-\Gamma_{K_+})w+w(\phi+\Phi)-w(\phi)\right. \nonumber\\
& & \qquad \left.+H_2(v_0,\phi+\Phi)+\mathfrak{G}_2(0,\phi)\right],\label{w+exp}\\
& & u_0^+ = P_1^{-1}(E_{n_1}+v_1)^{-1}P_1\left[-P_1^{-1}\partial_\phi v_0 P_2( \tilde{\omega}+w^+(\phi))+({\rm Id}-\Gamma_{K_+})u_0+u_0(\phi+\Phi)-u_0(\phi)\right. \nonumber\\
& & \qquad \left.+u_1(\phi+\Phi)v_0+H_1(v_0,\phi+\Phi)+\mathfrak{G}_1(0,\phi)\right],\label{u0+exp}\\
& & u_1^+ = P_1^{-1}(E_{n_1}+v_1)^{-1}P_1\left[-P_1^{-1}v_1P_1\widetilde{A}-P_1^{-1}\partial_\phi v_1 P_2( \tilde{\omega}+w^+(\phi))+({\rm Id}-\Gamma_{K_+})u_1\right. \nonumber\\
& & \qquad \left.+u_1(\phi+\Phi)-u_1(\phi)+u_1(\phi+\Phi)v_1+\partial_I H_1(v_0,\phi+\Phi)(E_{n_1}+v_1)\right. \nonumber\\
& & \qquad \left.+\partial_\rho\mathfrak{G}_1(0,\phi)-P_1^{-1}\partial_\phi v_0(E_{n_2}+\partial_\phi\Phi)^{-1} P_2Q \right],\label{u1+exp}\\
& & Q = \partial_I H_2(v_0,\phi+\Phi)(E_{n_1}+v_1)+\partial_\rho\mathfrak{G}_2(0,\phi),\nonumber\end{aligned}$$ $$\left(\begin{array}{c} H_1^+\\ H_2^+\end{array}\right) = \mathfrak{D}^+({\rm Id}-L)\left(\begin{array}{c} H_1\circ T^+ +\mathfrak{G}_1\\ H_2\circ T^+ +\mathfrak{G}_2 \end{array}\right)+
\left(\begin{array}{c}-P_1^{-1}(E_{n_1}+v_1)^{-1}(\partial_\phi v_1\rho) (E_{n_2}+\partial_\phi\Phi)^{-1} P_2Q\rho \\P_2^{-1}(E_{n_2}+\partial_\phi\Phi)^{-1} P_2Q\rho \end{array}\right),$$ or in another form, $$\begin{aligned}
& & H_2^+ = P_2^{-1}(E_{n_2}+\partial_\phi\Phi)^{-1}P_2\left[H_2\circ T^+(\rho,\phi)+ \mathfrak{G}_2(\rho,\phi)-H_2\circ T^+(0,\phi)-\mathfrak{G}_2(0,\phi)\right],\quad \label{H2+exp}\\
& & H_1^+ = P_1^{-1}(E_{n_1}+v_1)^{-1}P_1\left[({\rm Id}-L) (H_1\circ T^+(\rho,\phi) +\mathfrak{G}_1(\rho,\phi))\right. \nonumber \\
& & \qquad \left.+P_1^{-1}\partial_\phi v_0(E_{n_2}+\partial_\phi\Phi)^{-1} P_2Q\rho -P_1^{-1}(\partial_\phi v_0+\partial_\phi v_1\rho) P_2H_2^+\right].\label{H1+exp}\end{aligned}$$ By and , we have $$\label{Hiest}
|H_i|_{\mathcal{V},\alpha;s} \leq XC_0M\varepsilon_0+|H_i^0|_{\mathcal{U}_1,\alpha;\Pi}< \widetilde{C}_0C_0M\varepsilon_0,\quad i=1,2,$$ where $\widetilde{C}_0=2+\sum_{j\geq 1} \chi_j<\infty.$
Now, we proceed to prove (v.2) for $\nu+1$ and first estimate the three terms $u_0^+,u_1^+$ and $w^+$. We will use $C_\alpha$ to denote a constant only depending on $\alpha$. By using the Taylor expansions of $H_1$ and $H_2$, Lemma \[lemmaA6\] (ii), -, , , and the Cauchy inequality we find $$\begin{aligned}
|H_1(v_0,\phi+\Phi)|_{r_+,\alpha;s_+} &\leq & C_\alpha \widetilde{C}_0C_0M\varepsilon_0 r_+^{-2}|v_0|^2_{r_+,\alpha;s_+}\ll C_0M\varepsilon_0 r_+^{l-\alpha},\label{u0+est1}\\
|H_2(v_0,\phi+\Phi)|_{r_+,\alpha;s_+}&\leq & C_\alpha \widetilde{C}_0C_0M\varepsilon_0 r_+^{-1}|v_0|_{r_+,\alpha;s_+}\ll C_0M\varepsilon_0 r_+^{l-(\alpha+1)(\iota+2)},\label{w+est1}\\
|Q|_{r_+,\alpha;s_+} & \leq & C_\alpha \widetilde{C}_0C_0M\varepsilon_0 r_+^{-1} \label{Qest}\end{aligned}$$ and $$\label{u1+est1}
|\partial_I H_1(v_0,\phi+\Phi)(E_{n_1}+v_1) |_{r_+,\alpha;s_+}\leq C_\alpha \widetilde{C}_0C_0M\varepsilon_0 r_+^{-2}|v_0|_{r_+,\alpha;s_+}\ll C_0M\varepsilon_0 r_+^{l-(\alpha+1)(\iota+2)-1}$$ for sufficiently small $\varepsilon_0$. From the Cauchy inequality, Condition (v.2), Lemma \[lemmaA6\] (i) and , it follows $$\begin{aligned}
& & |u_0(\phi+\Phi)-u_0(\phi)|_{r_+,\alpha;s_+}\leq C_\alpha r_+^{-1}|u_0|_{r,\alpha;s}|\Phi|_{r_+,\alpha;s_+} \ll C_0M\varepsilon_0 r_+^{l-\alpha},\label{u0+est2}\\
& & |u_1(\phi+\Phi)-u_1(\phi)|_{r_+,\alpha;s_+}\leq C_\alpha r_+^{-1}|u_1|_{r,\alpha;s}|\Phi|_{r_+,\alpha;s_+}\ll C_0M\varepsilon_0 r_+^{l-(\alpha+1)(\iota+2)-1},\label{u1+est2}\\
& & |w(\phi+\Phi)-w(\phi)|_{r_+,\alpha;s_+}\leq C_\alpha r_+^{-1}|w|_{r,\alpha;s}|\Phi|_{r_+,\alpha;s_+}\ll C_0M\varepsilon_0 r_+^{l-(\alpha+1)(\iota+2)}.\label{w+est2}\end{aligned}$$ Lemma \[lemmaA5\], Condition (v.2) and the definition of $K_+$ imply $$\label{u01+est3}
|({\rm Id}-\Gamma_{K_+}) u_0|_{r_+,\alpha;s_+}\leq r_+ C_0M\varepsilon_0 r_+^{l-\alpha},\quad |({\rm Id}-\Gamma_{K_+}) u_1|_{r_+,\alpha;s_+}\ll C_0M\varepsilon_0 r_+^{l-(\alpha+1)(\iota+2)-1},$$ $$\label{w+est3}
|({\rm Id}-\Gamma_{K_+}) w|_{r_+,\alpha;s_+}\ll C_0M\varepsilon_0 r_+^{l-(\alpha+1)(\iota+2)},$$ and , , Condition (v.2) and , $$\begin{aligned}
& & |P_1^{-1}v_1P_1\widetilde{A}|_{r_+,\alpha;s_+}\leq C_\alpha |P_1^{-1}v_1P_1|_{r_+,\alpha;s_+}
|\widetilde{A}|_{\alpha;s_+}\ll C_0M\varepsilon_0 r_+^{l-(\alpha+1)(\iota+2)-1},\label{u1+est4}\\
& & |P_2^{-1}\partial_\phi\Phi P_2 \tilde{\omega}|_{r_+,\alpha;s_+}\leq C_\alpha |P_2^{-1}\partial_\phi\Phi P_2|_{r_+,\alpha;s_+}|w|_{r,\alpha;s}\ll C_0M\varepsilon_0 r_+^{l-(\alpha+1)(\iota+2)}. \label{w+est4}\end{aligned}$$ Combining the estimates , , , , and for $w^+$, we have $$\label{w+est}
|w^+|_{r_+,\alpha;s_+}\leq C_0M\varepsilon_0 r_+^{l-(\alpha+1)(\iota+2)}$$ by and choosing small $\varepsilon_0^*$. By the estimates , , , , and Condition (v.2), we also have $$\begin{aligned}
& & |P_1^{-1}\partial_\phi v_0 P_2( \tilde{\omega}+w^+)|_{r_+,\alpha;s_+}\leq C_\alpha |P_1^{-1}\partial_\phi v_0 P_2|_{r_+,\alpha;s_+}(|w|_{r,\alpha;s} + |w^+|_{r_+,\alpha;s_+}) \ll C_0M\varepsilon_0 r_+^{l-\alpha},\qquad \label{u0+est4}\\
& & |P_1^{-1}\partial_\phi v_1 P_2( \tilde{\omega}+w^+)|_{r_+,\alpha;s_+} \ll C_0M\varepsilon_0 r_+^{l-(\alpha+1)(\iota+2)-1} \label{u1+est6}\end{aligned}$$ and $$\label{u1+est5}
|P_1^{-1}\partial_\phi v_0(E_{n_2}+\partial_\phi\Phi)^{-1} P_2Q |_{r_+,\alpha;s_+}\leq C_\alpha |P_1^{-1}\partial_\phi v_0 P_2|_{r_+,\alpha;s_+} |Q|_{r_+,\alpha;s_+} \ll C_0M\varepsilon_0 r_+^{l-(\alpha+1)(\iota+2)-1}.$$ From Lemma \[lemmaA6\] (ii), Condition (v.2), and , it follows $$\begin{aligned}
& & |u_1(\phi+\Phi)v_0|_{r_+,\alpha;s_+}\leq C_\alpha |u_1(\phi+\Phi)|_{r_+,\alpha;s_+}|v_0|_{r_+,\alpha;s_+}\ll C_0M\varepsilon_0 r_+^{l-\alpha},\label{u0+est5}\\
& & |u_1(\phi+\Phi)v_1 |_{r_+,\alpha;s_+}\leq C_\alpha |u_1(\phi+\Phi)|_{r_+,\alpha;s_+} |v_1|_{r_+,\alpha;s_+} \ll C_0M\varepsilon_0 r_+^{l-(\alpha+1)(\iota+2)-1}.\label{u1+est7}\end{aligned}$$ On account of the estimates , , , , , - for $u_0^+$ and $u_1^+$, and -, and the expressions and , we can choose $\varepsilon_0^*$ so small that our estimates yield $$|u_0^+|_{r_+,\alpha;s_+}\leq 4 C_0M\varepsilon_0 r_+^{l-\alpha},\qquad
|u_1^+|_{r_+,\alpha;s_+}\leq C_0M\varepsilon_0 r_+^{l-(\alpha+1)(\iota+2)-1}.$$ To turn to the estimates of $H_1^+$ and $H_2^+$, by and we have $$\begin{aligned}
H_1^+-H_1& = & P_1^{-1}(E_{n_1}+v_1)^{-1}P_1\left[H_1\circ T^+(\rho,\phi)-H_1(\rho,\phi)-\partial_I H_1(v_0,\phi+\Phi)(E_{n_1}+v_1)\rho\right.\nonumber\\
& & \,\left.-H_1(v_0,\phi+\Phi)+\mathfrak{G}_1(\rho,\phi)-\mathfrak{G}_1(0,\phi)-\partial_\rho\mathfrak{G}_1(0,\phi)\rho\right.\nonumber\\
& & \left.+P_1^{-1}\partial_\phi v_0(E_{n_2}+\partial_\phi\Phi)^{-1} P_2Q\rho -P_1^{-1}(\partial_\phi v_0+\partial_\phi v_1\rho) P_2H_2^+-P_1^{-1}v_1P_1H_1\right] \label{H1+est}\end{aligned}$$ and $$\begin{aligned}
H_2^+-H_2& = & P_2^{-1}(E_{n_2}+\partial_\phi\Phi)^{-1}P_2\left[H_2\circ T^+(\rho,\phi)-H_2(\rho,\phi)-H_2(v_0,\phi+\Phi)\right.\nonumber \\
& & \,\left. +\mathfrak{G}_2(\rho,\phi)-\mathfrak{G}_2(0,\phi) -P_2^{-1}\partial_\phi\Phi P_2H_2(\rho,\phi) \right].\label{H2+est}\end{aligned}$$ After a short calculation, we find $$|H_1\circ T^+-H_1|_{\mathcal{V}_+,\alpha;s_+} \leq C_\alpha r_+^{-1} |H_1|_{\mathcal{V},\alpha;s}| T^+-{\rm Id}|_{\mathcal{V}_+,\alpha;s_+}\ll \chi_+C_0M\varepsilon_0$$ and $$|H_2\circ T^+-H_2|_{\mathcal{V}_+,\alpha;s_+} \ll \chi_+C_0M\varepsilon_0$$ by Lemma \[lemmaA6\] (i), the Cauchy inequality, and . It implies $$|\mathfrak{G}_1(\rho,\phi)-\mathfrak{G}_1(0,\phi)-\partial_\rho\mathfrak{G}_1(0,\phi)\rho|_{\mathcal{V}_+,\alpha;s_+} \ll \chi_+C_0M\varepsilon_0$$ and $$\quad |\mathfrak{G}_2(\rho,\phi)-\mathfrak{G}_2(0,\phi)|_{\mathcal{V}_+,\alpha;s_+} \ll \chi_+C_0M\varepsilon_0$$ by , $$|P_1^{-1}v_1P_1H_1|_{\mathcal{V}_+,\alpha;s_+} \leq C_\alpha |P_1^{-1}v_1P_1|_{r_+,\alpha;s_+} |H_1|_{\mathcal{V}_+,\alpha;s_+} \ll \chi_+C_0M\varepsilon_0$$ and $$\quad
|P_2^{-1}\partial_\phi\Phi P_2H_2|_{\mathcal{V}_+,\alpha;s_+} \ll \chi_+C_0M\varepsilon_0$$ by , and , and $$|P_1^{-1}\partial_\phi v_0(E_{n_2}+\partial_\phi\Phi)^{-1} P_2Q\rho |_{\mathcal{V}_+,\alpha;s_+} \leq C_\alpha |P_1^{-1}\partial_\phi v_0P_2|_{r_+,\alpha;s_+} |Q|_{r_+,\alpha;s_+} \ll \chi_+C_0M\varepsilon_0$$ by , and . The above estimates, , and the expression yield $$|H_2^+-H_2|_{\mathcal{V}_+,\alpha;s_+} \leq \chi_+C_0M\varepsilon_0,\qquad |H_2^+|_{\mathcal{V}_+,\alpha;s_+} \leq (X_++r_0^{l-1})C_0M\varepsilon_0 <\widetilde{C}_0C_0M\varepsilon_0,$$ which, together with and , implies $$|P_1^{-1}(\partial_\phi v_0+\partial_\phi v_1\rho) P_2H_2^+|_{\mathcal{V}_+,\alpha;s_+} \ll \chi_+C_0M\varepsilon_0.$$ Thus, the above estimates, , - and the expression also yield $$|H_1^+-H_1|_{\mathcal{V}_+,\alpha;s_+} \leq \chi_+C_0M\varepsilon_0,\qquad |H_1^+|_{\mathcal{V}_+,\alpha;s_+} \leq (X_++2r_0^{l-2})C_0M\varepsilon_0<\widetilde{C}_0C_0M\varepsilon_0.$$ Obviously, $$H_1^+=\varepsilon_0O_{\mathcal{V}_+,\alpha;s_+}(\rho^2),\qquad H_2^+=\varepsilon_0O_{\mathcal{V}_+,\alpha;s_+}(\rho).$$ This completes the proof of the Iteration Lemma. 0.4in $\blacksquare$
0.2in
5 Proof of Theorems 2 and 3 {#proof-of-theorems-2-and-3 .unnumbered}
===========================
By , (H1), (H2) and the Whitney extension theorem (see Lemma \[lemmaA.1\]), still using $\omega^\nu$ and $\Lambda^\nu$ to denote their extensions, we have $$||\Lambda^\nu||_{\alpha;\Pi}\leq 2c_1\varepsilon^{q_3}\leq 2c_1\varepsilon^{q_5},$$ and also $$\max_{0\leq \mu \leq \alpha}\parallel D^{\mu}\langle b,\omega^\nu(\xi,\varepsilon)\rangle\parallel \geq \frac{c_2}{2}\varepsilon^{q_5} \qquad {\rm in \, Case}\,\, n_{22}=0,$$ $$\label{nondeg2}
\max_{1\leq \mu \leq \alpha}\parallel D^{\mu}\langle b,\omega^\nu(\xi,\varepsilon)\rangle\parallel \geq \frac{c_2}{2}\varepsilon^{q_5} \qquad {\rm in \, Case}\,\, n_{22}\neq 0$$ by and , for sufficiently small $\varepsilon$, all $\xi\in \Pi, b\in \mathcal{S}_{n_2,1}, \nu=0,1,2,\cdots. $
Set $$f_{km}^\nu(\xi)=\langle k,\omega^\nu(\xi,\varepsilon)\rangle+\langle m,{\rm Im}\Lambda^\nu(\xi,\varepsilon)\rangle$$ for $0\neq k\in \mathbb{Z}^{n_2}, K_\nu<|k|_2\leq K_{\nu+1}; m={\rm col}(m_1,\cdots,m_{n_1})\in \mathbb{Z}^{n_1}, |m|_1\leq 2$ and $m_1+\cdots+m_{n_1}=0$ or $-1$, $\nu=0,1,2,\cdots $. Here ${\rm Im}\Lambda^\nu$ is the imaginary part of $\Lambda^\nu$. Then $$\mathfrak{R}_{km}^\nu(\gamma)\subset \left\{\xi\in \Pi_0:\, \mid f_{km}^{\nu-1}(\xi)\mid<\gamma \varepsilon^{q_5}|k|_2^{-\iota}\right\},\quad \nu=1,2,\cdots,$$ where $\Pi_0$ is the closed subset of $\Pi$ defined in Theorem 1.
\[lemma5.1\] If $|k|_2\geq \frac{16}{c_2}c_1|m|_1n_3^{\alpha/2}$, $0\neq k\in \mathbb{Z}^{n_2}$. Then $$\label{measest1}
{\rm meas}\mathfrak{R}_{km}^\nu(\gamma)\leq c_5({\rm diam}\Pi_0)^{n_3-1}(\gamma |k|_2^{-\iota-1})^{\frac{1}{\alpha}}$$ for some positive constant $c_5$, where ${\rm diam}\Pi_0$ represents the diameter of $\Pi_0$.
[**Proof**]{} We only give the proof for the case $n_{22}\neq 0$, the proof of the case $n_{22}=0$ is analogous and is omitted.
Due to the continuity of the derivatives and the compactness of $\Pi_0$ and $\mathcal{S}_{n_2,1}$, the non-degenerate condition implies that there exist finite covers $\{\Pi^i\}_{i=1}^{i_0}$ and $\{\mathcal{S}^j\}_{j=1}^{j_0}$ of $\Pi_0$ and $\mathcal{S}_{n_2,1}$, respectively, and $\mu_{ij}: 1\leq \mu_{ij}\leq \alpha, i=1,\cdots, i_0; j=1,\cdots, j_0$, $\Pi^i$ is chosen to be convex, such that $$\parallel D^{\mu_{ij}}\langle b,\omega^\nu(\xi,\varepsilon)\rangle\parallel \geq \frac{c_2}{4}\varepsilon^{q_5} \qquad {\rm for \, all}\,\,\xi\in\Pi^i, b\in\mathcal{S}^j.$$ Hence, for $ 0\neq k\in \mathbb{Z}^{n_2}, \frac{k}{|k|_2}\in \mathcal{S}^{j_k}$, we have $$\label{nondeg3}
\parallel D^{\mu_{ij_k}}f_{km}^\nu(\xi)\parallel\geq |k|_2\parallel D^{\mu_{ij_k}}\langle \frac{k}{|k|_2},\omega^\nu(\xi,\varepsilon)\rangle\parallel -2c_1|m|_1 \varepsilon^{q_5} n_3^{\frac{\mu_{ij_k}}{2}}\geq \frac{c_2}{8}\varepsilon^{q_5}|k|_2$$ for all $\xi\in \Pi^i, i=1,\cdots, i_0$, admitted $m$ and $\nu$ if $|k|_2\geq \frac{16}{c_2}c_1|m|_1n_3^{\alpha/2}$.
Now we estimate the measure of $\mathfrak{R}_{km}^\nu(\gamma)\bigcap \Pi^i$. It follows by and the definition of the norm (see Lemma \[lemmaA.3\]) that there is a vector $a\in\mathcal{S}_{n_3,1}$ such that $$\label{nondeg4}
\mid D^{\mu_{ij_k}}f_{km}^\nu(\xi)a^{\otimes\mu_{ij_k}}\mid \geq \frac{c_2}{8}\varepsilon^{q_5}|k|_2 \qquad {\rm for \, all}\,\,\xi\in\Pi^i.$$ Write $\xi=at+\zeta$ with $t\in \mathbb{R}, \zeta\in a^\perp$ and let $f(t)=f_{km}^\nu(at+\zeta)$, $I_{\zeta}=\{t\in \mathbb{R}:\, at+\zeta\in \Pi^i\}$. The inequality means $$\left|\frac{{\rm d}^{\mu_{ij_k}}f(t)}{{\rm d}t^{\mu_{ij_k}}}\right|\geq \frac{c_2}{8}\varepsilon^{q_5}|k|_2 \qquad {\rm for \, all}\,\,t\in I_{\zeta}.$$ By Fubini’s theorem and Lemma \[lemmaA.2\], it implies $${\rm meas}\left(\mathfrak{R}_{km}^\nu(\gamma)\bigcap \Pi^i\right)\leq c_6({\rm diam}\Pi_0)^{n_3-1}(\gamma |k|_2^{-\iota-1})^{\frac{1}{\alpha}},$$ where $c_6=4\max\{1,(4\alpha !/c_2)^{1/\alpha}\}$. Therefore $${\rm meas}\mathfrak{R}_{km}^\nu(\gamma)\leq \sum_{i=1}^{i_0}{\rm meas}\left(\mathfrak{R}_{km}^\nu(\gamma)\bigcap \Pi^i\right)\leq i_0 c_6({\rm diam}\Pi_0)^{n_3-1}(\gamma |k|_2^{-\iota-1})^{\frac{1}{\alpha}}.$$ The estimate is proved by setting $c_5=i_0c_6$. 0.4in $\blacksquare$ 0.2in Now, let $$\mathfrak{K}=\{(k,m)\in \mathbb{Z}^{n_2}\times \mathbb{Z}^{n_1}:\, 0<|k|_2<K^*, 1\leq |m|_1\leq 2, m_1+\cdots+m_{n_1}=0 \,{\rm or} \,-1\}.$$ By an analogous proof to Lemma \[lemma5.1\], we also have
\[lemma5.2\] If $(k,m)\in \mathfrak{K}$ and the condition (ii)’ in Theorem 3 holds, then there is a constant $c_7>0$ such that $$\label{measest2}
{\rm meas}\mathfrak{R}_{km}^\nu(\gamma)\leq c_7({\rm diam}\Pi_0)^{n_3-1}(\gamma |k|_2^{-\iota})^{\frac{1}{\alpha}}.$$
0.2in [**Remark 5.1**]{} If $q_3>q_5\geq 0$, then without the conditions (ii) and (ii)’, we can obtain $${\rm meas}\mathfrak{R}_{km}^\nu(\gamma)\leq c_5({\rm diam}\Pi_0)^{n_3-1}(2\gamma |k|_2^{-\iota-1})^{\frac{1}{\alpha}}\quad {\rm for\, all}\,\ (k,m)\in \mathfrak{K}.$$
In fact, for sufficiently small $\gamma$ (equivalently, sufficiently small $\varepsilon$), we have $$\mathfrak{R}_{km}^\nu(\gamma)\subset \left\{\xi\in \Pi_0:\, |\langle k, \omega^\nu(\xi,\varepsilon)\rangle|< 2\gamma \varepsilon^{q_5}|k|_2^{-\iota}\right\}$$ for all $(k,m)\in \mathfrak{K}$. From Lemma \[lemma5.1\] with $|m|_1=0$, it follows $$\begin{aligned}
{\rm meas}\mathfrak{R}_{km}^\nu(\gamma)&\leq & {\rm meas} \left\{\xi\in \Pi_0:\, |\langle k, \omega^\nu(\xi,\varepsilon)\rangle|< 2\gamma \varepsilon^{q_5}|k|_2^{-\iota}\right\}\\
&\leq & c_5({\rm diam}\Pi_0)^{n_3-1}(2\gamma |k|_2^{-\iota-1})^{\frac{1}{\alpha}}.\end{aligned}$$
0.2in [**Proof of Theorem 3**]{} By Lemmas \[lemma5.1\] and \[lemma5.2\], we obtain $$\begin{aligned}
{\rm meas}(\Pi_0\setminus \Pi_{\gamma})&\leq & \sum_{\nu=1}^{\infty}\sum_{K_{\nu-1}<|k|_2\leq K_\nu}\left({\rm meas}\mathfrak{R}_{k0}^\nu +\sum_{|m|_1=1} {\rm meas}\mathfrak{R}_{km}^\nu + \sum_{|m|_1=2} {\rm meas}\mathfrak{R}_{km}^\nu\right)\\
& \leq & \sum_{(k,m)\in \mathfrak{K}} {\rm meas}\mathfrak{R}_{km}^\nu + c_5({\rm diam}\Pi_0)^{n_3-1} \gamma^{\frac{1}{\alpha}} \left[ \sum_{0\neq k\in \mathbb{Z}^{n_2}} |k|_2^{-\frac{\iota+1}{\alpha}}\right.\\
& & + \left. n_1 \sum_{|k|_2\geq K^*} |k|_2^{-\frac{\iota+1}{\alpha}} + n_1(n_1-1) \sum_{|k|_2\geq K^*} |k|_2^{-\frac{\iota+1}{\alpha}}\right]\\
& \leq & c_{10}\gamma^{\frac{1}{\alpha}},\end{aligned}$$ where $c_{10}$ is a positive constant depending on $n_1, n_2, {\rm diam}\Pi, \omega_i$ and $\Lambda_i (i=1,2)$, and ${\rm meas}(\Pi\setminus \Pi_0)=O(\gamma)$. This proves Theorem 3. 0.4in $\blacksquare$
0.2in [**Proof of Theorem 2**]{} From the proof of Theorem 3, it is easy to see $${\rm meas}(\bigcup_{\nu=1}^{\infty}\bigcup_{(k,m)\notin \mathfrak{K}} \mathfrak{R}_{km}^\nu(\gamma))\leq c_{10}\gamma^{\frac{1}{\alpha}}\rightarrow 0 \qquad {\rm as}\, \gamma \rightarrow 0.$$ On the other hand, there is a $\nu_0$ such that $K_{\nu_0}\leq K^*$, hence $$\bigcup_{\nu=1}^{\infty}\bigcup_{(k,m)\in \mathfrak{K}} \mathfrak{R}_{km}^\nu(\gamma)\subset \bigcup_{\nu=1}^{\nu_0}\bigcup_{(k,m)\in \mathfrak{K}} \mathfrak{R}_{km}^\nu(\gamma).$$ By the condition (ii) in Theorem 2, the boundedness of $\Pi$, and , we have ${\rm meas}\mathfrak{R}_{km}^\nu(\gamma)\rightarrow 0$ as $\gamma\rightarrow 0$. Since $ \mathfrak{K}$ is finite, we also have $${\rm meas}(\bigcup_{\nu=1}^{\nu_0}\bigcup_{(k,m)\in \mathfrak{K}} \mathfrak{R}_{km}^\nu(\gamma))\rightarrow 0\qquad {\rm as}\, \gamma\rightarrow 0. \hskip 0.4in \blacksquare$$
0.2in
Appendix {#appendix .unnumbered}
=========
0.1in
A.1. Whitney extension theorem {#a.1.-whitney-extension-theorem .unnumbered}
-------------------------------
0.1in Let $\overline{\Omega}\subset \mathbb{R}^n$ be a closed set, $p$ be a non-negative integer, $p<l\leq p+1$. $C_W^l(\overline{\Omega})$ is the class of all collections $f=\{f^{(k)}\}_{|k|_1\leq p}$ of functions defined on $\overline{\Omega}$ which satisfy, for some finite $M$, $$\label{whitineq}
|f^{(k)}(x)|\leq M,\qquad |f^{(k)}(x)-P_k(x,y)|\leq M |x-y|^{l-|k|_1}$$ for all $x,y\in \overline{\Omega}$ and $|k|_1\leq p$, where $$P_k(x,y)=\sum_{|k+j|_1\leq p}\frac{1}{j!}f^{(k+j)}(y)(x-y)^j$$ is the analogue of the $k$-th Taylor polynomial. $f$ is called $C^l$ Whitney in $\overline{\Omega}$ with Whitney derivatives $D^k f=f^{(k)}$ for $|k|_1\leq p$. Define a norm $$||f||_{C_W^l(\overline{\Omega})}=\inf M$$ is the smallest $M$ for which both inequalities in hold. Then $C_W^l(\overline{\Omega})$ with the norm is a Banach space.
The following extension theorem indicates that a Whitney differentiable function has an extension to $\mathbb{R}^n$ which is differentiable in the standard sense.
\[lemmaA.1\] (Whitney extension theorem, [@Whit34; @Stein70; @Pos82]) Let $\overline{\Omega}$ be a closed set in $\mathbb{R}^n$, $p\in \mathbb{Z}_+$ and $p<l\leq p+1$. Then there exists a linear extension operator $$\mathfrak{E}: C_W^l(\overline{\Omega})\rightarrow C^l(\mathbb{R}^n), \qquad f=\{f^{(k)}\}_{|k|_1\leq p}\rightarrow F=\mathfrak{E}f$$ such that $$D^kF\mid_{\overline{\Omega}}=f^{(k)}, \qquad |k|_1\leq p$$ and $$||F||_{l;\mathbb{R}^n}\leq C ||f||_{C_W^l(\overline{\Omega})},$$ where the constant $C$ depends only on $l$ and the dimension $n$, but not on $\overline{\Omega}$. Moreover, if $\overline{\Omega}=\overline{\Omega}_1\times \mathbb{T}^{n_2}\subset \mathbb{R}^{n_1}\times \mathbb{T}^{n_2}$, then the extension can be chosen to be defined on $\mathbb{R}^{n_1}\times \mathbb{T}^{n_2}$, so that the periodicity is preserved.
0.2in
A.2. Measure estimate lemmas {#a.2.-measure-estimate-lemmas .unnumbered}
----------------------------
0.1in
\[lemmaA.2\] [@Russ01] Let $f:[a,b]\rightarrow \mathbb{R}$ with $a<b$ be an $\alpha$-times continuously differentiable function satisfying $$\left| \frac{{\rm d}^\alpha f(x)}{{\rm d}x^\alpha}\right| \geq c, \qquad x\in [a,b]$$ for some $\alpha \in \mathbb{N}$ and a constant $c>0$. Then we have the measure estimate $${\rm meas}\{x\in [a,b]:\, |f(x)|\leq \varepsilon\}\leq 4\left(\alpha !\frac{\varepsilon}{2c}\right)^{\frac{1}{\alpha}} \quad {\rm for\, all}\,\,\varepsilon>0.$$
\[lemmaA.3\] Let $\Pi\subset \mathbb{R}^p$ be a bounded closed set, $f_j: \Pi\rightarrow \mathbb{R}$ be of $C^{\alpha}$ on $\Pi$ with a positive integer $\alpha, j=1,\cdots,q$. Denote $f(\xi)={\rm col}(f_1(\xi),\cdots,f_q(\xi))$. Assume for $\xi\in \Pi$, $$\label{rankcon}
{\rm rank}\left( f(\xi), \frac{\partial^\beta f(\xi)}{\partial \xi^\beta}:\, 1\leq |\beta|_1\leq \alpha\right)=q \quad {\rm and} \quad 1+p+p^2+\cdots+p^{\alpha}\geq q.$$ Then there is a constant $c>0$ such that $$\max_{0\leq \mu\leq \alpha}||D^\mu\langle b, f(\xi)\rangle||\geq c \qquad {\rm for\, all}\,\,b\in \mathcal{S}_{q,1},\, \xi\in\Pi.$$ Here $D$ represents the differential operator with respect to the variable $\xi$, $$\mathcal{S}_{q,1}=\{b\in \mathbb{R}^q:\, |b|_2=1\}, \quad ||D^\mu\langle b, f(\xi)\rangle||=\max_{a\in\mathcal{S}_{p,1}} |D^\mu\langle b, f(\xi)\rangle a^{\otimes\mu}|,$$ $a^{\otimes\mu}=(a_1,a_2,\cdots,a_\mu)$ with $a_i=a, i=1,2,\cdots,\mu$.
[**Remark A.1**]{} Here, by the Whitney extension theorem we assume the continuous differentiability of a function $f$ with respect to the parameter variable $\xi$ on a closed set $\Pi$ means that $f$ is continuously differentiable in some neighbourhood of $\Pi$.
[**Proof**]{} Suppose such a constant $c$ does not exist. Then for any positive integer $n$, we can find $\xi_n\in\Pi$ and $b_n\in \mathcal{S}_{q,1}$ satisfying $$\max_{0\leq \mu\leq \alpha}||D^\mu\langle b_n, f(\xi_n)\rangle||<\frac{1}{n},\qquad n=1,2,\cdots.$$ Based on the compactness of $\Pi$ and $\mathcal{S}_{q,1}$ there are convergent subsequences of $\{b_n\}$ and $\{\xi_n\}$, respectively, still denoting by $\{b_n\}$ and $\{\xi_n\}$, such that $b_n\rightarrow b_0\in \mathcal{S}_{q,1}, \xi_n\rightarrow \xi_0\in \Pi$ as $n\rightarrow \infty$. Thus, the continuity of the derivatives implies $$||D^\mu\langle b_0, f(\xi_0)\rangle||=0 \qquad {\rm for\, all}\,\,0\leq \mu\leq \alpha.$$ Noting that $$|D^\mu\langle b_0, f(\xi_0)\rangle (a_1,\cdots,a_\mu)|\leq \frac{\mu^\mu}{\mu !}||D^\mu\langle b_0, f(\xi_0)\rangle||$$ for all $a_i\in \mathcal{S}_{p,1},i=1,\cdots,\mu$, we have $$b_0^T\left( f(\xi_0), \frac{\partial^\beta f(\xi_0)}{\partial \xi^\beta}:\, 1\leq |\beta|_1\leq \alpha\right)=0,$$ which implies $${\rm rank}\left( f(\xi_0), \frac{\partial^\beta f(\xi_0)}{\partial \xi^\beta}:\, 1\leq |\beta|_1\leq \alpha\right)<q$$ being in contradiction with the condition . The lemma is proved. 0.4in $\blacksquare$
[**Remark A.2**]{} From the proof of Lemma \[lemmaA.3\], it is easy to see that if the condition is replaced by $${\rm rank}\left( \frac{\partial^\beta f(\xi)}{\partial \xi^\beta}:\, 1\leq |\beta|_1\leq \alpha\right)=q \qquad {\rm and} \quad p+p^2+\cdots+p^\alpha\geq q,$$ then we also have $$\max_{1\leq \mu\leq \alpha}||D^\mu\langle b, f(\xi)\rangle||\geq c \qquad {\rm for\, all}\,\,b\in \mathcal{S}_{q,1},\, \xi\in\Pi.$$
0.2in
A.3. Properties of analytic smoothing operator {#a.3.-properties-of-analytic-smoothing-operator .unnumbered}
----------------------------------------------
0.1in Let $l>0, m\in \mathbb{N}$ and $C^l(\mathbb{R}^m)$ be the Hölder space defined in Definition \[def1\] without parameter variables, $u_0\in C_0^\infty(\mathbb{R})$ be an even function, vanishing outside the interval $[-1,1]$ and identically equal to 1 in a neighbourhood of 0, $u(x)=u_0(|x|_2^2)$ for $x\in \mathbb{R}^m$ and $$\tilde{u}(z)=\int_{\mathbb{R}^m} u(x)e^{\sqrt{-1}\langle z,x\rangle}dx \qquad {\rm for} \,\, z\in \mathbb{C}^m,$$ $$f_r(x)\coloneqq (\mathcal{S}_r f)(x)\coloneqq r^{-m}\int_{\mathbb{R}^m} \tilde{u}((x-y)/r)f(y)dy$$ for $x\in \mathbb{C}^m$ and $r\in (0,1]$.
\[lemmaA4\] The following assertions are valid
[(a)]{} $\int_{\mathbb{R}^m} \tilde{u}(x) dx=u(0)=1;$
[(b)]{} $(\mathcal{S}_r f)(x)=\int_{\mathbb{R}^m} \tilde{u}(y)f(x-ry)dy$ for $x\in \mathbb{R}^m$;
[(c)]{} $\int_{\mathbb{R}^m} x^k \tilde{u}(x) dx=0$ for $0\neq k\in \mathbb{Z}_+^m$;
[(d)]{} for any $p\in \mathbb{N}$, there is a constant $C_p>0$ such that $$\left|D^k\tilde{u}(z)\right|\leq \frac{C_p}{(1+|z|_2)^p} e^{|{\rm Im}z|_2} \qquad {\rm for \, all}\,\, |k|_1\leq p, k\in \mathbb{Z}_+^m,$$ 0.4in where $D^k=D_1^{k_1}\circ D_2^{k_2}\circ \cdots \circ D_{m}^{k_m}$, and $D_j^{k_j}=\frac{\partial^{k_j}}{\partial x_j^{k_j}}$;
[(e)]{} if $P$ is a polynomial, then $(\mathcal{S}_r P)(x)=P(x)$;
[(f)]{} there exists a constant $C_l>0$ such that $$\left|D^kf_r(x)-\sum_{|\beta|_1\leq l-|k|_1} D^{k+\beta}f({\rm Re} x)\frac{(\sqrt{-1}{\rm Im} x)^\beta}{\beta!}\right|\leq C_l r^{l-|k|_1}||f||_{l;\mathbb{R}^m},\quad |{\rm Im } x|\leq r \leq 1$$ for all $k\in \mathbb{Z}_+^m$ with $|k|_1\leq l$. In particular, for $x\in \mathbb{R}^m$ and $p\in \mathbb{Z}_+$, $$||f_r-f||_{p;\mathbb{R}^m}\leq C_{lp} r^{l-p}||f||_{l;\mathbb{R}^m},\qquad p\leq l$$ for a suitable constant $C_{lp}$ depending on $l,p$ and $m$.
[**Proof**]{} The definitions of $\tilde{u}$ and $\mathcal{S}_r$ imply (a) and (b), respectively. Noting that the $\tilde{u}$ is a Schwartz function (see (d)), and the Fourier transformation and differentiation can be exchanged, we have $$\int_{\mathbb{R}^m} x^k \tilde{u}(x) dx=\left.(\sqrt{-1})^{|k|_1}D_y^k\int_{\mathbb{R}^m} \tilde{u}(x)e^{-\sqrt{-1}\langle y,x\rangle} dx\right|_{y=0}= \left.(\sqrt{-1})^{|k|_1}D^ku(y)\right|_{y=0}=0,$$ which verifies (c). See Lemma 9, Proposition 8 and Remark 15 (i) in [@Chier03] for (d), (e) and (f), respectively, also see the proof of Lemma 2.1 in Part I of [@Zehn75] for (e). 0.4in $\blacksquare$
\[lemmaA5\] Let $K$ be a positive integer and $f$ be a bounded and analytic function in the strip $\{x:\,|{\rm Im} x|<r\}$ of $\mathbb{T}^n$, $f(x)=\sum_{k\in \mathbb{Z}^n} \hat{f}(k) e^{\sqrt{-1}\langle k,x\rangle}$. Define the truncation operator $\Gamma_K$ as follows $$\Gamma_K f=\sum_{|k|_2\leq K}\hat f(k)e^{\sqrt{-1}\langle k,x\rangle}.$$ If $K>(2\rho)^{-1}$, then we have $$|({\rm Id}-\Gamma_K)f|_{r-2\rho} \leq C(n) |f|_r\rho^{-n}e^{-\rho K},\qquad 0<2\rho \leq r,$$ where $C(n)=6(n!)n^ne^{-n}$.
[**Proof**]{} Set $\sigma=2\rho$. Based on the fact that the number of all $k$ with $|k|_1=m$ is bounded by $2nm^{n-1}$, we have $$\label{trun1}
|({\rm Id}-\Gamma_K)f|_{r-\sigma} \leq \sum_{|k|_2> K}|f|_r e^{-\sigma |k|_1} \leq \sum_{|k|_1> K}|f|_r e^{-\sigma |k|_1} \leq |f|_r\sum_{m> K}2nm^{n-1} e^{-\sigma m}.$$ Here we use Lemma A.1 in [@Poschel01]. Since the function $y^{n-1}e^{-\sigma y}$ is monotonically decreasing in the interval $[\frac{n-1}{\sigma},+\infty)$ and $K>\sigma^{-1}$, therefore, $$\begin{aligned}
\sum_{m> K}m^{n-1} e^{-\sigma m}<\int_{K}^{+\infty}y^{n-1}e^{-\sigma y}dy & = & \left(\frac{1}{\sigma}K^{n-1}+\frac{n-1}{\sigma^2}K^{n-2}+\cdots +\frac{(n-1)!}{\sigma^n}\right)e^{-\sigma K}\\
& < & 3(n-1)!K^ne^{-\sigma K}.\end{aligned}$$ Hence, by we obtain $$\label{trun2}
|({\rm Id}-\Gamma_K)f|_{r-\sigma} \leq 6 n! |f|_rK^n e^{-\sigma K}.$$ Noting that the maximum of the function $y^ne^{-y}$ on the interval $(0,+\infty)$ is $n^ne^{-n}$, implies $$|({\rm Id}-\Gamma_K)f|_{r-2\rho} \leq 6(n!)n^ne^{-n} |f|_r\rho^{-n}e^{-\rho K}. \hskip 0.4in \blacksquare$$ 0.2in
Let $\Omega_1$ and $\Omega_2$ be domains in $\mathbb{C}^n$, $\Pi$ be an open set in $\mathbb{R}^m$, $f(x,\xi)$ and $g(x,\xi)$ be analytic in $x\in (\Omega_1+r)$ and in $x\in \Omega_2$ respectively, and continuously differential up to order $\alpha$ in $\xi\in \Pi$, $g:\, \Omega_2\times \Pi\rightarrow \Omega_1$, where $r>0,\, \Omega_1+r=\{x\in \mathbb{C}^n:\, {\rm dist}(x,\Omega_1)<r\}$.
We introduce the notation for $1\leq \mu \leq \alpha$, $$|Df|_\mu \coloneqq \max_{1\leq |\beta|_1\leq \mu}\left|\partial_\xi^\beta Df\right|_{\Omega_1+r, 0; \Pi}, \qquad |g|_\mu \coloneqq \max_{1\leq |\beta|_1\leq \mu}\left|\partial_\xi^\beta g\right|_{\Omega_2, 0; \Pi},$$ where $Df$ represents the differential operator with respect to the coordinate variable $x$. Using the Chain Rule on differentiation of a composition of mappings and Cauchy inequality, we easily prove the following lemma.
\[lemmaA6\] Let $\beta \in \mathbb{Z}_+^m$ and $|\beta|_1=\mu, 1\leq \mu \leq \alpha$. Then
[(i)]{} $\left|\partial_\xi^\beta (f\circ g-f)\right|_{\Omega_2, 0; \Pi}\leq
\left\{\begin{array}{l}\left|(\partial_\xi^\beta f)\circ g-\partial_\xi ^\beta f\right|_{\Omega_2, 0; \Pi}+| Df|_{\Omega_1+r, 0; \Pi}|g|_1 \quad {\rm for} \quad \mu=1\\
\left|(\partial_\xi^\beta f)\circ g-\partial_\xi ^\beta f\right|_{\Omega_2, 0; \Pi}+| Df|_{\Omega_1+r, 0; \Pi}(|g|_\mu+\frac{(\mu-1)!}{r^{\mu-1}}|g|_1^\mu)\\
\,\, +\sum_{j=1}^{\mu-1}\frac{C_j}{r^{j-1}}|Df|_{\mu-1}|g|_{\mu-1}^j \qquad {\rm for } \quad 2\leq \mu \leq \alpha;
\end{array}\right.$
[(ii)]{} $\left|\partial_\xi^\beta (f\circ g)\right|_{\Omega_2, 0; \Pi}\leq \left|(\partial_\xi^\beta f)\circ g\right|_{\Omega_2, 0; \Pi}
+\sum_{j=1}^{\mu}\frac{C_j}{r^{j}}|f|_{\mu-1}|g|_{\mu}^j$,\
where $C_j(j=1,\cdots \mu)$ are nonnegative constants only depending on $\beta$.
0.2in
A.4. An estimate lemma for small divisors {#a.4.-an-estimate-lemma-for-small-divisors .unnumbered}
------------------------------------------
0.1in
\[lemmaA7\] Assume the frequency vector $\omega=(\omega_1,\cdots, \omega_n)$ satisfies the inequalities $$\label{smdivcon}
|\langle k, \omega\rangle|\geq \frac{\gamma}{|k|_2^\tau} \qquad {\rm and} \qquad |\langle k, \omega\rangle+ \lambda|\geq \frac{\gamma}{|k|_2^\tau}$$ for all integer vectors $0\neq k \in \mathbb{Z}^n$ with $|k|_2\leq K \leq \infty$, and some constants $\tau>n-1 \geq 1$, $K>0, \gamma>0$ and $\lambda \in \mathbb{R}$. Then the following inequalities hold $$\label{smdivest0}
\sum_{0\neq |k|_2\leq K} |k|_1^v|\langle k, \omega\rangle|^{-b}e^{-\sigma |k|_1}\leq C \gamma^{-b}\sigma^{-(\tau b+v+1)}$$ and $$\label{smdivest}
\sum_{0\neq |k|_2\leq K} |k|_1^v|\langle k, \omega\rangle + \lambda|^{-b}e^{-\sigma |k|_1}\leq C \gamma^{-b}\sigma^{-(\tau b+v+1)}$$ with $$C=15 \tau \sqrt{\tau b +v} 2^{2(n+b)-3} n^{\tau b +v+1}(\tau b -n+1)^{-1}\left(\frac{\tau b +v}{e}\right)^{\tau b+v},$$ where $v\geq 0, b\geq 1$ and $\sigma\in (0,1)$ are constants.
[**Proof**]{} The proof is based on the fact that only a few of the denominators $\langle k, \omega\rangle$ and $\langle k, \omega\rangle+ \lambda$ are small, which was used by Siegel [@Sieg], Arnol’d [@Arn63] and Moser [@Mos66]. For the sake of completeness, we present the proof for our situation and make the involved constants explicit in estimates. We only prove the inequality with $K=\infty$. The proof of and the case $K<\infty$ is analogous and is omitted. Set $$K(m,j)=\{k=(k_1,\cdots,k_n) \in \mathbb{Z}^n:\, |k|\coloneqq \max_{1\leq i \leq n}|k_i|=m,\, \gamma^{-1}2^j<|\langle k, \omega\rangle+ \lambda|^{-1}\leq \gamma^{-1}2^{j+1}\}$$ and let $K(m,j)^\#$ denote the number of points in $K(m,j)$. Then we have $$\label{Kindex}
K(m,j)^\#\leq (2n)^n(2m)^{n-1}2^{-\frac{1}{\tau}(n-1)(j-1)}.$$ In fact, if $k,k'\in K(m,j)$ are different points, then $$\gamma |k-k'|_2^{-\tau}\leq |\langle k-k', \omega\rangle|\leq |\langle k, \omega\rangle+ \lambda|+|\langle k', \omega\rangle+ \lambda|< \gamma 2^{1-j},$$ which implies $$|k-k'|\geq n^{-\frac{1}{2}}|k-k'|_2 > n^{-1}2^{\frac{j-1}{\tau}}\coloneqq 2\rho_j.$$ Noting $|k-k'|\leq 2m$ we get $\rho_j\leq m$. If we encircle every point $k\in K(m,j)$ by a cube $\mathfrak{C}_k:\, |x-k|\leq \rho_j$, then these cubes are mutually disjoint. The intersections of these cubes $\mathfrak{C}_k$ with the curved surface $|x|=m$ are disjoint $n-1$ dimensional sets with $n-1$ dimensional volume $\geq \rho_j^{n-1}$. As the $n-1$ dimensional volume of the curved surface $|x|=m$ is $2n(2m)^{n-1}$, we obtain $$K(m,j)^\#\leq \frac{2n(2m)^{n-1}}{\rho_j^{n-1}},$$ which verifies the inequality . Thus we have $$\sum_{K(m,j)}|\langle k, \omega\rangle + \lambda|^{-b}\leq \gamma^{-b}2^{b(j+1)}K(m,j)^\#\leq 2^{2(n+b)-1}n^n m^{n-1}\gamma^{-b}2^{(b-\frac{n-1}{\tau})(j-1)}\coloneqq C_1 2^{(b-\frac{n-1}{\tau})(j-1)}.$$ Let $j^*$ be the greatest occurring $j$ for which $K(m,j)\neq \emptyset$. Then the facts that $$\gamma^{-1}2^{j^*}<|\langle k, \omega\rangle+ \lambda|^{-1}\leq \gamma^{-1}|k|_2^\tau \leq \gamma^{-1}(nm)^\tau$$ and $$\{ k \in \mathbb{Z}^n:\, |k|=m\}^\#=(2m+1)^n-(2m-1)^n<2n(4m)^{n-1}$$ imply $$\begin{aligned}
\sum_{|k|=m}|\langle k, \omega\rangle + \lambda|^{-b} & \leq & \sum_{|k|=m,|\langle k, \omega\rangle + \lambda|^{-1}\leq 2\gamma^{-1}}|\langle k, \omega\rangle + \lambda|^{-b}+\sum_{j=1}^{j^*}\sum_{K(m,j)}|\langle k, \omega\rangle + \lambda|^{-b}\\
& \leq & n 2^{2n+b-1}m^{n-1}\gamma^{-b}+C_1\sum_{j=1}^{j^*}2^{(b-\frac{n-1}{\tau})(j-1)}\\
& \leq & \tau 2^{2(n+b)-1}n^{\tau b +1}(\tau b -n+1)^{-1}\gamma^{-b}m^{\tau b}\coloneqq C_2m^{\tau b}.\end{aligned}$$ Therefore, $$\begin{aligned}
& & \sum_{0\neq k\in \mathbb{Z}^n} |k|_1^v|\langle k, \omega\rangle + \lambda|^{-b}e^{-\sigma |k|_1} \leq \sum_{m=1}^\infty \sum_{|k|=m} (nm)^v|\langle k, \omega\rangle + \lambda|^{-b}e^{-\sigma m}\nonumber \\
& & \qquad \leq C_2n^v\sum_{m=1}^\infty m^{\tau b+v}e^{-\sigma m}. \label{sum1}\end{aligned}$$ Noting that the function $g(x)=x^{\tau b+v}e^{-\sigma x}$ on the interval $[1,\infty)$ gets its maximum at $x_0=\frac{\tau b +v}{\sigma}$, moreover is strictly increasing and decreasing on $[1,x_0)$ and $(x_0,\infty)$, respectively. Denote the integer part of $\frac{\tau b +v}{\sigma}$ by $m_0$. Then $m_0\geq 1$ and $$\begin{aligned}
\sum_{m=1}^\infty m^{\tau b+v}e^{-\sigma m} & \leq & \int_1^{m_0}x^{\tau b+v}e^{-\sigma x}dx+ g(\frac{\tau b +v}{\sigma}) +\int_{m_0}^\infty x^{\tau b+v}e^{-\sigma x}dx \nonumber\\
& \leq & g(\frac{\tau b +v}{\sigma}) + \sigma^{-(\tau b +v+1)}\int_{0}^\infty y^{\tau b+v}e^{-y}dy \nonumber\\
& = & \left(\frac{\tau b+v}{e\sigma}\right)^{\tau b +v}+\sigma^{-(\tau b +v+1)}\Gamma(\tau b +v+1).\label{sum2}\end{aligned}$$ By the Stirling formula of the gamma function, we have $$\label{gammaest}
\Gamma(\tau b +v+1)<\frac{11}{4}\sqrt{\tau b +v}\left(\frac{\tau b+v}{e}\right)^{\tau b +v}.$$ Combining -, we obtain the estimate . The proof of the lemma is complete. 0.2in $\blacksquare$ 0.2in
[**Remark A.3**]{} From the proof of Lemma \[lemmaA7\] it is easy to see that if the norm $|k|_2$ in the condition is replaced by the norm $|k|_1$, then the estimates and are still valid.
0.3in
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[^1]: This work is supported by the NNSF(11371132,11671392) of China, by Key Laboratory of High Performance Computing and Stochastic Information Processing. $^a$email: lixuemei$\[email protected], $^b$ email: [email protected]
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---
abstract: |
We show that if a language is recognized within certain error bounds by constant-depth quantum circuits over a finite family of gates, then it is computable in (classical) polynomial time. In particular, our results imply $$\EQNC^0 \subseteq {{\bf P}},$$ where $\EQNC^0$ is the constant-depth analog of the class $\EQP$.
On the other hand, we adapt and extend ideas of Terhal & DiVincenzo [@TD:constant-depth] to show that, for any family $\cal F$ of quantum gates including Hadamard and CNOT gates, computing the acceptance probabilities of depth-five circuits over $\cal F$ is just as hard as computing these probabilities for arbitrary quantum circuits over $\cal F$. In particular, this implies that $$\NQNC^0 = \NQACC = \NQP = {{\rm co}{{{{\bf C}_{=}{{\bf P}}}}}},$$ where $\NQNC^0$ is the constant-depth analog of the class $\NQP$. This essentially refutes a conjecture of Green et al. that $\NQACC
\subseteq \TC^0$ [@GHMP:QAC].
author:
- 'S. Fenner[^1]'
- 'F. Green[^2]'
- 'S. Homer[^3]'
- 'Y. Zhang'
bibliography:
- 'eqnc.bib'
title: 'Bounds on the Power of Constant-Depth Quantum Circuits'
---
Introduction {#sec:intro}
============
Quantum decoherence is a major obstacle to maintaining long quantum computations. The first working quantum computers will almost certainly be limited to realizing shallow—i.e., small-depth—quantum circuits. This dilemma has inspired much theoretical interest in the capabilities of these circuits, particularly circuits that have constant depth and polynomial size.
Recently, people have found that much can be done with $O(\log
n)$-depth circuits. For example, Cleve & Watrous were able to approximate the Quantum Fourier Transform over modulus $2^n$ with $O(\log n)$-depth circuits [@CW:fourier]. Log-depth seems to present a barrier for many computational problems, however; getting significantly shallower circuits appears difficult if not impossible—unless gates of unbounded width (i.e., number of qubits, or fan-in) are allowed. This has led to the study of constant-depth quantum circuits that can contain certain classes of unbounded fan-in gates.
There are a number of unbounded-width gate classes studied in the literature, most being defined in analogy to classical Boolean gates. The generalized Toffoli gate (see Section \[sec:prelims-gates\]) is the quantum equivalent of the unbounded Boolean AND-gate. Likewise, there are quantum equivalents of Mod-gates and threshold gates. One particular quantum gate corresponds to something taken almost completely for granted in Boolean circuits—fan-out. A fan-out gate copies the (classical) value of a qubit to several other qubits at once.[^4] Using these gates, one can define quantum versions of various classical circuit classes: $\QNC^k$ (Moore & Nilsson [@MN:quantum]), $\QAC^k$ and $\QACC^k$ (Moore [@Moore:fanout], Green et al. [@GHMP:QAC]), and $\QTC^k$ are analogous to $\NC^k$, $\AC^k$, $\ACC$, and $\TC^k$, respectively. The case of particular interest is when $k=0$. All these classes are allowed constant-width gates drawn from a finite family. The classes differ in the additional gates allowed. $\QNC^0$ is the most restrictive class; all gates must have bounded width. $\QAC^k$ circuits are allowed generalized Toffoli gates, and $\QACC^k$ circuits are allowed ${{\rm Mod}}_q$-gates, where $q$ is kept constant in each circuit family. $\QTC^k$ circuits are allowed quantum threshold gates. See Section \[sec:prelims-gates\] for detailed definitions of most of these classes.
Although quantum classes are defined analogously to Boolean classes, their properties have turned out to be quite different from their classical versions. A simple observation of Moore [@Moore:fanout] shows that the $n$-qubit fan-out gate and the $n$-qubit parity (${{\rm Mod}}_2$) gate are equivalent up to constant depth, i.e., each can be simulated by a constant-depth circuit using the other. This is completely different from the classical case, where parity cannot be computed even with $\AC^0$ circuits, where fan-out is unrestricted [@Ajtai:AC0; @FSS:AC0]. Later, Green et al. showed that quantum ${{\rm Mod}}_q$-gates are constant-depth equivalent for all $q>1$, and are thus all equivalent to fan-out. Thus, for any $q>1$, $$\QNC^0_f = \QACC^0(q) = \QACC^0.$$ (The $f$ subscript means, “with fan-out.”) The classical analogs of these classes are provably different. In particular, classical ${{\rm Mod}}_p$ and ${{\rm Mod}}_q$ gates are not constant-depth equivalent if $p$ and $q$ are distinct primes, and neither can be simulated by $\AC^0$ circuits [@Razborov:ACC; @Smolensky:ACC].
Using $\QNC^0$ circuits with unbounded fan-out gates, H[ø]{}yer & Špalek managed to parallelize a sequence of commuting gates applied to the same qubits, and thus greatly reduced the depth of circuits for various purposes [@HS:fanout]. They showed that threshold gates can be approximated in constant depth this way, and they can be computed exactly if Toffoli gates are also allowed. Thus $\QTC^0_f = \QACC^0$ as well. Threshold gates, and hence fanout gates, are quite powerful; many important arithmetic operations can be computed in constant depth with threshold gates [@SBKH:threshold]. This implies that the quantum Fourier transform—the quantum part of Shor’s factoring algorithm—can be approxmated in constant depth using fanout gates.
All these results rely for their practicality on unbounded-width quantum gates being available, especially fan-out or some (any) Mod gate. Unfortunately, making such a gate in the lab remains a daunting prospect; it is hard enough just to fabricate a reliable CNOT gate. Much more likely in the short term is that only one- and two-qubit gates will be available, which brings us back to the now more interesting question of $\QNC^0$. How powerful is this class? Can $\QNC^0$ circuits be simulated classically, say, by computing their acceptance probabilities either exactly or approximately? Is there *anything* that $\QNC^0$ circuits can compute that cannot be computed in classical polynomial time? The present paper addresses these questions.
A handful of hardness results about simulating constant-depth quantum circuits with constant-width gates were given recently by Terhal & DiVincenzo [@TD:constant-depth]. They showed that if one can classically efficiently simulate, via sampling, the acceptance probability of quantum circuits of depth at least three using one- and two-qubit gates, then $\BQP \subseteq \AM$. They also showed that the polynomial hierarchy collapses if one can efficiently compute the acceptance probability exactly for such circuits. (Actually, a much strong result follows from their proof, namely, ${{\bf P}}= \PP$.) Their technique uses an idea of Gottesman & Chuang for teleporting CNOT gates [@GC:teleportation] to transform an arbitrary quantum circuit with CNOT and single-qubit gates into a depth-three circuit whose acceptance probability is proportional to, though exponentially smaller than, the original circuit. Their results, however, only hold on the supposition that depth-three circuits with *arbitrary* single-qubit and CNOT gates are simulatable. We build on their techniques, making improvements and simplifications. We weaken their hypothesis by showing how to produce a depth-three circuit with essentially the same gates as the original circuit. In addition, we can get by with only with simple qubit state teleportation [@BBCJPW:teleportation]. Our results immediately show that the class $\NQNC^0$ (the constant-depth analog of $\NQP$, see below), is actually the same as $\NQP$, which is known to be as hard as the polynomial hierarchy [@FGHP:NQP]. We give this result in Section \[sec:main-lower-bound\]. It underscores yet another drastic difference between the quantum and classical case: while $\AC^0$ is well contained in ${{\bf P}}$, $\QNC^0$ circuits (even just depth-three) can have amazingly complex behavior. Our result is also tight; Terhal & DiVincenzo showed that the acceptance probabilities of depth-two circuits over one- and two-qubit gates are computable in polynomial time.
In Section \[sec:main-upper-bound\], we give contrasting upper bounds for $\QNC^0$-related language classes. We show that various bounded-error versions of $\QNC^0$ (defined below) are contained in ${{\bf P}}$. Particularly, $\EQNC^0 \subseteq {{\bf P}}$, where $\EQNC^0$ is the constant-depth analog of the class $\EQP$ (see below). Our proof uses elementary probability theory, together with the fact that single output qubit measurement probabilities can be computed directly, and the fact that output qubits are “largely” independent of each other. In hindsight, it is not too surprising that $\EQNC^0\subseteq{{\bf P}}$. $\EQNC^0$ sets a severe limitation on the behavior of the circuit: it must accept with certainty or reject with certainty. This containment is more surprising (to us) for the bounded-error $\QNC^0$ classes.
We give open questions and suggestions for further research in Section \[sec:open\].
Preliminaries {#sec:prelims}
=============
Gates and circuits {#sec:prelims-gates}
------------------
We assume prior knowledge of basic concepts in computational complexity: polynomial time, ${{\bf P}}$, $\NP$, as well as the counting class ${{\#{{\bf P}}}}$ [@Valiant:NumP]. Information can be found, for example, in Papadimitriou [@Papadimitriou:complexity]. The class ${{{\bf C}_{\neq}{{\bf P}}}}$ (${{\rm co}{{{{\bf C}_{=}{{\bf P}}}}}}$) was defined by Wagner [@Wagner:C=P]. One way of defining ${{{\bf C}_{\neq}{{\bf P}}}}$ is as follows: a language $L$ is in ${{{\bf C}_{\neq}{{\bf P}}}}$ iff there are two $\#P$ functions $f$ and $g$ such that, for all $x$, $x\in L \iff f(x)\neq g(x)$. ${{{\bf C}_{\neq}{{\bf P}}}}$ was shown to be hard for the polynomial hierarchy by Toda & Ogihara [@TO:PHinBPGapP].
We will also assume some (but less) background in quantum computation and the quantum circuit model. See Nielsen and Chuang [@NC:quantumbook] for a good reference of basic concepts and notation.
We review some standard quantum (unitary) gates. Among the single-qubit gates, we have the Pauli gates $X$, $Y$, and $Z$, the Hadamard gate $H$, and the $\pi/8$ gate $T$, which are defined thus, for $b\in{\{0,1\}}$: $$\begin{aligned}
X{|{b}\rangle} & = & {|{\neg b}\rangle}, \\
Y{|{b}\rangle} & = & i(-1)^b{|{\neg b}\rangle}, \\
Z{|{b}\rangle} & = & (-1)^b{|{b}\rangle}, \\
H{|{b}\rangle} & = & ({|{0}\rangle} + (-1)^b{|{1}\rangle})/\sqrt{2}, \\
T{|{b}\rangle} & = & e^{i\pi b/4}{|{b}\rangle}.\end{aligned}$$ For $n \geq 1$, the $(n+1)$-qubit *generalized Toffoli gate* $T_n$ satisfies $$T_n{|{x_1,\ldots,x_n,b}\rangle} = {|{x_1,\ldots,x_n,b\oplus
\bigwedge_{i=1}^n x_i}\rangle}.$$ Here $b$ is the *target qubit* and $x_1,\ldots,x_n$ are the *control qubits*. $T_n$ is a kind of multiply controlled $X$-gate (or NOT-gate), and is the quantum analog of the Boolean AND-gate with fanin $n$. $T_2$ is known simply as the Toffoli gate. $T_1$ is also known as the controlled NOT (CNOT) gate and is depicted below. Here, $a,b\in{\{0,1\}}$.
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A gate closely related to $T_n$ is the controlled $Z$-gate defined by $$Z_n{|{x_1,\ldots,x_n}\rangle} = (-1)^{\bigwedge_{i=1}^n
x_i}{|{x_1,\ldots,x_n}\rangle}.$$ Since $HXH = Z$, the gate $Z_{n+1}$-gate can be implemented by placing $H$-gates on either side of a $T_n$ gate on its target qubit.
The $(n+1)$-qubit fan-out gate $F_n$ is defined as follows: $$F_n{|{x_1,\ldots,x_n,b}\rangle} = {|{x_1\oplus b,\ldots,x_n\oplus
b,b}\rangle}.$$
For $q>1$, the $(n+1)$-qubit ${{\rm Mod}}_q$-gate acts on a basis state ${|{x_1,\ldots,x_n,b}\rangle}$ by flipping the target qubit $b$ iff $x_1+\cdots +x_n \not\equiv 0 \pmod{q}$. The control qubits $x_1,\ldots,x_n$ are left alone. The ${{\rm Mod}}_2$ gate is also known as the parity gate. The *width* of a gate is the number of qubits on which it acts.
Our notion of quantum circuits is fairly standard (again see, for example, [@NC:quantumbook]): a series of quantum gates, drawn from some specified set of unitary operators, acting on some specified number of qubits, labeled $1,\ldots,m$. The first few qubits are considered *input* qubits, which are assumed to be in some basis state initially (i.e., classical input); the rest are ancillæ, each assumed to be in the ${|{0}\rangle}$ state initially. Thus the initial state of the qubits is ${|{x,00\cdots 0}\rangle}$, for some binary string $x$. Some arbitrary set of qubits are specified as *output* qubits, and these qubits are measured in the computational basis at the final state. We assume that the sets of input and output qubits are part of the description of the circuit. The circuit *accepts* its input if all the output qubits are observed to be 0 in the final state. Otherwise the circuit rejects. We let $\Pr[C(x)]$ denote the probability that $C$ accepts input $x$.
If $C$ is any quantum circuit, it will be convenient for us to define $|C|$, the *size* of $C$, to be the number of output qubits plus the number of “contact points” between qubits and gates, so for example, a single-qubit gate counts one towards the size, while a two-qubit gate counts two, etc. $C$ may be laid out by partitioning its gates into *layers* $1,\ldots,d$, such that (i) gates in the same layer all act on pairwise disjoint sets of qubits, and (ii) all gates in layer $i$ are applied before any gates in layer $i+1$, for $1\leq i < d$. The *depth* of $C$ is then the smallest possible value of $d$. The *width* of $C$ is the number of qubits in $C$.
The standard quantum complexity classes can be defined in terms of quantum circuit families. A quantum circuit family is a sequence $\{C_n\}_{n\geq 0}$ of quantum circuits, where each $C_n$ has $n$ inputs. We say that $\{C_n\}$ is *uniform* if there is a (classical) polynomial-time algorithm that outputs a description of $C_n$ on input $0^n$.
Let $L$ be a language.
- $L\in\EQP$ iff there is a uniform quantum circuit family $\{C_n\}$ such that, for all $x$ of length $n$, $$\begin{aligned}
x\in L & \implies & \Pr[C_n(x)] = 1, \\
x\notin L & \implies & \Pr[C_n(x)] = 0.\end{aligned}$$
- $L\in\BQP$ iff there is a uniform quantum circuit family $\{C_n\}$ such that, for all $x$ of length $n$, $$\begin{aligned}
x\in L & \implies & \Pr[C_n(x)] \geq 2/3, \\
x\notin L & \implies & \Pr[C_n(x)] < 1/3.\end{aligned}$$
- $L\in\NQP$ iff there is a uniform quantum circuit family $\{C_n\}$ such that, for all $x$ of length $n$, $$\begin{aligned}
x\in L & \implies & \Pr[C_n(x)] > 0, \\
x\notin L & \implies & \Pr[C_n(x)] = 0.\end{aligned}$$
It is known that ${{\bf P}}\subseteq \EQP \subseteq \BQP$. It was shown in [@FGHP:NQP; @YY:NQP] that $\NQP = {{{\bf C}_{\neq}{{\bf P}}}}$, and is thus hard for the polynomial hierarchy.
Complexity classes using $\QNC$ circuits
----------------------------------------
The circuit class $\QNC$ was first suggested by Moore and Nilsson [@MN:quantum] as the quantum analog of the class $\NC$ of bounded fan-in Boolean circuits with polylogarithmic depth and polynomial size. We define the class $\QNC^k$ in the same fashion as definitions in Green, Homer, Moore, & Pollett [@GHMP:QAC] with some minor modifications.
$\QNC^k$ is the class of quantum circuit families $\{C_n\}_{n\geq 0}$ for which there exists a polynomial $p$ such that each $C_n$ contains $n$ input qubits and at most $p(n)$ many ancillæ. Each $C_n$ has depth $O(\log^k n)$ and uses only single-qubit gates and CNOT gates. The single-qubit gates must be from a fixed finite set.
Next we define the language classes $\NQNC^k$ and $\EQNC^k$. These are $\QNC^k$ analogs of the classes $\NQP$ and $\EQP$, respectively.
Let $k\geq 0$ be an integer.
- $\NQNC^k$ is the class of languages $L$ such that there is a uniform $\{C_n\}\in\QNC^k$ such that, for all $x$, $$x\in L \iff \Pr[C_{|x|}(x)]>0.$$
- $\EQNC^k$ is the class of languages $L$ such that there is a uniform $\{C_n\}\in\QNC^k$ such that, for all $x$, $\Pr[C_{|x|}(x)]\in\{0,1\}$ and $$x\in L \iff \Pr[C_{|x|}(x)] = 1.$$
#### Remark.
Green, Homer, Moore, & Pollett implicitly consider the output qubits of $C_n$ to be all the qubits in $C_n$ [@GHMP:QAC]. In our model we allow any subset of qubits to be the output qubits of $C_n$, and we do not restrict our circuits to be clean, i.e., the non-output qubits could end up in an arbitrary state, possibly entangled with the output qubits. The reason we define our circuits this way is based on the observation that, in their model, if a language $L$ is in $\EQNC^k$ (or $\BQNC^k_{\epsilon,\delta}$ for large enough $\delta$), then $L$ can contain no more than one string of each length.
Bounded-error $\QAC^k$ classes were mentioned in [@GHMP:QAC], and one can certainly ask about similar classes for $\QNC^k$ circuits. It is not obvious that there is one robust definition of $\BQNC^0$—perhaps because it is not clear how to reduce error significantly by amplification in constant depth.[^5] In the next definition, we will try to be as general as possible while still maintaining our assumption that $\vec{0}$ is the only accepting output.
Let $\epsilon$ and $\delta$ be functions mapping (descriptions of) quantum circuits into real numbers such that, for all quantum circuits $C$, $0 < \epsilon(C) \leq \delta(C) \leq 1$. We write $\epsilon_C$ and $\delta_C$ to denote $\epsilon(C)$ and $\delta(C)$, respectively. $\BQNC^k_{\epsilon,\delta}$ is the class of languages $L$ such that there is a uniform $\{C_n\}\in\QNC^k$ such that for any string $x$ of length $n$, $$\begin{aligned}
x\in L & \implies & \Pr[C_n(x)] \geq \delta_{C_n}, \\
x\notin L & \implies & \Pr[C_n(x)] < \epsilon_{C_n}. \end{aligned}$$
An interesting special case is when $\epsilon_C = \delta_C = 1$, that is, the input is accepted iff the circuit accepts with probability 1, and there is no promise on the acceptance probability. One might expect that, by the symmetry of the definitions, this class $\BQNC^0_{1,1}$ is the same as $\NQNC^0$, but it is almost certainly not, as we will see.
Other classes of constant-depth quantum circuits
------------------------------------------------
Let $k \geq 0$ and $q > 1$ be integers.
- $\QAC^k$ is the same as $\QNC^k$ except that generalized Toffoli gates are allowed in the circuits.
- $\QACC(q)$ is the same as $\QNC^0$ except that ${{\rm Mod}}_q$ gates are allowed in the circuits.
- $\QACC = \bigcup_{q>1} \QACC(q)$.
Main results
============
Simulating $\QNC^0$ circuits exactly is hard {#sec:main-lower-bound}
--------------------------------------------
\[thm:lower-bound\] $\NQNC^0 = \NQP = {{{\bf C}_{\neq}{{\bf P}}}}$.
As a corollary, we essentially solve an open problem of Green et al.[@GHMP:QAC]. They conjectured that $\NQACC \subseteq \TC^0$, the class of constant-depth Boolean circuits with threshold gates.
For any $k\geq 0$, $$\NQNC^0 = \NQNC^k = \NQAC^k = \NQACC = {{{\bf C}_{\neq}{{\bf P}}}}.$$ Thus, $\NQACC \not\subseteq \TC^0$ unless ${{{\bf C}_{\neq}{{\bf P}}}}= \TC^0$.
Let $B$ be the two-qubit Bell gate, defined as
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Also let
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which produces the EPR state $({|{00}\rangle} + {|{11}\rangle})/\sqrt{2}$. We prove the following lemma, from which the theorem follows quickly.
For any quantum circuit ${{\cal C}}$ using gates drawn from any family ${\cal
F}$, there is a depth-three quantum circuit ${{\cal C}}'$ of size linear in $|{{\cal C}}|$ using gates drawn from ${\cal F}\cup\{B,{{{B}^{\dagger}}}\}$ such that for any input $x$ of the appropriate length, $$\Pr[{{\cal C}}'(x)] = 2^{-m} \Pr[{{\cal C}}(x)],$$ for some $m \leq 2|{{\cal C}}|$ depending only on ${{\cal C}}$. The middle layer of ${{\cal C}}'$ contains each gate in ${{\cal C}}$ exactly once and no others. The third layer contains only ${{{B}^{\dagger}}}$-gates, and the first layer contains only $B$-gates, which are used only to create EPR states.
Our construction is a simplified version of the main construction in Terhal & DiVincenzo [@TD:constant-depth], but ours is stronger in one crucial respect discussed below: it does not significantly increase the family of gates used. To construct ${{\cal C}}'$, we start with ${{\cal C}}$ and simply insert, for each qubit $q$ of ${{\cal C}}$, a simplified teleportation module (shown in Figure \[fig:teleport\])
(0,0)![The nonadaptive teleportation module [@TD:constant-depth]. The state in qubit $q$ is teleported correctly iff the qubits $r_1$ and $r_2$ are both observed to be 0.[]{data-label="fig:teleport"}](figure04.ps "fig:")
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between any two consecutive quantum gates of ${{\cal C}}$ acting on $q$. No further gates involve the qubits $r_1$ and $r_2$ to the right of the ${{{B}^{\dagger}}}$-gate. This module, which lacks the usual corrective Pauli gates, is a nonadaptive version of the standard single-qubit teleportation circuit [@BBCJPW:teleportation]. It faithfully teleports the state if and only if the observed output of the ${{{B}^{\dagger}}}$-gate on the right is $00$. After inserting each teleportation circuit, the gates acting before and after it are now acting on different qubits. Further, it is important to note that any entanglement the qubit state has with other qubits is easily seen to be preserved in the teleported qubit. The input qubits of ${{\cal C}}'$ are those of ${{\cal C}}$. The output qubits of ${{\cal C}}'$ are of two kinds: output qubits corresponding to outputs of ${{\cal C}}$ are the *original outputs*; the other outputs are the qubits (in pairs) coming from the added ${{{B}^{\dagger}}}$-gates. We’ll call the measurement of each such pair a *Bell measurement*, even though it is really in the computational basis.
In addition to the gates in ${{\cal C}}$, ${{\cal C}}'$ uses only $B$-gates to make the initial EPR pairs and ${{{B}^{\dagger}}}$-gates for the Bell measurements. A sample transformation is shown in Figure \[fig:sample-trans\].
(0,0)![A sample transformation from ${{\cal C}}$ to ${{\cal C}}'$. The circuit ${{\cal C}}$ on the left has five gates: $S$, $T$, $U$, $V$, and $W$, with subscripts added to mark which qubits each gate is applied to. The qubits in ${{\cal C}}'$ are numbered corresponding to those in ${{\cal C}}$.[]{data-label="fig:sample-trans"}](figure05.ps "fig:")
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${{\cal C}}'$ has depth three since it uses the first layer to make the initial EPR states and the third layer to rotate the Bell basis back to the computational basis. All the gates of ${{\cal C}}$ appear on the second layer. From the above constuction and the properties of the teleportation module, it is not hard to see that for all $x$ of the appropriate length, $$\begin{aligned}
\Pr[{{\cal C}}(x)] & = & \Pr[\mbox{all original outputs of ${{\cal C}}'$ are 0} \mid
\mbox{all qubit states are teleported correctly}] \\
& = & \Pr[\mbox{all original outputs of are 0} \mid \mbox{all
Bell measurement results are 00}] \\
& = & \frac{\Pr[{{\cal C}}'(x)]}{\Pr[\mbox{all Bell measurement results are
00}]},\end{aligned}$$ since the Bell measurements are among the output measurements of ${{\cal C}}'$. Let $k$ be the number of ${{{B}^{\dagger}}}$-gates on layer 3. Clearly, $k \leq |{{\cal C}}|$, and it is well-known that each Bell measurement will give 00 with probability $1/4$, independent of all other measurements. So the lemma follows by setting $m = 2k$.
[Theorem \[thm:lower-bound\]]{} As mentioned before, $\NQP$ [@ADH:quantum] is defined as the class of languages recognized by quantum Turing machines (equivalently, uniform quantum circuit families over a finite set of gates) where the acceptance criterion is that the accepting state appear with nonzero probability. It is known [@FGHP:NQP; @YY:NQP] that $\NQP = {{{\bf C}_{\neq}{{\bf P}}}}$, which contains $\NP$ and is hard for the polynomial hierarchy. Since $\QNC^0$ circuit families must also draw their gates from some finite set, we clearly have $\NQNC^0 \subseteq \NQP$. The reverse containment follows from our construction: an arbitrary circuit ${{\cal C}}$ is transformed into a depth-three circuit ${{\cal C}}'$ *with the same gates* as ${{\cal C}}$ plus $B$ and ${{{B}^{\dagger}}}$. Moreover, ${{\cal C}}'$ accepts with nonzero probability iff ${{\cal C}}$ does. Thus an $\NQP$ language $L$ recognized by a uniform quantum circuit family over a finite set of quantum gates is also recognized by a uniform depth-three circuit family over a finite set of quantum gates, and so $L\in\NQNC^0$.
Using the gate teleportation apparatus of Gottesmann and Chuang [@GC:teleportation], Terhal & DiVincenzo also construct a depth-three[^6] quantum circuit ${{\cal C}}'$ out of an arbitrary circuit ${{\cal C}}$ (over CNOT and single-qubit gates) with a similar relationship of acceptance probabilities. However, they only teleport the CNOT gate, and their ${{\cal C}}'$ may contain single-qubit gates formed by compositions of arbitrary numbers of single-qubit gates from ${{\cal C}}$. (Such gates may not even be approximable in constant depth by circuits over a fixed finite family of gates.) When their construction is applied to each circuit in a uniform family, the resulting circuits are thus not generally over a finite gate set, even if the original circuits were.
Our construction solves this problem by teleporting every qubit state in between all gates involving it. Besides $B$ and ${{{B}^{\dagger}}}$, we only use the gates of the original circuit. We also are able to bypass the CNOT gate teleportation technique of [@GC:teleportation], using instead basic single-qubit teleportation [@BBCJPW:teleportation], which works with arbitrary gates.
Simulating $\QNC^0$ circuits approximately is easy {#sec:main-upper-bound}
--------------------------------------------------
In this section we prove that $\BQNC^0_{\epsilon,\delta} \subseteq {{\bf P}}$ for certain $\epsilon,\delta$. For convenience we will assume that all gates used in quantum circuits are either one- or two-qubit gates that have “reasonable” matrix elements—algebraic numbers, for instance. Our results can apply more broadly, but they will then require greater care to prove.
For a quantum circuit ${{\cal C}}$, we define a dependency graph over the set of its output qubits.
Let ${{\cal C}}$ be a quantum circuit and let $p$ and $q$ be qubits of ${{\cal C}}$. We say that *$q$ depends on $p$* if there is a forward path in ${{\cal C}}$ starting at $p$ before the first layer, possibly passing through gates, and ending at $q$ after the last layer. More formally, we can define dependence by induction on the depth of ${{\cal C}}$. For depth zero, $q$ depends on $p$ iff $q = p$. For depth $d>0$, let ${{\cal C}}'$ be the same as ${{\cal C}}$ but missing the first layer. Then $q$ depends on $p$ (in ${{\cal C}}$) iff there is a qubit $r$ such that $q$ depends on $r$ (in ${{\cal C}}'$) and either $p = r$ or there is a gate on the first layer of ${{\cal C}}$ that involves both $p$ and $r$.
For ${{\cal C}}$ a quantum circuit and $q$ a qubit of ${{\cal C}}$, define $$D_q = \{ p \mid \mbox{$q$ depends on $p$}\}.$$ If $S$ is a set of qubits of ${{\cal C}}$, define $D_S = \bigcup_{q\in S}
D_q$. Let the *dependency graph of ${{\cal C}}$* be the undirected graph with the output qubits of ${{\cal C}}$ as vertices, and with an edge between two qubits $q_1$ and $q_2$ iff $D_{q_1} \cap D_{q_2} \neq
\emptyset$.
If ${{\cal C}}$ has depth $d$, then it is easy to see that the degree of its dependency graph is less than $2^{2d}$. The following lemma is straightforward.
\[lem:independence\] Let ${{\cal C}}$ be a quantum circuit and let $S$ and $T$ be sets of output qubits of ${{\cal C}}$. Fix an input $x$ and bit vectors $u$ and $v$ with lengths equal to the sizes of $S$ and $T$, respectively. Let $E_{S=u}$ (respectively $E_{T=v}$) be the event that the qubits in $S$ (respectively $T$) are observed to be in the state $u$ (respectively $v$) in the final state of ${{\cal C}}$ on input $x$. If $D_S \cap D_T =
\emptyset$, then $E_{S=u}$ and $E_{T=v}$ are independent.
For an algebraic number $a$, we let $\|a\|$ be the size of some reasonable representation of $a$.
The results in this section follow from the next theorem.
\[thm:upper-bound\] There is a deterministic decision algorithm $A$ which takes as input
1. a quantum circuit ${{\cal C}}$ with depth $d$ and $n$ input qubits,
2. a binary string $x$ of length $n$, and
3. an algebraic number $t \in [0,1]$,
and behaves as follows: Let $D$ be one plus the degree of the dependency graph of ${{\cal C}}$. $A$ runs in time ${{\rm Poly}}(|{{\cal C}}|,2^{2^d},\|t\|)$, and
- if $\Pr[{{\cal C}}(x)] \geq 1-t$, then $A$ accepts, and
- if $\Pr[{{\cal C}}(x)] < 1-Dt$, then $A$ rejects.
Note that since $D \leq 2^{2d}$, if $t < 2^{-2d}$, then $A$ will reject when $\Pr[{{\cal C}}(x)] < 1-2^{2d}t$.
[Theorem \[thm:upper-bound\]]{} On input $({{\cal C}},x,t)$ as above,
1. $A$ computes the dependency graph $G = (V,E)$ of ${{\cal C}}$ and its degree, and sets $D$ to be the degree plus one.
2. $A$ finds a $D$-coloring ${{c}:{V}\rightarrow{\{1,\ldots,D\}}}$ of $G$ via a standard greedy algorithm.
3. For each output qubit $q\in V$, $A$ computes $P_q$—the probability that 0 is measured on qubit $q$ in the final state (given input $x$).
4. For each color $i\in\{1,\ldots,D\}$, let $B_i = \{ q\in V \mid c(q) =
i \}$. $A$ computes $$P_{B_i} = \prod_{q\in B_i} P_q,$$ which by Lemma \[lem:independence\] is the probability that all qubits colored $i$ are observed to be 0 in the final state.
5. If $P_{B_i} \geq 1-t$ for all $i$, the $A$ accepts; otherwise, $A$ rejects.
We first show that $A$ is correct. If $\Pr[{{\cal C}}(x)] \geq 1-t$, then for each $i\in\{1,\ldots,D\}$, $$1-t \leq \Pr[{{\cal C}}(x)] \leq P_{B_i},$$ and so $A$ accepts. On the other hand, if $\Pr[{{\cal C}}(x)] < 1-Dt$, then $$Dt < 1 - \Pr[{{\cal C}}(x)] \leq \sum_{i=1}^D \left(1-P_{B_i}\right),$$ so there must exist an $i$ such that $t < 1 - P_{B_i}$, and thus $A$ rejects.
To show that $A$ runs in the given time, first we show that the measurement statistics of any output qubit can be calculated in time polynomial in $2^{2^d}$. Pick an output qubit $q$. By looking at ${{\cal C}}$ we can find $D_q$ in time ${{\rm Poly}}(|{{\cal C}}|)$. Since ${{\cal C}}$ has depth $d$ and uses gates on at most two qubits each, $D_q$ had cardinality at most $2^d$. Then we simply calculate the measurement statistics of output qubit $q$ from the input state restricted to $D_q$, i.e., with the other qubits traced out. This can be done by computing the state layer by layer, starting with layer one, and at each layer tracing out qubits when they no longer can reach $q$. Because of the partial traces, the state will in general be a mixed state so we maintain it as a density operator. We are multiplying matrices of size at most $2^{2^d}\times 2^{2^d}$ at most $O(d)$ times. All this will take time polynomial in $2^{2^d}$, provided we can show that the individual field operations on the matrix elements do not take too long.
Since there are finitely many gates to choose from, their (algebraic) matrix elements generate a field extension $F$ of ${{\mathbb Q}}$ with finite index $r$. We can thus store values in $F$ as $r$-tuples of rational numbers, with the field operations of $F$ taking polynomial time. Furthermore, one can show that for $a,b\in F$, $\|ab\| = O(\|a\| +
\|b\|)$ and $\left\|\sum_{i=1}^n a_i\right\| = O(n \cdot \max_i
\|a_i\|)$ for any $a_1,\ldots,a_n\in F$. A bit of calculation then shows that the intermediate representations of numbers do not get too large.
The dependency graph and its coloring can clearly be computed in time ${{\rm Poly}}(|{{\cal C}}|)$. The only things left are the computation of the $P_{B_i}$ and their comparison with $1-t$. For reasons similar to those above for matrix multiplication, this can be done in time ${{\rm Poly}}(|{{\cal C}}|,2^{2^d},\|t\|)$.
\[cor:master\] Suppose $\epsilon$ and $\delta$ are polynomial-time computable, and for any quantum circuit $C$ of depth $d$, $\delta_C = 1 -
2^{-2d}(1-\epsilon_C)$. Then $$\BQNC^0_{\epsilon,\delta} \subseteq {{\bf P}}.$$
For each $C$ of depth $d$ in the circuit family and each input $x$, apply the algorithm $A$ of Theorem \[thm:upper-bound\] with $t =
1-\delta_C = 2^{-2d}(1-\epsilon_C)$, noting that $D \leq 2^{2d}$.
The following few corollaries are instances of Corollary \[cor:master\].
For quantum circuit $C$, let $\delta_C = 1 - 2^{-(2d+1)}$, where $d$ is the depth of $C$. Then $$\BQNC^0_{(1/2),\delta} \subseteq {{\bf P}}.$$
Apply algorithm $A$ to each circuit, setting $t = 2^{-(2d+1)}$.
\[cor:BQNC0\] $\BQNC^0_{1,1} \subseteq {{\bf P}}$.
Apply algorithm $A$ to each circuit, setting $t = 0$.
\[cor:EQNC0\] $\EQNC^0 \subseteq {{\bf P}}$.
Clearly, $\EQNC^0 \subseteq \BQNC^0_{1,1}$.
Corollaries \[cor:BQNC0\] and \[cor:EQNC0\] can actually be proven more directly. We simply compute, for each output, its probability of being 0. We accept iff all probabilities are 1.
We observe here that by a simple proof using our techniques, one can show that the generalized Toffoli gate cannot be simulated by a $\QNC^0$ circuit, since the target of the Toffoli gate can only depend on constantly may input qubits.
Conclusions, open questions, and further research {#sec:open}
=================================================
Our upper bound results in Section \[sec:main-upper-bound\] can be improved in certain ways. For example, the containment in ${{\bf P}}$ is easily seen to apply to $(\log\log n + O(1))$-depth circuits as well. Can we increase the depth further? Another improvement would be to put $\BQNC^0_{\epsilon,\delta}$ into classes smaller than ${{\bf P}}$. LOGSPACE seems managable. How about $\NC^1$?
There are some general questions about whether we have the “right” definitions for these classes. For example, the accepting outcome is defined to be all outputs being 0. One can imagine more general accepting conditions, such as arbitrary classical polynomial-time postprocessing. If we allow this, then all our classes will obviously contain ${{\bf P}}$. If we allow arbitrary classical polynomial-time *pre*processing, then all our classes will be closed under ptime $m$-reductions (Karp reductions).
Finally, there is the question of the probability gap in the definitions of $\BQNC^0$. Certainly we would like to narrow this gap (ideally, to $1/{\rm poly}$) and still get containment in ${{\bf P}}$.
Acknowledgments {#acknowledgments .unnumbered}
===============
We would like to thank David DiVincenzo and Mark Heiligman for helpful conversations on this topic. This work was supported in part by the National Security Agency (NSA) and Advanced Research and Development Agency (ARDA) under Army Research Office (ARO) contract numbers DAAD 19-02-1-0058 (for S. Homer, and F. Green) and DAAD 19-02-1-0048 (for S. Fenner and Y. Zhang).
[^1]: Dept. of CS and Eng., University of South Carolina, Columbia, SC 29208, $\{$fenner$|$zhang29$\}[email protected]
[^2]: Dept. of Math and CS, Clark University, Worcester, MA 01610, [email protected]
[^3]: Computer Science Department, Boston University, Boston, MA 02215, [email protected]
[^4]: There is no violation of the No-Cloning Theorem here; only the classical value is copied.
[^5]: One can always reduce error classically by just running the circuit several times on the same input. In this case, the best definition of $\BQNC^0$ may be that the gap between the allowed accept and reject probabilities should be at least $1/{\rm poly}$.
[^6]: They count the depth as four, but they include the final measurement as an additional layer whereas we do not.
|
{
"pile_set_name": "ArXiv"
}
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---
abstract: 'We propose a spin filter scheme using a $T$-stub waveguide. By applying a moderate magnetic field at the tip of the sidearm, this device can produce both large electric and spin current. The direction, polarization of the output spin current can be further adjusted electronically by a remote gate which tunes the length of the sidearm. The device is robust against the disorder.'
author:
- 'X. Y. Feng'
- 'J. H. Jiang'
- 'M. Q. Weng'
title: 'Remote-control spin filtering through a $T$-type structure'
---
[^1]
Spintronic devices have many advantages over the traditional electronic devices such as higher operation speed, lower power consumption.[@datta; @wolf; @das] Tremendous efforts have been devoted in the past few years to overcome the fundamental obstacles in the realization of spintronic devices, such as spin generation, control and detection. Direct injection of spin current from ferromagnetic metal or semiconductor was first proposed to realize the spin injection.[@fiederling_1999; @oestreich_1999] Despite the great efforts, the highest spin injection rate reported is about $90\%$[@fiederling_1999] for p-typed and $57\%$[@jiang] for n-typed semiconductor. A great number of schemes of spin filter have been proposed to produce highly polarized spin current using various of structures such as electronic waveguide,[@zhai; @zhou_apl_2004] double-bend structure,[@zhou_apl_2005] Aharonov-Bohm (AB) ring,[@frustaglia]resonant tunneling diode,[@koga] quantum dot,[@recher; @karkka; @folk] ferromagnetic electrode.[@grundler_2001_prb]More recently, the spin filter for the hole system was brought to public.[@wu_2005_prb] Dynamical spin generation in semiconductor using oscillation field is also proposed to avoid the difficulty of spin injection through interface.[@cheng_2005_apl; @zhang]
In this paper, we propose a spin filter scheme which can produce high spin polarization (SP) and also provides remote control of spin current magnetically as well as electronically. Our scheme is a localized magnetic field modulated 2-dimensional (2D) T-stub waveguide as shown in Fig. \[fig1\]. The waveguide is composed of a longitudinal conductor with length $L$, width $N_y$ and a sidearm of width $N_x$ attached to the center of the conductor. The effective conducting length of sidearm $L_s$ can be controlled by the remote gate voltage $V_g$ which changes $L_d$, the length of depletion area (area B in Fig. \[fig1\]. The far edge of sidearm (gray areas A and B in the Fig. \[fig1\]) is modulated by an applied magnetic field which gives Zeeman splitting of $2 V_0$. T-stub geometry has long been proposed as quantum modulated transistor (QMT).[@sols_jap_1989; @aihara_apl_1993] The conductance of this kind of devices is determined by the quantum interference effect between different Feynman paths and oscillates with the Fermi energy. When a modulate magnetic field is applied, the electrons of different spins will have different phase shifts transpassing through the modulated area and therefore have different conductances. In this way, the non-spin-polarized electron current pass through this device will produce SP. The spin current can also be controlled electronically by the remote gate by adjusting the effective sidearm length $L_s$. It is worth noting that our spin filter is different from other implementations in the fact that our device can be [*remotely*]{} controlled by both magnetical and electronic methods while the other ones can only be controlled locally. Moreover our device has more energy windows to generate high polarized spin current than the device made by a single quantum dot.[@folk]
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We describe the T-stub geometry by the tight-binding Hamiltonian with nearest-neighbor approximation: $$\begin{aligned}
H&=&\sum_{l,m,\sigma} \varepsilon_{l,m,\sigma} c^\dagger_{l,m,\sigma}
c_{l,m,\sigma} +
t_0 \sum_{l,m,\sigma} (c^\dagger_{l+1,m,\sigma}c_{l,m,\sigma} \nonumber \\
&&+ c^\dagger_{l,m+1,\sigma}c_{l,m,\sigma} +h.c.)\\end{aligned}$$ in which $l$ and $m$ denote the “lattice” site index along the $x$- and $y$-axis respectively. The on-site energy $\varepsilon_{l,m,\sigma}=\varepsilon_0+\sigma V_0$ when $(l,m)$ locates in the modulated regime (the gray area in the Fig. \[fig1\]) and $\varepsilon_0$ otherwise. $\varepsilon_0=-4t_0$ and $t_0=-\hbar/(2m^{\ast}a^2)$ is the hopping energy with $m^{\ast}$ and $a$ standing for the effective mass and the “lattice” constant respectively. $\sigma V_0$ is the Zeeman energy of spin $\sigma$ in the modulate magnetic field and $\sigma=\pm 1$ for spin-up and -down electrons respectively.
The two-terminal spin-dependent conductance is obtained by using Landauer-Büttiker[@buttiker] formula $G^{\sigma \sigma^\prime}(E)=(e^2/h)\mathtt{Tr}[\Gamma^{\sigma}_{1}
G^{\sigma\sigma^\prime+}_{1L}(E)\Gamma^{\sigma^\prime}_{N}G^{\sigma
^\prime\sigma -}
_{L1}(E)]$ with $\Gamma^{\sigma}_{1/N}$ representing the self-energy function for the isolated left/right ideal leads.[@datta_95] $G^{\sigma\sigma^\prime+}_{1L}(E)$ and $G^{\sigma\sigma^\prime-}_{L1}(E)$ are the full retarded and advanced Green functions for the conductor which have taken account the effect of the leads. $\mathtt{Tr}$ stands for the trace over the $y$-axis. The spin dependent current is given by $I_{\sigma}=\int_E^{E+\Delta}
G^{\sigma\sigma}(\varepsilon) d\varepsilon$ for energy window $[E,E+\Delta]$.
We first study the analytically solvable transport problem in quasi-one-dimensional ($N_x=N_y=1$) T-stub geometry system. By solving the time independent Schrödinger equation,[@xiong] the transmission coefficient in this system can be written as $$\begin{aligned}
&&t_{\sigma}=2i G^{\sigma\sigma}_{0,0}(E)t_0^3 \sin(ka)/\{
[\varepsilon_1-E+t_0e^{ika}+G^{\sigma\sigma}_{0,0}(E)t_0^2]
\nonumber \\
&&\times
[\varepsilon_{-1}-E+t_0e^{ika}+G^{\sigma\sigma}_{0,0}(E)t_0^2]
-[G^{\sigma\sigma}_{0,0}(E)t_0^2]^2\},\end{aligned}$$ in which $G^{\sigma\sigma^{\prime}}_{l,m}(E)
=(E-\hat{H})^{-1}_{l\sigma,m\sigma^{\prime}}$ and $\hat{H}$ are the Green function and the Hamiltonian of sidearm respectively. $E=2|t_0|(1-\cos(k\,a))$ is the energy of electron with wave-vector $k$. The corresponding conductance is $G^{\sigma}=(e^2/h)|t_{\sigma}|^2$. Without the modulation, the transmission coefficient drops to zero at the anti-resonance points $2|t_0|[1-\cos (n\pi/(L_s+1))] (n=1,2,\cdots)$ and reaches its maximum quickly at resonance points $2|t_0|[1-\cos
(n\pi/L_s)] (n=1,2,\cdots)$. When the modulate field turns on, the the (anti-)resonance points of spin-up and -down shift apart. In Fig. \[fig2\](a) we plot spin resolved conductance as a function of the electron energy for a typical quasi-one-dimensional T-stub device with sidearm length $L_s=63\,a$, magnetic field $V_0=0.001\,|t_0|$ and modulation length $L_m=10\,a$. Since a pair of anti-resonance and resonance points are very close to each other when the sidearm is relatively long, it is possible to choose the system parameters so that the resonance point of spin-up (-down) electron matches the anti-resonance point of spin-down (-up) electron under moderate magnetic field. In this way, we can obtain both large electric current and large spin current.
We now study the transport in T-stub waveguide with finite conductor and sidearm widths. We carry out a numerical calculation for a waveguide whose geometry parameters are $L=60a$, $N_x=10a$, $N_y=10a$. The leads are assumed to have perfect Ohmic contacts with the conductor. A hard wall potential is applied at the edge of the waveguide. This makes the lowest energy of $n$-th subband (mode) in leads be $\varepsilon_n=2|t_0|\{1-\cos[n\pi/(N_y+1)]\}$. The Fermi energy in our calculation is between $0.083|t_0|$ and $0.124|t_0|$ so that only the lowest mode contributes to the conductance. The Zeeman splitting energy is $V_0=0.001|t_0|$ corresponding to a few Teslas for the typical III-V semiconductors with lattice constance $a=20\AA$. In Fig. \[fig2\](b) we present the electron conductance as a function of the injection electron energy for a device with $L_s=60\,a$ and $L_m=15\,a$. Noted that the energy is count from bottom of the first mode. It is seen from the figure that many properties of quasi-one-dimensional T-stub waveguide survive in the finite width one: The conductance oscillates with the injection energy. It approaches to zero at the anti-resonance points and then quickly rise to about $e^2/h$ at the resonance points. And most importantly, an anti-resonance point is alway accompanied by a resonance point which is very close to it. In the presence of modulation magnetic field, the difference of conductances between these two spins is noticeably large when the inject energy locates in one of the anti-resonance/resonance energy windows. One can tune the Fermi Energy to obtain both large electric and spin current. For example in the energy windows $[0.11619|t_0|,0.11789|t_0|]x$, we obtain the largest spin current density with $I_{\uparrow}^{SP}=I_{\uparrow}-I_{\downarrow}\approx5.986nA$ for the spin-up current.
-0.3cm
-0.3cm
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-0.3cm
Like the original QMT, we can also use the a remote gate voltage to control the length of the sidearm. In Fig. \[fig3\](b) we present the conductances for different sidearm lengths $L_s$ but constant $L_s-L_m$. In this way we simulate the electronically controlling of the device without changing the magnetic modulation profile, [*i.e.*]{} the strength and the positions of the applied magnetic field. One can see from the figure that, the anti-resonance and the corresponding resonance points change with the length of the sidearm as they should have been. For a specified injection energy, when the length of the sidearm changes, not only the conductance but also the SP of output current change. For examples, in the energy window near $0.0871|t_0|$ one gets about $100\%$ polarized spin-down current when the sidearm is $60\,a$ long. Once the length of sidearm is adjusted to $75\,a$, one gets about $8\%$ polarized current but the direction of spin change to up. If one further adjust the length to $90\,a$, the output current is almost non-polarized. It is seen that with this filter one can control the strength, direction and polarization rate of spin current electronically.
In order to further check the robustness of the spin filter we propose, we now add Anderson disorder to the system and study its effect on the SP. In Fig. \[fig4\], the spin-dependent conductances $G^{\uparrow \uparrow}$ and $G^{\downarrow \downarrow}$ as well as the SP are plotted against the energy of the incident electrons with the Anderson disorder included. The strength of the disorder $W=0.005|t_0|$, five times of the modulated potential $V_0$. It is found that the disorder decreases the transmission coefficients, but only have slight effect on SP. In some energy windows, one can obtain SP as high as $80\%$. Therefore the scheme we propose is robust against the disorder.
In summary, we propose a spin filter scheme which enables the electrically and magnetically remote control the spin polarization of output current. The spin filter is a T-stub waveguide with a modulation magnetic field at the tip of the sidearm. In this device, electron conductance drops to the minimum when Fermi energy locates at the anti-resonance points and rises to the maximum at resonance points. With the modulation field, the (anti-)resonance points of spin-up and -down electrons shift apart. Since a pair of the anti-resonance and resonance points are very close to each other, one is able to use moderate magnetic field to produce both large electric and spin currents. Moreover one is able to control the direction and polarization of the output spin current of the T-stub waveguide via a remote gate which tunes the length of the sidearm and therefore realize the remote electronically control of spin current. We further shown that the device is robust against the disorder.
The authors would like to thank M. W. Wu for proposing the topic as well as directions during the investigation. This work was supported by the Natural Science Foundation of China under Grant No. 10574120, the National Basic Research Program of China under Grant No. 2006CB922005, the Knowledge Innovation Project of Chinese Academy of Sciences and SRFDP. One of the authors (X.Y.F.) thanks J. Zhou for valuable discussions.
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[^1]: Author to whom correspondence should be addressed
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We present an algorithm to decide the intruder deduction problem (IDP) for a class of locally stable theories enriched with normal forms. Our result relies on a new and efficient algorithm to solve a restricted case of higher-order associative-commutative matching, obtained by combining the *Distinct Occurrences of AC-matching* algorithm and a standard algorithm to solve systems of linear Diophantine equations. A translation between natural deduction and sequent calculus allows us to use the same approach to decide the *elementary deduction problem* for locally stable theories. As an application, we model the theory of blind signatures and derive an algorithm to decide IDP in this context, extending previous decidability results.'
author:
- 'Mauricio Ayala-Rincón [^1]'
- Maribel Fernández
- 'Daniele Nantes-Sobrinho[^2]'
nocite: '[@*]'
title: 'Elementary Deduction Problem for Locally Stable Theories with Normal Forms[^3]'
---
Introduction {#introduction .unnumbered}
============
There are different approaches to model cryptographic protocols and to analyse their security properties [@survey]. One technique consists of proving that an attack requires solving an algorithmically hard problem; another consists of using a process calculus, such as the spi-calculus [@spi], to represent the operations performed by the participants and the attacker. In recent years, the deductive approach of Dolev and Yao [@dolev], which abstracts from algorithmic details and models an attacker by a deduction system, has successfully shown the existence of flaws in well-known protocols. A deduction system under Dolev-Yao’s approach specifies how the attacker can obtain new information from previous knowledge obtained either by eavesdropping the communication between honest protocol participants (in the case of a passive attacker), or by eavesdropping and fraudulently emitting messages (in the case of an active attacker). The *intruder deduction problem* (IDP) is the question of whether a passive eavesdropper can obtain a certain information from messages observed on the network.
Abadi and Cortier’s approach [@AbadiCo2006] proposes conditions for analysing message deducibility and indistinguishability relations for security protocols modelled in the applied pi-calculus [@appliedpicalculus]. In particular, [@AbadiCo2006] shows that IDP is decidable for *locally stable* theories. However, to ensure the soundness of this approach, the definition of locally stable theories given in [@AbadiCo2006] needs to be modified (as confirmed via personal communication with the second author of [@AbadiCo2006]). In this work, we made the necessary modifications and propose a new approach to solve IDP in the context of locally stable theories.
Our notion of locally stable theory is based on the existence of a finite and computable saturated set, but, unlike [@AbadiCo2006], our saturated sets include normal forms[^4]. The new approach we propose in order to prove the decidability of IDP is based on an algorithm to solve a restricted case of higher-order associative-commutative matching (AC-matching). To design this algorithm we use well-known results for solving systems of linear Diophantine equations (SLDE) [@BouConDe; @ClauFor; @frumkin; @papadimitriou], which we combine with a polynomial algorithm to solve the DO-ACM problem (Distinct Occurrences of AC-Matching) [@narendran].
In the case where the signature of the equational theory contains, for each AC function symbol $\oplus$, its corresponding inverse $i_{\oplus}$, we obtain a decidability result which is polynomial with relation to the size of the saturated set (built from the initial knowledge of the intruder). Thanks to the use of the algorithm for solving SLDE over $\mathbb{Z}$, we avoid an exponential time search over the solution space in the case of AC symbols (improving over [@AbadiCo2006], where an exponential number of possible combinations have to be considered). For more details we refer the reader to the extended version of this paper [@AFSextended]. After introducing the class of locally stable theories and proving the decidability of the IDP for protocols in this class, we show that the Elementary Deduction Problem (EDP) introduced in [@TiGo2009] is also decidable in polynomial time with relation to the size of a saturated set of terms. EDP is stated as follows: given a set $\Gamma$ of messages and a message $M$, is there an $E$-context $C[\ldots]$ and messages $M_1,\ldots, M_k\in \Gamma$ such that $C[M_1,\ldots, M_k]\approx_{E}M$? Here, $E$ is the equational theory modelling the protocol. We use this approach to model theories with blind signatures. As an application, using a previous result that links the decidability of the EDP to the decidability of the IDP when the theory $E$ satisfies certain conditions, we obtain decidability of IDP for a subclass of locally stable theories combined with the theory $B$ of blind signatures. In this way, we generalise a result from [@AbadiCo2006] (Section 5.2.4): it is not necessary to prove that the combination of the theories $E$ and $B$ is locally stable.
**Related Work.** The analysis of cryptographic protocols has attracted a lot of attention in the last years and several tools are available to try to identify possible attacks, see Maude-NPA [@maudeNPA07], ProVerif [@proverif], CryptoVerif [@cryptoverif], Avispa [@avispa], Yapa [@YAPA].
Sequent calculus formulations of Dolev Yao intruders [@Tiu2007] have been used in a formulation of open bisimulation for the spi-calculus. In [@TiGo2009], deductive techniques for dealing with a protocol with blind signatures in mutually disjoint AC-convergent equational theories, containing a unique AC operator each, are considered. As an alternative approach, the intruder’s deduction capability is modelled inside a sequent calculus modulo a rewriting system, following the approach of [@BeC06]. Then, the IDP is reduced in polynomial time to EDP.
By combining the techniques in [@TiGo2009] and [@Bursucconstraints], the IDP formulation for an Electronic Purse Protocol with blind signatures was proved to reduce in polynomial time to EDP for an AC-convergent theory containing three different $AC$ operators and rules for exponentiation [@nantesayala], extending the previous results. However, no algorithm was provided to decide EDP. More precisely, assuming that EDP is solved in time $O(f(n))$, it was proved that IDP reduces polynomially to EDP with complexity $O(n^k \times f(n))$, for some constant $k$. Thus, whenever the former problem is polynomial, the IDP is also polynomial.
**Contributions.** We present a technique to decide EDP or IDP in AC-convergent equational theories. Our approach is based on a “local stability” property inspired by [@AbadiCo2006], instead of proving that the deduction rules are “local” in the sense of [@mcallester] as done in many previous works [@CoLuSh03; @De2006; @Laf07; @Bursucconstraints]. More precisely, the main contributions of this paper are:
- We adapt and refine the technique proposed in [@AbadiCo2006], where deducibility and indistinguishability relations are claimed to be decidable in polynomial time for locally stable theories. First, we changed the definition of locally stable theories, adding normal forms, which are needed to carry out the decidability proofs. Second, we designed a new algorithm to decide IDP in locally stable theories. The algorithm provided in [@AbadiCo2006] is polynomial for the class of subterm theories (Proposition 10 in [@AbadiCo2006]), but the proof does not extend directly to locally stable theories (despite the statement in Proposition 16). Our algorithm relies on solving a restricted case of higher-order AC-matching problem that is used to decide the deduction relation. It is a combination of two standard algorithms: one for solving the DO-ACM problem [@narendran] which has a polynomial bound in our case; and one for solving systems of Linear Diophantine Equations(SLDE), which is polynomial in $\mathbb{Z}$ [@BouConDe; @ClauFor; @frumkin; @papadimitriou]. Using this algorithm we prove that IDP is decidable in polynomial time with respect to the saturated set of terms, for locally stable theories with inverses.
- A decidability result for the EDP for locally stable theories, which extends the work of Tiu and Goré [@TiGo2009]. As an application, we present a strategy to decide IDP for locally stable theories combined with blind signatures. Here, the combination of theories does not need to be locally stable.
In order to get the polynomial decidability result claimed in [@AbadiCo2006] for locally stable theories, we had to restrict to theories that contain, for each $AC$ symbol in the signature, the corresponding inverse. The inverses are necessary when we interpret our term algebra inside the integers $\mathbb{Z}$ to solve SLDE (terms headed by the inverse function will be seen as negative integers). If the theory does not contain inverses, we would have to solve the SLDE for $\mathbb{N}$ which is a well known NP-complete problem.
Preliminaries
=============
Standard rewriting notation and notions are used (e.g. [@baader]). We assume the following sets: a countably infinite set $N$ of *names* (we use $a,b,c, m$ to denote names); a countably infinite set $X$ of *variables* (we use $x,y,z$ to denote variables); and a finite *signature* $\Sigma$, consisting of function names and their arities. We write $arity(f)$ for the arity of a function $f$, and let $ar(\Sigma)$ be the maximal arity of a function symbol in $\Sigma$.
The set of *terms* is generated by the following grammar: $$M,N:= a\,|\, x\,|\, f(M_1,\ldots, M_n)$$ where $f$ ranges over the function symbols of $\Sigma$ and $n$ matches the arity of $f$, $a$ denotes a name in $N$ (representing principal names, nonces, keys, constants involved in the protocol, etc) and $x$ a variable. We denote by $V(M)$ the set of variables occurring in $M$. A message $M$ is *ground* if $V(M)=\emptyset$. The *size* $|M|$ of a term $M$ is defined by $|u|=1$, if $u$ is a name or a variable; and $|f(M_1,\ldots, M_n)|=1+\sum_{i=1}^n|M_i|$.
The set of *positions* of a term $M$, denoted by $\mathcal{P}os(M)$, is defined by $\mathcal{P}os(M):= \{\epsilon\}$, if $M$ is a name or a variable; and $\mathcal{P}os(M):= \{\epsilon\} \cup \bigcup_{i=1}^n\{ip \,|\, p \in \mathcal{P}os(M_i)\}$, if $M=f(M_1, \ldots, M_n)$ where $f\in \Sigma$. The position $\epsilon$ is called the *root* position. The size of $|M|$ coincides with the cardinality of $\mathcal{P}os(M)$. The set of *subterms* of $M$ is defined as $st(M)=\{M|_p \,|\, p \in \mathcal{P}os(M)\}$, where $M|_p$ denotes the subterm of $M$ at position $p$. For a set $\Gamma$ of terms, the notion of subterm can be extended as usual: $st(\Gamma):= \bigcup_{M\in \Gamma}st(M)$. For $p \in \mathcal{P}os(M)$, we denote by $M[t]_p$ the term that is obtained from $M$ by replacing the subterm at position $p$ by $t$.
A term rewriting system (TRS) is a set $\mathcal{R}$ of oriented equations over terms in a given signature. For terms $s$ and $t$, $s\rightarrow_{\mathcal{R}} t$ denotes that $s$ rewrites to $t$ using an instance of a rewriting rule in $\mathcal{R}$. The transitive, reflexive-transitive and equivalence closures of $\rightarrow_{\mathcal{R}}$ are denoted by $\stackrel{+}{\rightarrow}_{\mathcal{R}}, \stackrel{*}{\rightarrow}_{\mathcal{R}}$ and $\stackrel{*}{\leftrightarrow}_{\mathcal{R}}$, respectively. The equivalence closure of the rewriting relation, $\stackrel{*}{\leftrightarrow}_{\mathcal{R}}$, is denoted by $\approx_{\mathcal{R}}$. Given a TRS $\mathcal{R}$ in which some function symbols are assumed to be AC, and two terms $s$ and $t$, $s\rightarrow_{\mathcal{R}\cup AC}t$ if there exists $w$ such that $s=_{AC}w$ and $w\rightarrow_{\mathcal{R}} t$, where $=_{AC}$ denotes equality modulo AC (according to the AC assumption on function symbols). For every term $s$, the set of normal forms $s\downarrow_{\mathcal{R}}$ (closed modulo AC) of $s$ is the set of terms $t$ such that $s\stackrel{*}{\rightarrow}_{\mathcal{R}\cup AC}t$ and $t$ is irreducible for $\rightarrow_{\mathcal{R}\cup AC}$. $\mathcal{R}$ is said to be AC-convergent whenever it is AC-terminating and AC-confluent.
We equip the signature $\Sigma$ with an equational theory $\approx_E$ induced by a set of $\Sigma$-equations $E$, that is, $\approx_E $ is the smallest equivalence relation that contains $E$ and is closed under substitutions and compatible with $\Sigma$-contexts. An equational theory $\approx_E$ is said to be equivalent to a TRS $\mathcal{R}$ whenever $\approx_{\mathcal{R}}\; =\; \approx_E$. An equational theory $\approx_E$ is AC-convergent when it has an equivalent rewrite system $\mathcal{R}$ which is AC-convergent. In the next sections, given an AC-convergent equational theory $\approx_E$, normal forms of terms are computed with respect to the TRS $\mathcal{R}$ associated to $\approx_E$, unless otherwise specified. To simplify the notation we will denote by $E$ the equational theory induced by the set of $\Sigma$-equations $E$. We will denote by $\Sigma_E$ the signature used in the set of equations $E$. The *size* $c_E$ of an equational theory $E$ with an associated TRS $\mathcal{R}$ consisting of rules $\bigcup_{i=1}^k\{l_i\rightarrow r_i\}$ is defined as $c_E=max_{1\leq i\leq k}\{|l_i|,|r_i|, ar(\Sigma)+1\}$. For $\mathcal{R}= \emptyset$, define $ c_E= ar(\Sigma)+1$.
Let $\square$ be a new symbol which does not yet occur in $\Sigma\cup X$. A $\Sigma$-*context* is a term $t\in T(\Sigma, X\cup \{ \square\})$ and can be seen as a term with “holes”, represented by $\square$, in it. Contexts are denoted by $C$. If $\{p_1,\ldots,p_n\}=\{p\in\mathcal{P}os(C) \,|\, C|_p = \square\}$, where $p_i$ is to the left of $p_{i+1}$ in the tree representation of $C$, then $C[T_1\ldots, T_n]:= C[T_1]_{p_1}\ldots [T_n]_{p_n}$. In what follows a context formed using only function symbols in $\Sigma_{E}$ will be called an $E$-*context* to emphasize the equational theory $E$.
A term $M$ is said to be an $E$-*alien* if $M$ is headed by a symbol $f\notin \Sigma_{E}$ or a private name/constant. We write $M==N$ to denote syntactic equality of ground terms.
In the rest of the paper, we use signatures, terms and equational theories to model protocols. *Messages* exchanged between participants of a protocol during its execution are represented by terms. Equational theories and rewriting systems are used to model the cryptographic primitives in the protocol and the algebraic capabilities of an intruder.
Deduction Problem {#sec:locallystable}
=================
Given a set $\Gamma$ that represents the information available to an attacker, we may ask whether a given ground term $M$ may be deduced from $\Gamma$ using equational reasoning. This relation is written $\Gamma \vdash M$ and axiomatised in a natural deduction like system of inference rules.
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Locally Stable Theories
-----------------------
Let $\oplus$ be an arbitrary function symbol in $\Sigma_E$ for an equational theory $E$. We write $\alpha \cdot_{\oplus} M$ for the term $M\oplus \ldots \oplus M$, $\alpha$ times ($\alpha \in \mathbb{N}$). Given a set $S$ of terms, we write $sum_{\oplus}(S)$ for the set of arbitrary sums of terms in $S$, closed modulo $AC$: $$sum_{\oplus}(S)=\{(\alpha_1 \cdot_{\oplus}T_1)\oplus \ldots\oplus(\alpha_n \cdot_{\oplus}T_n)\,|\, \alpha_i \geq 0, T_i\in S\}$$ Define $sum(S)= \bigcup_{i=1}^k sum_{\oplus_i}(S)$, where $\oplus_1,\ldots, \oplus_k$ are the AC-symbols of the theory.
For a rule $l\rightarrow r\in \mathcal{R}$ and a substitution $\theta$ such that
- either there exists a term $s_1$ such that $s=_{AC}s_1$, $s_1=_{AC}l\theta$ and $t=r\theta$;
- or there exist terms $s_1$ and $s_2$ such that $s=_{AC}s_1 \oplus s_2$, $s_1=_{AC}l\theta$ and $t=_{AC}r\theta \oplus s_2$.
we write $s\stackrel{h}{\rightarrow}t$ and say that the reduction occurs in the head.
As in [@AbadiCo2006] we associate with each set $\Gamma$ of messages, a set of subterms in $\Gamma$ that may be deduced from $\Gamma$ by applying only “small” contexts. The concept of small is arbitrary — in the definition below, we have bound the size of an $E$-context $C$ by $c_E$ and the size of $C'$ by $c_E^2$, but other bounds may be suitable. Notice that limiting the size of an $E$-context by $c_E$ makes the context big enough to be an instance of any of the rules in the TRS $\mathcal{R}$ associated to $E$.
\[locallystable\] An AC-convergent equational theory $E$ is *locally stable* if, for every finite set $\Gamma=\{M_1, \ldots,M_n\}$, where the terms $M_i$ are ground and in normal form, there exists a finite and computable set $sat(\Gamma)$, closed modulo $AC$, such that
1. $M_1, \ldots, M_n \in sat(\Gamma)$;\[rule1\]
2. if $M_1,\ldots,M_k \in sat(\Gamma)$ and $f(M_1,\ldots,M_k)\in st(sat(\Gamma))$ then $f(M_1,\ldots,M_k)\in sat(\Gamma)$, for $f\in \Sigma_E$;\[rule2\]
3. if $C[S_1, \ldots,S_l]\stackrel{h}{\rightarrow}M$, where $C$ is an $E$-context such that $|C|\leq c_{E}$, and $S_1, \ldots, S_l \in$ $sum_{\oplus}(sat(\Gamma))$, for some $AC$ symbol $\oplus$, then there exist an $ E$-context $C'$, a term $M'$, and terms $S_1', \ldots, S_k' \in sum_{\oplus}(sat(\Gamma))$, such that $|C'|\leq c_{E}^2$, and $M\stackrel{*}{\rightarrow}_{\mathcal{R}\cup AC}M'=_{AC}C'[S_1', \ldots, S_k']$;\[rule3\]
4. if $M\in sat(\Gamma)$ then $M\downarrow\in sat(\Gamma)$.\[rule4\]
5. if $M\in sat(\Gamma)$ then $\Gamma \vdash M$.\[rule5\]
Notice that the set $sat(\Gamma)$ may not be unique. Any set $sat(\Gamma)$ satisfying the five conditions is adequate for the results.
The addition of rule 4 in the Definition \[locallystable\] is necessary to prove case 1b of Lemma \[lemma:epcloseness\], where the rewriting reduction occurs in a term $M_i\in sat(\Gamma)$ in a position different from the “head”. Normal forms are strictly necessary in the set $sat(\Gamma)$, they are essential to lift the applications of rewriting rules in the head of “small” contexts to applications of rewriting rules in arbitrary positions of “small” contexts. With this additional condition, Lemma 11 in [@AbadiCo2006] can also be proved. This fact was confirmed via personal communication with the second author of [@AbadiCo2006].
The lemma and the corollary below, adapted from [@AbadiCo2006], are used in the proof of Theorem \[theorem:epdecidability\].
\[lemma:epcloseness\] Let $E$ be a locally stable theory and $\Gamma=\{M_1,\ldots,M_n\}$ a set of ground terms in normal form. For every $E$-context $C_1$, for every $M_i \in sat(\Gamma)$, for every term $T$ such that $C_1[M_1,\ldots,M_k]\rightarrow_{\mathcal{R}\cup AC} T$, there exist an $E$-context $C_2$, and terms $M_i' \in sat(\Gamma)$, such that $T \stackrel{*}{\rightarrow}_{\mathcal{R}\cup AC} C_2[M_1', \ldots, M_l']$.
Suppose that $C_1[M_1,\ldots,M_k]\rightarrow_{AC}T$, for an $E$-context $C_1$ and $M_i\in\,sat(\Gamma)$. The proof is divided in two cases:
1. The reduction happens inside one of the terms $M_i$:
1. if $M_i\stackrel{h}{\rightarrow}M_i'$ then by definition of $sat(\Gamma)$ (since $E$ is locally stable), there exist an $E$-context $C$ such that $|C|\leq c_E^2$ and $M_i'\stackrel{*}{\rightarrow}C[S_1,\ldots,S_l]$ where $S_j\in sum_{\oplus}(sat(\Gamma))$.
Each $S_j\in sum_{\oplus}(sat(\Gamma))$ is of the form $S_j=(\alpha_1\cdot_{\oplus}M_{j_1})\oplus \ldots\oplus (\alpha_n \cdot_{\oplus}M_{j_n}),$ for $M_{j_k}\in sat(\Gamma)$. That is, $S_j=C_j[M_{j_1},\ldots, M_{j_k}]$, for $1\leq j\leq l$. Therefore, $$\begin{split}
C_1[M_1,\ldots, M_i, \ldots,M_k]\stackrel{h}{\rightarrow}C_1[M_1,\ldots, M'_i, \ldots,M_k]&\stackrel{*}{\rightarrow}_{AC}C_1[M_1,\ldots, C[S_1,\ldots, S_l], \ldots,M_k]\\
&=_{AC}C_2[M_1^{''},\ldots, M_s^{''}],
\end{split}$$ where $M_t^{''} \in sat(\Gamma)$, for $1\leq t\leq s$.
2. if $M_i\rightarrow_{AC}M_i'$ in a position different from “head”, then \[case:correction\] $$C_1[M_1,\ldots, M_i, \ldots,M_k]\rightarrow C_1[M_1,\ldots, M'_i, \ldots,M_k]\stackrel{*}{\rightarrow}_{AC}C_1[M_1,\ldots, M_i\downarrow, \ldots,M_k].$$ By case 4 in Definition \[locallystable\], $M_i\downarrow \in sat(\Gamma)$.
2. The case where the reduction does not occur inside the terms $M_i$: this case if very technical and will be omitted here. The complete proof can be found in the extended version of this paper.
As a consequence we obtain the following Corollary:
\[corollary:epcloseness\] Let $E$ be a locally stable theory. Let $\Gamma=\{M_1,\ldots,M_n\}$ be a set of ground terms in normal form. For every $E$-context $C_1$, for every $M'_i\in sat(\Gamma)$, for every $T$ in normal form such that $C_1[M'_1,\ldots,M'_k]\stackrel{*}{\rightarrow}_{\mathcal{R}\cup AC}T$, there exist an $E$-context $C_2$ and terms $M_j{''}\in sat(\Gamma)$ such that $T=_{AC}C_2[M^{''}_1,\ldots,M_l^{''}]$.
The proof is the same as in [@AbadiCo2006].
In the following we show that any term $M$ deducible from $\Gamma$ is equal modulo AC to an $E$-context over terms in $sat(\Gamma)$.
\[lemma:deductioncontext\] Let $E$ be a locally stable theory. Let $\Gamma =\{M_1,\ldots,M_n\}$ be a finite set of ground terms in normal form, and $M$ be a ground term in normal form. Then $\Gamma \vdash M$ if and only if there exist an $E$-context $C$ and terms $M'_1, \ldots,M'_k\in sat(\Gamma)$ such that $M=_{AC}C[M^{'}_1,\ldots,M^{'}_n]$.
The proof is the same as in [@AbadiCo2006].
As a consequence of the previous results decidability of IDP for locally stable theories is obtained:
\[theo:IDPdecidable\] The Intruder Deduction Problem is decidable for locally stable theories.
In the next section we will provide a complexity bound for the decidability of the intruder deduction problem for a restricted case of locally stable theories.
Locally Stable Theories with Inverses
=====================================
In order to obtain the polynomial complexity bound of our decidability algorithm we will need to consider the existence of inverses for each $AC$ symbol in the signature of our equational theory. Our algorithm will rely on solving systems of linear Diophantine equations over $\mathbb{Z}$ and the inverses will be interpreted as *negative integers*.
(\*) *In the following results, let $E$ be a locally stable theory whose signature $\Sigma_E$ contains, for each $AC$ function symbol $\oplus$, its corresponding *inverse* $i_{\oplus}$.*
That is, the following results are related to equational theories $E$ containing the following equation: $$x\oplus i_{\oplus} (x) = e_{\oplus}$$ for each AC-symbol $\oplus$ in $\Sigma_E$, where $i_{\oplus}$ is the unary function symbol representing the inverse of $\oplus$ and $e_{\oplus}$ is the corresponding neutral element.
\[def:locallystableinverses\] An AC-convergent equational theory $E$ satisfying (\*) is *locally stable* if, for every finite set $\Gamma=\{M_1, \ldots,M_n\}$, where the terms $M_i$ are ground and in normal form, there exists a finite and computable set $sat(\Gamma)$, closed modulo $AC$, such that
1. $M_1, \ldots, M_n \in sat(\Gamma)$, $e_{\oplus} \in sat(\Gamma)$ for each $\oplus \in \Sigma_{E}$;
2. if $M_1,\ldots,M_k \in sat(\Gamma)$ and $f(M_1,\ldots,M_k)\in st(sat(\Gamma))$ then $f(M_1,\ldots,M_k)\in sat(\Gamma)$, for $f\in \Sigma_E$;
3. if $C[S_1, \ldots,S_l]\stackrel{h}{\rightarrow}M$, where $C$ is an $E$-context such that $|C|\leq c_{E}$, and $S_1, \ldots, S_l \in sum_{\oplus}(sat(\Gamma))$, for some $AC$ symbol $\oplus$, then there exist an $ E$-context $C'$, a term $M'$, and terms $S_1', \ldots, S_k' \in sum_{\oplus}(sat(\Gamma))$, such that $|C'|\leq c_{E}^2$, and $M\stackrel{*}{\rightarrow}_{\mathcal{R}\cup AC}M'=_{AC}C'[S_1', \ldots, S_k']$;
4. if $M\in sat(\Gamma)$ then $M\downarrow\in sat(\Gamma)$.
5. if $M\in sat(\Gamma)$ then $i_{\oplus}(M)\downarrow\in sat(\Gamma)$ for each AC symbol $\oplus$ in $E$.
6. if $M\in sat(\Gamma)$ then $\Gamma \vdash M$.
Based on a well-founded ordering over the symbols in the language, we prove that a restricted case of higher-order AC-matching (“is there an $E$-context $C$ such that $M=_{AC}C[M_1,\ldots,M_k]$ for some $M_1,\ldots,M_k\in sat(\Gamma)$?”) can be solved in polynomial time in $|sat(\Gamma)|$ and $|M|$. This AC-matching problem is solved using the DO-ACM (Distinct-Occurrences of AC-matching) [@narendran], where every variable in the term being matched occurs only once. In addition, we also use a standard and polynomial time algorithm for solving SLDE over $\mathbb{Z}$ [@BouConDe; @ClauFor; @frumkin; @papadimitriou].
To facilitate the description of the algorithm below we have considered only one AC-symbol $\oplus$ whose corresponding inverse will be denoted by $i$. The proof can be extended similarly for theories with multiple AC-symbols each one with its corresponding inverse.
\[lemma:acmatching\] Let $E$ be a locally stable theory satisfying (\*), $\Gamma=\{M_1,\ldots,M_n\}$ a finite set of ground messages in normal form and $M$ a ground term in normal form. Then the question of whether there exists an $E$-context $C$ and $T_1,\ldots,T_k\in sat(\Gamma)$ such that $M=_{AC}C[T_1,\ldots,T_k]$ is decidable in polynomial time in $|M|$ and $|sat(\Gamma)|$.
Given $\Gamma$, we construct the set $sat(\Gamma)=\{T_1,\ldots, T_s\}$, which is computable and finite by Definition \[locallystable\]. We can then check whether $M=^?_{AC}C[T_1,\ldots,T_k]$ for some $E$-context $C$ and terms $T_1,\ldots,T_k\in sat(\Gamma)$ using the following algorithm which is divided in its main component A), and procedures B) and C) for reducing linear Diophantine equations and selecting $T_i$’s from $sat(\Gamma)$, respectively.
A\) **Algorithm 1.**
1. For all positions $p$ in $M$ headed by $\oplus$ starting from the longest positions in decreasing order (positions seen as sequences) solve the *system of linear Diophantine equations* (see part B below) for $M|_p$ with $sat(\Gamma)\cup S$, where $S$ is built incrementally from $sat(\Gamma)$, starting with $S_0=\emptyset$, including all $M|_p$ that have solutions. In other words:
Let $\mathcal{P}'=\{p_1,\ldots,p_t\}$ be the set of positions of $M$ such that $M|_p$ is headed with $\oplus$, organised in decreasing order. For each $p_j \in \mathcal{P}'$ let $M|_{p_j}$ be the subterm of $M$ such that $$M|_{p_j}=n_{j_1}\oplus \ldots \oplus n_{j_{kj}}\,\, (j=1,\ldots, t)$$
Recursively find, but suppressing step 1 in this recursive call, solutions for the arguments $n_{j_{i_1}},\ldots, n_{j_{i_l}}$ of $M|_{p_j}$ with $n_{j_{im}} \in \{n_{j_1},\ldots,n_{j_{k_j}} \}$ with respective $E$-contexts $C_{j_{i_1}},\ldots, C_{j_{i_l}}$ such that $$n_{j_{i_m}}=C_{j_{i_m}}[T_1,\ldots,T_{s_{i_m}}]$$ where $T_q \in sat(\Gamma)\cup S_{j-1}$, $q = 1,\ldots, s_{i_m}$.
Then one checks satisfiability of the SLDE generated from $M|_{p_j}$ and $sat(\Gamma)\cup S_{j-1} \cup \{n_{j_{i_1}},\ldots, n_{j_{k_l}}\}$ (see steps B and C).
If there is a solution then $S_j:= S_{j-1} \cup \{n_{j_{i_1}},\ldots, n_{j_{k_l}}\}\cup \{M|_{p_j}\}$
2. Let $S:= S_t$. Classify the terms in $sat(\Gamma)\cup S$ by size.
3. For each term $T_i \in sat(\Gamma) \cup S$ (from terms of maximal size to terms of minimal size) check:
- For each position $q\in \mathcal{P}os(M)$ such that $T_i=_{AC}M|_q$ do
Check whether the path between $T_i$ and the root of $M$ contains a $\oplus$:
- if NOT, then delete $M|_q$ from $M $ and move to $T_{i+1}$.
- if YES (there is a $\oplus$) then $M$ has a subterm $ N$ such that $N=n_1\oplus \ldots \oplus n_j[T_i]\oplus \ldots \oplus n_k$ and $N$ cannot be constructed from $sat(\Gamma)\cup S$. Therefore, $M$ cannot be written as an $E$-context with terms from $sat(\Gamma)$.
4. Check whether the remaining part of $M$ still contains $E$-aliens. If it is not the case, we have found an $E$-context $C$ and terms $M_1,\ldots,M_k \in sat(\Gamma)$ and $M=_{AC}C[M_1,\ldots,M_k]$; otherwise such an $E$-context does not exist.
B\) **Reduction to linear Diophantine equations.**
First, notice that, for each position $p$ such that $M|_p$ is headed with $\oplus$ we have $$\label{Eq:eq1}
M|_p=\alpha_1 m_1\oplus \ldots \oplus \alpha_r m_r\, , \, \alpha_j \in \mathbb{N}$$ where $m_j$ is not headed with $\oplus$ and $\alpha_jm_j$ counts for $\underbrace{m_j\oplus \ldots \oplus m_j}_{\alpha_j-times}$.
We want to prove that there are $\beta_1,\ldots,\beta_q \in \mathbb{N}$ such that $$\label{Eq:eq2}
\beta_1T_1\oplus \ldots \oplus \beta_qT_q =_{AC} M|_p=\alpha_1 m_1\oplus \ldots \oplus \alpha_r m_r$$ This AC-equality is only possible when $T_i=\gamma_{1i}m_1\oplus \ldots\oplus \gamma_{ri}m_r $ for each $i$, $1\leq i \leq q\leq s$ and $\gamma_{j_i}\in \mathbb{N}$.
That is, $\beta_1T_1\oplus \ldots \oplus \beta_qT_q=_{AC}\alpha_1 m_1\oplus \ldots \oplus \alpha_r m_r$ if and only if $$\begin{split}
&\beta_1(\gamma_{1_1}m_1\oplus \ldots\oplus \gamma_{r_1}m_r)\oplus \beta_2(\gamma_{1_2}m_1\oplus \ldots\oplus \gamma_{r_2}m_r)\oplus\ldots \\
&\ldots\oplus \beta_q(\gamma_{1_q}m_1\oplus \ldots\oplus \gamma_{r_q}m_r)=\alpha_1 m_1\oplus \ldots \oplus \alpha_r m_r
\end{split}$$
if and only if $$\begin{split}
&(\gamma_{1_1}\beta_1 \oplus \gamma_{1_2}\beta_2 \ldots\oplus \gamma_{1_q}\beta_q)m_1\oplus (\gamma_{2_1}\beta_1 \oplus \gamma_{2_2}\beta_2 \ldots\oplus \gamma_{2_q}\beta_q)m_2\oplus \ldots\\
&\ldots(\gamma_{r_1}\beta_1 \oplus \gamma_{r_2}\beta_2 \ldots\oplus \gamma_{r_q}\beta_q)m_r=\alpha_1 m_1\oplus \ldots \oplus \alpha_r m_r
\end{split}$$ if and only if
$$S=\left\{
\begin{split}
\gamma_{1_1}\beta_1 \oplus \gamma_{1_2}\beta_2 \ldots\oplus \gamma_{1_q}\beta_q&=\alpha_1\\
\gamma_{2_1}\beta_1 \oplus \gamma_{2_2}\beta_2 \ldots\oplus \gamma_{2_q}\beta_q&=\alpha_2\\
\vdots \hspace{1cm} &\\
\gamma_{r_1}\beta_1 \oplus \gamma_{r_2}\beta_2 \ldots\oplus \gamma_{r_q}\beta_q&=\alpha_r\\
\end{split}
\right.$$
where $S$ is a system of linear Diophantine equations over $\mathbb{Z}$ which can be solved in polynomial time [@BouConDe; @ClauFor; @frumkin; @papadimitriou].
We will interpret the equations \[Eq:eq1\] and \[Eq:eq2\] inside integer arithmetic. If there exists an index $j$ such that $m_j= i(m_j')$ and $m_j' $ is not headed with $i$ then $\alpha_j m_j=\alpha_j (i(m_j'))$ and we will take it as $(-\alpha_j)m_j'$. Therefore, we can take $\alpha_j \in \mathbb{Z}$, for all $j$. We can use the same reasoning to conclude that $\beta_j \in \mathbb{Z}$, for all $1\leq j\leq q$ and $\gamma_{j_i} \in \mathbb{Z}$, for all $i$ and $j$.
C\) **Selecting the $T_j's$ from $sat(\Gamma)$.**
For each $T_i \in sat(\Gamma)$, $1\leq i\leq s$ we want to check if $T_i=\gamma_{1_i}m_1\oplus \ldots\oplus \gamma_{r_i}m_r $.
**Algorithm 2:**
For each $T_i \in sat(\Gamma)$, $1\leq i \leq s$, solve the equation $T_i \oplus x_i =_{AC} \alpha_{1}m_1 \oplus \ldots\oplus \alpha_{r}m_r $ where $x_i$ is a fresh variable.
Since the $T_i's$ and $M$ are ground terms, this equation can be seen as an instance of the DO-ACM matching problem which can be solved in time $\mathcal{O}(|T_i\oplus x_i|.|M|_p|)$ [@narendran]. If there exists $T_i \in sat(\Gamma)$ such that $T_i=\gamma_{1_i}^*m_1\oplus \ldots\oplus \gamma_{r_i}^*m_r\oplus u $, where $u$ is not empty, $\gamma_{i_j}^*\in \mathbb{N} $ and the **Algorithm 2** can no longer be applied then $T_i$ will not be selected.
Notice that each step of the algorithm can be done in polynomial time in $|M|$ and $|sat(\Gamma)|$. Therefore, the whole procedure is polynomial in $|M|$ and $sat(\Gamma)$.
For the proof we can adopt an ordering in which, for instance, variables are smaller than constants, constants smaller than function symbols, and function symbols are also ordered, but other suitable order can be used. Terms are compared by the associated lexicographical ordering built from this ordering on symbols.
We consider the theory of Abelian Groups where the signature is $\Sigma_{AG}=\{+,0,i\}$ for $i$ the inverse function and $+$ the AC group operator. The equational theory $E_{AG}$ is: $$E_{AG}=\left\{
\begin{array}{l@{\hspace{1cm}}c @{\hspace{1cm}}r}
\begin{array}{rcl}
x+(y+z)&=&(x+y)+z\\
x+y&=&y+x\\
i(x+y)&=&i(y)+i(x)\\
\end{array}
&
\begin{array}{rcl}
x+0&=&x\\
x+i(x)&=&0
\end{array}
&
\begin{array}{rcl}
i(i(x))&=&x\\
i(0)&=&0
\end{array}
\end{array}
\right.$$ We define $\mathcal{R}_{AG} $ by orienting the equations from left to right (excluding the equations for associativity and commutativity). $\mathcal{R}_{AG}$ is AC-convergent. The size $c_{E_{AG}}$ of the theory is at least 5. In the following prove that $E_{AG}$ is locally stable with inverses for finite models, i.e., we define a set $sat(\Gamma)$ satisfying the properties in the Definition \[locallystable\]. For a given set $\Gamma=\{M_1,\ldots,M_k\}$ of ground terms in normal form, $sat(\Gamma)$ is the smallest set such that:
1. $M_1,\ldots,M_k\in sat(\Gamma)$;
2. $M_1,\ldots,M_k\in sat(\Gamma)$ and $f(M_1,\ldots,M_k)\in st(sat(\Gamma))$ then $f(M_1,\ldots,M_k)\in sat(\Gamma)$, $f\in \Sigma_{AG}$;
3. if $M_i,M_j \in sat(\Gamma)$ and $M_i+M_j\stackrel{h}{\rightarrow}M$ via rule $x+i(x)\rightarrow 0$ then $M\downarrow\in sat(\Gamma)$;
4. if $M_j \in sat(\Gamma)$ then $i(M_j)\downarrow \in sat(\Gamma)$;
5. if $M_i=_{AC}M_j$ and $M_i \in sat(\Gamma)$ then $M_j \in sat(\Gamma).$
The set $sat(\Gamma)$ defined for Finite Abelian Groups is finite.
Although it was said in [@AbadiCo2006] that the theory of Abelian Groups is locally stable, no proof of such fact was found in the literature. With the proviso that the Abelian Group under consideration is finite, we have demonstrated that $|sat(\Gamma)|$ is exponential in the size of $|\Gamma|$.
These results give rise to the decidability of deduction for locally stable theories. Notice that polynomiality on $|sat(\Gamma)|$ relies on the use of the AC-matching algorithm proposed in Lemma \[lemma:acmatching\]. Unlike [@AbadiCo2006], we do not need to compute of the congruence class modulo AC of $M$ (which may be exponential). This gives us a slightly different version of the decidability theorem:
\[theorem:epdecidability\] Let $E$ be a locally stable theory satisfying (\*). If $\Gamma=\{M_1,\ldots,M_n\}$ is a finite set of ground terms in normal form and $M$ is a ground term in normal form, then $\Gamma\vdash M$ is decidable in polynomial time in $|M|$ and $|sat(\Gamma)|$.
The result follows directly from Lemmas \[lemma:acmatching\] and \[lemma:deductioncontext\].
In the following example we consider the *Pure AC-theory* which can be proven to be locally stable but does not contain the inverse of the AC-symbol $+$.
$\Sigma_{AC}$ contains only constant symbols, the AC-symbol $\oplus$ and the equational theory contains only the $AC$ equations for $\oplus$: $$AC=\left\{
\begin{array}{l@{\hspace{3cm}}r}
x\oplus y= y\oplus x
&
x\oplus(y\oplus z) =(x\oplus y)\oplus z
\end{array}
\right\}$$ In this case, $E=AC$ and $ \mathcal{R}=\emptyset$ is the AC-convergent TRS associated to $E$. Let $\Gamma=\{M_1,\ldots,M_k\}$ be a finite set of ground terms in normal form. Let us define $sat(\Gamma)$ for the pure $AC$ theory as the smallest set such that
1. $M_1,\ldots, M_k\in sat(\Gamma)$;
2. if $M_i,M_j\in sat(\Gamma)$ and $M_i\oplus M_j \in st(sat(\Gamma))$ then $M_i\oplus M_j\in sat(\Gamma)$.
3. if $M_i=_{AC}M_j$ and $M_i\in sat(\Gamma)$ then $M_j\in sat(\Gamma)$.
The set $sat(\Gamma)$ is finite since we add only terms whose size is smaller or equal than the maximal size of the terms in $\Gamma$. It is easy to see that the set $sat(\Gamma)$ satisfies the rules \[rule1\],\[rule2\], \[rule4\] and \[rule5\]. Since $\mathcal{R}=\emptyset$ it follows that \[rule3\] is also satisfied. Therefore, $AC$ is locally stable.
*The size of $sat(\Gamma)$:*
- Steps 1 and 2: only subterms in $sat(\Gamma)$ are added.
- Step 3: for each $M_i\in sat(\Gamma)$ add $M_j=_{AC}M_i\in sat(\Gamma)$. Notice that the number of terms added in $sat(\Gamma)$, in this case, depends on the number of occurrences of $\oplus$ in $M_i$. Suppose that $M_i$ contains $n$ occurrences of $\oplus$: $$M_i=M_{i_1}\oplus \ldots \oplus M_{i_{n+1}}.$$ There are $(n+1)!$ terms $M_j$ such that $M_1=_{AC}M_j$.
Suppose that each $M_i$ in $\Gamma$ contains $n_i$ occurrences of $\oplus$.Then, $|M_i|=\displaystyle\sum_{j=1}^{n_i+1}|M_{i_j}|+n_i.$ Let $n=\max_{1\leq i\leq k}\{n_i\}$. There exists an index $r$ such that $M_r$ contains $n_r=n$ occurrences of $\oplus$. Since $|\Gamma|=\displaystyle\sum_{i=1}^k|M_i|$ it follows that $ n \leq |M_r|-\displaystyle\sum_{j=1}^{n+1}|M_{r_j}|\leq |\Gamma|.$ Then the number of terms added in step 3 is $\displaystyle\sum_{i=1}^k (n_i+1)!\leq (n+1)! \cdot k \leq (|\Gamma|+1)!\cdot k .$
In this case one can adapt Lemma \[lemma:acmatching\] such that the algorithm would rely on solving systems of linear Diophantine equations over $\mathbb{N}$ which is NP-complete [@papadimitriou]. Therefore, the complexity of IDP for pure AC would be exponential, agreeing with previous results [@lafourcade].
Elementary Deduction Problem for Locally Stable Theories
========================================================
To establish necessary concepts for the next results, we recall the well-known translation between natural deduction and sequent calculus systems to model the IDP as a proof search in sequent calculus, whose properties (such as cut or subformula) facilitate the study of decidability of deductive systems. For an AC-convergent equational theory E, the System $\mathcal{N}$ in Table \[equationalreasoning\] is equivalent to the $(id)$-rule of the sequent calculus (Table \[DeductionRulesForIntruder\]) introduced in [@TiGo2009]:
Consequently, IDP for System $\mathcal{N}$ is equivalent to the *Elementary Deduction Problem*:
Given an AC-convergent equational theory $E$ and a sequent $\Gamma \vdash M$ ground and in normal form, the *elementary deduction problem* (EDP) for $E$, written $\Gamma \Vdash_{E}M$, is the problem of deciding whether the $(id)$-rule is applicable in $\Gamma\vdash M$.
The theorem below decides EDP for locally stable theories :
\[theorem:edpisptime\] Let $E$ be a locally stable equational theory satisfying (\*). Let $\Gamma \vdash M$ be a ground sequent in normal form. The *elementary deduction problem* for the theory $E$ ($\Gamma \Vdash_E M$) is decidable in polynomial time in $|sat(\Gamma)|$ and $|M|$.
By Lemma \[lemma:acmatching\], the problem whether $M=_{AC}C[M_1,\ldots,M_k]$ for an $E$-context $C$ and terms $M_1,\ldots,M_k\in sat(\Gamma)$ is decidable in polynomial time in $|sat(\Gamma)|$ and $|M|$. If $M=_{AC}C[M_1,\ldots,M_k]$ for an $E$-context $C$ and terms $M_1,\ldots,M_k\in sat(\Gamma)$ then there exist an $ E$-context $C'$ and terms $M'_1,\ldots,M'_n\in \Gamma$ such that $C[M'_1,\ldots,M'_n]\stackrel{*}{\rightarrow}_{\mathcal{R}\cup AC} M.$ It is enough to observe that for all $T\in sat(\Gamma)$, $T$ can be constructed from the terms in $\Gamma$.
If there is no $E$-context $C$ and terms $M_1,\ldots,M_k\in sat(\Gamma)$ such that $M=_{AC}C[M_1,\ldots,M_k]$ then, by Corollary \[corollary:epcloseness\], there are no $\textsf E$-context and terms $M'_1, \ldots, M'_t\in sat(\Gamma)$ such that $C[M'_1, \ldots, M'_t]\stackrel{*}{\rightarrow}_{\mathcal{R}\cup AC} M.$ Therefore, there is no $E$-context $C''$ and terms $M''_1,\ldots,M''_l\in \Gamma$ such that $C''[M''_1, \ldots, M''_l]\stackrel{*}{\rightarrow}_{\mathcal{R}\cup AC} M.$ Thus, the EDP for $E$ is decidable in polynomial time in $|sat(\Gamma)|$ and $|M|$.
Extension with Blind Signatures {#extension}
-------------------------------
Blind signature is a basic cryptographic primitive in e-cash. This concept was introduced by David Chaum in [@Chaum] to allow a bank (or anyone) sign messages without seeing them. David Chaum’s idea was to use this homomorphic property in such a way that Alice can multiply the original message with a random (encrypted) factor that will make the resulting image meaningless to the Bank. If the Bank agrees to sign this random-looking data and return it to Alice, she is able to divide out the blinding factor such that the Bank’s signature in the original message will appear.
Given a locally stable equational theory $E$, we extend the signature $\Sigma_{E}$ with $\Sigma_C$, a set containing function symbols for “constructors” for blind signatures, in order to obtain decidability results for the extension of the IDP for System $\mathcal{N}$ taking into account some rules for blind signatures.
### Extended Syntax {#extended-syntax .unnumbered}
The signature $\Sigma$ consists of function symbols and is defined by the union of two sets: $\Sigma =\Sigma_C \cup \Sigma_{E}$ ( with $\Sigma_{\textsf{E}}\cap \Sigma_C=\emptyset$), where $$\Sigma_C=\left\{\textsf{pub}(\_), \textsf{sign}(\_\; ,\_), \textsf{blind}(\_\; ,\_), \left\{\_\right\}_{\_}, <\_\; ,\_>\right\}$$ represents the *constructors*, whose interpretations are: $\textsf{pub}(M)$ gives the public key generated from a private key $M$; $\textsf{blind}(M,N)$ gives $M$ encrypted with $N$ using blinding encryption; $\textsf{sign}(M,N)$ gives $M$ signed with a private key $N$; $\left\{M\right\}_N$ gives $M$ encrypted with the key $N$ using Dolev-Yao symmetric encryption; $\langle M,N\rangle$ constructs a pair of terms from $M$ and $N$. Then the extended grammar of the set of *terms* or messages is given as $$M,N \;:= a \;|\; x \;| f(M_1,\ldots, M_n)|\textsf{pub}(M) | \textsf{sign}(M,N) | \textsf{blind}(M,N)|\left\{M\right\}_N|\langle M, N\rangle$$
Notice that, with the extension an $E$-alien term $M$ is a term headed with $f\in \Sigma_C$ or $M$ is a private name/constant. An $E$-alien subterm $M$ of $N$ is said to be an $E$-*factor* of $N$ if there is another subterm $F$ of $N$ such that $M$ is an immediate subterm of $F$ and $F$ is headed by a symbol $f\in\Sigma_{E}$. This notion can be extended to sets in the obvious way: a term $M$ is an $E$-factor of $\Gamma$ if it is an $E$-factor of a term in $\Gamma$. These notions were introduced in [@TiGo2009].
The operational meaning of each constructor will be defined by their corresponding inference rules in the sequent calculus to be described.
### Extending the EDP to Model Blind Signatures {#extending-the-edp-to-model-blind-signatures .unnumbered}
Following the approach proposed in [@TiGo2009], we extend EDP with blind signatures using the sequent calculus $\mathcal{S}$ described in Table \[DeductionRulesForIntruder\]. In this way, we can model intruder deduction for the combination of a locally stable theory $E$ with blind signatures in a modular way: the theory $E$ is used in the $id$ rule, while blind signatures are modelled with additional deduction rules. As shown below, this approach has the advantage that we can derive decidability results for the intruder deduction problem without needing to prove that the combined theory is locally stable (in contrast with the results in the previous section and in [@AbadiCo2006]).
------------------------------------------------------------------------
------------------------------------------------------------------------
Analysing the system $\mathcal{S}$ one can make the following observations:
1. The rules $p_L, e_L$,$ \textsf{sign}_L$,$ \textsf{blind}_{L1}$, $\textsf{blind}_{L2}$ and $acut$ are called *left rules* with $\langle M,N\rangle$, $\{M\}_K$, $\textsf{sign}(M,K)$, $\textsf{blind}(M,K)$, $\textsf{sign}(\textsf{blind}((M,R),K)$ and $A$ as *principal term*, respectively. The rules $p_R, e_R,\textsf{sign}_R$ and $ \textsf{blind}_{R}$ are called *right rules*.
2. The rule $(acut)$, called *analytic cut* is necessary to prove cut rule *admissibility*. A complete proof can be found in [@TiGo2009; @nantesayala].
Considerations about locally stable theories with blind signatures:
1. All the results proved on Section \[sec:locallystable\] are valid under this extension with blind signatures since the results depend only on the equational theory $E$ and on the symbols in $\Sigma_E$. Unlike example 5.2.4 [@AbadiCo2006], the theory of Blind Signatures is not considered as part of the equational theory, the functions are abstracted in the set of constructors with the operational meaning represented in the sequent calculus.
2. In [@TiGo2009] it is shown that the intruder deduction problem for $\mathcal{S}$ is *polynomially reducible* to the EDP for $E$: *if the EDP problem in $E$ has complexity $f(m)$ then the deduction problem $\Gamma\vdash M$ in $\mathcal{S}$ has complexity $O(n^k.f(n))$ for some constant $k$*[^5]. This result was proved for an AC-convergent equational theory $E$ containing only one $AC$ symbol and extended to finite a combination of disjoint AC-convergent equational theories each one containing only one AC-symbol.
3. In [@nantesayala], it was proved that deduction in $\mathcal{S}$ reduces polynomially to $EDP$ in the case of the AC-convergent equational theory ${\textsf{EP}}$, which contains three different AC-symbols and rules for exponentiation and cannot be split into disjoint parts.
As a consequence of the results mentioned in the above remark, we can state the following result:
\[theo:eppblind\] Let $E$ be a locally stable theory satisfying (\*) containing only one AC-symbol or formed by a finite and disjoint combination of AC-symbols. Let $\Gamma$ a finite set of ground terms in normal form and $M$ a ground term in normal form. The IDP for the theory $E$ combined with blind signatures ($\Gamma\vdash M$) is decidable in polynomial time in $|sat(\Gamma)|$ and $|M|$.
Conclusion
==========
We have shown that the IDP is decidable for locally stable theories. In order to obtain the polynomiality result, a restriction on the equational theory is necessary: the theory must contain inverses of all AC-symbols. We have proposed an algorithm to solve a restricted case of higher-order AC-matching by using the DO-ACM matching algorithm combined with an algorithm to solve linear Diophantine equations over $\mathbb{Z}$. Based on this algorithm, we obtain a polynomial decidability result for IDP for a class of locally stable theories with inverses. Our algorithm does not need to compute the set of normal forms modulo AC of a given term (which may be exponential). Therefore, we can conclude that the deducibility relation is decidable in polynomial time for a very restricted class of equational theories, it does not work for all locally stable theories as [@AbadiCo2006] has claimed. It also decides the IDP for the combination of locally stable theories with the theory of blind signatures, using a translation between natural deduction and sequent calculus.
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[^1]: Author partially supported by CNPq.
[^2]: Corresponding author. Author supported by CNPq
[^3]: Work supported by grants from the CNPq/CAPES *Science without Borders* programme and FAPDF PRONEX.
[^4]: With this simple modification, the correctness proof in [@AbadiCo2006] can also be carried out, fixing a gap in Lemma 11.
[^5]: Here, $m$ is the size of the input of EDP and $n$ is the cardinality of the set $St(\Gamma\cup \{M\})$ defined in [@TiGo2009]
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- |
$^{a}$Jianwen Huang $^{b}$Wendong Wang$^{b,c}$Feng Zhang $^{c}$Jianjun Wang[^1]\
[$^{a}$School of Mathematics and Statistics, Tianshui Normal University, Tianshui, 741001, China]{}\
[$^{b}$School of Mathematics and Statistics, Southwest University, Chongqing, 400715, China]{}\
[$^c$College of Artificial Intelligence, Southwest University, Chongqing, 400715, China]{}
title: '**The perturbation analysis of nonconvex low-rank matrix robust recovery**'
---
> [**Abstract.**]{} In this paper, we bring forward a completely perturbed nonconvex Schatten $p$-minimization to address a model of completely perturbed low-rank matrix recovery. The paper that based on the restricted isometry property generalizes the investigation to a complete perturbation model thinking over not only noise but also perturbation, gives the restricted isometry property condition that guarantees the recovery of low-rank matrix and the corresponding reconstruction error bound. In particular, the analysis of the result reveals that in the case that $p$ decreases $0$ and $a>1$ for the complete perturbation and low-rank matrix, the condition is the optimal sufficient condition $\delta_{2r}<1$ [@Recht; @et; @al; @2010]. The numerical experiments are conducted to show better performance, and provides outperformance of the nonconvex Schatten $p$-minimization method comparing with the convex nuclear norm minimization approach in the completely perturbed scenario.
>
> [**Keywords.**]{} Low rank matrix recovery; Perturbation of linear transformation; Nonconvex Schatten $p$-minimization
Introduction {#sec1}
============
Low-rank matrix recovery (LMR) is a rapidly developing topic attracting the interest of numerous researchers in the field of optimization and compressed sensing. Mathematically, we can describe it as follows: $$\begin{aligned}
\label{eq.1}
y=\mathcal{A}(X)\end{aligned}$$ where $\mathcal{A}:~\mathbb{R}^{m\times n}\to\mathbb{R}^M$ is a known linear transformation (we suppose that $m\leq n$), $y\in\mathbb{R}^M$ is a given observation vector, and $X\in\mathbb{R}^{m\times n}$ is the matrix to be recovered. The objective of LMR is to find the lowest rank matrix based on $(y,~\mathcal{A})$. If the observation $y$ is corrupted by noise $z$, model (\[eq.1\]) is changed into the following form $$\begin{aligned}
\label{eq.2}
\hat{y}=\mathcal{A}(X)+z\end{aligned}$$ where $\hat{y}$ is the noisy measurement, and $z$ is the additive noise independent of the matrix $X$. However, more LMR models can be encountered where not only the linear measurement $y$ is contaminated by the noise vector $z$, but also the linear transformation $\mathcal{A}$ is perturbed by $\mathcal{E}$ for completely perturbed setting, namely, substitute the linear transformation $\mathcal{A}$ with $\mathcal{\hat{A}}=\mathcal{A}+\mathcal{E}$. The completely perturbed appearance arises in remote sensing[@Fannjiang; @et; @al; @2010], radar[@Herman; @and; @Strohmer; @2009], source separation[@Blumensath; @and; @Davies; @2007], etc. When $m=n$ and the matrix $X=\mbox{diag}(x)~(x\in\mathbb{R}^m)$ is diagonal, models (\[eq.1\]) and (\[eq.2\]) degenerates to the compressed sensing models $$\begin{aligned}
\label{eq.3}y&=Ax,\\
\label{eq.4}\hat{y}&=Ax+z\end{aligned}$$ where $A\in\mathbb{R}^{M\times m}$ is a measurement matrix and $x\in\mathbb{R}^m$ is an unknown sparse signal. We call the problem (\[eq.3\]) as the sparse signal recovery. For the completely perturbed model, the convex nuclear norm minimization is frequently considered [@Huang; @et; @al; @2019] as follows: $$\begin{aligned}
\label{eq.5}
\min_{\tilde{Z}\in\mathbb{R}^{m\times n}}\|\tilde{Z}\|_*~\mbox{s.t.}~\|\hat{\mathcal{A}}(\tilde{Z})-\hat{y}\|_2\leq\epsilon'_{\mathcal{A},r,y},\end{aligned}$$ where $\|\tilde{Z}\|_*$ is the nuclear norm of the matrix $\tilde{Z}$, that is, the sum of its singular values, and $\epsilon'_{\mathcal{A},r,y}$ is the total noise level. Problem (\[eq.5\]) can be reduced to the $l_1$-minimization [@Herman; @and; @Strohmer; @2010] $$\begin{aligned}
\label{eq.6}
\min_{\tilde{z}\in\mathbb{R}^{n_1}}\|\tilde{z}\|_1~\mbox{s.t.}~\|\hat{A}\tilde{z}-\hat{y}\|_2\leq\epsilon'_{A,r,y},\end{aligned}$$ where $\|\tilde{z}\|_1$ is the $l_1$-norm of the vector $\tilde{z}$, that is, the sum of absolute value of its coefficients.
Chartrand [@Chartrand; @2007] showed that fewer measurements are required for exact reconstruction if $l_1$-norm is substituted with $l_p$-norm. There exist many work regarding reconstructing $x$ via the $l_p$-minimization [@Chartrand; @and; @Staneva; @2008], [@Foucart; @and; @Lai; @2009], [@Lai; @and; @Liu; @2011], [@Lai; @et; @al; @2013], [@Wang; @Y; @et; @al; @2014], [@Song; @and; @Xia; @2014], [@Wang; @J; @et; @al; @2015], [@Wen; @J; @et; @al; @2015], [@Wen; @F; @et; @al; @2017], [@Gao; @et; @al; @2017], [@Zhang; @and; @Li; @2017], [@Wen; @F; @et; @al; @2018]. In [@Chartrand; @2007], numerical simulations demonstrated that fewer measurements are needed for exact reconstruction than when $p=1$.
In this paper, we are interested in the completely perturbed model for the nonconvex Schatten $p$-minimization ($0<p<1$) $$\begin{aligned}
\label{eq.7}
\min_{\tilde{Z}\in\mathbb{R}^{m\times n}}\|\tilde{Z}\|^p_p~\mbox{s.t.}~\|\hat{\mathcal{A}}(\tilde{Z})-\hat{y}\|_2\leq\epsilon'_{\mathcal{A},r,y},\end{aligned}$$ where $\|\tilde{Z}\|^p_p$ is the Schatten $p$ quasi-norm of the matrix $\tilde{Z}$, that is, $\|\tilde{Z}\|^p_p=(\sum_i\sigma^p_i(\tilde{Z}))^{1/p}$ with $\sigma_i(\tilde{Z})$ being $i$th singular value of $\tilde{Z}$. Problem (\[eq.7\]) can be returned to the $l_p$-minimization [@Ince; @and; @Nacaroglu; @2014] $$\begin{aligned}
\label{eq.8}
\min_{\tilde{z}\in\mathbb{R}^{M\times m}}\|\tilde{z}\|^p_p~\mbox{s.t.}~\|\hat{A}\tilde{z}-\hat{y}\|_2\leq\epsilon'_{A,r,y},\end{aligned}$$ where $\|\tilde{z}\|^p_p=(\sum_i\tilde{z}^p_i)^{1/p}$ is the $l_p$-quasi-norm of the vector $\tilde{z}$. To the best of our knowledge, recently researches are considered only in unperturbed situation ($\mathcal{E}=0$), that is, the linear transformation $\mathcal{A}$ is not perturbed by $\mathcal{E}$ (for related work, see [@Mohan; @and; @Fazel; @2010], [@Dvijotham; @and; @Fazel; @2010], [@Zhang; @et; @al; @2013], [@Kong; @and; @Xiu; @2013], [@Wang; @and; @Li; @2013], [@Chen; @and; @Li; @2015], [@Gao; @Y; @et; @al; @2017b], [@Wang; @W; @et; @al; @2019]). From the perspective of application, it is more practical to investigate the recovery of low-rank matrices in the scenario of complete perturbation.
In this paper, based on restricted isometry property (RIP), the performance of low-rank matrices reconstruction is showed by the nonconvex Schatten $p$-minimization in completely perturbed setting. The main contributions of this paper are as follows. First, we present a sufficient condition for reconstruction of low-rank matrices via the nonconvex Schatten $p$-minimization. Second, the estimation accurateness between the optimal solution and the original matrix is described by a total noise and a best $r$-rank approximation error. The result reveals that stable and robust performance concerning reconstruction of low-rank matrices in existence of total noise. Third, numerical experiments are conducted to sustain the gained results, and demonstrate that the performance of nonconvex Schatten $p$-minimization can be better than that of convex nuclear norm minimization in completely perturbed model.
The rest of this paper is constructed as follows.
Notation and main results
=========================
Before presenting the main results, we first introduce the notion of RIC of a linear transformation $\mathcal{A}$, which is as follows.
\[def.1\] The restricted isometry constant (RIC) $\delta_r$ of a linear transformation $\mathcal{A}$ is the smallest constant such that $$\begin{aligned}
\label{eq.9}
(1-\delta)\|X\|^2_F\leq\|\mathcal{A}(X)\|_2^2\leq (1+\delta)\|X\|^2_F\end{aligned}$$ holds for all $r$-rank $X\in\mathbb{R}^{m\times n}$ (i.e., $rank(X)\leq r$), where $\|X\|_F:=\sqrt{\left<X,X\right>}=\sqrt{trace(X^{\top}X)}$ is the Frobenius norm of the matrix $X$.
Then we provide some notations similar to [@Huang; @et; @al; @2019], which quantifying the perturbations $\mathcal{E}$ and $z$ with the bounds: $$\begin{aligned}
\label{eq.10}
\frac{\|\mathcal{E}\|_{op}}{\|\mathcal{A}\|_{op}}\leq\epsilon_{\mathcal{A}},~
\frac{\|\mathcal{E}\|^{(r)}_{op}}{\|\mathcal{A}\|^{(r)}_{op}}\leq\epsilon^{(r)}_{\mathcal{A}},~
\frac{\|z\|_2}{\|y\|_2}\leq\epsilon_{y},\end{aligned}$$ where $\|\mathcal{A}\|_{op}=\sup\{\|\mathcal{A}(X)\|_2/\|X\|_F:~X\in\mathbb{R}^{m\times n}\setminus\{0\}\}$ is the operator norm of linear transformation $\mathcal{A}$, and $\|\mathcal{A}\|^{(r)}_{op}=\sup\{\|\mathcal{A}(X)\|_2/\|X\|_F:~X\in\mathbb{R}^{m\times n}\setminus\{0\}~\mbox{and}~\mbox{rank}(X)\leq r\}$, and representing $$\begin{aligned}
\label{eq.11}
t_r=\frac{\|X_{[r]^c}\|_F}{\|X_{[r]}\|_F},~s_r=\frac{\|X_{[r]^c}\|_*}{\sqrt{r}\|X_{[r]}\|_F},~
\kappa^{(r)}_{\mathcal{A}}=\frac{\sqrt{1+\delta_r}}{\sqrt{1-\delta_r}},~
\alpha_{\mathcal{A}}=\frac{\|\mathcal{A}\|_{op}}{\sqrt{1-\delta_r}}.\end{aligned}$$ Here $X_{[r]}$ is the best $r$-rank approximation of the matrix $X$, its singular values are composed of $r$-largest singular values of the matrix $X$, and $X_{[r]^c}=X-X_{[r]}$. With notations and symbols above, we present our results for reconstruction of low-rank matrices via the completely perturbed nonconvex Schatten $p$-minimization.
\[the.1\] For given relative perturbations $\epsilon_{\mathcal{A}}$, $\epsilon^{(r)}_{\mathcal{A}}$, $\epsilon^{(2r)}_{\mathcal{A}}$, and $\epsilon_{y}$ in (\[eq.10\]), suppose the RIC for the linear transformation $\mathcal{A}$ fulfills $$\begin{aligned}
\label{eq.12}
\delta_{2ar}<\frac{2+\sqrt{2}a^{1/2-1/p}}{(1+\sqrt{2}a^{1/2-1/p})(1+\epsilon^{(2ar)}_{\mathcal{A}})^2}-1\end{aligned}$$ for $a>1$ and that the general matrix $X$ meets $$\begin{aligned}
\label{eq.13}
t_r+s_r<\frac{1}{\kappa^{(r)}_{\mathcal{A}}}.\end{aligned}$$ Then a minimizer $X^*$ of problem (\[eq.7\]) approximates the true matrix $X$ with errors $$\begin{aligned}
\label{eq.14}\|X-X^*\|^p_F&\leq C_1(\epsilon'_{\mathcal{A},r,y})^p+C_2\frac{\|X_{[r]^c}\|^p_p}{r^{1-p/2}},\\
\label{eq.15}\|X-X^*\|^p_p&\leq C'_1r^{1-p/2}(\epsilon'_{\mathcal{A},r,y})^p+C'_2\|X_{[r]^c}\|^p_p,\end{aligned}$$ where the total noise is $$\begin{aligned}
\label{eq.16}
\epsilon'_{\mathcal{A},r,y}=\left[\frac{\epsilon^{(r)}_{\mathcal{A}}\kappa^{(r)}_{\mathcal{A}}
+\epsilon_{\mathcal{A}}\alpha_{\mathcal{A}}t_r}{1-\kappa^{(r)}_{\mathcal{A}}(t_r+s_r)}+\epsilon_{y}\right]\|y\|_2,\end{aligned}$$ and $$\begin{aligned}
\label{eq.17}
C_1&=\frac{2^{p}(1+a^{p/2-1})
(1+\hat{\delta}_{(a+1)r})^{p/2}}{(1-\hat{\delta}_{(a+1)r})^{p}-a^{p/2-1}(\hat{\delta}_{(a+1)r}^2+\hat{\delta}_{2ar}^2)^{p/2}},\\
\label{eq.18}
C_2&=2a^{p/2-1}[1+\frac{(1+a^{p/2-1})
(\hat{\delta}_{(a+1)r}^2+\hat{\delta}_{2ar}^2)^{p/2}}{(1-\hat{\delta}_{(a+1)r})^{p}-a^{p/2-1}(\hat{\delta}_{(a+1)r}^2+\hat{\delta}_{2ar}^2)^{p/2}}],\\
\label{eq.19}
C'_1&=\frac{2^{p+1}(1+a)^{1-p/2}
(1+\hat{\delta}_{(a+1)r})^{p/2}}{(1-\hat{\delta}_{(a+1)r})^{p}-a^{p/2-1}(\hat{\delta}_{(a+1)r}^2+\hat{\delta}_{2ar}^2)^{p/2}},\\
\label{eq.18}
C'_2&=2+\frac{4(1+a)^{1-p/2}a^{p/2-1}
(\hat{\delta}_{(a+1)r}^2+\hat{\delta}_{2ar}^2)^{p/2}}{(1-\hat{\delta}_{(a+1)r})^{p}-a^{p/2-1}(\hat{\delta}_{(a+1)r}^2+\hat{\delta}_{2ar}^2)^{p/2}},\end{aligned}$$ where $\hat{\delta}_{(a+1)r}=(1+\delta_{(a+1)r})(1+\epsilon_{A}^{((a+1)r)})^2-1,~~\hat{\delta}_{2ar}=(1+\delta_{2ar})(1+\epsilon_{A}^{(2ar)})^2-1$.
Theorem \[the.1\] gives a sufficient conditions for the reconstruction of low-rank matrices via nonconvex Schatten $p$-minimization in completely perturbed scenario. Condition (\[eq.12\]) of the Theorem extends the assumption of $l_p$ situation in [@Ince; @and; @Nacaroglu; @2014] to the nonconvex Schatten $p$-minimization. Observe that as the value of $p$ becomes large, the bound of RIC $\delta_{2ar}$ reduces, which reveals that smaller value of $p$ can induce weaker reconstruction guarantee. Particularly, when $p\to 0~(a>1)$ ((\[eq.7\]) degenerates to the rank minimization: $\min_{\tilde{Z}\in\mathbb{R}^{m\times n}}\mbox{rank}(\tilde{Z})~\mbox{s.t.}~\|\hat{\mathcal{A}}(\tilde{Z})-\hat{y}\|_2\leq\epsilon'_{\mathcal{A},r,y}$), it leads to the RIP condition $\delta_{2r}<2/(1+\epsilon^{(2ar)}_{\mathcal{A}})^2-1$ for reconstruction of low-rank matrices via the rank minimization, to the best of our knowledge, the current optimal recovery condition about RIP is $\delta_{2r}<1$ to ensure exact reconstruction for $r$-rank matrices via rank minimization [@Recht; @et; @al; @2010], therefore the Theorem extends that condition to the scenario of presence of noise and $r$-rank matrices. Furthermore, when $m=n$ and the matrix $X=\mbox{diag}(x)~(x\in\mathbb{R}^m)$ is diagonal, the Theorem reduces to the case of compressed sensing given by [@Ince; @and; @Nacaroglu; @2014].
Under the requirement (\[eq.12\]), one can easily check that the condition (\[eq.13\]) is satisfied. Besides, when $\mbox{rank}(X)\leq r$, the condition (\[eq.13\]) holds. Additionally, the inequalities (\[eq.14\]) and (\[eq.15\]) in Theorem \[the.1\] which exploit two kinds of metrics provide upper bound estimations on the reconstruction of nonconvex Schatten $p$-minimization. The estimations evidence that reconstruction accurateness can be controlled by the best $r$-rank approximation error and the total noise. In particular, when there aren’t noise (i.e., $\mathcal{E}=0$ and $z=0$), they clear that the $r$-rank matrix can be accurately reconstructed via the nonconvex Schatten $p$-minimization. In (\[eq.14\]), both the error bound noise constant $C_1$ and the error bound compressibility constant $C_2$ may rely on the value of $p$. Numerical simulations reveal that when we fix the other independent parameters, a smaller value of $p$ will produce a smaller $C_1$ and a smaller $C_2/r^{1-p/2}$. For more details, see Fig. \[fig.1\].
When the matrix $X$ is a strictly $r$-rank matrix (i.e., $X=X_{[r]}$), a minimizer $X^*$ of problem (\[eq.7\]) approximates the true matrix $X$ with errors $$\begin{aligned}
\notag\|X-X^*\|_F&\leq C^{1/p}_1\epsilon'_{\mathcal{A},r,y},\\
\notag\|X-X^*\|_p&\leq C'^{1/p}_1r^{1/p-1/2}\epsilon'_{\mathcal{A},r,y},\end{aligned}$$ where $$\begin{aligned}
\notag\epsilon'_{\mathcal{A},r,y}=[\epsilon^{(r)}_{\mathcal{A}}\kappa^{(r)}_{\mathcal{A}}+\epsilon_{y}]\|y\|_2.\end{aligned}$$ In the case of $\mathcal{E}=0$, that is, there doesn’t exist perturbation in the linear transformation $\mathcal{A}$, then $\hat{\delta}_{(a+1)r}=\delta_{(a+1)r},$ $\hat{\delta}_{2ar}=\delta_{2ar}$. In the case that $m=n$, the matrix $X=\mbox{diag}(x)~(x\in\mathbb{R}^m)$ is diagonal (i.e., the results of Theorem reduce to the case of compressed sensing), $p=1$ and $a=1$, our result contains that of Theorem $2$ in [@Herman; @and; @Strohmer; @2010].
Proofs of the main results
==========================
In this part, we will provide the proofs of main results. In order to prove our main results, we need the following auxiliary lemmas. Firstly, we give Lemma \[lem.1\] which incorporates an important inequality associating with $\delta_r$ and $\hat{\delta}_r$.
\[lem.1\](RIP for $\mathcal{\hat{A}}$ [@Huang; @et; @al; @2019]) Given the RIC $\delta_r$ related with linear transformation $\mathcal{A}$ and the relative perturbation $\epsilon^{(r)}_{\mathcal{A}}$ corresponded with linear transformation $\mathcal{E}$, fix the constant $\hat{\delta}_{r,\max}=(1+\delta_r)(1+\epsilon^{(r)}_{\mathcal{A}})^2-1$. Then the RIC $\hat{\delta}_r\leq\hat{\delta}_{r,\max}$ for $\mathcal{\hat{A}}=\mathcal{A}+\mathcal{E}$ is the smallest nonnegative constant such that $$\begin{aligned}
\label{eq.20}
(1-\hat{\delta}_r)\|X\|^2_F\leq\|\mathcal{\hat{A}}(X)\|_2^2\leq (1+\hat{\delta}_r)\|X\|^2_F\end{aligned}$$ holds for all matrices $X\in\mathbb{R}^{m\times n}$ that are $r$-rank.
We will employ the fact that $\mathcal{\hat{A}}$ maps low-rank orthogonal matrices to nearly sparse orthogonal vectors, which is given by [@Candes; @and; @Plan; @2011].
\[lem.2\]([@Candes; @and; @Plan; @2011]) For all $X,~Y$ satisfying $\left<X,Y\right>=0$, and $\mbox{rank}(X)\leq r_1$, $\mbox{rank}(Y)\leq r_2$, $$\begin{aligned}
\label{eq.22}
\left|\left<\mathcal{\hat{A}}(X),\mathcal{\hat{A}}(Y)\right>\right|\leq\hat{\delta}_{r_1+r_2}\|X\|_F\|Y\|_F.\end{aligned}$$
Moreover, the following lemma will be utilized in the proof of main result, which combines with Lemma $2.3$ [@Recht; @et; @al; @2010] and Lemma $2.2$ [@Kong; @and; @Xiu; @2013].
\[lem.3\] Assume that $X,~Y\in\mathbb{R}^{m\times n}$ obey $X^{\top}Y=0$ and $XY^{\top}=0$. Let $0<p\leq 1$. Then $$\begin{aligned}
\label{eq.23}
\|X+Y\|^p_p=\|X\|^p_p+\|Y\|^p_p,~\|X+Y\|_p\geq\|X\|_p+\|Y\|_p,\end{aligned}$$ where $\|X\|^p_p$ and $\|X\|_p$ stand for the nuclear norm of matrix $X$ in the case of $p=1$.
For any matrix $X\in\mathbb{R}^{m\times n}$, we represent the singular values decomposition (SVD) of $X$ as $$\begin{aligned}
\notag
X=U\mbox{diag}(\sigma(X))V^{\top},\end{aligned}$$ where $\sigma(X):=(\sigma_1(X),\cdots,\sigma_m(X))$ is the vector of the singular values of $X$, $U$ and $V$ are respectively the left and right singular value matrices of $X$.
Let $X$ denote the original matrix to be recovered and $X^*$ denote the optimal solution of (\[eq.7\]). Let $Z=X-X^*$, and based on the SVD of $X$, its SVD is given by $$\begin{aligned}
\notag
U^{\top}ZV=U_1\mbox{diag}(\sigma(U^{\top}ZV))V^{\top}_1,\end{aligned}$$ where $U_1,~V_1\in\mathbb{R}^{m\times m}$ are orthogonal matrices, and $\sigma(U^{\top}ZV)$ stands for the vector comprised of the singular values of $U^{\top}ZV$. Let $T_0$ is the set composed of the locations of the $r$ largest magnitudes of elements of $\sigma(X)$. We adopt technology similar to the reference [@Ince; @and; @Nacaroglu; @2014] to partition $\sigma(U^{\top}ZV)$ into a sum of vectors $\sigma_{T_i}(U^{\top}ZV)~(i=0,1,\cdots,J)$, where $T_1$ is the set composed of the locations of the $ar$ largest magnitudes of entries of $\sigma_{T^c_0}(U^{\top}ZV)$, $T_2$ is the set composed of the locations of the second $ar$ largest magnitudes of entries of $\sigma_{T^c_0}(U^{\top}ZV)$, and so forth (except possibly $T_J$). Then $Z=\sum_{i=0}^JZ_{T_i}$ where $Z_{T_i}=UU_1\mbox{diag}(\sigma_{T_i}(U^{\top}ZV))(VV_1)^{\top}$, $i=0,1,\cdots,J$. One can easily verify that $Z^{\top}_{T_i}Z_{T_j}=0$ and $Z_{T_i}Z^{\top}_{T_j}=0$ for all $i\neq j$, and $\mbox{rank}(Z_{T_0})\leq r$, $\mbox{rank}(Z_{T_j})\leq ar$, $i=0,1,\cdots,J$. For simplicity, denote $T_{01}=T_0\bigcup T_1$. Then, we have (see (22) in [@Zhang; @et; @al; @2013], Lemma $2.6$ [@Chen; @and; @Li; @2015]) $$\begin{aligned}
\label{eq.24}
\|Z_{T^c_0}\|^p_p\leq\|Z_{T_0}\|^p_p+2\|X_{[r]^c}\|^p_p.\end{aligned}$$ By the decomposition of $Z$, for each $l\in T_i,~k\in T_{i-1},~i\geq 2$, $\sigma_{T_{i}}(U^{\top}ZV)[l]\leq\sigma_{T_{i-1}}(U^{\top}ZV)[k]$, it implies that $$\begin{aligned}
\label{eq.25}
(\sigma_{T_{i}}(U^{\top}ZV)[l])^p\leq \frac{\sum^{ar}_{k=1}(\sigma_{T_{i-1}}(U^{\top}ZV)[l])^p}{ar}=
\frac{\|\sigma_{T_{i-1}}(U^{\top}ZV)\|^p_p}{ar}=\frac{\|Z_{T_{i-1}}\|^p_p}{ar},\end{aligned}$$ which deduces $$\begin{aligned}
\label{eq.26}
\|Z_{T_{i}}\|^2_F\leq(ar)^{1-\frac{2}{p}}\|Z_{T_{i-1}}\|^2_p.\end{aligned}$$ Thereby, $$\begin{aligned}
\label{eq.27}
\|Z_{T_{i}}\|^p_F\leq(ar)^{\frac{p}{2}-1}\|Z_{T_{i-1}}\|^p_p.\end{aligned}$$ Notice that $Z^{\top}_{T_i}Z_{T_j}=0$ and $Z_{T_i}Z^{\top}_{T_j}=0$ for all $i\neq j$, due to Lemma \[lem.3\] and (\[eq.27\]), then we can get $$\begin{aligned}
\label{eq.28}
\sum_{i\geq2}\|Z_{T_{i}}\|^p_F\leq(ar)^{\frac{p}{2}-1}\sum_{i\geq2}\|Z_{T_{i-1}}\|^p_p
=(ar)^{\frac{p}{2}-1}\|Z_{T^c_0}\|^p_p.\end{aligned}$$ By the inequality $\|Z_{T_0}\|^p_F\leq\|Z_{T_{01}}\|^p_F$ and H$\ddot{o}$lder’s inequality, we get $$\begin{aligned}
\label{eq.29}
\|Z_{T_0}\|^p_p\leq r^{1-\frac{p}{2}}\|Z_{T_{01}}\|^p_F.\end{aligned}$$ From (\[eq.24\]), (\[eq.28\]), (\[eq.29\]) and the inequality that for every fixed $n\in\mathbb{N}$, and any $0<\alpha\leq1$, $(\sum_{i=1}^nx)^{\alpha}\leq\sum_{i=1}^nx^{\alpha}$ for every $x_i\geq0,~i=1,\cdots,n$, it follows $$\begin{aligned}
\label{eq.30}
\|Z_{T^c_{01}}\|^p_F=(\sum_{i\geq2}\|Z_{T_{i}}\|^2_F)^{\frac{p}{2}}\leq\sum_{i\geq2}\|Z_{T_{i}}\|^p_F
\leq(ar)^{\frac{p}{2}-1}(r^{1-\frac{p}{2}}\|Z_{T_{01}}\|^p_F+2\|X_{[r]^c}\|^p_p).\end{aligned}$$ Since $$\begin{aligned}
\notag\|\mathcal{\hat{A}}(Z_{T_{01}})\|^2_2&=<\hat{\mathcal{A}}(Z_{T_{01}}),\mathcal{\hat{A}}(Z_{T_{01}})>\\
\notag&=<\hat{\mathcal{A}}(Z_{T_{01}}),\mathcal{\hat{A}}(Z)>
-<\hat{\mathcal{A}}(Z_{T_{01}}),\sum_{i\geq2}\mathcal{\hat{A}}(Z_{T_i})>\\
\label{eq.31}&\leq\|\hat{\mathcal{A}}(Z_{T_{01}})\|_2\|\hat{\mathcal{A}}(Z)\|_2
+\sum_{i\geq2}|<\hat{\mathcal{A}}(Z_{T_{01}}),\mathcal{\hat{A}}(Z_{T_i})>|,\end{aligned}$$ we get $$\begin{aligned}
\label{eq.32}
\|\mathcal{\hat{A}}(Z_{T_{01}})\|^{2p}_2\overset{\text{(a)}}{\leq}\|\hat{\mathcal{A}}(Z_{T_{01}})\|^p_2\|\hat{\mathcal{A}}(Z)\|^p_2
+\sum_{i\geq2}|<\hat{\mathcal{A}}(Z_{T_{01}}),\mathcal{\hat{A}}(Z_{T_i})>|^p,\end{aligned}$$ where (a) follows from the fact that $(a+b)^p\leq a^p+b^p$ for nonnegative $a$ and $b$.
Additionally, by the minimality of $X^*$, we get $$\begin{aligned}
\label{eq.33}
\|\mathcal{\hat{A}}(Z)\|^2_2\leq\|\hat{y}-\mathcal{\hat{A}}(X)\|^2_2
+\|\hat{y}-\mathcal{\hat{A}}(X^*)\|^2_2\leq2\epsilon'_{\mathcal{A},r,y}.\end{aligned}$$ Since $Z_{T_{01}}$ is $(a+1)r$-rank and $Z_{T_i}$ is $ar$-rank, $i\geq2$, by applying the RIP of $\mathcal{\hat{A}}$ and combination with (\[eq.32\]) and (\[eq.33\]), we get $$\begin{aligned}
\label{eq.34}
\|\mathcal{\hat{A}}(Z_{T_{01}})\|^{2p}_2\leq(2\epsilon'_{\mathcal{A},r,y})^p
(1+\hat{\delta}_{(a+1)r})^{\frac{p}{2}}\|Z_{T_{01}}\|^p_F
+\sum_{i\geq2}|<\hat{\mathcal{A}}(Z_{T_{01}}),\mathcal{\hat{A}}(Z_{T_i})>|^p.\end{aligned}$$ Because $<Z_{T_i},Z_{T_j}>$ for all $i\neq j$, and $Z_{T_0}$ is $r$-rank, by Lemma \[lem.2\] and (\[eq.30\]), we get $$\begin{aligned}
\notag\|\mathcal{\hat{A}}(Z_{T_{01}})\|^{2p}_2
&\leq(2\epsilon'_{\mathcal{A},r,y})^p
(1+\hat{\delta}_{(a+1)r})^{\frac{p}{2}}\|Z_{T_{01}}\|^p_F
+(\hat{\delta}_{(a+1)r}\|Z_{T_0}\|_F+\hat{\delta}_{2ar}\|Z_{T_1}\|_F)^p\
\sum_{i\geq2}\|Z_{T_i}\|^p_F\\
\notag&\leq(2\epsilon'_{\mathcal{A},r,y})^p
(1+\hat{\delta}_{(a+1)r})^{\frac{p}{2}}\|Z_{T_{01}}\|^p_F\\
\label{eq.35}&\quad+(\hat{\delta}_{(a+1)r}\|Z_{T_0}\|_F+\hat{\delta}_{2ar}\|Z_{T_1}\|_F)^p
(ar)^{\frac{p}{2}-1}(r^{1-\frac{p}{2}}\|Z_{T_{01}}\|^p_F+2\|X_{[r]^c}\|^p_p)\end{aligned}$$ From (\[eq.12\]), one can easily check that $$\begin{aligned}
\label{eq.36}
a^{\frac{p}{2}-1}(\hat{\delta}_{(a+1)r}^2+\hat{\delta}_{2ar}^2)^{\frac{p}{2}}
<(1-\hat{\delta}_{(a+1)r})^{\frac{p}{2}}.\end{aligned}$$ By (\[eq.35\]), (\[eq.36\]) and the inequality $\|\mathcal{\hat{A}}(Z_{T_{01}})\|^{p}_2\geq(1-\hat{\delta}_{(a+1)r})^{\frac{p}{2}}\|Z_{T_{01}}\|^p_F$, one can get $$\begin{aligned}
\notag\|Z_{T_{01}}\|^p_F\leq&\frac{2^p(1+a^{\frac{p}{2}-1})(1+\hat{\delta}_{(a+1)r})^{\frac{p}{2}}}
{(1-\hat{\delta}_{(a+1)r})^{\frac{p}{2}}-a^{\frac{p}{2}-1}(\hat{\delta}_{(a+1)r}^2+\hat{\delta}_{2ar}^2)^{\frac{p}{2}}}
(\epsilon'_{\mathcal{A},r,y})^p\\
\notag&+\frac{2a^{\frac{p}{2}-1}(\hat{\delta}_{(a+1)k}^2+\hat{\delta}_{2ar}^2)^{\frac{p}{2}}}
{(1-\hat{\delta}_{(a+1)r})^{\frac{p}{2}}-a^{\frac{p}{2}-1}(\hat{\delta}_{(a+1)r}^2+\hat{\delta}_{2ar}^2)^{\frac{p}{2}}}
\frac{\|X_{[r]^c}\|^p_p}{r^{1-\frac{p}{2}}}\\
\label{eq.37}=&:\beta(\epsilon'_{\mathcal{A},r,y})^p+\gamma\frac{\|X_{[r]^c}\|^p_p}{r^{1-\frac{p}{2}}},\end{aligned}$$ consequently, $$\begin{aligned}
\label{eq.38}\|Z_{T_0}\|^p_p&\leq r^{1-\frac{p}{2}}\|Z_{T_0}\|^p_F\\
\notag&\leq\beta r^{1-\frac{p}{2}}(\epsilon'_{\mathcal{A},r,y})^p+\gamma\|X_{[r]^c}\|^p_p.\end{aligned}$$ Thus, from (\[eq.30\]) and (\[eq.37\]), we get $$\begin{aligned}
\notag\|Z\|^p_F&\leq\|Z_{T_{01}}\|^p_F+\|Z_{T^c_{01}}\|^p_F\\
\label{eq.39}&\leq C_1(\epsilon'_{\mathcal{A},r,y})^p+C_2\frac{\|X_{[r]^c}\|^p_p}{r^{1-\frac{p}{2}}};\end{aligned}$$ in addition, a combination of (\[eq.24\]) and (\[eq.38\]), one can get $$\begin{aligned}
\notag\|Z\|^p_p&\leq\|Z_{T_0}\|^p_p+\|Z_{T^c_0}\|^p_p\\
\label{eq.40}&\leq C'_1r^{1-\frac{p}{2}}(\epsilon'_{\mathcal{A},r,y})^p+C'_2\|X_{[r]^c}\|^p_p,\end{aligned}$$ where the constants $C_1$, $C_2$, $C'_1$ and $C'_2$ are defined in Theorem \[the.1\]. The proof is complete.
Numerical experiments
=====================
In this section, we carry out some numerical experiments to sustain verification of our theoretical results, we implement all experiments in MATLAB 2016a running on a PC with an Inter core i7 processor (3.6 GHz) with 8 GB RAM. In order to address the completely perturbed nonconvex Schatten $p$-minimization model, we employ the alternating direction method of multipliers (ADMM) method, which is often applied in compressed sensing and sparse approximation [@Lu; @C; @et; @al; @2018], [@Wen; @F; @et; @al; @TSP; @2017], [@Wang; @W; @et; @al; @2017], [@Wang; @and; @Wang; @2018]. The constrained optimization problem (\[eq.7\]) can be transformed into an equivalent unconstrained form $$\begin{aligned}
\label{eq.41}
\min_{\tilde{Z}\in\mathbb{R}^{m\times n}}\lambda\|\tilde{Z}\|^p_p+\frac{1}{2}\|\hat{A}\mbox{vec}(\tilde{Z})-\hat{y}\|^2_2,\end{aligned}$$ where $\hat{A}\in\mathbb{R}^{M\times mn}$, $\mbox{vec}(\tilde{Z})$ represents the vectorization of $\tilde{Z}$. Hence, $\hat{A}\mbox{vec}(\tilde{Z})$ presents the linear map $\hat{\mathcal{A}}(\tilde{Z})$. Then, introducing an auxiliary variable $W\in\mathbb{R}^{m\times n}$, the problem (\[eq.41\]) can be equivalently turned into
$$\begin{aligned}
\label{eq.42}
\min_{W,~\tilde{Z}\in\mathbb{R}^{m\times n}}\lambda\|W\|^p_p+\frac{1}{2}\|\hat{A}\mbox{vec}(\tilde{Z})-\hat{y}\|^2_2~\mbox{s.t.}~
\tilde{Z}=W.\end{aligned}$$
The augmented Lagrangian function is provided by $$\begin{aligned}
\label{eq.43}
L_{\rho}(\tilde{Z},W,Y)=\lambda\|W\|^p_p+\frac{1}{2}\|\hat{A}\mbox{vec}(\tilde{Z})-\hat{y}\|^2_2+
<Y,\tilde{Z}-W>+\frac{\rho}{2}\|\tilde{Z}-W\|^2_F,\end{aligned}$$ where $Y\in\mathbb{R}^{m\times n}$ is dual variable, and $\rho>0$ is a penalty parameter. Then, ADMM used to (\[eq.43\]) comprises of the iterations as follows $$\begin{aligned}
\label{eq.44}\tilde{Z}^{k+1}&=\arg\min_{\tilde{Z}}\frac{1}{2}\|\hat{A}\mbox{vec}(\tilde{Z})-\hat{y}\|^2_2+
\frac{\rho}{2}\|\tilde{Z}-(W^k-\frac{Y^k}{\rho})\|^2_F,\\
\label{eq.45}W^{k+1}&=\arg\min_{W}\lambda\|W\|^p_p+\frac{\rho}{2}\|\tilde{Z}^{k+1}-(W-\frac{Y^k}{\rho})\|^2_F,\\
\label{eq.46}Y^{k+1}&=Y^k+\rho(\tilde{Z}^{k+1}-W^{k+1}).\end{aligned}$$ All solving processes are concluded in Algorithm 4.1.
\[alg.1\]
Input $A\in\mathbb{R}^{M\times mn}$, $y\in\mathbb{R}^M$, perturbation $E\in\mathbb{R}^{M\times mn}$ with $\|E\|=\epsilon_A\|A\|$, $p\in(0,1]$. Initialize $\hat{\mathcal{Z}}^0=W^0=Y^0$, $\gamma=1.1$, $\lambda_0 = 10^{-6}$, $\lambda_{\max} = 10^{10}$, $\rho= 10^{-6}$, $\varepsilon=10^{-8}$, $k=0$. Updated $\tilde{Z}^{k+1}$ by $$\tilde{z}=(\hat{A}^{\top}\hat{A}+\rho I)^{-1}\left(\hat{A}^{\top}\hat{y}+\rho\mbox{vec}(W^k)-\mbox{vec}(Y^k)\right);$$ $\tilde{Z}^{k+1}\leftarrow\tilde{z}$: reshape $\tilde{z}$ to the matrix $\tilde{Z}^{k+1}$ of size $m\times n$. Update $W^{k+1}$ by $$\arg\min_{W}\lambda\|W\|^p_p+\frac{\rho}{2}\|\tilde{Z}^{k+1}-(W-\frac{Y^k}{\rho})\|^2_F;$$ Update $Y^{k+1}$ by $$Y^{k+1}=Y^k+\rho(\tilde{Z}^{k+1}-W^{k+1});$$ Update $\lambda_{j+1}$ by $\lambda_{j+1}=\min(\gamma\lambda_{j},\lambda_{\max});$ Check the convergence conditions $$\|\tilde{Z}^{k+1}-\tilde{Z}^{k}\|_{\infty}\leq\varepsilon,~\|W^{k+1}-W^{k}\|_{\infty}\leq\varepsilon,$$ $$\|\hat{A}\mbox{vec}(\tilde{Z}^{k+1})-\hat{y}\|_{\infty}\leq\varepsilon,~\|\tilde{Z}^{k}-W^{j+1}\|_{\infty}\leq\varepsilon.$$
In our experiments, we generate a measurement matrix $A\in\mathbb{R}^{M\times mn}$ with i.i.d. Gaussian $\mathcal{N}(0,1/M)$ elements. We generate $X\in\mathbb{R}^{m\times n}$ of rank $r$ by $X=PQ$, where $P\in\mathbb{R}^{m\times r}$ and $Q\in\mathbb{R}^{r\times n}$ are with its elements being zero-mean, one-variation Gaussian, i.i.d. random variables. We select $M=660$, $m=n=30$ and $r=0.2m$. With $X$ and $A$, the measurements $y$ are produced by $y=A\mbox{vec}(X)+z$, where $z$ is the Gaussian noise. The perturbation matrix $E$ is with its entries following Gaussian distribution, which fulfills $\|E\|=\epsilon_A\|A\|$, where $\epsilon_A$ denotes the perturbation level of $A$ and its value is not fixed. The perturbed matrix $\hat{A}$, $\hat{A}=A+E$, is used in (\[eq.44\]). To avoid the randomness, we perform 100 times independent trails as well as the average result in all test.
To look for a proper parameter $\lambda$ that derives the better recovery effect, we carry out two sets of trails. Fig. \[fig.2\] (a) and (b) respectively plot the parameter $\lambda$ and relative error (RelError, $\|X-X^*\|_F/\|X\|_F$) results in different $p$ values and perturbation level $\epsilon_A$. $\lambda$ ranges from $10^{-6}$ to $1$. Fig. \[fig.2\] (a) and (b) show that $\lambda\in[10^{-6},10^{-2}]$ is relatively suitable.
By giving $\lambda=10^{-6}$, we consider the convergence of Algorithm 4.1. Fig. \[fig.3\](a) presents the relationship between relative neighboring iteration error (RNIE, $r(k)=\|X^{k+1}-X^k\|_F/\|X^k\|_F$) and number of iterations $k$. One can easily see that with the increasing of iterations, RNIE decreases quickly, and when $k\geq250$, $r(k)<10^{-4}.$ The results that relative error versus the values of $p$ in different $\epsilon_A$ are showed in Fig. \[fig.3\](b). Fig. \[fig.3\](b) indicates that the proper choice of the size of $p$ will be helpful to facilitate the performance of nonconvex Schatten $p$-minimization.
The theoretical error bound and $\|X-X^*\|^p_F$ versus the values of $p$ with $a=2$, $\delta_{2ar}=\delta_{(a+1)r}=0.1$ and $r=6$ in different perturbation level $\epsilon_A$, the results are provided in Figs. \[fig.7\] (a) and (b). The values of $p$ vary from $0.1$ to $0.9$. From the observation of Fig. \[fig.7\], $\|X-X^*\|^p_F$ is smaller than the theoretical error bound.
In Fig. \[fig.4\], the relative error is plotted versus the number of measurements $M$ in different $\epsilon_A=0,~0.05,~0.10,~0.15,~0.20$ and $p=0.1,~0.3,~0.5,~0.7,~1$, respectively. From Fig. \[fig.4\], with the increase of number of measurements or the decrease of perturbation level, the recovery performance of nonconvex Schatten $p$-minimization gradually improves. Moreover, Fig. \[fig.4\](b) reveals that the performance of nonconvex Schatten $p$-minimization is better than that of convex nuclear norm minimization. In Fig. \[fig.5\], we plot the the relative error versus the rank $r$ of the matrix $X$ for different $\epsilon_A$ and $p$, respectively. The results indicate that the smaller the rank of the matrix, the better the recovery performance, and choosing a smaller perturbation level or the values of $p$ will improve the reconstruction effect of nonconvex Schatten $p$-minimization.
Furthermore, Fig. \[fig.6\] offers the results concerning the recovery performance of the nonconvex method and the convex method for the $\epsilon_A=0.05$. The curves of relationship between the relative error and the rank $r$ are described by nonconvex Schatten $p$-minimization and convex nuclear norm minimization, respectively. Fig. \[fig.6\] displays that the performance of nonconvex method is superior to that of the convex method.
[![Reconstruction performance of nonconvex Schatten $p$-minimization and convex nuclear norm minimization, varying rank $r$ for $\epsilon_A=0.05$[]{data-label="fig.6"}](RelVSrank_NonconvexVSconvex.eps "fig:"){width="50.00000%"}]{}
Conclusion
==========
In this paper, we investigate the completely perturbed problem employing the nonconvex Schatten $p$-minimization for reconstructing low-rank matrices. We derive a sufficient condition and the corresponding upper bounds of error estimation. The gained results reveal the nonconvex Schatten $p$-minimization has the stability and robustness for reconstructing low-rank matrices with the existence of a total noise. The practical meaning of gained results, not only can conduct the choice of the linear transformations for reconstructing low-rank matrices, that is, a linear transformation with a smaller RIC instead of a larger one can superior enhance the reconstruction performance, but also can also present a theoretical sustaining to approximation accurateness. Moreover, the numerical experiments further show the verification of our results, and the performance of nonconvex Schatten $p$-minimization is better than that of convex nuclear norm minimization in the complete perturbation situation.
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[^1]: Corresponding author, E-mail: [email protected], [email protected](J.J. Wang), E-mail: [email protected] (J. Huang)
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{
"pile_set_name": "ArXiv"
}
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---
abstract: 'Recommendation systems are emerging as an important business application with significant economic impact. Currently popular systems include Amazon’s book recommendations, Netflix’s movie recommendations, and Pandora’s music recommendations. In this paper we address the problem of estimating probabilities associated with recommendation system data using non-parametric kernel smoothing. In our estimation we interpret missing items as randomly censored observations and obtain efficient computation schemes using combinatorial properties of generating functions. We demonstrate our approach with several case studies involving real world movie recommendation data. The results are comparable with state-of-the-art techniques while also providing probabilistic preference estimates outside the scope of traditional recommender systems.'
author:
- |
Mingxuan Sun[^1] Guy Lebanon\
[Georgia Institute of Technology]{}\
[Atlanta GA 30332 USA]{}
- |
Paul Kidwell\
[Lawrence Livermore National Lab]{}\
[Livermore CA 94550 USA]{}
title: Estimating Probabilities in Recommendation Systems
---
Introduction {#sec:intro}
============
Recommendation systems are emerging as an important business application with significant economic impact. The data in such systems are collections of incomplete tied preferences across $n$ items associated with $m$ different users. Given an incomplete tied preference associated with an additional $m+1$ user, the system recommends unobserved items to that user based on the preference relations of the $m+1$ users. Currently deployed recommendation systems include book recommendations at amazon.com, movie recommendations at netflix.com, and music recommendations at pandora.com. Constructing accurate recommendation systems (that recommend to users items that are truly preferred over other items) is important for assisting users as well as increasing business profitability. It is an important unsolved goal in machine learning and data mining.
In most cases of practical interest the number of items $n$ indexed by the system (items may be books, movies, songs, etc.) is relatively high in the $10^3-10^4$ range. Perhaps due the size of $n$, it is almost always the case that each user observes only a small subset of the items, typically in the range 10-100. As a result the preference relations expressed by the users are over a small subset of the $n$ items.
Formally, we have $m$ users providing incomplete tied preference relations on $n$ items $$\begin{aligned}
\nonumber
S_1:\quad A_{1,1} \prec &A_{1,2}\prec \cdots \prec A_{1,k(1)} \\ \nonumber
S_2:\quad A_{2,1} \prec &A_{2,2} \prec \cdots \prec A_{2,k(2)} \\
&\vdots \label{eq:data} \\
S_m:\quad A_{m,1} \prec &A_{m,2} \prec \cdots \prec A_{m,k(m)}
\nonumber\end{aligned}$$ where $A_{i,j}\subset \{1,\ldots,n\}$ are sets of items (wlog we identify items with integers $1,\ldots,n$) defined by the following interpretation: user $i$ prefers all items in $A_{i,j}$ to all items in $A_{i,j+1}$. The notation $k(i)$ above is the number of such sets provided by user $i$. The data is incomplete since not all items are necessarily observed by each user i.e., $\bigcup_{j=1}^{k(i)} A_{i,j} \subsetneq \{1,\ldots,n\}$ and may contain ties since some items are left uncompared, i.e., $|A_{i,j}|>1$. Recommendation systems recommend items to a new user, denoted as $m+1$, based on their preference $$\begin{aligned}
\label{eq:activeUser}
S_{m+1}:A_{m+1,1} \prec &A_{m+1,2} \prec \cdots \prec A_{m+1,k(m+1)} \end{aligned}$$ and its relation to the preferences of the $m$ users .
As an illustrative example, assuming $n=9, m=3$, the data $$\begin{aligned}
&S_1:\qquad 1,8,9 \prec 4 \prec 2,3,7\\
&S_2:\qquad 4\prec 2,3 \prec 8\\
&S_3:\qquad 4,8\prec 2,6,9\end{aligned}$$ corresponds to $A_{1,1}=\{1,8,9\}$, $A_{1,2}=\{4\}$, $A_{1,3}=\{2,3,7\}$, $A_{2,1}=\{4\}$, $A_{2,2}=\{2,3\}$, $A_{2,3}=\{8\}$, $A_{3,1}=\{4,8\}$, $A_{3,2}=\{2,6,9\}$, and $k(1)=k(2)=3, k(3)=2$. From the data we may guess that item 4 is relatively popular across the board while some users like item 8 (users 1, 3) and some hate it (user 2). Given a new $m+1$ user issuing the preference $1 \prec 2,3,7$ we might observe a similar pattern of preference or taste as user 1 and recommend to the user item 8. We may also recommend item 4 which has broad appeal resulting in the augmentation $$1 \prec 2,3,7 \qquad \mapsto \qquad 1,4,8 \prec 2,3,7.$$
We note that in some cases the preference relations arise from users providing numeric scores to items. For example, if the users assign 1-5 stars to movies, the set $A_{i,j}$ contains all movies that user $i$ assigned $6-j$ stars to and $k(i)=5$ (assuming some movies were assigned to each of the 1, 2, 3, 4, 5 star levels). As pointed out by a wide variety of studies in economics and social sciences, such numeric scores are inconsistent among different users. We therefore proceed to interpret such data as ordinal rather than numeric.
A substantial body of literature in computer science has addressed the problem of constructing recommendation systems. We have attempted to outline the most important and successful approaches in the related work section towards the end of this paper. However, none of these previous approaches are fully satisfactory from a statistical perspective: there are no reasonable probability models assumed to generate the data and no clear meaningful statistical estimation procedures. We substantiate this argument more fully in the related work section.
In this paper we describe a non-parametric statistical technique for estimating probabilities on preferences based on the data . This technique may be used in recommendation systems in different ways. Its principal usage may be to provide a statistically meaningful estimation framework for issuing recommendations (in conjunction with decision theory). However, it also leads to other important applications including mining association rules, exploratory data analysis, and clustering items and users. Two key observations that we make are: (i) incomplete tied preference data may be interpreted as randomly censored permutation data, and (ii) using generating functions we are able to provide a computationally efficient scheme for computing the estimator in the case of triangular smoothing.
We proceed in the next sections to describe notations and our assumptions and estimation procedure, and follow with case studies demonstrating our approach on real world recommendation systems data.
Definitions and Estimation Framework
====================================
We describe the following notations and conventions for permutations, which are taken from [@Diaconis:88] where more detail may be found. We denote a permutation by listing the items from most preferred to least separated by a $\prec$ or $|$ symbol: $\pi^{-1}(1)\prec \pi^{-1}(2)\prec \cdots\prec \pi^{-1}(n)$, e.g. $\pi(1)=2,\pi(2)=3,\pi(3)=1$ is $3\prec 1\prec 2$. Ranking with ties occur when judges do not provide enough information to construct a total order. In particular, we define tied rankings as a partition of $\{1,\ldots,n\}$ to $k< n$ disjoint subsets $A_1,\ldots,A_k\subset \{1,\ldots,n\}$ such that all items in $A_i$ are preferred to all items in $A_{i+1}$ but no information is provided concerning the relative preference of the items among the sets $A_i$. We denote such rankings by separating the items in $A_i$ and $A_{i+1}$ with a $\prec$ or $|$ notation. For example, the tied ranking $A_1=\{3\}, A_2=\{2\}, A_3=\{1,4\}$ (items 1 and 4 are tied for last place) is denoted as $3\prec 2\prec 1,4$ or $3|2|1,4$.
Ranking with missing items occur when judges omit certain items from their preference information altogether. For example assuming a set of items $\{1,\ldots,4\}$, a judge may report a preference $3\prec 2\prec 4$, omitting altogether item 1 which the judge did not observe or experience. This case is very common in situations involving a large number of items $n$. In this case judges typically provide preference only for the $l\ll n$ items that they observed or experienced. For example, in movie recommendation systems we may have $n\sim 10^3$ and $l\sim 10^1$.
Rankings can be full (permutations), with ties, with missing items, or with both ties and missing items. In either case we denote the rankings using the $\prec$ or $|$ notation or using the disjoint sets $A_1,\ldots,A_k$ notation. We also represent tied and incomplete rankings by the set of permutations that are consistent with it. For example, $$\begin{aligned}
3\prec 2\prec 1,4 &= \{3\prec 2\prec 1\prec 4\}\cup\{ 3\prec 2\prec 4\prec 1\} \\
3\prec 2\prec 4 &=\{1\prec 3\prec 2\prec 4\}\cup\{3\prec 1\prec 2\prec 4\} \cup\{3\prec 2\prec 1\prec 4\}\cup\{3\prec 2\prec 4\prec 1\}\end{aligned}$$ are sets of two and four permutations corresponding to tied and incomplete rankings, respectively.
It is hard to directly posit a coherent probabilistic model on incomplete tied data such as . Different preferences relations are not unrelated to each other: they may subsume one another (for example $1\prec 2 \prec 3$ and $1 \prec 3$), represent disjoint events (for example $1\prec 3$ and $3\prec 1$), or interact in more complex ways (for example $1 \prec 2 \prec 3$ and $1 \prec 4 \prec 3$). A valid probabilistic framework needs to respect the constraints resulting from the axioms of probability, e.g., $p(1 \prec 2 \prec 3) \leq p(1 \prec 3)$.
Our approach is to consider the incomplete tied preferences as censored permutations. That is, we assume a distribution $p(\pi)$ over permutations $\pi\in\S_n$ ($\S_n$ is the symmetric group of permutations of order $n$) that describes the complete without-ties preferences in the population. The data available to the recommender system is sampled by drawing $m$ iid permutations from $p$: $\pi_1,\ldots,\pi_m\iid p$, followed by censoring to result in the observed preferences $S_1,\ldots,S_m$ $$\begin{aligned}
\pi_i &\sim p(\pi),\quad S_i\sim p(S|\pi_i), \quad i=1,\ldots,m+1 \\
p(\pi|S) &= \frac{I(\pi\in S)p(\pi)}{\sum_{\sigma\in S} p(\sigma)} \label{eq:condProb}
\\
p(S|\pi) &= p(\pi|S)p(S)/p(\pi) = \frac{I(\pi\in S) p(\pi) p(S)}{p(\pi)\sum_{\sigma\in S} p(\sigma)} =
\frac{I(\pi\in S) p(S)}{\sum_{\sigma\in S} p(\sigma)}\end{aligned}$$ where $p(S)$ is the probability of observing the censoring $S$ (specifically, it is not equal to $\sum_{\sigma\in S} p(\sigma)$).
Although many approaches for estimating $p$ given $S_1,\ldots,S_m$ are possible, experimental evidence point to the fact that in recommendation systems with high $n$, the distribution $p$ does not follow a simple parametric form such as the Mallows, Bradley-Terry, or Thurstone models [@Marden1996] (see Figure \[fig:heatMaps\] for a demonstration how parametric assumptions break down with increasing $n$). Instead, the distribution $p$ tends to be diffuse and multimodal with different probability mass regions corresponding to different types of judges (for example in movie preferences probability modes may correspond to genre as fans of drama, action, comedy, etc. having similar preferences).
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Heat map visualization of the density of ranked data using multidimensional scaling with expected Kendall’s Tau distance. The datasets are APA voting (left, $n=5$), Jester (middle, $n=100$), and EachMovie (right, $n=1628$) datasets. None of these cases show a simple parametric form, and the complexity of the density increases with the number of items $n$. This motivates the use of non-parametric estimators for modeling preferences over a large number of items.[]{data-label="fig:heatMaps"}](figure0001 "fig:") ![Heat map visualization of the density of ranked data using multidimensional scaling with expected Kendall’s Tau distance. The datasets are APA voting (left, $n=5$), Jester (middle, $n=100$), and EachMovie (right, $n=1628$) datasets. None of these cases show a simple parametric form, and the complexity of the density increases with the number of items $n$. This motivates the use of non-parametric estimators for modeling preferences over a large number of items.[]{data-label="fig:heatMaps"}](figure0002 "fig:") ![Heat map visualization of the density of ranked data using multidimensional scaling with expected Kendall’s Tau distance. The datasets are APA voting (left, $n=5$), Jester (middle, $n=100$), and EachMovie (right, $n=1628$) datasets. None of these cases show a simple parametric form, and the complexity of the density increases with the number of items $n$. This motivates the use of non-parametric estimators for modeling preferences over a large number of items.[]{data-label="fig:heatMaps"}](figure0003 "fig:")
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
We therefore propose to estimate the underlying distribution $p$ on permutations using non-parametric kernel smoothing. The standard kernel smoothing formula applies to the permutation setting as $$\hat p(\pi) = \frac{1}{m}\sum_{i=1}^m K_h(T(\pi,\pi_i))$$ where $\pi_1,\ldots,\pi_m\iid p$, $T$ a distance on permutations such as Kendall’s distance and $K_h(r)=h^{-1} K(r/h)$ a normalized unimodal function. In the case at hand, however, the observed preferences $\pi_i$ as well as $\pi$ are replaced with permutations sets $S_1,\ldots,S_m, R$ representing incomplete tied preferences $$\begin{aligned}
\hat p(R) = \sum_{\pi\in R} \hat p(\pi) = \frac{1}{m} \sum_{i=1}^m \sum_{\pi\in R} \sum_{\sigma\in S_i} q(\sigma|S_i) K_h(T(\pi,\sigma)) \label{eq:npEstimator}\end{aligned}$$ where $q(\sigma|S_i)$ serves as a surrogate for the unknown $p(\sigma|S_i)\propto I(\sigma\in S_i)p(\sigma)$ (see ). Selecting $q(\sigma|S_i)=p(\sigma|S_i)$ would lead to consistent estimation of $p(R)$ in the limit $h\to 0$, $m\to\infty$ assuming positive $p(\pi), p(S)$. Such a selection, however, is generally impossible since $p(\pi)$ and therefore $p(\sigma|S_i)$ are unknown.
In general the specific choice of the surrogate $q(\sigma|S)$ is important as it may influence the estimated probabilities. Furthermore, it may cause underestimation or overestimation of $\hat p(R)$ in the limit of large data. An exception occurs when the sets $S_1,\ldots,S_m$ are either subsets of $R$ or disjoint from $R$. In this case $\lim_{h\to 0} K_h(\pi,\sigma) = I(\pi=\sigma)$ resulting in the following limit (with probability 1 by the strong law of large numbers) $$\begin{aligned}
\lim_{m\to \infty}\,\, \lim_{h\to 0} \,\, \hat p(R) &=
\lim_{m\to \infty}\,\, \frac{1}{m} \sum_{i=1}^m I(S_i\subset R) \sum_{\sigma\in S_i} q(\sigma|S_i)\\
&= \lim_{m\to \infty}\,\, \frac{1}{m} \sum_{i=1}^m I(S_i\subset R)=
\lim_{m\to \infty}\,\, \frac{1}{m} \sum_{i=1}^m I(\pi_i\in R)= p(R).\end{aligned}$$ Thus, if we our data is comprised of preferences $S_i$ that are either disjoint or a subset of $R$ we have consistency *regardless* of the choice of the surrogate $q$. Such a situation is more realistic when the preference $R$ involves a small number of items and the preferences $S_i$, $i=1,\ldots,m$ involve a larger number of items. This is often the case for recommendation systems where individuals report preferences over 10-100 items and we are mostly interested in estimating probabilities of preferences over fewer items such as $i\prec j,k$ or $i\prec j,k\prec l$ (see experiment section).
The main difficulty with the estimator above is the computation of $\sum_{\pi\in R}\sum_{\sigma\in S_i} q(\sigma|S_i) K_h(T(\pi,\sigma))$. In the case of high $n$ and only a few observed items $k$ the sets $S_i,R$ grow factorially as $(n-k)!$ making a naive computation of intractable for all but the smallest $n$. In the next section we explore efficient computations of these sums for a triangular kernel $K_h$ and a uniform $q(\pi|S)$.
Computationally Efficient Kernel Smoothing
==========================================
In previous work [@Lebanon2008] the estimator is proposed for tied (but complete) rankings. That work derives closed form expressions and efficient computation for assuming a Mallows kernel [@Mallows1957] $$\begin{aligned}
\label{eq:MallowsKernel}
K_h(T(\pi,\sigma)) &= \exp\left( - \frac{T(\pi,\sigma)}{h}\right) \prod_{j=1}^n \frac{1-e^{-1/h}}{1-e^{-j/h}}\end{aligned}$$ where $T$ is Kendall’s Tau distance on permutations (below $I(x)=1$ for $x>0$ and 0 otherwise) $$\begin{aligned}
T(\pi,\sigma) &= \sum_{i=1}^{n-1}\sum_{l>i} I(\pi\sigma^{-1}(i)-\pi\sigma^{-1}(l)).\end{aligned}$$ Unfortunately these simplifications do not carry over to the case of incomplete rankings where the sets of consistent permutations $S_1,\ldots,S_m$ are not cosets of the symmetric group. As a result the problem of probability estimation in recommendation systems where $n$ is high and many items are missing is particularly challenging. However, as we show below replacing the Mallows kernel with a triangular kernel leads to efficient computation in some cases. Specifically, the triangular kernel on permutation is $$\begin{aligned}
\label{eq:triangularKernelRankedData} K_{h}(T(\pi,\sigma)) &= (1-h^{-1}T(\pi,\sigma)) \, I(h-T(\pi,\sigma)) \, /\, C\end{aligned}$$ where the bandwidth parameter $h$ represent both the support (the kernel is 0 for all larger distances) and the inverse slope of the triangle. As we show below the normalization term $C$ is a function of $h$ and may be efficiently computed using generating functions. Figure \[fig:kernelPlot\] (right panel) displays the linear decay of for the simple case of permutations over $n=3$ items.
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![Tricube, triangular, and uniform kernels on $\mathbb{R}$ with bandwidth $h=1$ (left) and $h=2$ (middle). Right: triangular kernel on permutations ($n=3$).[]{data-label="fig:kernelPlot"}](figure0004 "fig:"){width="\textwidth"}
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![Tricube, triangular, and uniform kernels on $\mathbb{R}$ with bandwidth $h=1$ (left) and $h=2$ (middle). Right: triangular kernel on permutations ($n=3$).[]{data-label="fig:kernelPlot"}](figure0005 "fig:"){width="\textwidth"}
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[c]{}
$\!\!K_{3}(\cdot,1\prec 2\prec 3)$ $\!\!\!\!\!\!K_{5}(\cdot,1\prec 2\prec 3)$
-------------------- ------------------------------------ --------------------------------------------
$1\prec 2 \prec 3$ 0.50 0.33
$1\prec 3 \prec 2$ 0.25 0.22
$2\prec 1 \prec 3$ 0.25 0.22
$3\prec 1 \prec 2$ 0 0.11
$2\prec 3 \prec 1$ 0 0.11
$3\prec 2 \prec 1$ 0 0
### Combinatorial Generating Function {#combinatorial-generating-function .unnumbered}
Generating functions, a tool from enumerative combinatorics, allow efficient computation of by concisely expressing the distribution of distances between permutations. Kendall’s tau $T(\pi,\sigma)$ is the total number of discordant pairs or inversions between $\pi,\sigma$ [@Stanley2000] and thus its computation becomes a combinatorial counting problem. We associate the following generating function with the symmetric group of order $n$ permutations $$\begin{aligned}
G_n(z)=\prod_{j=1}^{n-1} \sum_{k=0}^{j}z^k.\end{aligned}$$ As shown for example in [@Stanley2000] the coefficient of $z^k$ of $G_n(z)$, which we denote as $[z^k]G_n(z)$, corresponds to the number of permutations $\sigma$ for which $T(\sigma,\pi')=k$. For example, the distribution of Kendall’s tau $T(\cdot,\pi')$ over all permutations of 3 items is described by $G_3(z)=(1+z)(1+z+z^2)=1z^0+2z^1+2z^2+1z^3$ i.e., there is one permutation $\sigma$ with $T(\sigma,\pi')=0$, two permutations $\sigma$ with $T(\sigma,\pi')=1$, two with $T(\sigma,\pi')=2$ and one with $T(\sigma,\pi')=3$. Another important generating function is $$H_n(z)=\frac{G_n(z)}{1-z} = (1+z+z^2+z^3+\cdots)G_n(z)$$ where $[z^k]H_n(z)$ represents the number of permutations $\sigma$ for which $T(\sigma,\pi')\leq k$.
The normalization term $C(h)$ is given by $C(h) = [z^h] H_n(z)-h^{-1}[z^{h-1}]\frac{G_n'(z)}{1-z}.$
The proof factors the non-normalized triangular kernel $C K_h(\pi,\sigma)$ to $I(h-T(\pi,\sigma))$ and $h^{-1}T(\pi,\sigma)I(h-T(\pi,\sigma))$ and making the following observations. First we note that summing the first factor over all permutations may be counted by $[z^h]H_n(z)$. The second observation is that $[z^{k-1}]G_n'(z)$ is the number of permutations $\sigma$ for which $T(\sigma,\pi')=k$, multiplied by $k$. Since we want to sum over that quantity for all permutations whose distance is less than $h$ we extract the $h-1$ coefficient of the generating function $G_n'(z)\sum_{k\geq 0} z^k=G_n'(z)/(1-z)$. We thus have $$\begin{aligned}
C=\sum_{\sigma: T(\pi',\sigma)\leq h}1-h^{-1}\sum_{\sigma: T(\pi',\sigma)\leq h} T(\pi',\sigma) = [z^{h}]H_n(z)-h^{-1}[z^{h-1}]\frac{G_n'(z)}{1-z}.\end{aligned}$$
The complexity of computing $C(h)$ is $O(n^4)$.
We describe a dynamic programming algorithm to compute the coefficients of $G_n$ by recursively computing the coefficients of $G_k$ from the coefficients of $G_{k-1}$, $k=1,\ldots,n$. The generating function $G_k(z)$ has $k(k+1)/2$ non-zero coefficients and computing each of them (using the coefficients of $G_{k-1}$) takes $O(k)$. We thus have $O(k^3)$ to compute $G_k$ from $G_{k-1}$ which implies $O(n^4)$ to compute $G_k$, $n=1,\ldots,n$. We conclude the proof by noting that once the coefficients of $G_n$ are computed the coefficients of $H_n(z)$ and $G_n(z)/(1-z)$ are computable in $O(n^2)$ as these are simply cumulative weighted sums of the coefficients of $G_n$.
Note that computing $C(h)$ for one or many $h$ values may be done offline prior to the arrival of the rankings and the need to compute the estimated probabilities.
Denoting by $k$ the number of items ranked in either $S$ or $R$ or both, the computation of $\hat p(\pi)$ in requires $O(k^2)$ online and $O(n^4)$ offline complexity if either non-zero smoothing is performed over the entire data i.e., $\max_{\pi\in R}\max_{i=1}^n\max_{\sigma\in S_i} T(\sigma,\pi)<h$ or alternatively, we use the modified triangular kernel $K_h^*(\pi,\sigma)\propto (1-h^{-1})T(\pi,\sigma)$ which is allowed to take negative values for the most distant permutations (normalization still applies though).
For two sets of permutations $S,R$ corresponding to tied-incomplete rankings $$\begin{aligned}
\label{eq:expectedTau}
\frac{1}{|S||R|}\sum_{\pi\in S}\sum_{\sigma\in R} T(\pi,\sigma) &= \frac{n(n-1)}{4}-\frac{1}{2}\sum_{i=1}^{n-1}\sum_{j=i+1}^n (1-2p_{ij}(S))(1-2p_{ij}(R))\\
p_{ij}(U) &= \begin{cases}
I(\tau_U(j)-\tau_U(i)) & \mbox{$i$ and $j$ are ranked in $U$ with $\tau_U(i)\neq\tau_U(j)$}\\
1-\frac{\tau_U(i)+\frac{\phi_U(i)-1}{2}}{k+1} & \mbox{only $i$ is ranked in $U$}\\
\frac{\tau_U(j)+\frac{\phi_U(j)-1}{2}}{k+1} & \mbox{only $j$ is ranked in $U$}\\
1/2 & \mbox{otherwise}\end{cases}. \nonumber\end{aligned}$$ with $\tau_U(i)=\min_{\pi\in U} \pi(i)$, and $\phi_U(i)$ being the number of items that are tied to $i$ in $U$.
We note that is an expectation with respect to the uniform measure. We thus start by computing the probability $p_{ij}(U)$ that $i$ is preferred to $j$ for $U=S$ and $U=R$ under the uniform measure. Five scenarios exist for each of $p_{ij}(U)$ corresponding to whether each of $i$ and $j$ are ranked by $S,R$. Starting with the case that $i$ is not ranked and $j$ is ranked, we note that $i$ is equally likely to be preferred to any item or to be preferred to. Given the uniform distribution over compatible rankings item $j$ is equally likely to appear in positions $\tau_U(j),\ldots,\tau_U(j)+\phi_U(j)-1$. Thus $$\begin{aligned}
p_{ij}&=\frac{1}{\phi_U(j)}\frac{\tau_U(j)}{k+1}+\cdots+\frac{1}{\phi_U(j)}\frac{\tau_U(j)+\phi_U(j)-1}{k+1}=\frac{\tau_U(j)+\frac{\phi_U(j)-1}{2}}{k+1}\end{aligned}$$ Similarly, if $j$ is unknown and $i$ is known then $p_{ij}+p_{ji}=1$. If both $i$ and $j$ are unknown either ordering must be equally likely given the uniform distribution making $p_{ij}=1/2$. Finally, if both $i$ and $j$ are known $p_{ij}=1,1/2,0$ depending on their preference. Given $p_{ij}$, linearity of expectation, and the independence between rankings, the change in the expected number of inversions relative to the uniform expectation $n(n-1)/4$ can be found by considering each pair separately, $$\begin{aligned}
\mbox{E}T(i,j)&=&\frac{1}{2}P\left(\mbox{$i$ and $j$ disagree}\right)-\frac{1}{2}P\left(\mbox{$i$ and $j$ agree}\right)\\
&=&\frac{1}{2}(p_{ij}(\sigma)(1-p_{ij}(\pi))+(1-p_{ij}(\sigma))p_{ij}(\pi)) -p_{ij}(\sigma)p_{ij}(\pi)-(1-p_{ij}(\sigma))(1-p_{ij}(\pi)))\\
&=&\frac{-1}{2}\left(1-2p_{ij}(\sigma)\right)\left(1-2p_{ij}({\pi})\right).\end{aligned}$$ Summing the $n(n-1)/2$ components yields the desired quantity.
Denoting the number of items ranked by either $S$ or $R$ or both as $k$, and assuming either $h>\max_{\pi\in R}\max_{i=1}^n\max_{\sigma\in S_i} T(\sigma,\pi)$ or that the modified triangular kernel $K_h^*(\pi,\sigma)\propto (1-h^{-1})T(\pi,\sigma)$ is used, the complexity of computing $\hat p(R)$ in (assuming uniform $q(\pi|S_i)$) is $O(k^2)$ online and $O(n^4)$ offline.
\[corr:expDist\]
The proof follows from noting that reduces to $O(n^4)$ offline computation of the normalization term and $O(k^2)$ online computation of the form .
Applications and Case Studies
=============================
We divide our experimental study to three parts. In the first we examine the task of predicting probabilities. The remaining two parts use these probabilities for rank prediction and rule discovery.
In our experiments we used three datasets. The Movielens dataset[^2] contains one million ratings from $6040$ users over $3952$ movies. The EachMovie dataset[^3] contains $2.6$ million ratings from $74424$ users over $1648$ movies. The Netflix dataset [^4] contains 100 million movie ratings from $480189$ users on $17770$. In all of these datasets users typically rated only a small number of items. Histograms of the distribution of the number of votes per user, number of votes per item, and vote distribution appear in Figure \[fig:hists\].
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[Movielens]{} [Netflix]{} [EachMovie]{}
![Histograms of the number of user votes per movie (top row), number of movies ranked per user (middle row), and votes (bottom row). []{data-label="fig:hists"}](figure0006 "fig:") ![Histograms of the number of user votes per movie (top row), number of movies ranked per user (middle row), and votes (bottom row). []{data-label="fig:hists"}](figure0007 "fig:") ![Histograms of the number of user votes per movie (top row), number of movies ranked per user (middle row), and votes (bottom row). []{data-label="fig:hists"}](figure0008 "fig:")
![Histograms of the number of user votes per movie (top row), number of movies ranked per user (middle row), and votes (bottom row). []{data-label="fig:hists"}](figure0009 "fig:") ![Histograms of the number of user votes per movie (top row), number of movies ranked per user (middle row), and votes (bottom row). []{data-label="fig:hists"}](figure0010 "fig:") ![Histograms of the number of user votes per movie (top row), number of movies ranked per user (middle row), and votes (bottom row). []{data-label="fig:hists"}](figure0011 "fig:")
![Histograms of the number of user votes per movie (top row), number of movies ranked per user (middle row), and votes (bottom row). []{data-label="fig:hists"}](figure0012 "fig:") ![Histograms of the number of user votes per movie (top row), number of movies ranked per user (middle row), and votes (bottom row). []{data-label="fig:hists"}](figure0013 "fig:") ![Histograms of the number of user votes per movie (top row), number of movies ranked per user (middle row), and votes (bottom row). []{data-label="fig:hists"}](figure0014 "fig:")
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Estimating Probabilities
------------------------
We consider here the task of estimating $\hat p(R)$ where $R$ is a set of permutations corresponding to a tied incomplete ranking. Such estimates may be used to compute conditional estimates $\hat P(R|S_{m+1})$ which are used to predict which augmentations $R$ of $S_{m+1}$ are highly probable. For example, given an observed preference $3\prec 2 \prec 5$ we may want to compute $\hat p(8 \prec 3 \prec 2\prec 5 | 3\prec 2 \prec 5)=\hat p(8\prec 3\prec 2 \prec 5) / \hat p(3 \prec 2\prec 5)$ to see whether item $8$ should be recommended to the user.
For simplicity we focus in this section on probabilities of simple events such as $i\prec j$ or $i\prec j \prec k$. The next section deals with more complex events. In our experiment, we estimate the probability of $i\prec j$ for the $n=53$ most rated movies in Netflix and $m=10000$ users who rate most of these movies. The probability matrix of the pairs is shown in Figure \[fig:pairPrb1\] where each cell corresponds to the probability of preference between a pair of movies determined by row $j$ and column $i$. In the top left panel the rows and columns are ordered by average probability of a movie being preferred to others $r(i)=\frac{\sum_{j}\hat p(i\prec j)}{n}$ with the most preferred movie in row and column 1 (top right panel indicates the ordering according to $r(i)$). In the bottom left panel the movies were ordered first by popularity of genres and then by $r(i)$. The bottom right panel indicates that ordering. The names, genres, and both orderings of all 53 movies appear in Figure \[fig:netflix53stat\].
The three highest movies in terms of $r(i)$ are Lord of the Rings: The Return of the King, Finding Nemo, and Lord of the Rings: The Two Towers. The three lowest movies are Maid in Manhattan, Anger Management, and The Royal Tenenbaums. Examining the genre (colors in right panels of Figure \[fig:pairPrb1\]) we see that family and science fiction are generally preferred to others movies while comedy and romance generally receive lower preferences. The drama, action genres are somewhere in the middle.
Also interesting is the variance of the movie preferences within specific genres. Family movies are generally preferred to almost all other movies. Science fiction movies, on the other hand, enjoy high preference overall but exhibit a larger amount of variability as a few movies are among the least preferred. Similarly, the preference probabilities of action movies are widely spread with some movies being preferred to others and others being less preferred. More specifically (see bottom left panel of Figure \[fig:pairPrb1\]) we see that the decay of $r(i)$ within genres is linear for family and romance and nonlinear for science fiction, action, drama, and comedy. In these last three genres there are a few really “bad” movies that are substantially lower than the rest of the curve. Figure \[fig:netflix53stat\] shows the full information including titles, genres and orderings of the $53$ most popular movies in Netflix.
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![Left: The estimated probability of movie $i$ being preferred to movie $j$. Right: a plot of $r(i)=\sum_j \hat p(i\prec j)/n$ for all movies with color indicating genres. In both panels the movies were ordered by $r(i)$ (top row) and first by popularity of genres and then by $r(i)$ (bottom row).[]{data-label="fig:pairPrb1"}](figure0015 "fig:") ![Left: The estimated probability of movie $i$ being preferred to movie $j$. Right: a plot of $r(i)=\sum_j \hat p(i\prec j)/n$ for all movies with color indicating genres. In both panels the movies were ordered by $r(i)$ (top row) and first by popularity of genres and then by $r(i)$ (bottom row).[]{data-label="fig:pairPrb1"}](figure0016 "fig:")
![Left: The estimated probability of movie $i$ being preferred to movie $j$. Right: a plot of $r(i)=\sum_j \hat p(i\prec j)/n$ for all movies with color indicating genres. In both panels the movies were ordered by $r(i)$ (top row) and first by popularity of genres and then by $r(i)$ (bottom row).[]{data-label="fig:pairPrb1"}](figure0017 "fig:") ![Left: The estimated probability of movie $i$ being preferred to movie $j$. Right: a plot of $r(i)=\sum_j \hat p(i\prec j)/n$ for all movies with color indicating genres. In both panels the movies were ordered by $r(i)$ (top row) and first by popularity of genres and then by $r(i)$ (bottom row).[]{data-label="fig:pairPrb1"}](figure0018 "fig:")
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We plot the individual values of $\hat p(i\prec j)$ for three movies: Shrek (family), Catch Me If You Can (drama) and Napoleon Dynamite (comedy) (Figure \[fig:pairPrb3\]). Comparing the three stem plots we observe that Shrek is preferred to almost all other movies, Napoleon Dynamite is less preferred than most other movies, and Catch Me If You Can is preferred to some other movies but less preferred than others. Also interesting is the linear increase of the stem plots for Catch Me If You Can and Napoleon Dynamite and the non-linear increase of the stem plot for Shrek. This is likely a result of the fact that for very popular movies there are only a few comparable movies with the rest being very likely to be less preferred movies ($\hat p(i\prec j)$ close to 1).
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![The value $\hat p(i\prec j)$ for all $j$ for three movies: Shrek (left), Catch Me If You Can (middle) and Napoleon Dynamite (right).[]{data-label="fig:pairPrb3"}](figure0019 "fig:") ![The value $\hat p(i\prec j)$ for all $j$ for three movies: Shrek (left), Catch Me If You Can (middle) and Napoleon Dynamite (right).[]{data-label="fig:pairPrb3"}](figure0020 "fig:") ![The value $\hat p(i\prec j)$ for all $j$ for three movies: Shrek (left), Catch Me If You Can (middle) and Napoleon Dynamite (right).[]{data-label="fig:pairPrb3"}](figure0021 "fig:")
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[ll]{}
Titles Genre Order1 Order2
---------------------------------- ------- -------- --------
Finding Nemo 6 2 1
Shrek 6 4 2
The Incredibles 6 5 3
Monsters, Inc. 6 8 4
Shrek II 6 9 5
LOTR: The Return of the King 1 1 6
LOTR: The Two Towers 1 3 7
LOTR: The Fellowship of the Ring 1 6 8
Spider-Man II 1 12 9
Spider-Man 1 16 10
The Day After Tomorrow 1 36 11
Tomb Raider 1 46 12
Men in Black II 1 47 13
Pirates of the Caribbean I 3 7 14
The Last Samurai 3 10 15
Man on Fire 3 11 16
The Bourne Identity 3 13 17
The Bourne Supremacy 3 15 18
National Treasure 3 17 19
The Italian Job 3 19 20
Kill Bill II 3 23 21
Kill Bill I 3 25 22
Minority Report 3 31 23
S.W.A.T. 3 44 24
The Fast and the Furious 3 45 25
Ocean’s Eleven 2 14 26
I, Robot 2 20 27
&
Titles Genre Order1 Order2
---------------------------------- ------- -------- --------
Mystic River 2 21 28
Troy 2 22 29
Catch Me If You Can 2 24 30
Big Fish 2 28 31
Collateral 2 29 32
John Q 2 34 33
Pearl Harbor 2 35 34
Swordfish 2 39 35
Lost in Translation 2 48 36
50 First Dates 4 18 37
My Big Fat Greek Wedding 4 26 38
Something’s Gotta Give 4 27 39
The Terminal 4 30 40
How to Lose a Guy in 10 Days 4 32 41
Sweet Home Alabama 4 38 42
Sideways 4 41 43
Two Weeks Notice 4 43 44
Mr. Deeds 4 49 45
The Wedding Planner 4 50 46
Maid in Manhattan 4 53 47
The School of Rock 5 33 48
Bruce Almighty 5 37 49
Dodgeball: A True Underdog Story 5 40 50
Napoleon Dynamite 5 42 51
The Royal Tenenbaums 5 51 52
Anger Management 5 52 53
\
In a second experiment (see Figure \[fig:taskLoglikely\]) we compare the predictive behavior of the kernel smoothing estimator with that of a parametric model (Mallows model) and the empirical measure (frequency of event occurring in the $m$ samples). We evaluate the predictive performance of a probability estimator by separating the data to two parts: a training set that is used to construct the estimator and a testing set used for evaluation via its loglikelihood. A higher test set loglikelihood indicates that the model assigns high probability to events that occurred. Mathematically, this corresponds to approximating the KL divergence between nature and the model. Since the Mallows model is intractable for large $n$ we chose in this experiment small values of $n$: $3,4,5$.
We observe that the kernel estimator consistently achieves higher test set loglikelihood than the Mallows model and the empirical measure. The former is due to the breakdown of parametric assumptions as indicated by Figure \[fig:heatMaps\] (note that this happens even for $n$ as low as 3). The latter is due to the superior statistical performance of the kernel estimator over the empirical measure.
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[Movielens1M]{} [Netflix]{} [EachMovie]{}
![The test-set log-likelihood for kernel smoothing, Mallows model, and the empirical measure with respect to training size $m$ for a small number of items $n=3,4,5$ (top, middle, bottom rows) on three datasets. Both of the Mallows model (which is also intractable for large $n$ which is why $n\leq 5$ in the experiment) and the empirical measure perform worse than the kernel estimator $\hat p$.[]{data-label="fig:taskLoglikely"}](figure0022 "fig:") ![The test-set log-likelihood for kernel smoothing, Mallows model, and the empirical measure with respect to training size $m$ for a small number of items $n=3,4,5$ (top, middle, bottom rows) on three datasets. Both of the Mallows model (which is also intractable for large $n$ which is why $n\leq 5$ in the experiment) and the empirical measure perform worse than the kernel estimator $\hat p$.[]{data-label="fig:taskLoglikely"}](figure0023 "fig:") ![The test-set log-likelihood for kernel smoothing, Mallows model, and the empirical measure with respect to training size $m$ for a small number of items $n=3,4,5$ (top, middle, bottom rows) on three datasets. Both of the Mallows model (which is also intractable for large $n$ which is why $n\leq 5$ in the experiment) and the empirical measure perform worse than the kernel estimator $\hat p$.[]{data-label="fig:taskLoglikely"}](figure0024 "fig:")
![The test-set log-likelihood for kernel smoothing, Mallows model, and the empirical measure with respect to training size $m$ for a small number of items $n=3,4,5$ (top, middle, bottom rows) on three datasets. Both of the Mallows model (which is also intractable for large $n$ which is why $n\leq 5$ in the experiment) and the empirical measure perform worse than the kernel estimator $\hat p$.[]{data-label="fig:taskLoglikely"}](figure0025 "fig:") ![The test-set log-likelihood for kernel smoothing, Mallows model, and the empirical measure with respect to training size $m$ for a small number of items $n=3,4,5$ (top, middle, bottom rows) on three datasets. Both of the Mallows model (which is also intractable for large $n$ which is why $n\leq 5$ in the experiment) and the empirical measure perform worse than the kernel estimator $\hat p$.[]{data-label="fig:taskLoglikely"}](figure0026 "fig:") ![The test-set log-likelihood for kernel smoothing, Mallows model, and the empirical measure with respect to training size $m$ for a small number of items $n=3,4,5$ (top, middle, bottom rows) on three datasets. Both of the Mallows model (which is also intractable for large $n$ which is why $n\leq 5$ in the experiment) and the empirical measure perform worse than the kernel estimator $\hat p$.[]{data-label="fig:taskLoglikely"}](figure0027 "fig:")
![The test-set log-likelihood for kernel smoothing, Mallows model, and the empirical measure with respect to training size $m$ for a small number of items $n=3,4,5$ (top, middle, bottom rows) on three datasets. Both of the Mallows model (which is also intractable for large $n$ which is why $n\leq 5$ in the experiment) and the empirical measure perform worse than the kernel estimator $\hat p$.[]{data-label="fig:taskLoglikely"}](figure0028 "fig:") ![The test-set log-likelihood for kernel smoothing, Mallows model, and the empirical measure with respect to training size $m$ for a small number of items $n=3,4,5$ (top, middle, bottom rows) on three datasets. Both of the Mallows model (which is also intractable for large $n$ which is why $n\leq 5$ in the experiment) and the empirical measure perform worse than the kernel estimator $\hat p$.[]{data-label="fig:taskLoglikely"}](figure0029 "fig:") ![The test-set log-likelihood for kernel smoothing, Mallows model, and the empirical measure with respect to training size $m$ for a small number of items $n=3,4,5$ (top, middle, bottom rows) on three datasets. Both of the Mallows model (which is also intractable for large $n$ which is why $n\leq 5$ in the experiment) and the empirical measure perform worse than the kernel estimator $\hat p$.[]{data-label="fig:taskLoglikely"}](figure0030 "fig:")
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Rank Prediction
---------------
Our task here is to predict ranking of new unseen items for users. We follow the standard procedure in collaborative filtering: the set of users is partitioned to two sets, a training set and a testing set. For each of the test set users we further split the observed items into two sets: one set used for estimating preferences (together with the preferences of the training set users) and the second set to evaluate the performance of the prediction [@Pennock2000]. Given a loss function $L(i,j)$ which measures the loss of predicting rank $i$ when true rank is $j$ (rank here refers to the number of sets of equivalent items that are more or less preferred than the current item) we evaluate a prediction rule by the expected loss. We focus on three loss functions: $L_0(i,j)=0 \text{ if } i=j \text{ and } 1 \text{ otherwise}$, $L_1(i,j)=|i-j|$ which reduces to the standard CF evaluation technique described in [@Pennock2000], and an asymmetric loss function (rows correspond to estimated number of stars (0-5) and columns to actual number of stars (0-5) $$\begin{aligned}
L_e=\begin{pmatrix}
0 & 0 & 0 & 3 & 4 & 5 \\
0 & 0 & 0 & 2 & 3 & 4 \\
0 & 0 & 0 & 1 & 2 & 3 \\
9 & 4 & 1.5&0 & 0 & 0\\
12 & 6 & 3&0 & 0 & 0\\
15 & 8 & 4.5&0 & 0 & 0\\
\end{pmatrix}.\end{aligned}$$ In contrast to the $L_0$ and $L_1$ loss, $L_e$ captures the fact that recommending bad movies as good movies is worse than recommending good movies as bad.
For example, consider a test user whose observed preference is $3 \prec 4,5,6 \prec 10,11,12 \prec 23\prec 40,50,60\prec 100,101$. We may withhold the preferences of items $4,11$ for evaluation purposes. The recommendation systems then predict a rank of 1 for item 4 and a rank of 4 for item 11. Since the true ranking of these items are 2 and 3 the absolute value loss is $|1-2|=1$ and $|3-4|=1$ respectively.
In our experiment, we use the kernel estimator $\hat p$ to predict ranks that minimize the posterior loss and thus adapts to customized loss functions such as $L_e$. This is an advantage of a probabilistic modeling approach over more ad-hoc rule based recommendation systems.
Figure \[fig:taskItemPredict\] compares the performance of our estimator to several standard baselines in the collaborative filtering literature: two older memory based methods vector similarity (sim1), correlation (sim2) e.g., [@Breese1998], and a recent state-of-the-art non-negative matrix (NMF) factorization (gnmf) [@Lawrence09]. The kernel smoothing estimate performed similar to the state-of-the-art but substantially better than the memory based methods to which it is functionally similar.
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[Movielens]{} [Netflix]{} [EachMovie]{}
![The prediction loss (top row: 0/1 loss $L_0$, middle row: $L_1$ loss, bottom row: asymmetric loss $L_e$) with respect to training size on three datasets. The kernel smoothing estimate performed similar to the state-of-the-art gnmf (matrix factorization) but substantially better than the memory based methods to which it is functionally similar. []{data-label="fig:taskItemPredict"}](figure0031 "fig:") ![The prediction loss (top row: 0/1 loss $L_0$, middle row: $L_1$ loss, bottom row: asymmetric loss $L_e$) with respect to training size on three datasets. The kernel smoothing estimate performed similar to the state-of-the-art gnmf (matrix factorization) but substantially better than the memory based methods to which it is functionally similar. []{data-label="fig:taskItemPredict"}](figure0032 "fig:") ![The prediction loss (top row: 0/1 loss $L_0$, middle row: $L_1$ loss, bottom row: asymmetric loss $L_e$) with respect to training size on three datasets. The kernel smoothing estimate performed similar to the state-of-the-art gnmf (matrix factorization) but substantially better than the memory based methods to which it is functionally similar. []{data-label="fig:taskItemPredict"}](figure0033 "fig:")
![The prediction loss (top row: 0/1 loss $L_0$, middle row: $L_1$ loss, bottom row: asymmetric loss $L_e$) with respect to training size on three datasets. The kernel smoothing estimate performed similar to the state-of-the-art gnmf (matrix factorization) but substantially better than the memory based methods to which it is functionally similar. []{data-label="fig:taskItemPredict"}](figure0034 "fig:") ![The prediction loss (top row: 0/1 loss $L_0$, middle row: $L_1$ loss, bottom row: asymmetric loss $L_e$) with respect to training size on three datasets. The kernel smoothing estimate performed similar to the state-of-the-art gnmf (matrix factorization) but substantially better than the memory based methods to which it is functionally similar. []{data-label="fig:taskItemPredict"}](figure0035 "fig:") ![The prediction loss (top row: 0/1 loss $L_0$, middle row: $L_1$ loss, bottom row: asymmetric loss $L_e$) with respect to training size on three datasets. The kernel smoothing estimate performed similar to the state-of-the-art gnmf (matrix factorization) but substantially better than the memory based methods to which it is functionally similar. []{data-label="fig:taskItemPredict"}](figure0036 "fig:")
![The prediction loss (top row: 0/1 loss $L_0$, middle row: $L_1$ loss, bottom row: asymmetric loss $L_e$) with respect to training size on three datasets. The kernel smoothing estimate performed similar to the state-of-the-art gnmf (matrix factorization) but substantially better than the memory based methods to which it is functionally similar. []{data-label="fig:taskItemPredict"}](figure0037 "fig:") ![The prediction loss (top row: 0/1 loss $L_0$, middle row: $L_1$ loss, bottom row: asymmetric loss $L_e$) with respect to training size on three datasets. The kernel smoothing estimate performed similar to the state-of-the-art gnmf (matrix factorization) but substantially better than the memory based methods to which it is functionally similar. []{data-label="fig:taskItemPredict"}](figure0038 "fig:") ![The prediction loss (top row: 0/1 loss $L_0$, middle row: $L_1$ loss, bottom row: asymmetric loss $L_e$) with respect to training size on three datasets. The kernel smoothing estimate performed similar to the state-of-the-art gnmf (matrix factorization) but substantially better than the memory based methods to which it is functionally similar. []{data-label="fig:taskItemPredict"}](figure0039 "fig:")
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Rule Discovery
--------------
In the third task, we used the estimator $\hat p$ to detect noteworthy association rules of the type $i\prec j \Rightarrow k\prec l$ (if $i$ is preferred to $j$ than it is probably the case that $k$ is preferred to $l$). Such association rules are important for both business analytics (devising marketing and manufacturing strategies) and recommendation system engineering. Specifically, we used $\hat p$ to select sets of four items $i,j,k,l$ for which the mutual information $I(i\prec j\,; k\prec l)$ is maximized. After these sets are identified we detected the precise shape of the rule (i.e., $i\prec j \Rightarrow k\prec l$ rather than $j\prec i \Rightarrow k\prec l$ by examining the summands in the mutual information expectation).
Figure \[fig:task3\] (top) shows the top 10 rules that were discovered. These rules nicely isolate viewer preferences for genres such as fantasy, romantic comedies, animation, and action (note however that genre information was not used in the rule discovery). To quantitatively evaluate the rule discovery process we judge a rule $i\prec j \Rightarrow k\prec l$ to be good if $i,k$ are of the same genre and $j,l$ are of the same genre. This quantitative evaluation appears in Figure \[fig:task3\] (bottom) where it is contrasted with the same rule discovery process (maximizing mutual information) based on the empirical measure.
\
![Top: top 10 rules discovered by the kernel smoothing estimator on Netflix in terms of maximizing mutual information. Bottom: a quantitative evaluation of the rule discovery. The $x$ axis represents the number of rules discovered and the $y$ axis represents the frequency of good rules in the discovered rules. Here a rule $i\prec j \Rightarrow k\prec l$ is considered good if $i,k$ are of the same genre and $j,l$ are of the same genre.[]{data-label="fig:task3"}](figure0040 "fig:")
In another rule discovery experiment, we used $\hat p$ to detect association rules of the form $i \text{ ranked highest} \Rightarrow j \text{ ranked second highest}$ by selecting $i,j$ that maximize the score $\frac{p(\pi(i)=1,\pi(j)=2)}{p(\pi(i)=1)p(\pi(j)=2)}$ between pairs of movies in the Netflix data. We similarly detected rules of the form $i \text{ ranked highest} \Rightarrow j \text{ ranked lowest}$ by maximizing the scores $\frac{p(\pi(i)=1,\pi(j)=\text{last})}{p(\pi(i)=1)p(\pi(j)=\text{last})}$ between pairs of movies.
The left panel of Figure \[fig:tasks4preferPair\] shows the top 9 rules of 100 most rated movies, which nicely represents movie preference of similar type, e.g. romance, comedies, and action. The right of Figure \[fig:tasks4preferPair\] shows the top 9 rules which represents like and dislike of different movie types, e.g. like of romance leads to dislike of action/thriller.
In a third experiment, we used $\hat p$ to construct an undirected graph where vertices are items (Netflix movies) and two nodes $i$,$j$ are connected by an edge if the average score of the rule $i \text{ ranked highest} \Rightarrow j \text{ ranked second highest}$ and the rule $j \text{ ranked highest} \Rightarrow i \text{ ranked second highest}$ is higher than a certain threshold. Figure \[fig:tasks4cluster\] shows the graph for the 100 most rated movies in Netflix (only movies with vertex degree greater than 0 are shown). The clusters in the graph corresponding to vertex color and numbering were obtained using a graph partitioning algorithm and the graph is embedded in a 2-D plane using standard graph visualization technique. Within each of the identified clusters movies are clearly similar with respect to genre, while an even finer separation can be observed when looking at specific clusters. For example, clusters 6 and 9 both contain comedy movies, where as cluster 6 tends toward slapstick humor and cluster 9 contains romantic comedies.
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![A graph corresponding to the 100 most rated Netflix movies where edges represent high affinity as determined by the rule discovery process (see text for more details).[]{data-label="fig:tasks4cluster"}](figure0041)
Related Work
============
Collaborative filtering or recommendation system has been an active research area in computer science since the 1990s. The earliest efforts made a prediction for the rating of items based on the similarity of the test user and the training users [@Resnick1994; @Breese1998; @Herlocker1999]. Specifically, these attempts used similarity measures such as Pearson correlation [@Resnick1994] and Vector cosine similarity [@Breese1998; @Herlocker1999] to evaluate the similarity level between different users.
More recent work includes user and movie clustering [@Breese1998; @Ungar1998; @Xue2005], item-item similarities [@Sarwar2001], Bayesian networks [@Breese1998], dependence network [@Heckerman2000] and probabilistic latent variable models [@Pennock2000; @Hofmann2004; @marlin2004].
Most recently, the state of the art methods including the winner of the Netflix competition are based on non-negative matrix factorization of the partially observed user-rating matrix. The factorized matrix can be used to fill out the unobserved entries in a way similar to latent factor analysis [@Jester; @Rennie2005; @Lawrence09; @Koren2010].
Each of the above methods focuses exclusively on user ratings. In some cases item information is available (movie genre, actors, directors, etc) which have lead to several approaches that combine voting information with item information e.g., [@Basu1998; @Popescul2001; @Schein2002].
Our method differs from the methods above in that it constructs a full probabilistic model on preferences, it is able to handle heterogeneous preference information (not all users must specify the same number of preference classes) and does not make any parametric assumptions. In contrast to previous approaches it enables not only the prediction of item ratings, but also the discovery of association rules and the estimation of probabilities of interesting events.
Summary
=======
Estimating distributions from tied and incomplete data is a central task in many applications with perhaps the most obvious one being collaborative filtering. An accurate estimator $\hat p$ enables going beyond the traditional item-rank prediction task. It can be used to compute probabilities of interest, find association rules, and perform a wide range of additional data analysis tasks.
We demonstrate the first non-parametric estimator for such data that is computationally tractable i.e., polynomial rather than exponential in $n$. The computation is made possible using generating function and dynamic programming techniques.
We examine the behavior of the estimator $\hat p$ in three sets of experiments. The first set of experiments involves estimating probabilities of interest such as $p(i\prec j)$. The second set of experiments involves predicting preferences of held-out items which is directly applicable in recommendation systems. In this task, our estimator outperforms other memory based methods (to which it is similar functionally) and performs similarly to state-of-the-art methods that are based on non-negative matrix factorization. In the third set of experiments we examined the usage of the estimator in discovering association rules such as $i\prec j \Rightarrow k\prec l$.
[^1]: Corresponding author: [email protected]
[^2]: http://www.grouplens.org
[^3]: http://www.grouplens.org/node/76
[^4]: http://www.netflixprize.com/community
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{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We consider the problem of the symplectic realization of a Poisson-Nijenhuis manifold. By applying a new technique developed by M. Crainic and I. M$\check{\mathrm{a}}$rcu$\c{t}$ for the study of the above problem in the case of a Poisson manifold, we establish the existence, under a condition, of a nondegenerate Poisson-Nijenhuis structure on an open neighborhood of the zero-section of the cotangent bundle of the manifold, which symplectizes the initial structure. Additionally, we present some examples.'
author:
- |
Fani Petalidou\
\
*Department of Mathematics*\
*Aristotle University of Thessaloniki*\
*54124 Thessaloniki, Greece*\
\
*E-mail: [email protected]*
title: 'On the symplectic realization of Poisson-Nijenhuis manifolds'
---
1 cm
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*Dedicated to Professor Charles-Michel Marle,*
*with my deepest admiration and respect,*
*on the occasion of his 80$^{th}$ birthday.*
-- ------------------------------------------------
15 mm
[**[Keywords: ]{}**]{}[Bi-hamiltonian manifold, Poisson-Nijenhuis manifold, symplectic realization, contravariant connection, Poisson-spray.]{}
[**[MSC (2010):]{}**]{} 53D17, 53D05, 53D25, 37K10.
Introduction
============
A *bi-Hamiltonian manifold* is a smooth manifold $M$ endowed with a pair $(\Pi_0,\Pi_1)$ of compatible Poisson structures in the sense that $\Pi_0 + \Pi_1$ is still a Poisson structure. The last condition happens if and only if the Schouten bracket of $\Pi_0$ with $\Pi_1$ vanishes. *Poisson-Nijenhuis* manifolds is a particular class of bi-Hamiltonian manifolds which are characterized by the property that the pair $(\Pi_0, \Pi_1)$ possesses a Nijenhuis operator $N$ as a recursion operator. The first notion is due to F. Magri [@magri-1978] and the second has been introduced by F. Magri and C. Morosi [@magri-mo] in order to study the complete integrability of Hamiltonian dynamical systems. Franco Magri first discovered that if a dynamical system $X$ on $M$ can be written in Hamiltonian form in two different compatible ways, namely, there exist a bi-Hamiltonian structure $(\Pi_0, \Pi_1)$ on $M$ and $f_0,f_1 \in \C$ such that $$X = \Pi_0^\#(df_1) = \Pi_1^\#(df_0),$$ then it possesses an infinity of first integrals. More precisely, if $\Pi_0$ and $\Pi_1$ have Casimirs functions, then, they are first integrals of $X$ as well as the Casimirs of the Poisson pencil $\Pi_\lambda = \Pi_0 + \lambda \Pi_1$, $\lambda \in \R$. While, if $\Pi_0$ is nondegenerate, then the pair $(\Pi_0, \Pi_1)$ has a recursion operator $N = \Pi_1^\# \circ \Pi_0^{\#^{-1}}$ and the functions $I_k = \mathrm{Trace}(N^k)$, $k=1,\ldots, n=\displaystyle{\frac{1}{2}\dim M}$, are first integrals of $X$. Consequently, if enough of the obtained first integrals are functionally independent, then the system is completely integrable in the sense of Arnold-Liouville. This result is the origin of an increased interest in the study of bi-Hamiltonian and specially Poisson-Nijenhuis manifolds during the last 35 years. Many mathematicians have examined a plethora of problems related with these structures. Indicatively, we cite the works of Y. Kosmann-Schwarzbach and F. Magri [@yks-magri] where Poisson-Nijenhuis and related structures are studied under algebraic circumstances, [@yks-LMP] in which Y. Kosmann-Schwarzbach suggested a relation between Lie bialgebroids and Poisson-Nijenhuis structures, [@vai-N] and [@vai-Red-PN] in which I. Vaisman developed the theory of Poisson-Nijenhuis manifolds using Lie algebroids and studied the reduction problem of these manifolds, respectively. Moreover, we cite the works of I. M. Gel’fand and I. Zakharevich [@gel-zak], P. J. Olver [@olver], F. J. Turiel [@tu] and of the author [@petal-thesis], [@petal-coimbra] where the local classification of bi-Hamiltonian and Poisson-Nijenhuis structures is considered.
One of the most important problems in Poisson geometry from the point of view of integration and quantization theory of Poisson manifolds is that of *symplectic realization of a Poisson manifold* $(M,\Pi)$. It consists of constructing a surjective submersion $\Phi : (\tilde{M}, \tilde{\Pi}) \to (M, \Pi)$ from a symplectic-Poisson manifold $(\tilde{M}, \tilde{\Pi})$, i.e. $\tilde{\Pi}$ is nondegenerate, to $(M,\Pi)$ such that $\Phi$ is a Poisson map. The existence of a local symplectic realization for a given $(M, \Pi)$, i.e. in the neighborhood of a singular point of $\Pi$, and its universality was proven by A. Weinstein in [@wei] while the existence of a global symplectic realization for any Poisson manifold is established by A. Weinstein and his collaborators A. Coste and P. Dazord in [@cdw]. The same global result was obtained independently by M. Karasev [@karasev].
The analogous problem in the framework of bi-Hamiltonian manifolds is expressed as follows: *For a given bi-Hamiltonian manifold $(M, \Pi_0, \Pi_1)$ whose all the Poisson structures of the associated Poisson pencil $\Pi_\lambda = \Pi_0 + \lambda \Pi_1$, $\lambda \in \R$, are degenerate, construct a surjective submersion $\Phi : (\tilde{M}, \tilde{\Pi}_0, \tilde{\Pi}_1) \to (M, \Pi_0, \Pi_1)$ from a nondegenerate bi-Hamiltonian manifold $(\tilde{M}, \tilde{\Pi}_0, \tilde{\Pi}_1)$, i.e., at least one of the structures of $(\tilde{\Pi}_0, \tilde{\Pi}_1)$ is nondegenerate, to $(M, \Pi_0, \Pi_1)$ such that $\Phi$ is a Poisson map for the both pairs $(\tilde{\Pi}_0, \Pi_0)$ and $(\tilde{\Pi}_1, \Pi_1)$.* It is a difficult problem and it has been studied by the author in [@petal-sympl]. Her results are local and concern some special cases.
Recently, M. Crainic and I. M$\check{\mathrm{a}}$rcu$\c{t}$ have presented a new proof of the existence of a symplectic realization of a Poisson manifold $(M, \Pi)$ based on the theory of contravariant connections and of Poisson sprays. It consisted in the construction of a symplectic form on an open neighborhood of the zero-section of the cotangent bundle of $M$ [@cr-marcut]. By studying this paper the natural question which arises is: *Can we use the M. Crainic and I. M$\check{\mathrm{a}}$rcu$\c{t}$’s new technique in order to investigate the symplectic realization problem of a degenerate bi-Hamiltonian manifold $(M, \Pi_0, \Pi_1)$?*
The purpose of this paper is to study the above question. Because of the difficulties that we have encountered in the consideration of the general case, we restricted our study in the case of Poisson-Nijenhuis manifolds and we proved the following result: *For any Poisson-Nijenhuis manifold $(M, \Pi_0, N)$ endowed with a symmetric covariant connection $\nabla$ compatible with $N$, in a sense specified below, there exists a nondegenerate bi-Hamiltonian structure on an open neighborhood of the zero-section of the cotangent bundle of $M$ that symplectizes $(\Pi_0, N)$.*
The proof of the main result is given in Section \[section theorem\] and for its presentation we follow the notation of M. Crainic and I. M$\check{\mathrm{a}}$rcu$\c{t}$’s paper. Section \[preliminaries\] is devoted to the recall of some preliminary notions while in section \[section examples\] we give some examples.
Preliminaries
=============
We first fix our notation and recall some important notions and results needed in the following. Let $M$ be a smooth $n$-dimensional manifold, we denote by $TM$ and $T^\ast M$ its tangent and cotangent bundle, respectively, by $\Gamma(TM)$ and $\Gamma(T^\ast M)$ the corresponding spaces of smooth sections of $TM$ and $T^\ast M$, and by $\C$ the space of smooth functions on $M$. The canonical projection of $T^\ast M$ onto the base $M$ is denoted by $\pi: T^\ast M \to M$. Finally, for any local coordinate system $(x^1,\ldots, x^n)$ of $M$ we denote by $(x^1,\ldots,x^n,y_1,\ldots,y_n)$ or $(x,y)$ the adapted coordinate system on $T^\ast M$.
Poisson-Nijenhuis manifolds
---------------------------
We recall some basic definitions concerning Poisson structures and we give the formal definition of a Poisson-Nijenhuis manifold.
A *Poisson manifold* $(M, \Pi)$ is a smooth manifold $M$ equipped with a smooth bivector field $\Pi$ such that $[\Pi, \Pi]= 0$, where $[\cdot, \cdot]$ denotes the Schouten bracket, the unique natural extension of the Lie bracket between vector fields to multivector fields, [@lib-marle], [@vai-b], [@duf-zung]. The bivector field $\Pi$ defines a natural vector bundle morphism $\Pi^\# : T^\ast M \to TM$ whose the induced morphism on the space of smooth sections, also denoted by $\Pi^\#$, is defined, for any $\alpha, \beta \in \Gamma(T^\ast M)$, by $$\label{def-diese}
\langle \beta, \Pi^\#(\alpha)\rangle = \Pi(\alpha, \beta).$$ The map $\Pi^\#$ is a Lie algebra homomorphism from the Lie algebra $(\Gamma(T^\ast M), [\cdot,\cdot]_\Pi)$ to the Lie algebra $(\Gamma(TM), [\cdot,\cdot])$, where the Lie bracket $[\cdot,\cdot]_\Pi$ on $\Gamma(T^\ast M)$ is given by $$\label{bracket-Lie-forms}
[\alpha, \beta]_\Pi = \mathcal{L}_{\Pi^\#(\alpha)}\beta - \mathcal{L}_{\Pi^\#(\beta)}\alpha - d(\Pi(\alpha,\beta)).$$ In the particular case where $\alpha = df$, the vector field $\Pi^\#(df)$ is called the Hamiltonian vector field of $f$ with respect to $\Pi$ and it is denoted by $X_f$.
A differentiable map between two Poisson manifolds $\Phi : (M_1, \Pi_1) \to (M_2, \Pi_2)$ is called *Poisson map* or *Poisson morphism* if the vector bundle morphisms $\Pi_1^\# : T^\ast M_1 \to TM_1$ and $\Pi_2^\# : T^\ast M_2 \to TM_2$ satisfy, for all $x\in M_1$, $$\label{Poisson-map}
\Pi_{2_{\Phi(x)}}^\# = \Phi_{\ast_x} \circ \Pi_{1_x}^\# \circ \Phi_x^\ast.$$
A *Nijenhuis structure* on a manifold $M$ is a tensor field $N$ of type $(1,1)$, viewed also as a vector bundle map $N : TM \to TM$, with Nijenhuis torsion $T(N) : TM\times TM \to TM$ identically zero on $M$. This means that, for any pair $(X,Y)$ of vector fields on $M$, $$T(N)(X, Y) = [NX, NY] - N[NX, Y] - N[X, NY] + N^2[X,Y]\equiv 0.$$
A *Poisson-Nijenhuis* manifold is a Poisson manifold $(M,\Pi_0)$ equipped with a compatible Nijenhuis structure $N$ in the sense that $$\label{cond-N-Pi0}
N\circ \Pi_0^\# = \Pi_0^\#\circ \,^tN$$ and the Magri-Morosi’s concomitant $C(\Pi_0, N)$ of $\Pi_0$ and $N$ is identically zero on $M$. The concomitant $C(\Pi_0,N)$ is a $T^\ast M$-valued bivector field on $M$ defined, for any $(\alpha, \beta)\in \Gamma(T^\ast M) \times \Gamma(T^\ast M)$, by $$\label{def-concomitant}
C(\Pi_0, N)(\alpha, \beta) = (\mathcal{L}_{\Pi_0^\#(\alpha)}\,^tN)\beta - (\mathcal{L}_{\Pi_0^\#(\beta)}\,^tN)\alpha + \,^tN d(\Pi_0(\alpha, \beta)) - d(\Pi_1(\alpha,\beta)).$$ The condition (\[cond-N-Pi0\]) and the vanishing of $C(\Pi_0,N)$ ensures that $\Pi_1$, defined by $\Pi_1^\# = N\circ \Pi_0^\#$, is a bivector field which satisfies the relation $[\Pi_0, \Pi_1] =0$. Then, the vanishing of $T(N)$ implies that $\Pi_1$ is also Poisson, [@joana-marle]. Thus, $(\Pi_0,\Pi_1)$ is a bihamiltonian structure.
As is well-known [@yks-magri], a Poisson-Nijenhuis structure $(\Pi_0, N)$ defines a whole hierarchy $(\Pi_k)_{k\in \N}$ of Poisson structures, $\Pi_k^\# = N^k\circ \Pi_0^\#$, which are pairwise compatible, i.e., for all $k,l \in \N$, $[\Pi_k, \Pi_l]=0$. In the particular case where $N$ is nondegenerate, the hierarchy is defined for any $k\in \Z$, also.
Lifts to the cotangent bundle
-----------------------------
The theory of *lifts* of tensor fields from an arbitrary manifold $M$ to its cotangent bundle $T^\ast M$ is dealt with in the book by K. Yano and S. Ishihara [@Yano-Ishi]. In this subsection we shall present some results concerning the lifts on $T^\ast M$ of a Nijenhuis tensor and of a vector field on $M$.
Let $N$ be a Nijenhuis tensor field on $M$ which in a local coordinate system $(x^1, \ldots, x^n)$ of $M$ is written as $N = \nu^i_j \displaystyle{\frac{\partial}{\partial x^i}}\otimes dx^j$. We shall present two different ways of lifting $N$ on $T^\ast M$. In the first way, the *vertical lift of* $N$, we get a vertical vector field $N^v$ on $T^\ast M$ given by $N^v = y_i \nu^i_j \displaystyle{\frac{\partial}{\partial y_j}}$. In the second way, the *complete lift of* $N$, we obtain a tensor field $N^c$ of type $(1,1)$ again on $T^\ast M$; its local expression in the coordinates $(x^1, \ldots, x^n, y_1, \ldots, y_n)$ of $T^\ast M$ is $$N^c = \nu^i_j (\frac{\partial}{\partial x^i}\otimes dx^j + \frac{\partial}{\partial y_j}\otimes dy_i) + y_i(\frac{\partial \nu^i_j}{\partial x^k} - \frac{\partial \nu^i_k}{\partial x^j} )\frac{\partial}{\partial y_j}\otimes dx^k.$$ Moreover, $N^c$ viewed as a vector bundle map $N^c : T(T^\ast M) \to T(T^\ast M)$ has the matrix expression $$\label{expression N}
N^c = \begin{pmatrix} N & 0 \cr
A & \,^t N
\end{pmatrix},$$ where $A = (a^j_k)$ with $a^j_k = y_i\displaystyle{(\frac{\partial\nu^i_j}{\partial x^k} - \frac{\partial\nu^i_k}{\partial x^j})}$. We have that $T(N^c) = (T(N))^c$, where $(T(N))^c$ is the complete lift on $T^\ast M$ of the skew-symmetric $(1,2)$-tensor field $T(N)$ on $M$ [@Yano-Ishi]. Thus we conclude
The complete lift $N^c$ of a Nijenhuis operator $N$ on $M$ is a Nijenhuis operator on $T^\ast M$ and reciprocally.
Now, we assume that $M$ is endowed with a classical symmetric linear connection $\nabla$. The symmetry condition of $\nabla$ means that its torsion $T_{\nabla}$ is identically zero, i.e., for any $X,Y \in \Gamma(TM)$, $$\label{torsion-connection}
T_{\nabla}(X,Y) = \nabla_X Y - \nabla_Y X - [X, Y] \equiv 0.$$ In a coordinate system $(x^1,\ldots,x^n)$, the symmetry of $\nabla$ is expressed by the fact that $\Gamma_{ij}^k = \Gamma^k_{ji}$, where $\Gamma^k_{ij}$, $i,j,k = 1,\ldots,n$, are the coefficients (Christoffel symbols) of $\nabla_{\frac{\partial}{\partial x^i}}\displaystyle{\frac{\partial}{\partial x^j}} = \Gamma_{ij}^k\displaystyle{\frac{\partial}{\partial x^k}}$. For a vector field $X = \chi^i \displaystyle{\frac{\partial}{\partial x^i}}$ on $M$, we set $$X^h = \chi^i \frac{\partial}{\partial x^i} + y_k\Gamma^k_{ij}\chi^j \frac{\partial}{\partial y_i}$$ and we call the vector field $X^h$ the *horizontal lift of* $X$ on $T^\ast M$. The horizontal lifts of all vector fields on $M$ define the *horizontal bundle* $\mathcal{H}$ on $T^\ast M$ with respect to $\nabla$ which is a distribution on $T^\ast M$ complementary to its vertical distribution $\ker \pi_\ast$, namely $$T(T^\ast M) = \mathcal{H} \oplus \ker \pi_\ast,$$ [@sau]. The sections of $\mathcal{H}$ are called *horizontal vector fields* of $T^\ast M$. It is evident that a horizontal lift is indeed horizontal and, conversely, a *projectable horizontal vector field*, i.e., a horizontal vector field whose coefficients of $\displaystyle{\frac{\partial}{\partial x^i}}$ are functions pulled back from $M$, is the horizontal lift of its projection. $\mathcal{H}$ is involutive if $\nabla$ is of zero curvature.
We have that the complete lift $N^c$ of a Nijenhuis operator $N$ maps the projectable horizontal vector fields $X^h$, $X\in \Gamma(TM)$, to $$\label{im hor by N compl}
N^c X^h = (NX)^h + [\nabla N]_X^v,$$ where $[\nabla N]_X^v$ is the vertical lift of the $(1,1)$-tensor field $[\nabla N]_X$ on $M$ defined, for any $Y \in \Gamma(TM)$, by $$[\nabla N]_X Y = (\nabla_X N)Y - (\nabla_Y N)X,$$ [@Yano-Ishi]. Therefore, we get that $N^c$ conserves the horizontal distribution $\mathcal{H}$ if and only if $[\nabla N]_X = 0$, for any $X\in \Gamma(TM)$, or, equivalently, for any $X, Y \in \Gamma(TM)$, $$\label{cond-nabla-N}
(\nabla_X N)Y - (\nabla_Y N)X = 0.$$ In the following, equation (\[cond-nabla-N\]) will be referred as *compatibility condition between $N$ and $\nabla$*. An easy computation yields that, in local coordinates, (\[cond-nabla-N\]) is equivalent to the system of equations $$\label{cond-nabla-N locally}
\frac{\partial \nu^i_j}{\partial x^k} - \frac{\partial \nu^i_k}{\partial x^j} = \Gamma^i_{jl}\nu^l_k - \Gamma^i_{kl}\nu^l_j.$$
Contravariant connections
--------------------------
The fundamental concept of *contravariant connection* in Poisson geometry appeared firstly in R. L. Fernandes’ paper [@fer-conn] and from then it has an essential contribution in the study of the global properties of Poisson manifolds, [@fer-Lie; @Holo], [@cr-fer-int-Pois]. Its definition is inspired by that of a classical covariant connection and it is based on the general philosophical principle in Poisson geometry that *the cotangent bundle plays the role of the tangent bundle and the two are related by the bundle map* $\Pi^\#$.
Let $(E, \tau, M)$ be a vector bundle over a Poisson manifold $(M, \Pi)$ and $\Gamma(E)$ the space of the smooth sections of $E$. A *contravariant connection* on $E$ is a bilinear map $$\begin{aligned}
\nabla : \Gamma(T^\ast M) \times \Gamma(E) & \to & \Gamma(E) \nonumber \\
(\alpha, s) & \mapsto & \nabla_\alpha s\end{aligned}$$ satisfying, for any $(\alpha, s) \in \Gamma(T^\ast M) \times \Gamma(E)$ and $f\in \C$, the following properties: $$\nabla_{f \alpha} s = f \nabla_\alpha s \quad \quad \mathrm{and} \quad \quad \nabla_\alpha (fs) = f \nabla_\alpha s + (\mathcal{L}_{\Pi^\#(\alpha)}f) s.$$
In order to introduce the corresponding notion of *contravariant derivative along a path* for a contravariant connection $\nabla$ on $E$, we define a suitable notion of *cotangent path* (or $T^\ast M$-path in the sense of [@cr-fer-int-Lie]) as follows.
A *cotangent path $a$ with base path $\gamma$* is a curve $a : [0,1] \to T^\ast M$ sitting above some curve $\gamma : [0,1] \to M$, that means that $\pi(a(t)) = \gamma(t)$, such that $$\frac{d\gamma}{dt}(t) = \Pi^\#(a(t)).$$
Hence, given a $T^\ast M$-path $(a, \gamma)$ and a path $u : [0,1] \to E$ on $E$ above $\gamma$, i.e., $\tau (u(t)) = \gamma(t)$, the *contravariant derivative of $u$ along $a$*, denoted by $\nabla_a u$, is well defined. We choose a time-dependent section $s$ of $E$ such that $s(t, \gamma(t)) = u(t)$ and we set $$\nabla_a u (t) = \nabla_a s_t(\gamma(t)) + \frac{d s_t}{dt}(\gamma(t)).$$
In what follows we are interested in contravariant connections on the cotangent bundle $(T^\ast M, \pi, M)$ of a Poisson manifold $(M, \Pi)$ induced by a classical covariant connection $\nabla$ on $M$. We know that any such connection $\nabla$ on $(M, \Pi)$ induces a contravariant connection $\bar{\nabla}$ on $TM$ and a contravariant connection $\tilde{\nabla}$ on $T^\ast M$ which are defined, for any $\alpha, \beta \in \Gamma(T^\ast M)$ and $X\in \Gamma(TM)$, respectively, by $$\label{def-contravariant connection}
\bar{\nabla}_\alpha X = \Pi^\#(\nabla_X \alpha) + [\Pi^\#(\alpha), \, X] \quad \quad \mathrm{and} \quad \quad \tilde{\nabla}_\alpha \beta = \nabla_{\Pi^\#(\beta)}\alpha +[\alpha, \beta]_{\Pi},$$ where $[\cdot, \cdot]_\Pi$ is the Lie bracket (\[bracket-Lie-forms\]) on $\Gamma(T^\ast M)$ defined by $\Pi$. Because $$\Pi^\# : (\Gamma(T^\ast M), [\cdot, \cdot]_\Pi) \to (\Gamma(TM), [\cdot,\cdot])$$ is a Lie algebra homomorphism, the two connections are related by the formula $$\bar{\nabla}_\alpha (\Pi^\#(\beta)) = \Pi^\#(\tilde{\nabla}_\alpha \beta).$$ Moreover, in the case where $\nabla$ is without torsion, they are, also, related as it is indicated in the following Lemma.
([@cr-marcut]) Let $\nabla$ be a linear torsion-free connection on $M$ and $a : [0,1] \to T^\ast M$ a cotangent path with base path $\gamma$. Then, for any smooth paths $\theta : [0,1] \to T^\ast M$ and $u: [0,1] \to TM$, both above $\gamma$, the following identity holds: $$\label{lemma 1.3 cr-mrc}
\langle \tilde{\nabla}_{a_t} \theta_t, \, u_t\rangle + \langle \theta_t, \, \bar{\nabla}_{a_t} u_t\rangle = \frac{d}{dt}\langle \theta_t, u_t\rangle.$$
We close this subsection by calculating the coefficients $\tilde{\Gamma}^{ij}_k$ of $\tilde{\nabla}$ in a local coordinate system $(x^1,\ldots,x^n)$ of $M$. Let $\Gamma_{ij}^k$ be the Christoffel symbols of $\nabla$ in these coordinates. We consider the extension of $\nabla$ on $T^\ast M$, also denoted by $\nabla$, and we calculate its coefficients in $(x^1,\ldots,x^n)$: $$\label{conn-ext-T*M}
\nabla_{\frac{\partial}{\partial x^j}}dx^i = -\Gamma_{jk}^i dx^k.$$ Thus, $$\begin{aligned}
\tilde{\nabla}_{dx^i} dx^j & \stackrel{(\ref{def-contravariant connection})}{=} & \nabla_{\Pi^\#(dx^j)}dx^i +[dx^i, dx^j]_{\Pi} \, \stackrel{(\ref{bracket-Lie-forms})}{=} \, \nabla_{\Pi^{jl}\frac{\partial}{\partial x^l}}dx^i + \frac{\partial \Pi^{ij}}{\partial x^k}dx^k \\
& = & \Pi^{jl}\nabla_{\frac{\partial}{\partial x^l}}dx^i + \frac{\partial \Pi^{ij}}{\partial x^k}dx^k \, \stackrel{(\ref{conn-ext-T*M})}{=} \, \Pi^{jl}(-\Gamma_{lk}^idx^k)+ \frac{\partial \Pi^{ij}}{\partial x^k}dx^k \\
& = & (\Gamma_{lk}^i \Pi^{lj}+ \frac{\partial \Pi^{ij}}{\partial x^k})dx^k.\end{aligned}$$ Hence we get $$\label{Christ - T*M}
\tilde{\Gamma}_k^{ij} = \Gamma^i_{kl}\Pi^{lj}+ \frac{\partial \Pi^{ij}}{\partial x^k}.$$
Poisson sprays {#section poisson spray}
--------------
The contravariant analogue of the classical notion of a *spray* ([@lang]) is the one of *Poisson spray* that is defined in [@cr-marcut] as follows.
A *Poisson spray* on a Poisson manifold $(M, \Pi)$ is a vector field $\mathcal{V}_\Pi$ on $T^\ast M$ that satisfies the following properties:
1. For any $\xi \in T^\ast M$, $\xi = (x,y)$, $$\label{property-1}
\pi_{{\ast}_\xi} (\mathcal{V}_{\Pi_\xi}) = \Pi^\#(\xi);$$
2. $$\label{property-2}
m_{{t_\ast}_\xi}(\mathcal{V}_{\Pi_\xi}) = \frac{1}{t}\mathcal{V}_{\Pi_{m_t(\xi)}},$$
where $m : \R^\ast \times T^\ast M \to T^\ast M$ is the action by dilatation of $\R^\ast$ on the fibers of $T^\ast M$, i.e., for any $\xi = (x,y) \in T^\ast M$, $m_t(\xi) = (x, ty)$.
By studying the above properties of $\mathcal{V}_\Pi$ we conclude that, in a local coordinate system $(x,y)$ of $T^\ast M$, $\mathcal{V}_\Pi$ is written as $$\label{expression Poisson-spray}
\mathcal{V}_\Pi(x,y) = \sum_{i,j}\Pi^{ij}(x)y_i\frac{\partial}{\partial x^j} + \sum_{i,j,k}F_k^{ij}(x)y_i y_j\frac{\partial}{\partial y_k}, \quad \quad F_k^{ij}\in \C.$$
In the following, we describe the local behavior of the flow $\varphi$ of a Poisson spray $\mathcal{V}_\Pi$ on $(M, \Pi)$ in the neighborhood of the points of the zero-section of $T^\ast M$ because it plays a fundamental role in the proof of our main result (Theorem \[THEOREM\]). We note that, at each $x\in M$, the zero-section corresponds the zero-covector $0_x = (x,0)$ of $T^\ast_x M$, and, at such points we have $T_{0_x}(T^\ast M) = T_x M \oplus T_{0_x}(T_x^\ast M)\cong T_xM \oplus T_x^\ast M$, since $T_{0_x}(T_x^\ast M)$ is canonically identified with $T_x^\ast M$. Therefore, each element $u$ of $T_{0_x}(T^\ast M)$ can be written as $u = (\bar{u}, \theta_u)$, where $\bar{u} = \pi_{{\ast}_{0_x}}(u)$ and $\theta_u$ is the projection of $u$ on $T_{0_x}(T_x^\ast M) \cong T_x^\ast M$. On the other hand, taking into account the local expression (\[expression Poisson-spray\]) of $\mathcal{V}_\Pi$, we have that the points $0_x = (x,0)$, $x\in M$, are singular points of $\mathcal{V}_{\Pi}$. So, the maximal integral curves of $\mathcal{V}_{\Pi}$ through these points are the same ones, i.e., $\varphi_t(0_x) = 0_x$, for all $t\in \R$ and all $x\in M$. Hence, $\varphi_t$ is well defined on a neighborhood of the zero-section of $T^\ast M$ for all $t\in \R$ and in particular for $t\in [0,1]$. Its tangent map at these points, $\varphi_{t_{{\ast}_{0_x}}} : T_{0_x}(T^\ast M) \to T_{0_x}(T^\ast M)$, is defined by $\varphi_{t_{{\ast}_{0_x}}} = \exp(t \dot{\mathcal{V}}_{{\Pi}_{0_x}})$. In the last relation $\dot{\mathcal{V}}_{{\Pi}_{0_x}}$ is the composition of the tangent map $\mathcal{V}_{{\Pi}_{\ast_{0_x}}} : T_{0_x}(T^\ast M) \to T_{0_{0_x}}(T(T^\ast M))$ of $\mathcal{V}_{\Pi}$ at $0_x$, as soon as it is viewed as a smooth section of $T(T^\ast M)$, with the canonical projection of $T_{0_{0_x}}(T(T^\ast M))$, which is identified with $T_{0_x}(T^\ast M) \oplus T_{0_{0_x}}(T_{0_x}(T^\ast M)) \cong T_{0_x}(T^\ast M) \oplus T_{0_x}(T^\ast M)$, on its second (vertical) summand $T_{0_x}(T^\ast M)$. (For more details, see Proposition 22.3 in [@abr-rob].) From (\[expression Poisson-spray\]), we obtain that, in the coordinates $(x,y)$, $\mathcal{V}_{{\Pi}_{\ast_{0_x}}}$ has the matrix expression $$\mathcal{V}_{{\Pi}_{\ast_{0_x}}} = \begin{pmatrix} I & 0 \cr
0 & I \cr
0 & \Pi_x \cr
0 & 0 \end{pmatrix}.$$ Consequently, for any $u = (\bar{u}, \theta_u) \in T_{0_x}(T^\ast M)$, $\dot{\mathcal{V}}_{{\Pi}_{0_x}}(u)=\dot{\mathcal{V}}_{{\Pi}_{0_x}}(\bar{u}, \theta_u) = (-\Pi^\# (\theta_u), 0)$ and $$\label{im flow u_0}
\varphi_{t_{{\ast}_{0_x}}}(u) = \exp \big(t \dot{\mathcal{V}}_{{\Pi}_{0_x}}(u)\big) = (\bar{u} - t\Pi_0^\# (\theta_u), \theta_u).$$
Finally, we remark that the contravariant analogue of the classical notion of *geodesic spray* can also be defined in the framework of Poisson geometry. Given a contravariant connection $\nabla$ on $T^\ast M$ of $(M, \Pi)$ with coefficients $\Gamma^{ij}_k$, the notion of *geodesics of* $\nabla$ on $T^\ast M$ is defined, as usual, as the cotangent paths $a : [0,1] \to T^\ast M$ whose contravariant derivative along itself is identically zero: $\nabla_{a} a(t) = 0$. In local coordinates $(x,y)$, a geodesic can be regarded as a curve $(\gamma, a) : [0,1] \to M\times T^\ast M$, $(\gamma(t), a(t)) = (x^1(t), \ldots, x^n(t), y_1(t), \ldots, y_n(t))$, which satisfies the following system of ode’s: $$\label{system ode's}
\left\{
\begin{array}{l}
\frac{dx^i}{dt} = \Pi^{ki}(x(t))y_k(t)\\
\\
\frac{dy_i}{dt} = - \Gamma_i^{jk}(x(t))y_j(t)y_k(t)
\end{array}, \quad \quad
i = 1,\ldots,n.
\right.$$ The system (\[system ode’s\]) determines the vector field $\mathcal{V}_\Pi$ on $T^\ast M$, that is given by $$\mathcal{V}_\Pi = \Pi^{ki}y_k\frac{\partial}{\partial x^i} - \Gamma_i^{jk}y_jy_k \frac{\partial}{\partial y_i}$$ and it is called the *geodesic Poisson spray*, which, clearly, has the properties (\[property-1\]) and (\[property-2\]) of Poisson sprays. The above discussion ensures us the existence of Poisson sprays on $(M,\Pi)$.
We have that, if the considered contravariant connection on $T^\ast M$ is the connection $\tilde{\nabla}$ defined by a linear symmetric connection $\nabla$ on $TM$ as in (\[def-contravariant connection\]), the corresponding geodesic Poisson spray takes the form $$\begin{aligned}
\label{Geodesic-Poisson-spray}
\mathcal{V}_{\Pi} & = & \sum_{i,j}\Pi^{ij}y_i\frac{\partial}{\partial x^j} - \sum_{i,j,k}\tilde{\Gamma}_k^{ij}y_i y_j\frac{\partial}{\partial y_k} \nonumber \\
& \stackrel{(\ref{Christ - T*M})}{=} & \sum_{i,j}\Pi^{ij}y_i\frac{\partial}{\partial x^j} - \sum_{i,j,k,l}(\Gamma^i_{kl}\Pi^{lj}+ \frac{\partial \Pi^{ij}}{\partial x^k})y_i y_j\frac{\partial}{\partial y_k} \nonumber \\
& = & \sum_{i,j}\Pi^{ij}y_i\frac{\partial}{\partial x^j} - \sum_{i,j,k,l}\Gamma^i_{kl}\Pi^{lj}y_i y_j\frac{\partial}{\partial y_k} \nonumber \\
& = & \sum_i y_i (\Pi_0^\#(dx^i))^h.\end{aligned}$$ Note that, for any $k = 1, \ldots, n$, the sum $\sum_{i,j}\displaystyle{\frac{\partial \Pi^{ij}}{\partial x^k}y_i y_j}$ is annulled because of the skew-symmetry of $\Pi$. In this special case, at every point $\xi \in T^\ast M$, $\mathcal{V}_{\Pi_\xi}$ coincides with the horizontal lift of $\Pi^\#(\xi)$ on $T^\ast M$ with respect to $\nabla$. Therefore, $\mathcal{V}_{\Pi}$ is a section of the horizontal subbundle $\mathcal{H}$ of $T(T^\ast M)$ defined by $\nabla$.
Symplectic realization of a Poisson manifold {#realization Crainic-Marcut}
--------------------------------------------
In this subsection we present, briefly, the basic steps of the proof of M. Crainic and I. M$\check{\mathrm{a}}$rcu$\c{t}$’ theorem [@cr-marcut] on the *symplectic realization of a Poisson manifold:*
\[theorem-crainic-marcut\]([@cr-marcut]) Given a Poisson manifold $(M, \Pi)$ and a Poisson spray $\mathcal{V}_\Pi$, there exists an open neighborhood $\mathcal{U}\subset T^\ast M$ of the zero-section so that $$\Omega: = \int_0^1 \varphi_t^\ast \omega_{can}dt,$$ where $\varphi$ is the flow of $\mathcal{V}_\Pi$, is a symplectic structure on $\mathcal{U}$ and the canonical projection $\pi : (\mathcal{U}, \Omega) \to (M, \Pi)$ is a symplectic realization.
Firstly, they calculate the value of $\Omega$ on vectors tangent to $T^\ast M$ at zeros $0_x \in T_x^\ast M$. By identifying the space $T_{0_x}(T^\ast M)$ with $T_x M \oplus T_x^\ast M$, they prove that, for all $u=(\bar{u}, \theta_u), w=(\bar{w}, \theta_w) \in T_{0_x}(T^\ast M) \cong T_x M \oplus T_x^\ast M$, $$\label{values Omega 0}
\Omega_{0_x}(u,w) = \langle \theta_w, \bar{u}\rangle - \langle \theta_u, \bar{w}\rangle - \Pi (\theta_u, \theta_w).$$ The above formula implies that the closed $2$-form $\Omega$ is nondegenerate at the points of the zero-section of $T^\ast M$. So, there exists a neighborhood $\mathcal{U}$ of the zero-section in $T^\ast M$ on which $\varphi_t$ is well defined for any $t\in [0,1]$ and $\Omega \vert_{\mathcal{U}}$ is symplectic. In the second step, by fixing such a $\mathcal{U}$, they consider a torsion-free covariant connection $\nabla$ on $TM$, in order to handle vectors tangent to $T^\ast M$, and they establish a generalization of (\[values Omega 0\]) at arbitrary points $\xi \in \mathcal{U}$. Precisely, they prove (Lemma 2.3 in [@cr-marcut]) that, for any pair $(u,w)$ of elements of $T_{\xi}(T^\ast M)$, $$\Omega(u,w) = \big(\langle \tilde{\theta}_{w_t}, \bar{u}_t\rangle - \langle \tilde{\theta}_{u_t}, \bar{w}_t \rangle - \Pi (\tilde{\theta}_{u_t}, \tilde{\theta}_{w_t}) \big)\big\vert_0^1,$$ where $u_t = \varphi_{t_{\ast}}(u)$ (resp. $w_t = \varphi_{t_{\ast}}(w)$), $\bar{u}_t = \pi_\ast(u_t)$ (resp. $\bar{w}_t = \pi_\ast(w_t)$), $\theta_{u_t} = u_t - \bar{u}_t^h$ (resp. $\theta_{w_t} = w_t - \bar{w}_t^h$) and $\tilde{\theta}_{u_t}$ (resp. $\tilde{\theta}_{w_t}$) is a solution of the differential equation $\tilde{\nabla}_{a_t}\tilde{\theta}_{u_t} = \theta_{u_t}$ (resp. $\tilde{\nabla}_{a_t}\tilde{\theta}_{w_t} = \theta_{w_t}$), with $a_t$ being the path on $\mathcal{U}$ given by $a_t = \varphi_t(\xi)$. In the third and last step, they show that the projection $\pi\vert_{\mathcal{U}} : \mathcal{U} \to M$ push-down the bivector associated to $\Omega$ to the Poisson tensor $\Pi$, i.e., that $\pi$ is a Poisson map.
Sympectic realization of Poisson-Nijenhuis manifolds {#section theorem}
====================================================
Let $(M, \Pi_0, N)$ be a Poisson-Nijenhuis manifold. Without loss of generality we assume that $N$ is nondegenerate. We can make this assumption because, if $\det N = 0$, we can replace $N$ with the nondegenerate Nijenhuis operator $N'= I + N$ which produces with $\Pi_0$ the same bi-Hamiltonian structure viewed as a bi-parametric family $\Pi_{\kappa, \lambda} =\kappa \Pi_0 + \lambda \Pi_1$, $\kappa, \lambda \in \R$, of pairwise compatible Poisson structures, with $\Pi_1^\# = N\circ \Pi_0^\#$. In this case, an hierarchy $(\Pi_k)_{k \in \Z}$, $\Pi_k^\# = N^k\circ \Pi_0^\#$, of pairwise compatible Poisson structures is also defined on $M$. We consider the complete lift $N^c$ of $N$ on $T^\ast M$ and the pair of sympectic forms $(\omega_{can}, \omega_1)$, where $\omega_{can}$ is the canonical symplectic form on $T^\ast M$ and $\omega_1$ is the symplectic $2$-form defined by $\omega_1 (\cdot, \cdot) = \omega_{can} (N^c \cdot, \cdot) = \omega_{can}( \cdot, N^c \cdot)$. In the local coordinate system $(x,y)$ of $T^\ast M$, they have, respectively, the matrix expression $$\omega_{can} = \begin{pmatrix} 0 & -I \cr
I & 0
\end{pmatrix}
\quad \quad \mathrm{and} \quad \quad \omega_1 = \begin{pmatrix} -A & - \,^tN \cr
N & 0
\end{pmatrix}.$$ Since $N^c$ is a Nijenhuis operator, $\omega_{can}$ and $\omega_1$ are Poisson-compatible in the sense of [@tu; @magri-mo]. Furthermore, we consider a Poisson spray $\mathcal{V}_{\Pi_0}$ on $T^\ast M$ associated to $\Pi_0$ and we denote by $\varphi$ its flow. Thereafter, we endow $T^\ast M$ with the pair of closed $2$-forms $$\label{Omega0-Omega1}
\Omega_0 = \int_0^1 \varphi_t^\ast \omega_{can} dt \quad \quad \mathrm{and} \quad \quad \Omega_1 = \int_0^1 \varphi_t^\ast \omega_1 dt$$ and we remember that $\Omega_0$ is symplectic [@cr-marcut].
\[lemma compatible\] The pair $(\Omega_0,\Omega_1)$ is a pair of Poisson-compatible $2$-forms.
By definition [@tu], $(\Omega_0, \Omega_1)$ is Poisson-compatible if its recursion operator $R$ defined by $R = \Omega_0^{\flat^{-1}}\circ \Omega_1^\flat$ is a Nijenhuis operator.[^1] It is well known that it is true if and only if the 2-form $\Omega_2$ defined by the relation $\Omega_2^\flat = \Omega_1^\flat \circ R = \Omega_0^\flat \circ R^2$ is closed, [@br]. We have that, for all $u,w \in \Gamma(T(T^\ast M))$, $$\Omega_1(u,w) = \int_0^1 (\varphi_t^\ast \omega_1)(u,w) dt = \int_0^1 \omega_1(\varphi_{t \ast}u, \, \varphi_{t \ast}w) dt = \int_0^1 \omega_{can}(N^c (\varphi_{t \ast}u), \, \varphi_{t \ast}w)$$ and $$\Omega_1(u,w) = \Omega_0(Ru, w) = \int_0^1 \varphi_t^\ast \omega_{can} (Ru, w) dt = \int_0^1 \omega_{can} ( \varphi_{t_\ast}Ru, \varphi_{t_\ast}w) dt.$$ Thus, for all $u,w \in \Gamma(T(T^\ast M))$, $$\label{two expressions of Omega-2}
\int_0^1 \omega_{can} ( \varphi_{t_\ast}Ru, \varphi_{t_\ast}w) dt = \int_0^1 \omega_{can}(N^c (\varphi_{t_\ast}u), \, \varphi_{t_\ast}w).$$ We consider on $T^\ast M$ the closed $2$-form $\omega_2$ defined by $\omega_2^\flat = \omega_0^\flat \circ (N^c)^2$ (since $T(N^c)=0$, $d\omega_2 =0$) and we calculate, for all $u,w \in \Gamma(T(T^\ast M))$, $$\begin{aligned}
\Omega_2(u,w) & = & \Omega_0(R^2u,w) = \int_0^1 \varphi_t^\ast \omega_{can} (R^2u, w) dt = \int_0^1 \omega_{can} ( \varphi_{t_\ast}R^2u, \varphi_{t_\ast}w) dt \nonumber \\
& \stackrel{(\ref{two expressions of Omega-2})}{=} & \int_0^1 \omega_{can}(N^c (\varphi_{t_\ast}(Ru)), \, \varphi_{t_\ast}w)dt = \int_0^1 \omega_{can}(\varphi_{t_\ast}(Ru), \, N^c \varphi_{t_\ast}w)dt \nonumber \\
& = & \int_0^1 \omega_{can}(\varphi_{t_\ast}(Ru), \, \varphi_{t_\ast}(\varphi_{-t_\ast}N^c \varphi_{t_\ast}w))dt \nonumber \\
& \stackrel{(\ref{two expressions of Omega-2})}{=} & \int_0^1 \omega_{can}(N^c (\varphi_{t_\ast}u), \, \varphi_{t_\ast}(\varphi_{-t_\ast}N^c \varphi_{t_\ast}w))dt \nonumber \\
& = & \int_0^1 \omega_{can}(N^c (\varphi_{t_\ast}u), \, N^c (\varphi_{t_\ast}w))dt = \int_0^1 \omega_{can}((N^c)^2 (\varphi_{t_\ast}u), \, \varphi_{t_\ast}w)dt \nonumber \\
& = & \int_0^1 \omega_{2}(\varphi_{t_\ast}u, \, \varphi_{t_\ast}w)dt = \int_0^1 \varphi_t^\ast \omega_2(u,w) dt.\end{aligned}$$ Therefore, $$\Omega_2 = \int_0^1 \varphi_t^\ast \omega_2 dt \quad \quad \mathrm{and} \quad \quad d\Omega_2 =0.$$ Hence we get the compatibility of $\Omega_0$ with $\Omega_1$.
For the follow of our study we have also need the next lemma.
\[lemma nabla N\] Let $(M,\Pi_0,N)$ be a Poisson-Nijenhuis manifold equipped with a torsion-free covariant connection $\nabla$ compatible with $N$. Then, for any $\alpha, \beta \in \Gamma(T^\ast M)$, the following identity holds: $$\label{cond-tilde-nabla-N}
\tilde{\nabla}_{\alpha}(\,^tN \beta) = \,^tN(\tilde{\nabla}_{\alpha}\beta),$$ where $\tilde{\nabla}$ is the contravariant connection (\[def-contravariant connection\]) on $T^\ast M$ induced by $\nabla$ and $\Pi_0$.
Effectively, for any $\alpha, \beta \in \Gamma(T^\ast M)$ and $X\in \Gamma(TM)$, we have $$\begin{aligned}
\langle \tilde{\nabla}_{\alpha}(\,^tN \beta), X\rangle & \stackrel{(\ref{def-contravariant connection})}{=} & \langle \nabla_{\Pi_0^\#(\,^tN \beta)}\alpha + [\alpha, \,^tN\beta]_{\Pi_0},\, X \rangle \nonumber \\
& \stackrel{(\ref{bracket-Lie-forms})}{=} & \langle \nabla_{\Pi_0^\#(\,^tN \beta)}\alpha , X\rangle + \langle \mathcal{L}_{\Pi_0^\#(\alpha)}(\,^tN \beta) - \mathcal{L}_{\Pi_0^\#(\,^tN \beta)}\alpha -d(\Pi_1(\alpha,\beta)),\, X\rangle \nonumber \\
&\stackrel{(\ref{cond-N-Pi0})}{=} & \Pi_1^\#(\beta) \langle \alpha, \, X\rangle - \langle \alpha, \nabla_{N\Pi_0^\#(\beta)}X\rangle + \Pi_0^\#(\alpha)\langle \,^tN \beta,\, X\rangle \nonumber \\
& & -\, \langle \,^tN \beta,\, \mathcal{L}_{\Pi_0^\#(\alpha)}X\rangle -\Pi_1^\#(\beta)\langle \alpha, \,X\rangle + \langle \alpha, \, \mathcal{L}_{\Pi_1^\#(\beta)}X\rangle \nonumber \\
& & - \,\langle d(\Pi_1(\alpha,\beta)),\, X\rangle \nonumber \\
& \stackrel{(\ref{torsion-connection})}{=} & -\, \langle \alpha, [N\Pi_0^\#(\beta), X] + \nabla_X(N\Pi_0^\#(\beta))\rangle + \Pi_0^\#(\alpha)\langle \,^tN \beta,\, X\rangle \nonumber \\
& & -\, \langle \,^tN \beta,\, \mathcal{L}_{\Pi_0^\#(\alpha)}X\rangle + \langle \alpha, [\Pi_1^\#(\beta), X]\rangle - \langle d(\Pi_1(\alpha,\beta)),\, X\rangle \nonumber \\
& = & - \,\langle \alpha, (\nabla_XN)\Pi_0^\#(\beta) + N \nabla_X \Pi_0^\#(\beta)\rangle + \Pi_0^\#(\alpha)\langle \,^tN \beta,\, X\rangle \nonumber \\
& & - \, \langle \,^tN \beta,\, \mathcal{L}_{\Pi_0^\#(\alpha)}X\rangle - \langle d(\Pi_1(\alpha,\beta)),\, X\rangle.\end{aligned}$$ On the other hand, $$\begin{aligned}
\langle \,^tN (\tilde{\nabla}_{\alpha}\beta), \, X\rangle & \stackrel{(\ref{def-contravariant connection})}{=} & \langle \nabla_{\Pi_0^\#(\beta)}\alpha + [\alpha, \beta]_{\Pi_0}, \, NX\rangle \nonumber \\
& \stackrel{(\ref{bracket-Lie-forms})}{=} & \langle \nabla_{\Pi_0^\#(\beta)}\alpha + \mathcal{L}_{\Pi_0^\#(\alpha)}\beta - \mathcal{L}_{\Pi_0^\#(\beta)}\alpha - d(\Pi_0(\alpha, \beta)), \, NX \rangle \nonumber \\
& = & \Pi_0^\#(\beta)\langle \alpha, \,NX\rangle - \langle \alpha, \,\nabla_{\Pi_0^\#(\beta)}(NX)\rangle + \Pi_0^\#(\alpha)\langle \beta, \, NX\rangle \nonumber \\
& &-\, \langle \beta, \, \mathcal{L}_{\Pi_0^\#(\alpha)}(NX)\rangle - \Pi_0^\#(\beta) \langle \alpha, NX\rangle + \langle \alpha, \mathcal{L}_{\Pi_0^\#(\beta)}(NX)\rangle \nonumber \\
& & -\, \langle d(\Pi_0(\alpha, \beta)),\, NX\rangle \nonumber \\
& = & -\, \langle \alpha, \, (\nabla_{\Pi_0^\#(\beta)}N)X + N \nabla_{\Pi_0^\#(\beta)}X\rangle + \Pi_0^\#(\alpha)\langle \beta, \, NX\rangle \nonumber \\
& & -\, \langle \beta, \, (\mathcal{L}_{\Pi_0^\#(\alpha)}N)X + N \mathcal{L}_{\Pi_0^\#(\alpha)}X \rangle + \langle \alpha, \mathcal{L}_{\Pi_0^\#(\beta)}N(X) + N \mathcal{L}_{\Pi_0^\#(\beta)}X\rangle \nonumber \\
& & - \, \langle d(\Pi_0(\alpha, \beta)),\, NX\rangle \nonumber \\
& = & - \langle \alpha, \, (\nabla_{\Pi_0^\#(\beta)}N)X \rangle - \langle \,^tN\alpha, \, \nabla_{\Pi_0^\#(\beta)}X\rangle + \Pi_0^\#(\alpha)\langle \beta, \, NX\rangle \nonumber \\
& &- \,\langle (\mathcal{L}_{\Pi_0^\#(\alpha)}\,^tN)\beta, \, X \rangle - \langle \,^tN \beta, \, \mathcal{L}_{\Pi_0^\#(\alpha)}X\rangle + \langle (\mathcal{L}_{\Pi_0^\#(\beta)}\,^tN)\alpha, \,X\rangle \nonumber \\
& & + \,\langle \,^tN \alpha, \, \mathcal{L}_{\Pi_0^\#(\beta)}X\rangle -\langle \,^tN d(\Pi_0(\alpha, \beta)),\, X\rangle.\end{aligned}$$ Hence, taking into account the facts that $N$ is compatible with the symmetric connection $\nabla$ and $C(\Pi_0,N)=0$, we obtain $$\begin{aligned}
\langle \tilde{\nabla}_{\alpha}(\,^tN \beta) - \,^tN (\tilde{\nabla}_{\alpha}\beta), \,X \rangle & = & - \langle \alpha, (\nabla_XN)\Pi_0^\#(\beta) - (\nabla_{\Pi_0^\#(\beta)}N)X \rangle \nonumber \\
& & + \langle \,^tN \alpha, \nabla_{\Pi_0^\#(\beta)}X - \nabla_X \Pi_0^\#(\beta) - [\Pi_0^\#(\beta), \,X]\rangle \nonumber \\
& & + \,\langle - d(\Pi_1(\alpha,\beta)) + (\mathcal{L}_{\Pi_0^\#(\alpha)}\,^tN)\beta - (\mathcal{L}_{\Pi_0^\#(\beta)}\,^tN)\alpha \nonumber \\
& & + \,^tN d(\Pi_0(\alpha, \beta)),\, X\rangle \nonumber \\
&\stackrel{(\ref{cond-nabla-N}, \ref{torsion-connection}, \ref{def-concomitant})}{=}& 0,\end{aligned}$$ for any $X\in \Gamma(TM)$. So, equality (\[cond-tilde-nabla-N\]) is established. The above result means that, under the conditions of compatibility of $N$ with $(\Pi_0, \nabla)$, $\,^t N$ is parallel with respect to $\tilde{\nabla}$.
Now, we proceed with the proof of the central theorem of this work.
\[THEOREM\] Let $(M, \Pi_0, N)$ be a Poisson-Nijenhuis manifold, with $N$ nondegenerate, equipped with a torsion-free covariant connection $\nabla$ compatible with $N$. Let, also, $(\Pi_k)_{k\in \Z}$, $\Pi_k^\# = N^k\circ \Pi_0^\#$, be the associated hierarchy of pairwise compatible Poisson structures on $M$, $\mathcal{V}_{\Pi_0}$ a Poisson spray corresponding to $\Pi_0$ and $\varphi$ its flow. Then, there exists an open neighborhood $\mathcal{U}$ of the zero-section in $T^\ast M$ such that the canonical projection $\pi : (\mathcal{U}, \Omega_0, \Omega_1) \to (M, \Pi_0, \Pi_{-1})$ is a symplectic realization of $(M, \Pi_0, \Pi_{-1})$, where $(\Omega_0, \Omega_1)$ is the pair of Poisson-compatible symplectic structures on $T^\ast M$ defined by (\[Omega0-Omega1\]).
The result of M. Crainic and I. M$\check{\mathrm{a}}$rcu$\c{t}$ [@cr-marcut] (see, also, subsection \[realization Crainic-Marcut\]) ensures the existence of a neighborhood $\mathcal{U}$ in $T^\ast M$ of the zero-section of $T^\ast M$ such that $\varphi_t$ is defined for all $t\in [0,1]$, $\Omega_0\vert_{\mathcal{U}}$ is symplectic and $\pi : (\mathcal{U}, \tilde{\Pi}_0) \to (M,\Pi_0)$, where $\tilde{\Pi}_0$ is the symplectic Poisson structure defined by $\Omega_0$, is a Poisson map. Our aim is to prove the claims that $\Omega_1\vert_{\mathcal{U}}$ is symplectic and $\pi : (\mathcal{U}, \tilde{\Pi}_1) \to (M,\Pi_{-1})$ is also a Poisson map, where $\tilde{\Pi}_1$ is the symplectic Poisson structure defined by $\Omega_1$.
*First step:* We start by evaluating $\Omega_1$ on vectors tangent to $T^\ast M$ at the points $0_x = (x,0)$ of the zero-section of $T^\ast M$. As we noted in the subsection \[section poisson spray\], $T_{0_x}(T^\ast M) = T_x M \oplus T_{0_x}(T_x^\ast M)$ is canonically identified with $T_xM \oplus T_x^\ast M$ and each element $u$ of $T_{0_x}(T^\ast M)$ is identified with $u = (\bar{u}, \theta_u)$, where $\bar{u} = \pi_{{\ast}_{0_x}}(u)$ and $\theta_u$ is the projection of $u$ on $T_{0_x}(T_x^\ast M) \cong T_x^\ast M$. Hence, for any pair $(u,w)$ of elements of $T_{0_x}(T^\ast M)$, $u = (\bar{u}, \theta_u)$ and $w = (\bar{w}, \theta_w)$, we have $$\begin{aligned}
\label{omega1_0}
\omega_{1_{0_x}}(u,w)& = &\begin{pmatrix}\bar{w} & \theta_w \end{pmatrix}\begin{pmatrix} A & -\,^tN \cr
N & 0 \end{pmatrix}_{0_x}\begin{pmatrix} \bar{u} \cr \theta_u \end{pmatrix} = \begin{pmatrix}\bar{w} & \theta_w \end{pmatrix}\begin{pmatrix} 0 & -\,^tN \cr
N & 0 \end{pmatrix}\begin{pmatrix} \bar{u} \cr \theta_u \end{pmatrix} \nonumber \\
& = & \langle \theta_w, N\bar{u}\rangle - \langle \,^tN\theta_u, \bar{w}\rangle = \langle \theta_w, N\bar{u}\rangle - \langle \theta_u, N\bar{w}\rangle.\end{aligned}$$ Furthermore, since $\varphi_t(0_x) = 0_x$, $$\begin{aligned}
(\varphi_t^\ast \omega_1)_{0_x}(u,w) & = & \omega_{1_{0_x}} \big((\varphi_t)_{{\ast}_{0_x}}(u), (\varphi_t)_{{\ast}_{0_x}}(w) \big)\nonumber \\
& \stackrel{(\ref{im flow u_0})}{=}& \omega_{1_{0_x}}\big((\bar{u} - t\Pi_0^\# (\theta_u), \theta_u),(\bar{w} - t\Pi_0^\# (\theta_w), \theta_w) \big) \nonumber \\
& \stackrel{(\ref{omega1_0})}{=} & \langle \theta_w, N\bar{u} - tN\Pi_0^\# (\theta_u)\rangle - \langle \theta_u, N\bar{w} - tN\Pi_0^\# (\theta_w)\rangle \nonumber \\
& = & \langle \theta_w, N\bar{u}\rangle - \langle \theta_u, N\bar{w} \rangle - 2t\Pi_1 (\theta_u, \theta_w).\end{aligned}$$ Consequently, $$\begin{aligned}
\label{Omega_1 - 0}
\Omega_{1_{\,0_x}}(u,w)& = & \int_0^1 (\varphi_t^\ast \omega_1)_{0_x}(u,w)dt = \int_0^1 \big(\langle \theta_w, \bar{u}\rangle - \langle \theta_u, \bar{w} \rangle - 2t\Pi_1 (\theta_u, \theta_w) \big)dt \nonumber \\
& = & \langle \theta_w, N\bar{u}\rangle - \langle \theta_u, N\bar{w} \rangle - \Pi_1 (\theta_u, \theta_w).\end{aligned}$$ The last expression implies that $\Omega_1$ is nondegenerate at the points $0_x$, $x\in M$, of $T^\ast M$. Since $\varphi_t$ is defined on $\mathcal{U}$ for any $t\in [0,1]$, we conclude that $\Omega_1$ is symplectic on $\mathcal{U}$.
*Second step:* In this step we evaluate $\Omega_1$ on vectors tangent to $T^\ast M$ at arbitrary points $\xi$ of $\mathcal{U}$ and we establish a formula analogous of (\[Omega\_1 - 0\]) which we shall use in the proof of the assertion that $\pi : (\mathcal{U}, \Omega_1) \to (M,\Pi_{-1})$ is a Poisson morphism. In order to describe the sections of $T(T^\ast M)$, we assume that $M$ is equipped with a symmetric covariant connection $\nabla$ compatible with $N$ in the sense of (\[cond-nabla-N\]).[^2] Then, the tangent bundle of $T^\ast M$ is decomposed, with respect to $\nabla$, as $T(T^\ast M) = \mathcal{H}\oplus \ker \pi_{\ast}$, where $\mathcal{H}$ is the horizontal distribution on $T^\ast M$ defined by $\nabla$. Hence, any tangent vector $u$ of $T^\ast M$ at $\xi$ is written as $$u = \bar{u}^h + \theta_u,$$ where $\bar{u}^h$ is the horizontal lift at $\xi$ of the projection $\bar{u} = \pi_{\ast_{\xi}}(u)$ of $u$ on $T_{\pi(\xi)}M$ and $\theta_u$ is the projection of $u$ on $\ker \pi_{{\ast}_\xi} \cong T^\ast_{\pi(\xi)}M$ parallel to $\bar{u}^h$. Clearly, at the points $\xi = 0_x$, the decomposition $T_\xi(T^\ast M) = \mathcal{H}_\xi\oplus \ker \pi_{{\ast}_\xi}$ coincides with the one described in the previous step. The fact that $\nabla$ is torsion-free ensures that the distribution $\mathcal{H}$ is Lagrangian with respect to $\omega_{can}$. The extra condition (\[cond-nabla-N\]) of compatibility of $N$ with $\nabla$ implies that $\mathcal{H}$ is conserved by $N^c$. Hence, $\mathcal{H}$ is also Lagrangian with respect to $\omega_1$, i.e., $\mathcal{H}$ is a bi-Lagrangian distribution with respect to $(\omega_{can}, \omega_1)$. Indeed, for all $u,w \in T_\xi(T^\ast M)$, $$\label{ypolo Nc}
N^cu = N^c\bar{u}^h + N^c\theta_u \stackrel{(\ref{im hor by N compl},\ref{cond-nabla-N})}{=} (N\bar{u})^h + \,^tN\theta_u,$$ and $$\begin{aligned}
\label{omega1_xi}
\omega_1(u,w) & = & \omega_{can}(N^cu, w) \nonumber \\
& \stackrel{(\ref{ypolo Nc})}{=} & \omega_{can}((N\bar{u})^h + \,^tN\theta_u , \bar{w}^h + \theta_w) \nonumber \\
& = & \omega_{can}((N\bar{u})^h, \bar{w}^h) + \omega_{can}((N\bar{u})^h, \theta_w) + \omega_{can}(\,^tN\theta_u, \bar{w}^h) + \omega_{can}(\,^tN\theta_u, \theta_w) \nonumber \\
& = & \langle \theta_w, N\bar{u}\rangle - \langle \,^tN\theta_u, \bar{w}\rangle = \langle \theta_w, N\bar{u}\rangle - \langle \theta_u, N\bar{w}\rangle,\end{aligned}$$ because $\mathcal{H}$ and $\ker \pi_\ast$ are Lagrangian distributions on $T^\ast M$ with respect to $\omega_{can}$. The above formula is a generalization of (\[omega1\_0\]) at an arbitrary point $\xi \in T^\ast M$.
In the following, we fix $\xi$ in $\mathcal{U}$ and we consider the cotangent path $a : [0,1] \to \mathcal{U}$, $a_t : = \varphi_t(\xi)$, which is the integral curve of $\mathcal{V}_{\Pi_0}$ through $\xi$, and we denote by $\gamma = \pi \circ a$ its base path on $M$. By pushing forward a tangent vector $u \in T_\xi \mathcal{U}$ by $\varphi_{{t_\ast}_\xi} : T_\xi\mathcal{U} \to T_{a_t}\mathcal{U}$, $t\in [0,1]$, we obtain a smooth path $u_t : = \varphi_{{t_\ast}_\xi}(u)$ of vectors along $a$; its projection $\bar{u}_t = \pi_{{\ast}_{a_t}}(u_t)$ on $TM$ yields a path of vectors along $\gamma$ while its projection $\theta_{u_t}$ on the vertical space $\ker \pi_{{\ast}_{a_t}}$ parallel to $\bar{u}_t^h$ defines a path of covectors along $\gamma$, also. I.e., we have $\bar{u}_t \in T_{\gamma(t)}M$ and $\theta_{u_t} \in T^\ast_{\gamma(t)}(M)$. Accordingly to Lemma 2.2 of [@cr-marcut], the two paths are related by $$\label{lemma 2.2 cr-mrc}
\bar{\nabla}_{a_t}\bar{u}_t = \Pi_0^\#(\theta_{u_t}).$$ The action of $N^c$ on $u_t$ produces another path of vectors along $a$ given by $$N^c u_t \stackrel{(\ref{ypolo Nc})}{=} (N\bar{u}_t)^h + \,^tN\theta_{u_t}.$$ Its corresponding paths on $TM$ and $T^\ast M$ are $N\bar{u}_t$ and $\,^tN\theta_{u_t}$, respectively. Hence, taking into account (\[lemma 2.2 cr-mrc\]), we get $$\label{rel bar nabla Pi1}
\bar{\nabla}_{a_t}(N \bar{u}_t) = \Pi_0^\#(\,^tN\theta_{u_t}) = \Pi_1^\#(\theta_{u_t}).$$
Now, we can establish a generalization of (\[Omega\_1 - 0\]) at an arbitrary point $\xi$ of $\mathcal{U}$. We consider a pair $(u,w)$ of elements of $T_\xi \mathcal{U}$ and its associated pairs of paths $(u_t, w_t)$ on $T\mathcal{U}$ over $a$, $(\bar{u}_t, \bar{w}_t)$ on $TM$ and $(\theta_{u_t}, \theta_{w_t})$ on $T^\ast M$, both over the base path $\gamma$ of $a$. Let $\tilde{\theta}_{u_t}$ (resp. $\tilde{\theta}_{w_t}$) be a path in $T^\ast M$ solution of the differential equation $$\label{diff-equation theta}
\tilde{\nabla}_{a_t}\tilde{\theta}_{u_t} = \theta_{u_t} \quad \quad (\mathrm{resp.} \quad \tilde{\nabla}_{a_t}\tilde{\theta}_{w_t} = \theta_{w_t}).$$ We will show that $$\label{expression Omega 1 xi}
\Omega_1(u,w) = \big(\langle \tilde{\theta}_{w_t}, N\bar{u}_t\rangle - \langle \tilde{\theta}_{u_t}, N\bar{w}_t \rangle - \Pi_1 (\tilde{\theta}_{u_t}, \tilde{\theta}_{w_t}) \big)\big\vert_0^1.$$ We have $$\Omega_1(u,w) = \int_0^1 (\varphi_t^\ast \omega_1)(u,v)dt = \int_0^1 \omega_1 (u_t, w_t)dt \stackrel{(\ref{omega1_xi})}{=} \int_0^1 \big(\langle \theta_{w_t}, N\bar{u}_t\rangle - \langle \theta_{u_t}, N\bar{w}_t\rangle \big)dt.$$ Hence, it is enough to prove that $$\label{1}
\langle \theta_{w_t}, N\bar{u}_t\rangle - \langle \theta_{u_t}, N\bar{w}_t\rangle = \frac{d}{dt}\big( \langle \tilde{\theta}_{w_t}, N\bar{u}_t\rangle - \langle \tilde{\theta}_{u_t}, N\bar{w}_t \rangle - \Pi_1 (\tilde{\theta}_{u_t}, \tilde{\theta}_{w_t}) \big).$$ In fact, we have $$\begin{aligned}
\langle \theta_{w_t}, \, N\bar{u}_t\rangle - \langle \theta_{u_t}, \, N\bar{w}_t\rangle & = & \langle \tilde{\nabla}_{a_t}\tilde{\theta}_{w_t}, \, N\bar{u}_t \rangle - \langle \tilde{\nabla}_{a_t}\tilde{\theta}_{u_t}, \, N\bar{w}_t \rangle \nonumber \\
& \stackrel{(\ref{lemma 1.3 cr-mrc})}{=} & \frac{d}{dt} \big(\langle \tilde{\theta}_{w_t}, \, N\bar{u}_t \rangle - \langle \tilde{\theta}_{u_t}, \, N\bar{w}_t\rangle \big) \nonumber \\
& & - \langle \tilde{\theta}_{w_t}, \,\bar{\nabla}_{a_t}(N \bar{u}_t) \rangle + \langle \tilde{\theta}_{u_t}, \,\bar{\nabla}_{a_t}(N \bar{w}_t) \rangle.\end{aligned}$$ Taking into account (\[rel bar nabla Pi1\]) and Lemma \[lemma nabla N\], the last two terms yield $$\begin{aligned}
\langle \tilde{\theta}_{w_t}, \,\bar{\nabla}_{a_t}(N \bar{u}_t) \rangle - \langle \tilde{\theta}_{u_t}, \,\bar{\nabla}_{a_t}(N \bar{w}_t) \rangle & = & \langle \tilde{\theta}_{w_t}, \, \Pi_1^\#(\theta_{u_t})\rangle - \langle \tilde{\theta}_{u_t}, \, \Pi_1^\#(\theta_{w_t})\rangle \nonumber \\
& = & \langle \tilde{\theta}_{w_t}, \, N\Pi_0^\#(\tilde{\nabla}_{a_t}\tilde{\theta}_{u_t})\rangle - \langle \tilde{\theta}_{u_t}, \, \Pi_0^\# \,^tN(\tilde{\nabla}_{a_t}\tilde{\theta}_{w_t} )\rangle \nonumber \\
& = & \langle \,^tN \tilde{\theta}_{w_t}, \,\bar{\nabla}_{a_t}\Pi_0^\#(\tilde{\theta}_{u_t})\rangle + \langle \,^tN(\tilde{\nabla}_{a_t}\tilde{\theta}_{w_t} ), \, \Pi_0^\#(\tilde{\theta}_{u_t})\rangle \nonumber \\
& \stackrel{(\ref{cond-tilde-nabla-N})}{=} & \langle \,^tN \tilde{\theta}_{w_t}, \,\bar{\nabla}_{a_t}\Pi_0^\#(\tilde{\theta}_{u_t})\rangle + \langle \tilde{\nabla}_{a_t}(\,^tN\tilde{\theta}_{w_t} ), \, \Pi_0^\#(\tilde{\theta}_{u_t})\rangle \nonumber \\
& = & \frac{d}{dt} \langle \,^tN \tilde{\theta}_{w_t}, \, \Pi_0^\#(\tilde{\theta}_{u_t}) \rangle \nonumber \\
& = & \frac{d}{dt}(\Pi_1 (\tilde{\theta}_{u_t}, \, \tilde{\theta}_{w_t})).\end{aligned}$$ Thus, relation (\[1\]) is true and, consequently, formula (\[expression Omega 1 xi\]) is also satisfied.
*Third step:* In the third and last step, we shall prove that the projection $\pi$ is a Poisson map for the pair $(\tilde{\Pi}_1, \Pi_{-1})$. For this, we firstly remark that $$\mathrm{orth}_{\Omega_1}(\ker \pi_\ast) = \mathrm{orth}_{\Omega_0}(R\ker \pi_\ast) \quad \quad \mathrm{and} \quad \quad R\mathrm{orth}_{\Omega_1}(\ker \pi_\ast) = \mathrm{orth}_{\Omega_0}(\ker \pi_\ast),$$ where $\mathrm{orth}_{\Omega_i}(\cdot)$ is the orthogonal distribution of $(\cdot)$ with respect to the symplectic structure $\Omega_i$, $i=0,1$, and $R$ is the recursion operator of $(\Omega_0,\Omega_1)$. We will show that $$\label{equation orth}
\mathrm{orth}_{\Omega_1}(\ker \pi_\ast) = \ker(\pi_1)_\ast,$$ where $\pi_1 = \pi\circ \varphi_1$. Since $\varphi_1$ is a diffeomorphism of $\mathcal{U}$, $\dim \ker(\pi_1)_\ast = \dim \ker \pi_\ast =n$ and $\dim \mathrm{orth}_{\Omega_1}(\ker \pi_\ast) =n$. So, it suffices to show that $\ker(\pi_1)_\ast \subseteq \mathrm{orth}_{\Omega_1}(\ker \pi_\ast)$. We fix $\xi \in \mathcal{U}$ and we consider a vector $u\in \ker \pi_{\ast_\xi}$ and a vector $w \in \ker(\pi_1)_{\ast_\xi}$, then $\bar{u}_0 = \pi_{\ast_\xi} (u)=(\pi \circ \varphi_0)_{\ast_\xi} (u)=0$, since $\varphi_0 = id$, and $\bar{w}_1 =0$. In view of $N^c \ker \pi_\ast \subseteq \ker \pi_\ast$, we have $(\overline{N^cu})_0 = N\bar{u}_0=0$. On the other hand, we remark that the differential equation (\[diff-equation theta\]), as an equation on $\tilde{\theta}_{u_t}$ (resp. $\tilde{\theta}_{w_t}$), is a linear ordinary differential equation having solutions defined for any $t\in [0,1]$ and satisfying any given initial condition. So, we can choose solutions satisfying the conditions $\tilde{\theta}_{u_0}=0$ and $\tilde{\theta}_{w_1} =0$. Thus, $$\begin{aligned}
\Omega_1(u,w) & \stackrel{(\ref{expression Omega 1 xi})}{=} & \big( \langle \tilde{\theta}_{w_t}, N\bar{u}_t\rangle - \langle \tilde{\theta}_{u_t}, N\bar{w}_t \rangle - \Pi_1 (\tilde{\theta}_{u_t}, \tilde{\theta}_{w_t}) \big)\vert_0^1 \nonumber \\
& = & \langle \tilde{\theta}_{w_1}, N\bar{u}_1\rangle - \langle \tilde{\theta}_{u_1}, N\bar{w}_1 \rangle - \Pi_1 (\tilde{\theta}_{u_1}, \tilde{\theta}_{w_1}) \nonumber \\
& & -\langle \tilde{\theta}_{w_0}, N\bar{u}_0\rangle + \langle \tilde{\theta}_{u_0}, N\bar{w}_0 \rangle + \Pi_1 (\tilde{\theta}_{u_0}, \tilde{\theta}_{w_0}) \nonumber \\
& = & 0,\end{aligned}$$ whence we conclude that $w \in \mathrm{orth}_{\Omega_{1_\xi}}(\ker \pi_{\ast_\xi})$ and that (\[equation orth\]) is valid. But, $\ker(\pi_1)_\ast = \mathrm{orth}_{\Omega_0}(\ker \pi_\ast)$ ([@cr-marcut]) and $\mathrm{orth}_{\Omega_1}(\ker \pi_\ast) = \mathrm{orth}_{\Omega_0}(R\ker \pi_\ast)$, therefore, $\mathrm{orth}_{\Omega_0}(R\ker \pi_\ast) = \mathrm{orth}_{\Omega_0}(\ker \pi_\ast)$, which means that $\ker \pi_\ast$ is invariant by $R$.
Now we will calculate the projection of $\tilde{\Pi}_1$ by $\pi$ using the two expressions of $\Omega_1$ (see, Lemma \[lemma compatible\]). We consider a point $\xi\in \mathcal{U}$ and an arbitrary covector $\theta \in T_x^\ast M$, where $x=\pi(\xi)$, and we denote by $u$ the unique vector in $T_\xi \mathcal{U}$ defined by the relation $$\Omega_1^\flat(u) = \pi^\ast \theta.$$ We can easily remark that $u$ is a point of $\mathrm{orth}_{\Omega_{1_\xi}}(\ker \pi_\ast{_\xi}) = \ker(\pi_1)_{\ast_\xi}$, thus $\bar{u}_1 =0$. Furthermore, $\Omega_1^\flat(u) = \pi^\ast \theta \Leftrightarrow \Omega_0^\flat(Ru) = \pi^\ast \theta$, which yields that $Ru \in \mathrm{orth}_{\Omega_{0_\xi}}(\ker \pi_{\ast_\xi}) = \ker(\pi_1)_{\ast_\xi}$, so $(\overline{Ru})_1 =0$. Consequently, for any $w\in T_\xi \mathcal{U}$, we have $$\begin{aligned}
\Omega_1(u,w) & = & \Omega_0(Ru,w) = \big( \langle \tilde{\theta}_{w_t}, (\overline{Ru})_t\rangle - \langle \tilde{\theta}_{(Ru)_t}, \bar{w}_t \rangle - \Pi_0 (\tilde{\theta}_{(Ru)_t}, \tilde{\theta}_{w_t}) \big)\vert_0^1 \nonumber \\
& = & \langle \tilde{\theta}_{w_1}, (\overline{Ru})_1\rangle - \langle \tilde{\theta}_{(Ru)_1}, \bar{w}_1 \rangle - \Pi_0 (\tilde{\theta}_{(Ru)_1}, \tilde{\theta}_{w_1}) \nonumber \\
& & -\langle \tilde{\theta}_{w_0}, (\overline{Ru})_0\rangle + \langle \tilde{\theta}_{(Ru)_0}, \bar{w}_0 \rangle + \Pi_0 (\tilde{\theta}_{(Ru)_0}, \tilde{\theta}_{w_0})\end{aligned}$$ and, by choosing $\tilde{\theta}_{(Ru)_t}$ verifying the initial condition $\tilde{\theta}_{(Ru)_1} =0$, we obtain $$\begin{aligned}
\Omega_1(u,w) & = & -\langle \tilde{\theta}_{w_0}, (\overline{Ru})_0\rangle + \langle \tilde{\theta}_{(Ru)_0}, \bar{w}_0 \rangle + \Pi_0 (\tilde{\theta}_{(Ru)_0}, \tilde{\theta}_{w_0})\, \Leftrightarrow \nonumber \\
\langle \theta, \bar{w}_0\rangle & = & -\langle \tilde{\theta}_{w_0}, (\overline{Ru})_0\rangle + \langle \tilde{\theta}_{(Ru)_0}, \bar{w}_0 \rangle + \Pi_0 (\tilde{\theta}_{(Ru)_0}, \tilde{\theta}_{w_0}).\end{aligned}$$ The last equation holds for any $w\in T_\xi \mathcal{U}$ and for any initial value of $\tilde{\theta}_w$. Thus, $\tilde{\theta}_{(Ru)_0} = \theta$ and $\Pi_0^\#(\theta) = (\overline{Ru})_0$. On the other hand, from the second expression of $\Omega_1$ we take $$\begin{aligned}
\Omega_1(u,w) & = & \big( \langle \tilde{\theta}_{w_t}, N\bar{u}_t\rangle - \langle \tilde{\theta}_{u_t}, N\bar{w}_t \rangle - \Pi_1 (\tilde{\theta}_{u_t}, \tilde{\theta}_{w_t}) \big)\vert_0^1 \nonumber \\
& = & \langle \tilde{\theta}_{w_1}, N\bar{u}_1\rangle - \langle \tilde{\theta}_{u_1}, N\bar{w}_1 \rangle - \Pi_1 (\tilde{\theta}_{u_1}, \tilde{\theta}_{w_1}) \nonumber \\
& & -\langle \tilde{\theta}_{w_0}, N\bar{u}_0\rangle + \langle \tilde{\theta}_{u_0}, N\bar{w}_0 \rangle + \Pi_1 (\tilde{\theta}_{u_0}, \tilde{\theta}_{w_0}).\end{aligned}$$ Then, by choosing $\tilde{\theta}_{u_1} =0$ and because of $N\bar{u}_1 =0$, we get $$\langle \theta, \bar{w}_0\rangle = -\langle \tilde{\theta}_{w_0}, N\bar{u}_0\rangle + \langle \,^tN\tilde{\theta}_{u_0}, \bar{w}_0 \rangle + \Pi_0 (\,^tN\tilde{\theta}_{u_0}, \tilde{\theta}_{w_0}),$$ which holds for any $w\in T_\xi \mathcal{U}$ and for any initial value of $\tilde{\theta}_w$. Therefore, $\,^tN\tilde{\theta}_{u_0} = \theta$ and $\Pi_0^\#(\,^tN\tilde{\theta}_{u_0}) = N\bar{u}_0$. Hence, $$\label{Pi-1}
\Pi_0^\#(\theta) = \Pi_0^\#(\,^tN\tilde{\theta}_{u_0})= N\bar{u}_0 \Leftrightarrow N^{-1}\Pi_0^\#(\theta) = \bar{u}_0 \Leftrightarrow \Pi_{-1}^\#(\theta) = \bar{u}_0.$$ However, $$\begin{aligned}
\pi^\ast \theta = \Omega_1^\flat(u) & \Leftrightarrow & \tilde{\Pi}_1^\# (\pi^\ast \theta) = u \,\Rightarrow \, \pi_\ast (\tilde{\Pi}_1^\# (\pi^\ast \theta)) = \pi_\ast(u) \nonumber \\
& \Leftrightarrow & (\pi_\ast \circ \tilde{\Pi}_1^\# \circ \pi^\ast)(\theta)=\bar{u}_0 \, \stackrel{(\ref{Pi-1})}{\Leftrightarrow} \, (\pi_\ast \circ \tilde{\Pi}_1^\# \circ \pi^\ast)(\theta) = \Pi_{-1}^\#(\theta).\end{aligned}$$ Therefore, $$\pi_\ast \circ \tilde{\Pi}_1^\# \circ \pi^\ast = \Pi_{-1}^\#,$$ which means that $\pi$ is a Poisson map for the pair $(\tilde{\Pi}_1, \Pi_{-1})$, also.
*Let $\tilde{\Pi}_k$ be the Poisson structure defined, for any $k\in \Z$, by the symplectic form* $$\Omega_k = \int_0^1 \varphi_t^\ast \omega_k dt, \quad \quad \mathrm{where} \quad \quad \omega_k(\cdot,\cdot) = \omega_{can}((N^c)^k\cdot,\cdot).$$ *By considering on $\mathcal{U}$ the hierarchy $(\tilde{\Pi}_k)_{k \in \Z}$ of pairwise compatible Poisson structures, we can easily prove that $\pi$ is a Poisson map for any pair $(\tilde{\Pi}_k, \Pi_{-k})$, $k\in \Z$.*
\[remark-existence-connection\] *In the second step of the proof of the above theorem, we have assumed the existence of a symmetric covariant connection $\nabla$ on $M$ compatible with $N$. In general, it is difficult to establish the conditions under which a given tensor field $S$ on a smooth manifold $M$ admits a “compatible”, in a certain sense, symmetric covariant connection. In local coordinates, finding a torsionless $\nabla$ “compatible” with $S$ reduces to determining the existence of solutions for non-homogeneous $\C$-linear systems with unknowns the Christoffel symbols of $\nabla$. In our case, for given $N$, the corresponding systems are the ones given by (\[cond-nabla-N locally\]):* $$\label{system gamma}
\Gamma^i_{jl}\nu^l_k - \Gamma^i_{kl}\nu^l_j = \frac{\partial \nu^i_j}{\partial x^k} - \frac{\partial \nu^i_k}{\partial x^j}.$$ *By calling, for each $i=1,\ldots,n$, $\Gamma^i$ the symmetric matrix with elements the unknown functions $\Gamma^i_{jk}$, (\[system gamma\]) is written as* $$\Gamma^iN - \,^tN\Gamma^i = \big(\frac{\partial \nu^i_j}{\partial x^k} - \frac{\partial \nu^i_k}{\partial x^j} \big).$$ *It is a $\C$-linear system of $\displaystyle{\frac{n^2-n}{2}}$ equations with $\displaystyle{\frac{n^2+n}{2}}$ unknowns, the functions $\Gamma^i_{jk}$. Such a system, it is either incompatible, or, if it admits a solution, then it admits an infinity of solutions whose difference is a solution of the corresponding homogeneous system.*
Examples {#section examples}
========
In this section we give two examples of Poisson-Nijenhuis structures for which our result is applicable.
Pair of diagonal quadratic Poisson structures: *Let $V$ be a finite dimensional (real) vector space and $(x^1,\ldots, x^n)$ a system of linear coordinates for $V$. We recall that any bivector field $\Pi$ on $V$ of type* $$\Pi = \sum_{i<j}\varpi^{ij}x^ix^j\frac{\partial}{\partial x^i}\wedge \frac{\partial}{\partial x^j}, \quad \quad \mathrm{with} \quad \quad \varpi^{ij} \in \R,$$ *is Poisson and it is called *diagonal quadratic Poisson structure*, [@duf-zung], [@cam-pol-an]. Therefore, any pair $(\Pi_0, \Pi_1)$ of quadratic Poisson structures is compatible in the sense of Magri-Morosi and it defines a bi-Hamiltonian structure on $V$. We consider such a pair and we suppose that its elements are related by a recursion operator $N$, i.e., $\Pi_1^\# = N\Pi_0^\# = \Pi_0^\# \,^tN$. Precisely, if* $$\Pi_0 = \sum_{i<j}\varpi_0^{ij}x^ix^j\frac{\partial}{\partial x^i}\wedge \frac{\partial}{\partial x^j} \quad \quad \mathrm{and} \quad \quad \Pi_1 = \sum_{i<j}\varpi_1^{ij}x^ix^j\frac{\partial}{\partial x^i}\wedge \frac{\partial}{\partial x^j},$$ *then the components $\nu^i_j$ of $N$ must be of type $\nu^i_j = n^i_j \displaystyle{\frac{x^i}{x^j}}$ with $n^i_j \in \R$ and $\varpi_1^{ij} = n^i_l \varpi_0^{lj} = \varpi_0^{il}n_l^j$. Because $(\Pi_0, \Pi_1)$ are Poisson compatible, $N$ is necessary a Nijenhuis operator. Our problem consists of finding a symmetric covariant connection $\nabla$ on $V$ such that its Christoffel symbols $\Gamma^i_{jk}$ verify condition (\[cond-nabla-N locally\]). We can easily check that the connection with Christoffel symbols $\Gamma^i_{ii} = - \displaystyle{\frac{1}{x^i}}$ and all the other $\Gamma_{ij}^k$ zero gives a solution to our problem. Hence, we conclude that any Poisson-Nijenhuis structure of the considered type is symplectically realizable.*
*We consider the pair $(\Pi_0,\Pi_1)$ of compatible Poisson structures on $\R^6$, where $\Pi_0$ is the linear Poisson structure associated with the periodic Toda lattice of $3$-particles and $\Pi_1$ is the Poisson structure constructed in [@dam-pet] that has the same Casimir invariants with $\Pi_0$ (the functions $C = a_1a_2a_3$ and $C'=b_1+b_2+b_3$) and whose first part is the quadratic Poisson bracket associated to Volterra lattice. In Flaschka’s coordinate system $(a_1,a_2,a_3, b_1,b_2,b_3)$,* $$\Pi_0 = \sum_{i=1}^3 a_i\frac{\partial}{\partial a_i}\wedge (\frac{\partial}{\partial b_i} - \frac{\partial}{\partial b_{i+1}}) \quad \quad \mathrm{and} \quad \quad \Pi_1 = \sum_{i=1}^3(a_ia_{i+1}\frac{\partial}{\partial a_i}\wedge \frac{\partial}{\partial a_{i+1}} + \frac{\partial}{\partial b_i}\wedge \frac{\partial}{\partial b_{i+1}}),$$ *with the convention $(a_{i+3},b_{i+3}) = (a_i,b_i)$. The pair possesses an infinity of recursion operators of type* $$N = \begin{pmatrix} 0 & 0 & 0 & f & f & f + a_1 \cr
0 & 0 & 0 & g + a_2 & g & g \cr
0 & 0 & 0 & h & h+a_3 & h \cr
\displaystyle{\frac{a_3}{a_1}}F & \displaystyle{\frac{a_3}{a_2}F+\frac{1}{a_2}} & F & 0 & 0 & 0 \cr
\displaystyle{\frac{a_2}{a_1}G} & G & \displaystyle{\frac{a_2}{a_3}G+\frac{1}{a_3}} & 0 & 0 & 0 \cr
\displaystyle{\frac{a_3}{a_1}H + \frac{1}{a_1}} & \displaystyle{\frac{a_3}{a_2}H} & H & 0 & 0 & 0
\end{pmatrix},$$ *where $f,g,h,F,G,H\in C^\infty(\R^6)$. Since, $\Pi_1^\# = N\circ \Pi_0^\#$ is Poisson and compatible with $\Pi_0$, $N$ is a Nijenhuis operator. In the case where $F=G=H=0$ and $f=g=h=C$, there exists an infinity of symmetric connections compatible with $N$. Such a $\nabla$ is defined by the functions* $$\begin{aligned}
\Gamma^1_{44}=\Gamma^1_{45} = \Gamma^1_{54}=\Gamma^1_{46} = \Gamma^1_{64}=\Gamma^1_{55}=\Gamma^1_{56} = \Gamma^1_{65}= C & \mathrm{and} & \Gamma^1_{66} = C + a_1, \\
\Gamma^2_{45}= \Gamma^2_{54}=\Gamma^2_{46} = \Gamma^2_{64} =\Gamma^2_{55}=\Gamma^2_{56} = \Gamma^2_{65}=\Gamma^2_{66}= C & \mathrm{and} & \Gamma^2_{44} = C + a_2, \\
\Gamma^3_{44}=\Gamma^3_{45} = \Gamma^3_{54} =\Gamma^3_{46}=\Gamma^3_{64} =\Gamma^3_{56} = \Gamma^3_{65} =\Gamma^3_{66}= C & \mathrm{and} & \Gamma^3_{55} = C + a_3,\end{aligned}$$ *and all the other $\Gamma^i_{jk}$ are zero. Then, by applying Theorem \[THEOREM\] we conclude that $(\Pi_0, \Pi_1)$ is symplectizable.*
**Open problem:** Our first approach to the study of the problem mentioned in the Introduction was the following. Let $(M, \Pi_0, \Pi_1)$ be a bi-Hamiltonian manifold endowed with a symmetric covariant connection $\nabla$. We consider the convex linear combination $\Pi_s = (1-s)\Pi_0 + s\Pi_1$, $s\in [0,1]$, of $\Pi_0$ and $\Pi_1$ which produces on $M$ an $1$-parameter family of pairwise compatible Poisson structures. Then the Poisson sprays and the contravariant connections corresponding to the above structures have the following nice properties.
- If $\mathcal{V}_{\Pi_0}$ is a Poisson spray of $\Pi_0$ and $\mathcal{V}_{\Pi_1}$ is a Poisson spray of $\Pi_1$, then the vector field $$\mathcal{V}_{\Pi_s} = (1-s)\mathcal{V}_{\Pi_0} + s\mathcal{V}_{\Pi_1}$$ is a Poisson spray of $\Pi_s = (1-s)\Pi_0 + s\Pi_1$.
- If $\bar{\nabla}_i$ and $\tilde{\nabla}_i$, $i=0,1$, are the contravariant connections on $TM$ and $T^\ast M$, respectively, defined by the pair $(\nabla, \Pi_i)$, $i=0,1$, as in (\[def-contravariant connection\]), then
- $\bar{\nabla}_s = (1-s)\bar{\nabla}_0 + s \bar{\nabla}_1$ is a contravariant connection on $TM$,
- $\tilde{\nabla}_s = (1-s)\tilde{\nabla}_0 + s \tilde{\nabla}_1$ is a contravariant connection on $T^\ast M$,
and the two are related by the formula $$(\bar{\nabla}_s)_\alpha \Pi_s^\# (\beta) = \Pi_s^\#((\tilde{\nabla}_s)_\alpha\beta).$$
Hence, if $(\varphi_s)_t$ is the flow of $\mathcal{V}_{\Pi_s}$, by applying the M. Crainic and I. M$\check{\mathrm{a}}$rcu$\c{t}$’s technique we construct on a convenable neighborhood $\mathcal{U}$ of the $0$-section of $T^\ast M$ an $1$-parameter family of symplectic forms $$\Omega_s = \int_0^1 (\varphi_s)_t^\ast \omega_{can}dt, \quad \quad s\in [0,1],$$ such that $\pi : (\mathcal{U}, \tilde{\Pi}_s) \to (M, \Pi_s)$, where $\tilde{\Pi}_s = \Omega_s^{-1}$, is a Poisson map for any $s\in [0,1]$. The question which arises is: *Are the Poisson structures $\tilde{\Pi}_s$, $s\in [0,1]$, compatible between them? If not, under what conditions can this happen?*
**Acknowledgments** This work is partially supported by the project DRASI A of AUTH Research Committee.
[45]{}
[^1]: We recall that $\Omega_i^\flat$, $i=0,1$, denotes the vector bundle map from $TM$ to $T^\ast M$ whose the induced map on the space of smooth sections, also denoted by $\Omega_i^\flat$, is defined as follows: for any $X,Y \in \Gamma(TM)$, $\langle \Omega_i^\flat(X),Y\rangle = - \Omega_i(X,Y)$.
[^2]: For some comments on the existence of a such connection, see Remark [\[remark-existence-connection\]]{}.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We present sub-millimeter statistical detections of galaxies discovered in the $5''\times5''$ Spitzer Early Release Observations (to $\sim4-15\mu$Jy $5\sigma$ at $3.6-8\mu$m, $170\mu$Jy at $24\mu$m) through a stacking analysis of our reanalysed SCUBA 8mJy survey maps, and a Spitzer identification of a new sub-millimeter point source in the 8mJy survey region. For sources detected at $5.8\mu$m or $8\mu$m ($154$ and $111$ sources respectively), we detect positive skews in the sub-millimeter flux distributions at $99.2-99.8\%$ confidence using Kolmogorov-Smirnov tests, at both $850\mu$m and $450\mu$m. We also marginally detect the [*Spitzer*]{} $24\mu$m galaxies at $850\mu$m at $97\%$ confidence, and place limits on the mean sub-millimeter fluxes of the $3.6\mu$m and $4.5\mu$m sources. Integrating the sub-millimeter fluxes of the [*Spitzer*]{} populations, we find the $5.8\mu$m galaxies contribute $0.12\pm0.05$ nW m$^{-2}$ sr$^{-1}$ to the $850\mu$m background, and $2.4\pm0.7$ nW m$^{-2}$ sr$^{-1}$ to the $450\mu$m background; similar contributions are made by the $8\mu$m-selected sample. We infer that the populations dominating the $5.8\mu$m and $8\mu$m extragalactic background light also contribute around a quarter of the $850\mu$m background and the majority of the $450\mu$m background.'
author:
- 'S. Serjeant, A.M.J. Mortier, R.J. Ivison, E. Egami, G.H. Rieke, S.P. Willner, D. Rigopoulou, A. Alonso-Herrero, P. Barmby, L. Bei, H. Dole, C.W. Engelbracht, G.G. Fazio, E. Le Floc’h, K.D. Gordon, T. R. Greve, D.C. Hines, J.-S. Huang, K.A. Misselt, S. Miyazaki, J.E. Morrison, C. Papovich, P.G. Pérez-González, M.J. Rieke, J. Rigby, G. Wilson'
title: 'Sub-millimeter detections of [*Spitzer Space Telescope*]{} galaxy populations'
---
Introduction
============
The early pioneering sub-millimeter surveys in lensing clusters (Smail, Ivison & Blain 1997) and in blank fields (Hughes et al. 1998, Barger et al. 1998) demonstrated the feasibility of deep extragalactic surveys exploiting the favorable K-corrections in the sub-millimeter. Deep $850\mu$m imaging has now resolved around half of the $850\mu$m extragalactic background (Hughes et al. 1998, Blain et al. 1999, Cowie, Barger & Kneib 2002). These galaxies are often called SCUBA galaxies after the instrument with which they were first detected (Holland et al. 1999). The $14''$ $850\mu$m SCUBA beam makes identifications at other wavelengths difficult; nevertheless, $\sim50\%$ of $850\mu$m sources are identifiable in $\mu$Jy $1.4$GHz imaging (Ivison et al. 2002). These radio identifications have led to optical identifications, morphologies and ultimately spectroscopic redshifts in multiwavelength follow-up campaigns (e.g. Chapman et al. 2003). Furthermore, the non-detection of SCUBA galaxies in hard X-ray imaging (e.g. Alexander et al. 2003) suggests that the bulk of the population has far-infrared luminosities dominated by star formation. The morphologies, redshifts, clustering and molecular gas contents are so far consistent with at least some of the SCUBA population being the progenitors of giant elliptical galaxies (e.g. Dunlop 2002), though other alternatives are still viable (Efstathiou & Rowan-Robinson 2003) and the SCUBA population is heterogeneous (e.g. Ivison et al. 1998, 2000). Finally, the K-correction effects in the sub-millimeter make the sub-millimeter extragalactic background sensitive to contributions from the far-infrared luminous energy densities at all redshifts $z\stackrel{<}{_\sim}10$. The populations which contribute to the $850\mu$m extragalactic background are necessarily also significant contributors to the cosmic history of dust-shrouded star formation.
Following the [*IRAS*]{} mid-infrared surveys of the local Universe (e.g. Rush, Malkan & Spinoglio 1993), the mid-infrared was first made accessible to deep extragalactic surveys by the Infrared Space Observatory ([*ISO*]{}, Kessler et al. 1996) which conducted a suite of surveys with a variety of depths and areal coverages (e.g. Genzel & Cesarsky 2000 and refs. therein, Rowan-Robinson et al. 2004). The rapid upturn in the $15\mu$m extragalactic source counts clearly demonstrated the existence of a strongly evolving population of obscured starbursts and active galaxies (e.g. Serjeant et al. 2000, Franceschini et al. 2001, Gruppioni et al. 2002). It has also been argued that the populations dominating the $15\mu$m extragalactic background light, which are resolved by [*ISO*]{}, are also largely the same populations which dominate the unresolved $140\mu$m background (Elbaz et al. 2002). If correct, this is a significant breakthrough in determining the populations which dominated the far-infrared luminous energy density throughout the relatively recent history of the Universe (e.g. $z<1$).
Finding the population that supplies the luminous energy density at $z>1$ requires understanding the sub-millimeter background light. However, it has been difficult to find sub-millimeter source counterparts in the mid-infrared. Very few sub-millimeter-selected sources have been detected by [*ISO*]{} in the mid-infrared (e.g. Eales et al. 2000, Webb et al. 2003b, Sato et al. 2002). The reverse procedure of looking for mid-infrared sources in the sub-millimeter via stacking analyses have not fared much better. Serjeant et al. (2003a) found no excess $850\mu$m flux at the locations of $15\mu$m sources in the HDF North. Lyman break galaxies, in contrast, are detectable statistically (e.g. Peacock et al. 2000, Webb et al. 2003a). If SCUBA galaxies are extreme star-forming galaxies in the most massive high-redshift halos, then their anomalously faint K-band identifications imply heavy obscuration in the observed-frame near-infrared (Serjeant et al. 2003b), suggesting that SCUBA galaxies may be detectable in $\mu$Jy-level mid-infrared imaging.
The [*Spitzer Space Telescope*]{} (hereafter [*Spitzer*]{}, Werner et al. 2004) is an enormous advance over [*ISO*]{} in terms of mapping speed, sensitivity, and wavelength coverage. It may now be possible to resolve the bulk of the extragalactic background light at $3.6-8\mu$m in exposures of the order $15$ minutes, an equivalent depth achieved at $6.7\mu$m in $23$ hours with [*ISO*]{} by Sato et al. 2003. In this paper we present statistical sub-millimeter detections of galaxies selected in a new [*Spitzer*]{} survey. The [*Spitzer*]{} identifications of previously-published sub-millimeter sources are discussed by Egami et al. (2004). Identifications of MAMBO sources are discussed by Ivison et al. (2004).
Observations
============
The [*Spitzer*]{} Early Release Observations survey is one of the first extragalactic surveys conducted by [*Spitzer*]{}. Besides the pragmatic goal of characterising the survey capabilities of the facility, the survey has the science goals of making the first constraints on the populations which dominate the extragalactic backgrounds in the shorter wavelength [*Spitzer*]{} bands, and the links between these galaxies and other known populations. Accordingly, the survey field was selected to lie in the Lockman Hole, an area with abundant multi-wavelength survey coverage, and in particular with $7$ galaxies from the $850\mu$m 8mJy survey $>3.5\sigma$ catalog (Scott et al. 2002, Fox et al. 2002, Ivison et al. 2002; see below).
The [*Spitzer*]{} imaging is described by Egami et al. (2004) and Huang et al. (2004). In summary, IRAC (Fazio et al. 2004) imaged a $\sim5'\times5'$ field for $750$s per sky pixel at all four bands, resulting in median $1\sigma$ depths of $0.77\mu$Jy, $0.78\mu$Jy, $2.75\mu$Jy and $1.67\mu$Jy at $3.6\mu$m, $4.5\mu$m, $5.8\mu$m and $8\mu$m respectively, and sources were extracted to $>4.7\sigma$. MIPS (Rieke et al. 2004) observed the field in photometry mode at $24\mu$m for $250$s per sky pixel, resulting in a typical $1\sigma$ depth of $30\mu$Jy. Source confusion and blending make the MIPS completeness and reliability more problematic than for SCUBA or IRAC. MIPS source catalogs were extracted to $>170\mu$Jy ($80\%$ complete, $>90\%$ reliable) and to $>120\mu$Jy ($50\%$ complete, $50\%$ reliable). We conservatively assume the catalogs are all extragalactic, as expected at these flux densities; the deletion of any (hypothetical) contaminant population of galactic stars with negligible sub-millimeter fluxes would improve the confidence levels of our statistical detections.
The SCUBA 8mJy survey data (which covers all our [*Spitzer*]{} field) is described in detail by Scott et al. (2002). We have reanalysed this data, modelling the intra-night SCUBA gain variations, improving the extinction corrections using improved fits to the $225$GHz skydips monitored at the Caltech Sub-millimeter Observatory, and removing the cross-talking $450\mu$m bolometers A7 and A16 (see Mortier et al. 2004). Additional sources are detectable in the zero-sum chopped/nodded sub-millimeter maps. After point source detection using noise-weighted PSF convolution (Serjeant et al. 2003a) including the effect of negative chop/nod positions, the median $1\sigma$ depth (total flux) in the [*Spitzer*]{} Early Release Observations field is $1.6$mJy at $850\mu$m, and $14.4$mJy at $450\mu$m.
Results {#sec:ids}
=======
Our reanalysis of the SCUBA 8mJy survey uncovered new candidate sub-millimeter sources (Mortier et al. 2004). Figure \[fig:ids\] shows the [*Spitzer*]{} identifications of the most significant of these, at $\alpha=10^{\rm
h}$ $51^{\rm m}$ $53.94^{\rm s}$, $\delta=+57^\circ$ $25'$ $05.5''$ (J2000), detected with a $3.7\sigma$ $850\mu$m flux of $4.3$mJy. At $450\mu$m the source is not detected, though the $450\mu$m flux at its position is $17.9$mJy ($1.6\sigma$); there is also a hint of positive flux at $1.8\sigma$ significance at $1250\mu$m (Greve et al. 2004) at the position of this source. There are hints that the source is extended at $850\mu$m, suggesting a blend of more than one source, and indeed there are two candidate identifications in the [*Spitzer*]{} imaging. The probabilities of a random association ($P=1-\exp(-n(>s)\pi r^2)$ where $n(>s)$ is the surface density of catalogued objects brighter than the identification at distance $r$) are $0.033$ and $0.155$ for the brighter and fainter $4.5\mu$m flux respectively. Interestingly, the low-significance contours of $450\mu$m emission are coincident with the brighter of the two candidates. The [*Spitzer*]{} fluxes of this source are $16.6$, $26.7$, $14.9$, $18.9\mu$Jy and $216\mu$Jy at $3.6\mu$m, $4.5\mu$m, $5.8\mu$m, $8\mu$m and $24\mu$m respectively. The fluxes of the second candidate identification are $6.2\mu$Jy and $4.6\mu$Jy at $3.6\mu$m, and $4.5\mu$m, but the source is not detected at longer wavelength [*Spitzer*]{} bands. While deblending the fluxes from the two identifications is beyond the scope of this paper, it is interesting to note that either identification would have a sub-millimeter:[*Spitzer*]{} flux ratio which is redder than in the population as a whole, derived below.
The $>3.5\sigma$ sub-millimeter sources are already known to have [*Spitzer*]{} identifications (Egami et al. 2004), so we construct a mask to remove the sources at these positions from the stacking analysis. At $850\mu$m we masked a $9.9''$ radius ($\sqrt{2}\times$ the SCUBA FWHM, i.e. the FWHM of point sources in our PSF-convolved maps) around the $>3.5\sigma$ $850\mu$m point sources from Scott et al. (2002), as well as the new point sources from our reanalysis (Mortier et al. 2004). At $450\mu$m the masking radius was $5.2''$, which we also applied to new $450\mu$m sources from our reanalysis. We also masked regions with high noise levels, arbitrarily selected as $>5$mJy at $850\mu$m, or $>100$mJy at $450\mu$m, and also masked all regions in the sub-millimeter maps without [*Spitzer*]{} data. The unmasked area at $850\mu$m ($450\mu$m) is $22.1$ ($22.7$) arcmin$^2$ for the IRAC catalogues. For MIPS, the corresponding area is $27.6$ ($28.5$) arcmin$^2$.
The sub-millimeter maps give the best-fit point source flux at each location (equation A4 of Serjeant et al. 2003a), so we measure the values of the sub-millimeter images at the [*Spitzer*]{} galaxy positions. Figure \[fig:histograms\] shows the sub-millimeter signal-to-noise ratios (equation A6 of Serjeant et al. 2003a) at the positions of the [*Spitzer*]{} Early Release Observations source catalogs at $5.8\mu$m and $8\mu$m, which lie in the unmasked regions. These figures also show histograms for the whole of the unmasked maps. Note the clear positive skews in the sub-millimeter fluxes at the [*Spitzer*]{} source positions, relative to the maps as a whole. We used Kolmogorov-Smirnov tests to determine the confidence level of these relative positive skews, listed in table \[tab:coadd\_results\]. There are significant detections at $5.8\mu$m and $8\mu$m, and marginal detections at $24\mu$m. No significant skew was found at $3.6\mu$m or $4.5\mu$m. We also calculated the mean fluxes in the sub-millimeter maps at the positions of the [*Spitzer*]{} galaxies. These are almost all positive and are listed in table \[tab:coadd\_results\]. This statistic is less efficient than the Kolmogorov-Smirnov test (which does not just use the first moment), so the uncertainties on the mean fluxes are larger than suggested by the Kolmogorov-Smirnov confidence levels. The [*Spitzer*]{} galaxies are individually undetected in the sub-millimeter, so one must be careful in interpreting the stacking analysis; for example, we do not use the [*noise-weighted*]{} mean fluxes because they can give false positives, as noted by Serjeant et al. 2003a. Also, many fluctuations in the sub-millimeter maps are blends (e.g. Scott et al. 2002), so is there a risk of overestimating the flux of any given [*Spitzer*]{} galaxy by also counting its neighbors? The answer is no, as Peacock et al. (2000) showed: for a Poissonian sampling of the sub-millimeter map, the expected total sub-millimeter flux from all neighbors equals the mean flux of the map, which is exactly zero for the chopped, nodded SCUBA maps. (The chopped/nodded PSF is also zero-sum, so the PSF-convolved images are still zero-sum.) Blending slightly degrades the uncertainty on the mean, but it does not affect the mean fluxes. Another demonstration of this is to note that the total flux of any given source is exactly zero in these chopped, nodded maps. Even confusing sources within the same SCUBA beam will have no [*net*]{} effect, provided that there are also similar sources uniformly distributed over the rest of the map.
The histograms of the whole map in figure \[fig:histograms\] effectively act as the control sample. Nevertheless, as a test of the stacking analysis methodology, we performed the same tests on simulated [*Spitzer*]{} catalogs of the same size as the observed catalogs, with a Poissonian distribution in the unmasked regions. No positive skews in the sub-millimeter flux distributions were detected relative to the maps as a whole. These simulations also verified that the mean sub-millimeter flux around randomly chosen positions is consistent with zero.
Discussion
==========
Although our [*Spitzer*]{} $5.8\mu$m and $8\mu$m catalogues are small ($154$ and $111$ respectively), we can use our statistical detections to obtain constraints on the contribution to the sub-millimeter extragalactic background light from the [*Spitzer*]{} populations, by multiplying the mean sub-millimeter flux of a single [*Spitzer*]{} galaxy (table \[tab:coadd\_results\]) with the observed surface density of [*Spitzer*]{} galaxies. At $850\mu$m, the integrated contribution from the $5.8\mu$m population is $0.12\pm0.05$ nW m$^{-2}$ sr$^{-1}$, and from the (overlapping) $8\mu$m population $0.10\pm0.04$ nW m$^{-2}$ sr$^{-1}$. This is already comparable to the total $850\mu$m extragalactic background reported by Lagache et al. (2000) of $\sim0.5$ nW m$^{-2}$ sr$^{-1}$. At $450\mu$m the [*Spitzer*]{} contributions can, within the uncertainties, account for all the observed $\sim3$ nW m$^{-2}$ sr$^{-1}$ extragalactic background: $2.4\pm0.7$ nW m$^{-2}$ sr$^{-1}$ from the $5.8\mu$m-selected sample, and $2.2\pm0.6$ nW m$^{-2}$ sr$^{-1}$ from the $8\mu$m galaxies. Our sub-millimeter faint source density is consistent with those in the deepest $850\mu$m maps (Cowie et al. 2002).
The sub-millimeter:[*Spitzer*]{} flux ratios in sub-millimeter point sources (Egami et al. 2004, see above) are much higher than average in the population (table \[tab:coadd\_results\]). Despite the commonality between the SCUBA and [*Spitzer*]{} galaxy populations, sub-millimeter galaxies are not at all representative of the [*Spitzer*]{} population. The sub-millimeter background independently suggests the same result: if the $24\mu$m population had sub-millimeter:[*Spitzer*]{} flux ratios as high as SCUBA galaxies (e.g. median $S_{850\mu{\rm m}}/S_{24\mu{\rm m}}=34.4$ for the Egami et al. (2004) sample), the $>170\mu$Jy $24\mu$m sources alone would over-predict the $850\mu$m background by $\sim60\%$. Our weak sub-millimeter detection of the $24\mu$m population is less surprising in that context. One possible interpretation is that the SCUBA population has fewer (or more heavily obscured) active nuclei than the [*Spitzer*]{} population as a whole.
Is our stacking analysis signal due to dusty galaxies, or to high-$z$ galaxies, or both? Our samples are too small to distinguish these possibilities, but stacking analyses of subsets show that either option is credible. Almost all the $5.8\mu$m galaxies are detected at $3.6\mu$m, and as with the $5.8\mu$m population as whole (table \[tab:coadd\_results\]) this subset ($153$ galaxies) is detected statistically. The $299$ remaining $3.6\mu$m galaxies are not detected in the stacking analyses (e.g., only $37\%$ confidence in the $450\mu$m signal:noise maps), so the redder galaxies are more likely to be sub-millimeter emitters. The lack of a stacking signal at $3.6\mu$m and $4.5\mu$m is at least in part due to dilution of the signal from the high density of bluer galaxies detected at these wavelengths. The red colors could be due to heavy obscuration, but the colors also become redder quickly with increasing redshift. Simpson & Eisenhardt (1999) show that $F_{4.5\mu\rm m}/F_{3.6\mu\rm m}>1.32$ or $F_{8\mu\rm m}/F_{5.8\mu\rm m}>1.32$ selects redshifts $z\stackrel{>}{_\sim}1.5$. There are $47$ galaxies satisfying either criteria, with detections in the relevant bands, and these galaxies are detected in the stacking analysis (e.g. $99.1\%$ confidence at $450\mu$m); the corresponding $400$ foreground $z\stackrel{<}{_\sim}1.5$ galaxies are not detected (e.g. only $71\%$ confidence at $450\mu$m). Further work on larger samples is needed to discriminate between high-redshift and high-obscuration sub-populations.
The sub-millimeter backgrounds are sensitive to contributions from the far-infrared luminous energy densities throughout the redshift range of favorable K-corrections. It follows that the [*Spitzer*]{} $5.8\mu$m and $8\mu$m-selected galaxies are necessarily significant contributors to the comoving volume averaged star formation density. Many of the $5.8\mu$m and $8\mu$m galaxies are challenging targets for 8-10m-class optical/near-infrared spectroscopy, but redshift surveys of these populations, together with 2-D and 3-D clustering, would discriminate between competing semi-analytic descriptions of galaxy evolution, and determine whether (or which) SCUBA/[*Spitzer*]{} galaxies are the sites of the assembly of giant ellipticals. Finally, it is worth stressing that the conclusions in this paper could only have been reached with co-ordinated multi-wavelength surveys.
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Sub-millimeter stacking analysis of [*Spitzer*]{} source catalogs
[llllllllllll]{} Wave- & $450\mu$m & $450\mu$m & $450\mu$m & $\langle S_{450}\rangle$ & $\langle S_{450}\rangle/$ & $850\mu$m & $850\mu$m & $850\mu$m & $\langle S_{850}\rangle$ & $\langle S_{850}\rangle/$\
length & $N_{\rm src}$ & K-S (S:N) & K-S (Flux) & mJy & $\langle S_{\rm\it Spitzer}\rangle$ & $N_{\rm src}$ & K-S (S:N) & K-S (Flux) & mJy & $\langle S_{\rm\it Spitzer}\rangle$\
$3.6\mu$m & $452$ & $71.5\%$ & $69.3\%$ & $1.32\pm0.78$ & $38$ & $437$ & $67.3\%$ & $61.8\%$ & $0.07\pm0.10$ & $2.0$\
$4.5\mu$m & $436$ & $76.9\%$ & $70.1\%$ & $1.58\pm0.79$ & $64$ & $421$ & $76.2\%$ & $85.2\%$ & $0.09\pm0.10$ & $3.4$\
$5.8\mu$m & $154$ & $99.72\%$ & $99.80\%$ & $4.19\pm1.33$ & $95$ & $141$ & $98.66\%$ & $99.2\%$ & $0.39\pm0.17$ & $8.8$\
$8\mu$m & $111$ & $99.80\%$ & $99.76\%$ & $5.51\pm1.57$ & $98$ & $98$ & $99.16\%$ & $99.65\%$ & $0.48\pm0.20$ & $9.4$\
$24\mu$m$^a$ & $83$ & $82.5\%$ & $84.6\%$ & $2.24\pm2.08$ & $9.3$ & $78$ & $98.42\%$ & $97.65\%$ & $0.30\pm0.24$ & $1.25$\
$24\mu$m$^b$ & $42$ & $88.3\%$ & $91.51\%$ & $4.35\pm2.92$ & $12.9$& $38$ & $97.87\%$ & $97.32\%$ & $0.55\pm0.35$ & $1.68$\
|
{
"pile_set_name": "ArXiv"
}
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$F_{\pi}^0$ (MeV) 87.2 93.3
--------------------------------------- -------------- --------------
$\epsilon_{\rm g}$ (GeV$^4$) $- 0.00565 $ $ - 0.0074 $
$\epsilon_{\rm g}$ (GeV$^4$) $- 0.00565 $ $ - 0.0074 $
$\langle \bar{G}^2 \rangle$ (GeV$^4$) $0.0226$ $0.0296 $
$\langle {G}^2 \rangle$ (GeV$^4$) $0.016$ $0.0215 $
$\chi_{\rm t}^{1/4}$ (MeV) [ ]{} [ ]{}
(NSVZ) $135$ $144.4 $
(HZ) 110.22 118
$m_{\eta'}$ (MeV) [ ]{} $ { } $
(NSVZ) $512$ $547.4 $
(HZ) $341.25$ $365.56 $
: Scheme A. ZME’s chiral QCD topology
$F_{\pi}^0$ (MeV) 87.2 93.3
--------------------------------------- -------------- -------------
$\epsilon_{\rm g}$ (GeV$^4$) $- 0.00688 $ $ - 0.009 $
$\langle \bar{G}^2 \rangle$ (GeV$^4$) $0.02752$ $0.036 $
$\langle {G}^2 \rangle$ (GeV$^4$) $0.02$ $0.026 $
$\chi_{\rm t}^{1/4}$ (MeV) [ ]{} [ ]{}
(NSVZ) $141.8$ $151.16 $
(HZ) 115.78 123.8
$m_{\eta'}$ (MeV) [ ]{} $ { } $
(NSVZ) $564.8$ $599.88 $
(HZ) $376.5$ $402.4 $
: Scheme B. ZME’s chiral QCD topology
[ ]{} RILM
--------------------------------- ------------ ------------ ------------
$n$ (fm$^{-4}$) 1.0 1.3 1.6
$\epsilon_{\rm I}$ (GeV$^4$) $-0.00417$ $-0.00542$ $-0.00667$
$\langle G^2 \rangle$ (GeV$^4$) 0.01213 0.01577 0.01941
$\chi_{\rm t}^{1/4}$ (MeV) [ ]{}
(NSVZ) 125.1 133.6 140.7
(HZ) 102.2 109.1 114.9
$m_{\eta'}$ (MeV) [ ]{} [ ]{} [ ]{}
(NSVZ) 411.0 468.6 519.8
(HZ) 274.0 312.4 346.6
: Instanton’s chiral QCD topology
$F^0_{\pi} = 93.3 \ MeV$ ZME + RILM
----------------------------------- ------------ ------------ ------------
$\epsilon_{\rm t}$ (GeV$^4$) $0.01157 $ $0.01282 $ $0.01407 $
$\langle {G}^2 \rangle$ (GeV$^4$) 0.0336 0.0373 0.04
$\chi_{\rm t}^{1/4}$ (MeV) [ ]{}
(NSVZ) 161.5 165.7 169.6
(HZ) 131.8 135.3 138.45
$m_{\eta'}$ (MeV) [ ]{} [ ]{} [ ]{}
(NSVZ) 709 720.8 755.2
(HZ) 456 480.6 503.24
: Scheme A. Chiral QCD topology
$F^0_{\pi} = 93.3 \ MeV$ ZME + RILM
----------------------------------- ------------ ------------ ------------
$\epsilon_{\rm t}$ (GeV$^4$) $0.01317 $ $0.01442 $ $0.01567 $
$\langle {G}^2 \rangle$ (GeV$^4$) 0.038 0.042 0.0456
$\chi_{\rm t}^{1/4}$ (MeV) [ ]{}
(NSVZ) 166.8 170.6 174.2
(HZ) 136.2 139.3 142.2
$m_{\eta'}$ (MeV) [ ]{} [ ]{} [ ]{}
(NSVZ) 730.4 764.1 796.7
(HZ) 487 509.4 530.8
: Scheme B. Chiral QCD topology
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We investigate an array of identical phase oscillators non-locally coupled without time delay, and find that chimera state with two coherent clusters exists which is only reported in delay-coupled systems previously. Moreover, we find that the chimera state is not stationary for any finite number of oscillators. The existence of the two-cluster chimera state and its time-dependent behaviors for finite number of oscillators are confirmed by the theoretical analysis based on the self-consistency treatment and the Ott-Antonsen ansatz.'
address:
- '1 School of Science, Beijing University of Posts and Telecommunications, Beijing, 100876, People’s Republic of China'
- '2 Physics Department, Beijing Normal University, Beijing, 100875, People’s Republic of China'
author:
- 'Yun Zhu$^1$, Yuting Li$^1$, Mei Zhang$^2$, Junzhong Yang$^1$'
title: 'The oscillating two-cluster chimera state in non-locally coupled phase oscillators'
---
Introduction
============
An array of identical oscillators has been used to model a wide range of systems, such as neural networks, convecting fluids, laser arrays and coupled biochemical oscillators. These systems exhibit rich collective behaviors including synchrony and spatiotemporal chaos [@1; @2; @3; @4]. Most of the earlier theoretical works on these systems assume either local coupling (nearest-neighbor interactions) or global coupling (infinite-range interactions); a third type named non-locally coupling began to be explored in the past years, which is somewhere between local coupling and global coupling. In non-local coupled systems, the oscillators interact with all others and the strength between oscillators varies with the distance between them.
Chimera state is a spatiotemporal pattern in which some of the identical oscillators are coherent and synchronous while others remain incoherent [@5; @6; @7; @9; @10]. They usually appear in systems with non-local coupling and could only be built for proper initial conditions [@15]. Chimera state does not relate to the partially synchronized states observed in populations of nonidentical oscillators with dispersive frequencies in which the splitting of the population roots in the inhomogeneity of the oscillator themselves and the intrinsically fastest or slowest oscillators remain desynchronized. Its emergence cannot be ascribed to a supercritical instability of the spatially uniform oscillation, because it occurs even if the uniform state is stable. Chimera may be a paradigm to study unihemispheric sleep in neurology which states a fact that many creatures sleep with only half of their brain while the other half is still active at the same time. [@mat06].
In the year 2002, Chimera state was first reported by Kuramoto and his colleagues [@5; @6] when simulating the non-locally coupled complex Ginzburg-Landau equation. They showed that identical oscillators with non-locally symmetrical coupling could self-organize into chimera states. Soon, spiral wave chimera [@13; @14] was discovered in two-dimensional arrays of non-locally coupled oscillators. In succession, Abrams and Strogatz [@7] found an exact solution for this state in a ring of phase oscillators coupled by a cosine kernel. Recently, two interesting findings on chimera state are reported. Firstly, in the study of the non-locally coupled oscillators with time delay [@11; @12], clustered chimera state that has spatially distributed phase coherence separated by incoherence with adjacent coherent regions in antiphase, was found. The observed clustered chimera state in these systems is stationary and its pattern does not change with time. Secondly, Abrams and Strogatz [@8] found a breathing chimera state in a model consisting of two interacting subpopulations of oscillators. Pikovsky and Rosenblum [@16] considered oscillators ensembles consisting of several subpopulations of identical units, with a general heterogeneous coupling between subpopulations, through which they acquired quasiperiodic chimera states. Laing [@17; @19] summarized chimera states in several heterogeneous networks of coupled phase oscillators, in the mean time, he analyzed chimera state applying the Ott-Antonsen ansatz [@18] in one-dimensional and two-dimensional systems. He pointed out that, in one-dimensional system, when parameter heterogeneity is introduced, a breathing chimera state exists.
In this work, we study a one-dimensional array of non-locally coupled identical phase oscillators. We find that, in the absence of time delay, a two-cluster chimera state could exist. We also find that, in the absence of parameter heterogeneity, the two-cluster chimera state is not stationary but oscillating. Different from Laing’s results [@18], we find that the oscillation of the two-cluster chimera state only exists for the system with a finite number of oscillators. Both the two-clustered chimera state and finite size oscillations of the chimera state in this model are analyzed based on the Ott-Antonsen ansatz.
Model
=====
The array of non-locally coupled phase oscillators can be described in a concise form as $$\begin{aligned}
\label{eq:1}
\frac{\partial\phi}{\partial
t}=\omega-\int_{-k}^kG(x-x^{\prime})\sin[\phi(x,t)-\phi(x^\prime,t)+\alpha]dx^\prime.\end{aligned}$$ Here, $\phi(x,t)$ is the phase of the oscillator at position $x$ at time $t$. The space variable $x$ is in the range $[-k,k]$ $(0<k\leq\pi)$. The periodic boundary condition is imposed for $k=\pi$, otherwise the no-flux boundary condition is imposed on the system. $\omega$ is the natural frequency (same for all oscillators), which plays no role in the dynamics. Without losing generality, we can set $\omega=0$. The angle $\alpha$ $(0\leq\alpha\leq\frac{\pi}{2})$ is a tunable parameter. The kernel $G(x-x^\prime)$ provides non-local coupling between oscillators. $G(x)$ is non-negative, even, decreasing with $|x|$ along the array, and normalized to have unit integral. Following Abrams and Strogatz [@7], we make use of the cosine kernel $$\begin{aligned}
\label{eq:kernel}
G(x)=\frac{1}{2(k+A\sin k)}(1+A\cos x)\end{aligned}$$ where $0\leq A\leq 1$. When $k=\pi$, Abrams and Strogatz found a chimera state with only one coherent cluster [@7]. However, we find a novel chimera state which has two coherent clusters and is oscillating for any finite number of oscillators. As mentioned above, clustered chimera state is only found in the time-delay coupled systems [@11; @12] and oscillating chimera state is found in the systems with subpopulations [@8; @16] or the systems with parameter heterogeneity \[i.e., $\alpha=\alpha(x)$\] [@17].
Let $\Omega$ denotes the angular frequency of a rotating frame whose dynamics are simplified as much as possible, and let $\theta=\phi-\Omega t$ denotes the phase of an oscillator relative to this frame. The key idea behind the analysis of chimera stats is the introduction of a mean-field-like quantity, namely, a complex order parameter $Re^{i\Theta}$ [@5; @6; @7] which is defined as $$\begin{aligned}
\label{eq:3}
R(x,t)e^{i\Theta(x,t)}=\int_{-k}^{k}G(x-x^\prime)e^{i\theta(x^\prime,t)}dx^\prime\end{aligned}$$ Then Eq (1) becomes $$\begin{aligned}
\label{eq:4}
\frac{\partial\theta}{\partial
t}=\omega-\Omega-R\sin(\theta-\Theta+\alpha).\end{aligned}$$
For stationary state, $R$ and $\Theta$ are time-independent and only depend on space variable $x$. Let $\Delta=\omega-\Omega$ where $\Omega$ is the angular velocity for the oscillators in coherent regions, the oscillators with $\Delta\leq R$ are in coherent regions and are phase-locked to $\theta=\arcsin(\frac{\Delta}{R})+\Theta-\alpha$ [@5; @6; @7].
Simulate and results
====================
 (a) Phase pattern for two-cluster chimera state when the steady state is reached. Eq. (1) is integrated using the Runge-Kutta method with fixed time step $dt=0.1$ with oscillator number $N=256$, $k=\pi$, $\beta=0.10$ and $A=0.995$. (b) The triangle (green) symbol is the modulus $R$ of the complex order parameter, the square (black) symbol is the distribution of $\langle \dot{\theta}(x)\rangle$ of individual oscillators averaged over 200 time units. The circle (red) symbol is the fluctuation $\sigma^2(x)$ of $\dot\theta(x)$. (c) three types of distribution of the phase $\Theta$ of the order parameter at different time. (d), (e) and (f) show the contour graphs of $R(x)$, $\Theta(x)/\pi$ and $\theta(x)/\pi$, respectively. The horizontal axis is position $x$ and vertical axis is time $t$. ](Graph1.eps){width="6.0in"}
For parameters $\beta=\frac{\pi}{2}-\alpha=0.10$, $A=0.995$, $k=\pi$, the system with $N=256$ phase oscillators could evolve to a two-cluster chimera state under the initial conditions as follows: $$\begin{aligned}
\label{intial condition}
\phi(x,0)=\left\{
\begin{array}{l l}
2\pi re^{-2.76x^2}, & x\leq 0\\
2\pi re^{-2.76x^2}+\pi, & x>0
\end{array}\right.\end{aligned}$$ where $r$ is a random variable from a uniform distribution of $[-0.5,0.5]$. As shown in Fig. 1(a), there exist two coherent clusters in which all oscillators are synchronized. Oscillators in the same cluster are nearly in phase yet in different clusters are in antiphase. On the other hand, the oscillators between the two coherent clusters are de-synchronized and their phases are randomly distributed in $[-\pi,\pi]$.
 (a) Phase pattern for two-cluster chimera. The parameters $N=256$, $k=0.8\pi$, $\beta=0.10$ and $A=0.995$. (b) The triangle (green) symbol is the modulus $R$ of the complex order parameter, the square (black) symbol is the distribution of $\langle \dot{\theta}(x)\rangle$ of individual oscillators averaged over 200 time units. The circle (red) symbol is the fluctuation $\sigma^2$ of $\dot\theta(x)$. (c) the distribution of the phase $\Theta$ of order parameter. (d), (e) and (f) show the contour graphs of $R(x)$, $\Theta(x)/\pi$ and $\theta(x)/\pi$, respectively.](Graph2.eps){width="6.0in"}
Then we consider three quantities characterizing a chimera state: the modulus $R$ of the complex order parameter at an arbitrary time, the angular velocity $\langle \dot{\theta}(x)\rangle$ averaged in a time interval of 200 units, and the fluctuation of the instantaneous angular velocity which is defined as $\sigma(x)=\sqrt{\langle (\dot{\theta}(x)-\langle
\dot{\theta}(x)\rangle)^2\rangle}$. The quantities against the locations of oscillators are presented in Fig. 1(b). Clearly, there are two plateaus on the curve of $\langle
\dot{\theta}(x)\rangle$ which refer to the coherent clusters in the chimera state. The zero $\sigma$ in the coherent clusters means that the oscillators in the coherent clusters all move on the same instantaneous angular velocity. Further, nonzero $\sigma$ outside the coherent clusters refers to the fluctuation of angular velocities for the oscillators outside the coherent clusters and indicates desynchronization. Fig. 1(b) reveals two features on $R$ for the two-cluster chimera state. Firstly, the oscillators can be divided into two domains which join at the minimum of $R$ and the curve of $R$ against the locations of oscillators does not show symmetry about the minimum of $R$. Further explorations show that $R$ is a function of time. As shown in Fig. 1(d) where the spatiotemporal evolution of $R$ is featured, $R$ is oscillating in each coherent cluster. Especially, when $R$ reaches its minimum in one domain, $R$ in the other one reaches its maximum. Secondly, the boundaries of the coherent clusters are not determined by the condition of $\Delta=R(x)$ and the coherent regimes are narrower than those expected according to $\Delta=R(x)$ in most of the time, which are different from the stationary chimera state. The reason for this observation roots in the time-dependent order parameter $R$. Actually, for a forced phase oscillator obeying $\dot{\theta}=\omega+R(t)\sin\theta$, the synchronization of the phase oscillator by the force just requires the phase of the oscillator to be confined within $(0,2\pi)$ but not to a fixed value, which leads the onset of the synchronization of the oscillator not to obey the condition of $\Delta=R(t)$ and the onset of synchronization strongly depends on the details of the functional form of $R(t)$. To be noted, even though $R$ is time-dependent, the oscillators in the coherent clusters still have the same angular velocity which does not fluctuate as exhibited by zero $\sigma$. The spatiotemporal evolution of $R$ in Fig. 1(d) shows another feature: the two-cluster chimera state displays an irregular motion along the ring, for example, the locations of coherent clusters vary with time. Similar phenomenon is also observed for the chimera state with a single cluster [@omel2010]. Furthermore, the snapshot and the time evolution of $\Theta$ presented in Fig. 1(c) and (e) show that $\Theta$ is almost uniform in each domain except for those near the junction between domains and there is a phase difference of $\pi$ for $\Theta$ in different domains. To be stressed, the features on $\Theta$ revealed by Fig. 1(c) and (e) could be used as a more general scheme for the initial conditions to generate an oscillating two-cluster chimera state. That is, the initial conditions in Eq. \[intial condition\] could be changed to be $\phi(x,0)=\pi$ for $x\leq 0$ and $\phi(x,0)=0$ for $x>0$. To get a better illustration, we present the evolution of $\theta(x)$ in Fig. 1(f) which shows that, resulting from the oscillation of $R$, the territories of the coherent clusters alter with time. Furthermore, it should be pointed out that, though a two-cluster chimera state could be generated for the system characterized by Eq. \[eq:1\] in the absence of time delay under proper initial conditions, we have not detected other chimera states with more than two coherent clusters.
![\[Rmax\] (color online) The maxima of $R$ in one domain varies with time for different number of oscillators. (a, b, c, d) are for $N=128$, $256$, $512$ and $1024$, respectively. Other parameters: $A=0.995$, $\beta=0.1$ and $k=0.8\pi$. The (red) lines denote the growth of $R$ in the transient and the upper or lower bound of the maxima of $R$ in the steady state. The inset in (d) shows the amplitude of the oscillation of $R_{max}$, $\Delta
R_{max}$, against the system size $N$, which indicates a power law of $\Delta R_{max}\sim N^{-0.5}$.](Graph3.eps){width="5.0in"}
The two-cluster oscillating chimera state can exist when $k\neq\pi$. In comparison with the case with $k=\pi$ where oscillators locate on a ring, here the oscillators locate on a chain. We take $k=0.8\pi$ as an example. The results are given in Fig. \[fig2\]. The differences with those in Fig. 1 are that the pattern of two-cluster chimera state becomes stationary in space and, no matter where the chimera state is initialized, it will adjust its pattern to be symmetrical about the center of the chain where the minimum of $R$ appears.
The analysis above are made on the system with $N=256$. One question is how the observed two-cluster oscillating chimera state depends on the number of oscillators. For this aim, we focus on one domain and record the maximum value of $R$ in this domain at any time instance (we denote it as $R_{max}$). The time-dependent behavior of the two-cluster chimera state can be reflected by the time evolution of $R_{max}$. For example, a constant $R_{max}$ indicates a stationary chimera state, otherwise an oscillating one. To avoid the influences induced by the irregular motion of the chimera state along the ring in the system with periodic boundary condition, we let $k=0.8\pi$. The results for different $N$ are presented in Fig. \[Rmax\]. One remarkable feature revealed by the figure is that, before the oscillating chimera state is established, the system first evolves to a two-cluster chimera state which looks like a stationary one due to the weak oscillation of $R_{max}$. However, the “stationary” state is not stable and its instability leads to the appearance of an oscillating chimera state. Interestingly, the stationary value of $R_{max}$ is independent of the size of the system and the time consumed for the system to build an oscillating state becomes much longer as the number of the oscillator $N$ increases. Another feature in Fig.\[Rmax\] is that larger $N$ seems to weaken the oscillation of $R_{max}$ in the two-cluster oscillating chimera state, which is prominent by the comparison between Fig. \[Rmax\] (a) ($N=128$) and (b) ($N=256$). Since the transient time to build an oscillating two-cluster chimera state for large $N$ becomes extremely long, we just give a rough estimate based on the data presented in the inset of Fig. \[Rmax\](d) that the oscillation amplitude $\Delta R_{max}\sim N^{-0.5}$, which means that the two-cluster oscillating chimera state becomes a stationary one in the thermodynamic limit, which is different from the Laing’s results [@18].
Analysis
========
 The modulus $R$ and the phase $\Theta$ of the order parameter and $\Delta$ by solving of Eq. (\[KBFInal\]) via an iterative scheme as presented in the context. (a) The square (black) symbol denotes the distribution of $R$ and the circle (red) the value of $\Delta$. (b) Distribution of $\Theta$. Other parameters: $A=0.995$, $\beta=0.1$ and $k=\pi$. ](Graph4.eps){width="4in"}
The above results can be understood theoretically. First, the two-cluster stationary chimera state in the thermodynamic limit can be explained in terms of Kuramoto-Battogtokh self-consistency equation[@5] as follows: $$\begin{aligned}
\label{KB}
R(x)e^{i\Theta(x)}&=&e^{i\beta}\int_{-\pi}^{\pi}G(x-x^\prime)e^{i\Theta(x^\prime)}
\times\frac{\Delta-\sqrt{\Delta^2-R(x^\prime)^2}}{R(x^\prime)}dx^\prime\end{aligned}$$ note that there are three unknown quantities (the real-valued functions $R(x)$, $\Theta(x)$ and the real number $\Delta$) in terms of the assumed choices of $\beta$ and the kernel $G(x-x^\prime)$. We take $k=\pi$ as an example. Considering the patterns in Figs. 1(b) and (c), the modulus $R$ and the phase $\Theta$ of the order parameter approximately satisfy $$\begin{aligned}
\label{symmetry}
\left\{
\begin{array}{l l}
R(x+\pi)=R(x)\\\Theta(x+\pi)=\Theta(x)+\pi.
\end{array}\right .\end{aligned}$$ We substitute Eqs. (\[symmetry\]) and (\[eq:kernel\]) into Eq. (\[KB\]) and get $$\begin{aligned}
R(x)e^{i\Theta(x)}=e^{i\beta}\int_{-\pi}^0\frac{1}{2\pi}[1+A\cos(x-
x^\prime)e^{i\Theta(x^\prime)}H(x^\prime)dx^\prime\end{aligned}$$ $$\begin{aligned}
\label{KB2}
+e^{i\beta}\int_{0}^{\pi}\frac{1}{2\pi}[1+A\cos(x-
x^\prime)]e^{i\Theta(x^\prime)}H(x^\prime)dx^\prime,\end{aligned}$$ where $H(x)=(\triangle-\sqrt{\Delta^2-R(x)^2})/R(x)$. Under the transformation $x^\prime-\pi\rightarrow x^\prime$ in the second term of the right hand side of Eq. (\[KB2\]), the self-consistency equation changes into $$\begin{aligned}
\label{KBFInal}
R(x)e^{i\Theta(x)}=e^{i\beta}\int_{-\pi}^0\frac{A\cos(x-x^\prime)}{\pi}
e^{i\Theta(x^\prime)}H(x^\prime)dx^\prime.\end{aligned}$$
 Phase pattern for two-cluster chimera when the steady state is reached. Eq. (\[eq:12\]) and (\[eq:13\]) is integrated using the Runge-Kutta method with fixed time step $dt=0.005$ and the oscillator number $N=256$. (a, c) are the modulus $R$ and the phase $\Theta/\pi$ of the complex order parameter with $k=\pi$. (b, d) are the modulus $R$ and the phase $\Theta/\pi$ of the complex order parameter with $k=0.8\pi$. Other parameters: $\beta=0.1$, $A=0.995$. ](Graph5.eps){width="5.0in"}
To solve Eq. (\[KBFInal\]), we first determinate the value of $\Delta$. Because Eq. (\[KBFInal\]) is left unchanged by any rigid rotation $\Theta(x)\rightarrow\Theta(x)+\Theta_0$, we can specify the value of $\Theta(x)$ at any point $x$ we like. We set $\Theta(\frac{\pi}{2})=0$. Now we can get $\Delta$. Then we take $R(x)$ and $\Theta(x)$ obtained from the dynamical simulations as initial guesses and use an iterative scheme to determinate $R(x)$ and $\Theta(x)$ in function space, behind which the idea is that the current estimates of $R(x)$ and $\Theta(x)$ can be entered into the right-hand side of (\[KBFInal\]), and used to generate the new estimates appearing on the left-hand side. Figure \[tukb\] shows the results obtained from Eqs.(9) and (2). To be stressed, without the requirement of Eq.(7), the self-consistency equation for any finite system always yields to a one-cluster chimera state, which also evidences that the stationary two-cluster chimera state here is not stable.
From Fig. \[tukb\], we could notice the order parameter of stationary state, which has a little difference from Fig. \[pi\](b) and (c) in the vicinity of the junction between two domains. The transition of the former case performances like a step function while the latter one is continuous. The difference probably originates from the finite size effect in Fig. \[pi\].
The oscillating characteristic could be interpreted with the assistance of the Ott-Antonsen ansatz [@17; @18]. Following the line in [@8; @16; @17], we assume that there is a probability density function $f(x,\omega,\theta,t)$ characterizing the state of the system. This function satisfies the continuity equation [@8; @16; @17; @19]
$$\begin{aligned}
\label{eq:5}
\frac{\partial f}{\partial
t}+\frac{\partial}{\partial\theta}(fv)=0\end{aligned}$$
where $$\begin{aligned}
\label{eq:6}
v=\omega-\int_{-k}^{k}G(x-x^{\prime})
\int_{-\infty}^{\infty}\int_{-\pi}^{\pi}\sin(\theta-\theta^{\prime}+\alpha)f(x^{\prime},\omega,\theta^{\prime},t)d\theta^{\prime}d\omega
dx^{\prime}\end{aligned}$$ with $\theta=\theta(x)$ and $\theta^{\prime}=\theta(x^{\prime})$. The complex order parameter can be formulated as $$\begin{aligned}
\label{eq:7}
Z\equiv Re^{i\Theta}=\int_{-k}^{k}G(x-x^{\prime})
\int_{-\infty}^{\infty}\int_{-\pi}^{\pi}e^{i\theta^{\prime}}f(x^{\prime},\omega,\theta^{\prime},t)d\theta^{\prime}d\omega
dx^{\prime}.\end{aligned}$$ In terms of complex order parameter $Z$, Eq. (\[eq:6\]) can be rewritten as $$\begin{aligned}
\label{eq:8}
v=\omega-\frac{1}{2}[Ze^{-i(\theta-\beta)}+Z^{*}e^{i(\theta-\beta)}]\end{aligned}$$ where $Z^{*}$ denotes the complex conjugate of $Z$ and $\beta=\frac{\pi}{2}-\alpha$. Following Ott and Antonsen [@17; @18], we have
 The boundary of the two-cluster chimera state plotted on $k-\beta$ parameter plane. Below the curves, the two-cluster chimera state is stable. The square (black) symbol and circle (red) symbol are plotted at the parameter values $A=0.995$ and $A=0.95$, respectively. ](Graph6.eps){width="4.0in"}
$$\begin{aligned}
\label{eq:9}
f(x,\omega,\theta,t)=\frac{g(\omega)}{2\pi}\{1+\sum_{n=1}^{\infty}[(a(x,\omega,t)e^{i\theta})^n+c.c.]\}\end{aligned}$$
where $c.c.$ is the complex conjugate of the previous term and $g(\omega)$ is the distribution of natural frequency. In this work, we assign all oscillators a same natural frequency ($\omega=0$), so $g(\omega)=\delta(\omega)$. Substituting Eqs.(\[eq:8\]) and (\[eq:9\]) into Eqs. (\[eq:5\]) and (\[eq:7\]), we obtain $$\begin{aligned}
\label{eq:10}
\frac{\partial a(x,\omega,t)}{\partial
t}=\frac{i}{2}[Z^{*}e^{-i\beta}+Ze^{i\beta}a^2]\end{aligned}$$ $$\begin{aligned}
\label{eq:11}
Z=\int_{-k}^{k}G(x-x^\prime)\int_{-\infty}^{\infty}g(\omega)a^*(x,\omega,t)d\omega
dx^{\prime}.\end{aligned}$$ Letting $\hat{a}(x,t)=a(x,0,t)$, we have $$\begin{aligned}
\label{eq:12}
\frac{\partial \hat{a}(x,t)}{\partial
t}=\frac{i}{2}[Z^{*}e^{-i\beta}+Ze^{i\beta}\hat{a}^2]\end{aligned}$$ $$\begin{aligned}
\label{eq:13}
Z=\int_{-k}^{k}G(x-x^\prime)\hat{a}^*(x^\prime,t)dx^{\prime}.\end{aligned}$$
 Two-cluster chimera under the condition of different types of nonlocal coupling kernel $G(x)$. The top panels are the phase patterns for two-cluster chimera and the bottom are the modulus $R$ (black) and the phase $\Theta$ (red) of the order parameters. (a, b) is computed under the assumption that $G(x)$ takes exponential form with parameters: $A=4$, $k=1$ and $\beta=0.10$ while $G(x)$ in (c, d) is steplike function with parameters: $d=0.5$, $k=1$ and $\beta=0.10$. All above take no-flux boundary condition and the initial conditions are as following: $\phi(x,0)=0$ for $x<0$ and $\phi(x,0)=\pi$ for $x>0$](Graph7.eps){width="6.0in"}
By numerically simulating these two equations, we have the time evolutions of $R(x)$ and $\Theta (x)$. The results for $k=\pi$ and $k=0.8\pi$ are presented in Fig. \[Fig3\], respectively. Clearly, the main features in Figs. (1) and (2), such as the oscillating nature of the two-cluster chimera state, the movement of the pattern (or the frozen pattern) of $R$ and $\Theta$ for $k=\pi$ (or for $k=0.8\pi$), and the uniform distribution of $\Theta (x)$ in different domains, are reproduced in Fig. \[Fig3\].
Using Eqs. (\[eq:12\]) and (\[eq:13\]), we may probe into the regime for the existence of the two-cluster chimera state on $k-\beta$ parameter plane. The results are presented in Fig. \[Fig4\] at $A=0.995$ and $A=0.95$. As shown in this plot, the two-cluster chimera state is not favorable at large $\beta$, and either small $k$ or large $k$ tends to be harmful for the two-cluster chimera state. And the smaller $A$ is, the smaller the domain of two-clustered chimera exists. if $A<0.8$, the domain does not exist any longer. Beyond the stable regime for the two-cluster chimera, i.e., above the curves in Fig. \[Fig4\], the initial two-cluster chimera state tends to become a normal chimera state with only one coherent cluster. The two-cluster oscillating chimera state may be detected in coupled oscillators with other types of non-locally coupling kernel $G(x)$ which could be exemplified by two cases. In the first one, $G(x)$ takes an exponentially decaying function, $G_{exp}(x)=\frac{Ae^{-A|x|}}{2(1-e^{-Ak})}$. In the other case, $G(x)$ takes a steplike function, $G_{step}(x,d)=\frac{1}{2d}$ for $|x|\leq d$ and $G_{step}(x,d)=0$ for $|x|>d$. Figure 7 shows the results for the systems with these two types of non-locally coupling and, clearly, the two-cluster oscillating chimera states are reproduced.
Conclusion
==========
In summary, we study a one-dimensional system consisting of non-locally coupled phase oscillators, which is a prototype for studying chimera states. By numerically simulating this simplest system, we find the existence of a two-cluster oscillating chimera state in the absence of time delay coupling and parameter heterogeneity. The numerical results are confirmed by the theoretical analysis based on the self-consistency treatment and the Ott-Antonsen ansatz.
Acknowledgments {#acknowledgments .unnumbered}
===============
The work was supported by National Natural Science Foundation of China under Grant No. 90921015 and No. 10775022.
References {#references .unnumbered}
==========
Y. Braiman, J. F. Lindener and W. L. Ditto, Nature **378**, 465 (1995). L. Kocarev and U. Parlitz, Phys. Rev. Lett. **77**, 2206 (1996). H. G. Winful and L. Rahman, Phys. Rev. Lett. **65**, 1575 (1990). L. Kocarev, U. Parlitz, T. Stojanovski and P. Janjic, Phys. Rev. E **56**, 1238 (1997). Y. Kuramoto and D. Battogtkh, Nonlinear Phenomena in Complex Systems **5**, 380 (2002). D. Tanaka and Y. Kuramoto, Phys. Rev. E **68**, 026219 (2003). D. M. Abrams and S. H. Strogatz, Phys. Rev. Lett. **93**, 174102 (2004). Y. Kawamura, Phys. Rev. E **75**, 056204 (2007). O. E. Omel’chenko, Y. L. Maistrenko and P. A. Tass, Phys. Rev. Lett. **100**, 044105 (2008). A. E. Motter, Nat. Phys. **6**, 164 (2010) C. G. Mathews, J. A. Lesku, S. L. Lima, and C. J. Amlaner, Ethology 112, 286 (2006). S. I. Shima and Y. Kuramoto, Phys. Rev. E **69**, 036213 (2004). E. A. Martens, C. R. Laing and S. H. Strogatz, Phys. Rev. Lett. **104**, 044101 (2010). J. H. Sheeba, V. K. Chandrasekar and M. Lakshmanan, Phys. Rev. E **81**, 046203 (2010). G. C. Sethia, A. Sen and F. M. Atay, Phys. Rev. Lett. **100**, 144102 (2008). D. M. Abrams, R. Mirollo, S. H. Strogatz and D. A. Wiley, Phys. Rev. Lett. **101**, 084103 (2008). A. Pikovsky and M. Rosenblum, Phys. Rev. Lett. **101**, 264103 (2008). C. R. Laing, Physica D **238**, 1569 (2009). C. R. Laing, Chaos **19**, 013113 (2009). E. Ott and T. M. Antonsen, Chaos **18**, 037113 (2009). O. E. Omel’chenko, M. Wolfrum, and Y. L. Maistrenko, Phys. Rev. E 81, 065201 (2010).
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{
"pile_set_name": "ArXiv"
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Meson Exchange Currents in $(e,e'p)$ recoil polarization observables
[F. Kazemi Tabatabaei$^1$, J.E. Amaro$^1$ and J.A. Caballero$^2$]{}
$^1$ [*Departamento de Física Moderna, Universidad de Granada, Granada 18071, Spain*]{}\
$^2$ [*Departamento de Física Atómica, Molecular y Nuclear\
Universidad de Sevilla, Apdo. 1065, Sevilla 41080, Spain* ]{}\
1.5cm
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[**Abstract**]{}
A study of the effects of meson-exchange currents and isobar configurations in $A(\vec{e},e'\vec{p})B$ reactions is presented. We use a distorted wave impulse approximation (DWIA) model where final-state interactions are treated through a phenomenological optical potential. The model includes relativistic corrections in the kinematics and in the electromagnetic one- and two-body currents. The full set of polarized response functions is analyzed, as well as the transferred polarization asymmetry. Results are presented for proton knock-out from closed-shell nuclei, for moderate to high momentum transfer.
------------------------------------------------------------------------
[*PACS:*]{} 25.30.Fj; 24.10.Eq; 24.70.+s 24.10.Jv\
[*Keywords:*]{} electromagnetic nucleon knockout; polarized beam; nucleon recoil polarization; final state interactions; meson exchange currents; structure response functions.
Introduction
============
For the last decades coincidence (e,e$'$p) reactions on complex nuclei have provided precise information on bound nucleon properties, which have made it possible to test carefully the validity of present nuclear models [@Fru84; @Bof93; @Kel96; @Bof96]. Although the analysis of these processes, making use of different distorted wave approaches and coupled-channel models, has been extremely useful, there are still uncertainties associated to the various ingredients that enter in the description of the reaction mechanism: treatment of final-state interactions (FSI), nuclear correlations, off-shell effects, Coulomb distortion of the electrons, relativistic degrees of freedom, meson-exchange currents (MEC), etc. All of these ingredients affect the evaluation of the differential cross section and hence lead to ambiguities in the extraction of the spectroscopic factors. The origin of this uncertainty is directly connected with the complexity of the dynamics of the reaction and the different approaches to handle it, which produce different cross sections. It is clear that a reliable determination of spectrospic factors requires an accurate description of the reaction mechanism. Important efforts in this direction have been done in recent works [@Yim94; @Gai00; @Udi93; @Udi01; @Kel94].
The measurement of the separate nuclear response functions and asymmetries imposes additional restrictions over the theory. The exclusive response functions, which include different components of the hadronic tensor taken along the longitudinal (L) or transverse (T) directions with respect to the momentum transfer ${{\bf q}}$, may present very different sensitivities to the different aspects of the reaction. In this sense, it is interesting to point out that MEC are shown to contribute mainly to the transverse components [@Ama93; @Ama94; @MEC], while relativistic degrees of freedom play a crucial role in the interference TL response [@Udi01; @Udi99]. Thus, a joint analysis of cross sections and response functions, comparing the experimental data with the theoretical predictions, can provide very relevant and complementary information on the reaction mechanism. Separate response functions and the TL asymmetry have been measured for $^{16}$O(e,e$'$p) at moderate [@Chi91; @Spa93] and high [@Gao00] $q$-values. The asymmetry $A_{TL}$, obtained from the difference of cross sections measured at opposite azimuthal angles (with respect to ${{\bf q}}$) divided by the sum, results particularly relevant because it does not depend on the spectroscopic factors. For high missing momentum $p\ge 300$ MeV/c, $A_{TL}$ presents an oscillatory structure that has been shown to be consistent with predictions of ‘dynamical’ relativistic calculations [@Udi99; @Kel99b; @Deb00; @Cab01].
The advent of longitudinally polarized beams [@Man94] and recoil polarization measurements [@Woo98; @Mal00] has importantly enlarged the number of observables which can be accessible with this type of experiments, a fact that is welcome to challenge the theory strongly. In recent experiments carried out at MIT-Bates and Jefferson Lab, the induced (${{\bf P}}$) and transferred (${{\bf P}}'$) polarization asymmetries were measured for complex nuclei, $^{12}$C [@Woo98] and $^{16}$O [@Mal00], respectively. In both cases $(q,\omega)$-constant kinematics has been selected with $q\approx 760$ MeV/c, $\omega\approx 290$ MeV at MIT-Bates and $q\approx 1000$ MeV/c, $\omega\approx 450$ MeV at TJlab. Since the transfer momentum values are high enough, relativistic degrees of freedom should be incorporated in a consistent description of these reactions. After the pioneering work in [@Pic87; @Pic89], a detailed study on the induced normal polarization $P_n$ has been presented in [@Joh99; @Udi00] within the framework of the relativistic distorted wave impulse approximation (RDWIA). A comparison with non-relativistic analyses was also discussed. The sensitivity of polarized observables to channel coupling in final state interactions was analyzed in [@Kel96; @Kel99], while in [@Kel99b] the study was focussed on the effects of spinor distortion over the transfer polarization ratio $P'_x/P'_z$. In [@Ito97] the whole eighteen recoil nucleon polarized responses were computed from intermediate to high momentum transfer in the Dirac eikonal formalism. A comparison between the predictions of the Glauber and eikonal models for $P_n$ was presented in [@Deb02] with the aim of bridge the gap between the low- and high-energy description of FSI. More recently a theoretical study of kinematical and dynamical relativistic effects over polarized response functions and polarization asymmetries has been performed in [@Mar02a; @Mar02b] within the relativistic plane-wave impulse approximation (RPWIA). A general analysis of all the polarized observables within the RDWIA is at present in progress [@Cris03].
Our main aim in this work is to explore in depth the role played by the two-body currents in recoil nucleon polarization observables. Some previous analyses on this subject have been done by the Pavia group [@Bof90; @Bof91] and the Gent group [@Ryc01; @Ryc99]. The calculation of MEC in [@Bof90; @Bof91] makes use of an effective one-body operator leading to results which, in the unpolarized case, differ significantly from those obtained with other approaches that describe properly the two-body currents [@Slu94; @Ama99]. Recently the unpolarized model of [@Bof91] has been improved in [@Giu02], but differences with other MEC calculations still persist [@Ama03b]. In [@Ryc99] the induced and transferred polarization asymmetries $P_n$, $P'_l$ and $P'_t$ were evaluated for different kinematical situations. The model considered did not rely on any empirical input with respect to the FSI, describing the bound and scattering states as the solutions of the Schrödinger equation with a mean field potential obtained from a Hartree-Fock calculation. MEC were included based on the formalism developed in [@Slu94] which also differs from the MEC analysis performed in [@Ama99; @Ama03b]. In addition, in [@Ryc99] results for high momentum transfer (up to $q=1$ GeV/c) were evaluated including relativistic corrections into the one-body current operator obtained through the Foldy-Wouthuysen method.
In this work we extend the DWIA+MEC model developed for unpolarized reactions in Refs. [@Ama99; @Ama03b] in order to describe the spin observables in $(\vec{e},e'\vec{p})$ processes from closed shell nuclei. This model takes care of relativistic degrees of freedom by making use of semi-relativistic (SR) operators for the one-body (OB) current [@Udi99; @Maz02; @Ama96a; @Ama96b] as well as for the two-body MEC [@Ama03b; @Ama98a; @Ama02c; @Ama03a]. The SR currents are obtained by a direct Pauli reduction of the corresponding relativistic operators by expanding only in missing momentum over the nucleon mass while treating the transferred energy and momentum exactly. Relativistic kinematics for the ejected nucleon is assumed throughout this work. Finally FSI are incorporated through a phenomenological optical potential which, for high momentum transfer, is taken as the Schrödinger-equivalent form of a S-V Dirac optical potential. The goal of this work is to use the SR approach to evaluate the importance of MEC effects upon the spin observables and their dependence on the FSI for intermediate to high momentum transfer. As a complete relativistic distorted wave analysis of MEC in (e,e$'$p) processes is still lacking —the only study in this direction has been performed taking into account only the contact current [@Meu02]— the use of the SR model becomes, as a starting point, a convenient way of implementing relativistic effects in existing non relativistic descriptions of the reaction mechanism in order to explore the high momentum region.
The paper is organized as follows: in Section 2 we outline the DWIA formalism describing in detail the multipole expansion done for the separate response functions. In Section 3 we present our results for the polarized response functions and transferred polarization asymmetries for selected kinematics near the quasielastic peak. Finally our conclusions are drawn in Secion 4.
DWIA model of $(\vec{e},e'\vec{p})$
====================================
Cross section and response functions
------------------------------------
The general formalism for coincidence electron scattering on nuclei involving polarization degrees of freedom has been presented in detail in Refs. [@Bof96; @Pic87; @Pic89; @Ras89]. In this section we simply provide the basic description of our DWIA model focusing on the development of the multipole expansion used to compute the response functions. For this end we follow closely the multipole formalism developed in [@Ama98b] for polarized nuclei.
We consider the process depicted in Fig. 1, in which an incident electron with four-momentum $K^\mu_e=(\epsilon_e,{{\bf k}}_e)$ and helicity $h$ interacts with a nucleus $A$, scatters through an angle $\theta_e$ to four-momentum $K'{}^\mu_e=(\epsilon',{{\bf k}}'_e)$ and is detected in coincidence with a nucleon with momentum ${{\bf p}}'$ and energy $E'$. The four-momentum transferred to the nucleus is $Q^\mu=K^\mu_e-K'{}^{\mu}_e=(\omega,{{\bf q}})$, verifying $Q^2=\omega^2-q^2<0$. The polarization of the final nucleon is measured along an arbitrary direction defined by the unitary vector ${\mbox{\boldmath{$s$}}}$. Assuming plane waves for the electrons and neglecting the nuclear recoil, the cross section can be written in the extreme relativistic limit (ERL) $m_e \ll \epsilon_e$, as [@Ras89] $$\frac{d\sigma}{d\epsilon'_e d\Omega'_e d{{\bf \hat{p}}}'}
= \Sigma + h \Delta \, ,
\label{cross-section}$$ where a separation has been made into terms involving polarized and unpolarized incident electrons. Using the general properties of the leptonic tensor it can be shown that both terms, $\Sigma$ and $\Delta$, have the following decompositions: $$\begin{aligned}
\Sigma
&=& K \sigma_M \left( v_LR^L+v_T
R^T+v_{TL}R^{TL}+v_{TT}R^{TT}\right),
\label{sigma}\\
\Delta
&=&
K \sigma_M \left( v_{TL'}R^{TL'}+v_{T'}R^{T'}\right) \, ,
\label{delta}\end{aligned}$$ $\sigma_M$ is the Mott cross section, the factor $K\equiv
m_Np'/(2\pi\hbar)^3$, with $m_N$ the nucleon mass, and the $v_\alpha$-coefficients are the usual electron kinematical factors[^1].
\#1\#2[1.0\#1]{}
The hadronic dynamics of the process is contained in the exclusive response functions $R^K$, which are given as $$\begin{aligned}
& R^L = W^{00},
& R^T = W^{xx}+W^{yy},
\label{rlrt}
\\
& R^{TL} = \sqrt{2}(W^{0x}+W^{x0}),
& R^{TT} = W^{yy}-W^{xx},
\\
& R^{T'} =i(W^{xy}-W^{yx}),
& R^{TL'} = i\sqrt{2}(W^{0y}+W^{y0}),
\label{rt'rtl'}\end{aligned}$$ with $W^{\mu\nu}$ the hadronic tensor $$\label{hadronic-tensor}
W^{\mu\nu}
=\frac{1}{K}
\sum_{M_B}
\langle {{\bf p}}'{\mbox{\boldmath{$s$}}},B| \hat{J}^{\mu}({{\bf q}})|A\rangle^*
\langle {{\bf p}}'{\mbox{\boldmath{$s$}}},B| \hat{J}^{\nu}({{\bf q}})|A\rangle$$ constructed from the matrix elements of the electromagnetic nuclear current operator $\hat{J}^{\mu}({{\bf q}})$ between the ground state of the target nucleus $|A\rangle$ (assumed to have zero total angular momentum), and the final hadronic states $|{{\bf p}}'{\mbox{\boldmath{$s$}}},B\rangle$. In what follows we assume the residual nucleus to be left in a bound state, hence its wave function can be written down in the form $|B\rangle=|J_B M_B\rangle$ with $J_B$ the total angular momentum. The state $|{{\bf p}}'{\mbox{\boldmath{$s$}}}\rangle$ represents the asymptotic distorted wave function of the ejected nucleon polarized along an arbitrary ${\mbox{\boldmath{$s$}}}$-direction, determined by the angles ($\theta_s,\phi_s)$ referred to the $xyz$ coordinate system of Fig. 1. It is given by $$|{{\bf p}}'{\mbox{\boldmath{$s$}}}\rangle =
\sum_{\nu=-1/2}^{1/2}{\cal D}_{\nu\frac12}^{(1/2)}(\theta_s,\phi_s,0)\
|{{\bf p}}' \nu\rangle \, ,
\label{rotation}$$ where $|{{\bf p}}' \nu\rangle$ is referred to the system with the quantization axis along ${{\bf q}}$ and the arguments of the rotation matrices are the Euler angles that specify the ${\mbox{\boldmath{$s$}}}$-direction.
Isolating the explicit dependences on the azimuthal angle of the ejected nucleon $\phi'=\phi$, the hadronic responses can be expressed in the form $$\begin{aligned}
R^L &=& W^L \label{RL}\\
R^T &=& W^T \\
R^{TL}&=& \cos\phi\ W^{TL} + \sin\phi\ \widetilde{W}^{TL}\\
R^{TT}&=& \cos2\phi\ W^{TT} + \sin2\phi\ \widetilde{W}^{TT}\\
R^{T'}&=& \widetilde{W}^{T'} \label{RT'}\\
R^{TL'}&=& \cos\phi\ \widetilde{W}^{TL'}+\sin\phi\ W^{TL'} \label{RTL'} \, ,\end{aligned}$$ where the functions $W^K$ and $\widetilde{W}^K$ are totally specified by four kinematical variables, for instance $\{E,\omega,q,\theta'\}$, and the polarization direction $\{\theta_s,\Delta\phi=\phi-\phi_s\}$. The responses with and without tilde refer to their dependence on the spin vector ${\mbox{\boldmath{$s$}}}$. As shown below, $\widetilde{W}^K$ are purely spin-vector, while $W^K$ present also a spin-scalar dependence, so only the latter survive when the polarization of the ejected nucleon is not measured.
In the case of $(\vec{e},e'\vec{N})$ processes, the hadronic response functions are usually given by referring the recoil nucleon polarization vector ${\mbox{\boldmath{$s$}}}$ to the baryocentric system defined by the axes (see Fig. 1): ${\mbox{\boldmath{$l$}}}$ (along the ${{\bf p}}'$ direction), ${\mbox{\boldmath{$n$}}}$ (normal direction to the plane defined by ${{\bf q}}$ and ${{\bf p}}'$, i.e., along ${{\bf q}}\times{{\bf p}}'$) and ${\mbox{\boldmath{$t$}}}$ (determined by ${\mbox{\boldmath{$n$}}}\times{\mbox{\boldmath{$l$}}}$)[^2]. It can be shown (see Refs. [@Pic87; @Pic89] for details) that a total of eighteen response functions enter in the analysis of $(\vec{e},e'\vec{N})$ reactions. These are given by the decomposition $$\begin{aligned}
W^K
&=&
\frac12 W_{unpol}^K + W_n^K s_n,
\kern 1cm K=L,T,TL,TT,TL',
\label{Wgeneral}
\\
\widetilde{W}^{K}
&=&
W_l^{K}s_l + W_t^{K}s_t, \kern 1cm K=TL,TT,T',TL' \, ,
\label{Wtgeneral}\end{aligned}$$ where, as mentioned above, only the $W_{unpol}^K$ responses survive within the unpolarized case. Moreover, $W_{unpol}^{TL'}$ (referred as fifth response) enters only when the polarization of the incident electron is measured.
Owing to the above decomposition, the response functions (\[RL\]–\[RTL’\]) can be expressed in the form $R^K=R^K_{unpol}/2 +{{\bf R}}^K\cdot{\mbox{\boldmath{$s$}}}$, and similarly, the cross section (\[cross-section\]–\[delta\]) can be written as a sum of unpolarized and spin-vector dependent terms $$\begin{aligned}
\frac{d\sigma}{d\epsilon'_e d\Omega'_e d{{\bf \hat{p}}}'}
&=&
\frac12\Sigma_{unpol} + {\mbox{\boldmath$\Sigma$}}\cdot{\mbox{\boldmath{$s$}}}+h\left( \frac12\Delta_{unpol} + {\mbox{\boldmath$\Delta$}}\cdot{\mbox{\boldmath{$s$}}}\right)
\\
&=&
\frac12\Sigma_{unpol}\left[1+{{\bf P}}\cdot{\mbox{\boldmath{$s$}}}+h( A + {{\bf P}}'\cdot{\mbox{\boldmath{$s$}}})\right] \, ,\end{aligned}$$ where the usual polarization asymmetries have been introduced [@Kel96]: $$\begin{aligned}
{{\bf P}}&=& {\mbox{\boldmath$\Sigma$}}/\left(\frac12\Sigma_{unpol}\right)
\kern 1cm \mbox{Induced polarization},
\\
{{\bf P}}' &=& {\mbox{\boldmath$\Delta$}}/\left(\frac12\Sigma_{unpol}\right)
\kern 1cm \mbox{Transferred polarization,}
\\
A &=& \Delta_{unpol}/\Sigma_{unpol}
\kern 1cm \mbox{Electron analyzing power.}\end{aligned}$$
Multipole analysis of response functions
----------------------------------------
In this section we present the multipole expansion of the response functions to be used in our DWIA model. The final expressions, where the sums over third components of angular momenta have been performed analytically, result convenient in the present work since the computational time can be considerably reduced, specially the calculation concerning the MEC. Note that the number of multipoles needed to get convergence increases with $q,\omega$ and up to $\sim
36$ multipoles are needed for $q=1$ GeV/c. The expansion is performed following the formalism developed in [@Ama98b] for exclusive reactions from polarized nuclei. A basic difference between the present work and that of ref. [@Ama98b] lies on the sums performed over the third components which are different when initial and/or final state polarizations are considered. Here we simply present the final expressions, referring to ref. [@Ama98b] for details on the expansion method and to the Appendix for an outline on the procedure used to perform the sum over third components in the present case.
In order to compute the hadronic tensor in our DWIA model we first perform a multipole expansion of the ejected nucleon wave function in partial waves. The final hadronic states may then be written $$|{{\bf p}}' \nu, B\rangle =
\sum_{lMjm_pJ_fM_f}
i^l Y^*_{lM}({{\bf \hat{p}}}')
\langle {\textstyle \frac12}\nu l M| jm_p\rangle
\langle j m_p J_BM_B|J_f M_f\rangle
|(lj)J_B,J_fM_f\rangle \, ,
\label{final-states}$$ where the partial waves $(lj)$ are coupled to the angular momentum $J_B$ of the residual nucleus to give a total angular momentum $J_f$ in the final states $|f\rangle=|(lj)J_B,J_fM_f\rangle$.
The electromagnetic charge and transverse current operators are expanded as sums involving Coulomb (C), electric (E) and magnetic (M) tensor operators, $$\begin{aligned}
\hat{\rho}(q) &=& \sqrt{4\pi}\sum_{J=0}^{\infty} i^J [J] \hat{M}_{J0}(q)
\label{m-rho}
\\
\hat{J}_{m} &=& -\sqrt{2\pi}\sum_{J=1}^{\infty} i^J [J]
\left[\hat{T}^{el}_{Jm}(q)+m\hat{T}^{mag}_{Jm}(q)
\right] \, ,
\kern 1cm m=\pm 1 \, ,
\label{m-J}\end{aligned}$$ where, as usual, we assume the transfer momentum ${{\bf q}}$ along the $z$-direction and $\hat{J}_{m}$ are the spherical components of the current operator $\hat{{{\bf J}}}$. We use the bracket symbol $[J]=\sqrt{2J+1}$ for angular momenta. Inserting (\[rotation\],\[final-states\],\[m-rho\],\[m-J\]) into the hadronic tensor (\[hadronic-tensor\]), the following expansion for the responses $W^K$ and $\widetilde{W}^K$ is obtained $$W^K =
\frac12 W^K_{unpol}
+2\pi P_1^{1}(\cos\theta_s) \sin(\Delta\phi)
\widetilde{W}^{K}_{11},
\label{Wexpansion}$$ for $K=L,T,TL,TT,TL'$, and $$\widetilde{W}^K
=
2\pi \alpha_K \left[ P_1^{0}(\cos\theta_s)W^{K}_{10}
+ P_1^{1}(\cos\theta_s) \cos(\Delta\phi)W^{K}_{11}
\right]
\label{Wtexpansion}$$ for $K=TL,TT,T',TL'$, with the coefficient $\alpha_K=-1$ for $TL,TT$ and $\alpha_K=1$ for $T',TL'$, and $P_{{{\cal J}}}^{{{\cal M}}}(\cos\theta_s)$ the Legendre functions.
The five response functions $W^K_{unpol}$ are the only ones that survive when summing over final spins $\pm {\mbox{\boldmath{$s$}}}$, in which case the 1/2 factor cancels and the unpolarized cross section is recovered. The spin dependence is determined from the thirteen reduced response functions $W^K_{1{{\cal M}}}$ (${{\cal M}}=0,1$) and $\widetilde{W}^K_{11}$ introduced above. Explicit expressions for these reduced responses can be written in terms of the reduced matrix elements of the current multipole operators $$\begin{aligned}
C_\sigma &=& \langle (lj)J_B,J\|\hat{M}_J\|0\rangle
\label{Csigma}
\\
E_\sigma &=& \langle (lj)J_B,J\|\hat{T}^{el}_J\|0\rangle
\label{Esigma}
\\
M_\sigma &=& \langle (lj)J_B,J\| i\hat{T}^{mag}_J\|0\rangle \, ,
\label{Msigma}\end{aligned}$$ where we have defined a multiple index $\sigma = (l,j,J)$ corresponding to the quantum numbers of the final states. Note that the initial state $|A\rangle=|0\rangle$ has total angular momentum equal to zero, so $J_f=J$. The response functions involve quadratic products of these multipole matrix elements which can be decomposed into their real ($R^K_{\sigma'\sigma}$) and imaginary ($I^K_{\sigma'\sigma}$) parts: $$\begin{aligned}
C^*_{\sigma'}C_{\sigma}
&=&
R^L_{\sigma'\sigma}+iI^L_{\sigma'\sigma}
\label{ccL}
\\
E^*_{\sigma'}E_{\sigma}
+M^*_{\sigma'}M_{\sigma}
&=&
R^{T1}_{\sigma'\sigma}+iI^{T1}_{\sigma'\sigma}
\\
E^*_{\sigma'}M_{\sigma}
-M^*_{\sigma'}E_{\sigma}
&=&
R^{T2}_{\sigma'\sigma}+iI^{T2}_{\sigma'\sigma}
\\
C^*_{\sigma'}E_{\sigma}
&=&
R^{TL1}_{\sigma'\sigma}+iI^{TL1}_{\sigma'\sigma}
\\
C^*_{\sigma'}M_{\sigma}
&=&
R^{TL2}_{\sigma'\sigma}+iI^{TL2}_{\sigma'\sigma}
\\
E^*_{\sigma'}E_{\sigma}
-M^*_{\sigma'}M_{\sigma}
&=&
R^{TT1}_{\sigma'\sigma}+iI^{TT1}_{\sigma'\sigma}
\\
E^*_{\sigma'}M_{\sigma}
+M^*_{\sigma'}E_{\sigma}
&=& R^{TT2}_{\sigma'\sigma}+iI^{TT2}_{\sigma'\sigma} \, .
\label{emTT2}\end{aligned}$$ Expressions for the unpolarized response functions $W^K_{unpol}$ in terms of (\[ccL\]–\[emTT2\]) are given in ref. [@Maz02], while the recoil nucleon polarized responses can be written as $$\begin{aligned}
\widetilde{W}^L_{11}
&=&
\frac{1}{K}
\sum_{{{\cal J}}'L}\tilde{h}^1_{1{{\cal J}}'L0}(\theta')
\sum_{\sigma'\sigma}P^+_{l+l'+{{\cal J}}'}\Phi_{\sigma'\sigma}
{\left( \begin{array}{ccc}
J' & J & L \\
0 & 0 & 0
\end{array}
\right)}
\xi^+_{J'-l',J-l}I^L_{\sigma'\sigma}
\label{WL11}
\\
\widetilde{W}^T_{11}
&=&
-\frac{1}{K}
\sum_{{{\cal J}}'L} P^+_{{{\cal J}}'+L} \tilde{h}^1_{1{{\cal J}}'L0}(\theta')
\sum_{\sigma'\sigma}P^+_{l+l'+{{\cal J}}'}\Phi_{\sigma'\sigma}
{\left( \begin{array}{ccc}
J' & J & L \\
1 & -1 & 0
\end{array}
\right)}
\nonumber\\
&&\mbox{}\times
\left(\xi^+_{J'-l',J-l}I^{T1}_{\sigma'\sigma}
+\xi^-_{J'-l',J-l}I^{T2}_{\sigma'\sigma}
\right)
\\
\widetilde{W}^{TL}_{11}
&=&
-\frac{1}{K}
2\sqrt{2}\sum_{{{\cal J}}'L}(-1)^{{{\cal J}}'+L}\tilde{h}^1_{1{{\cal J}}'L1}(\theta')
\sum_{\sigma'\sigma}P^+_{l+l'+{{\cal J}}'}\Phi_{\sigma'\sigma}
{\left( \begin{array}{ccc}
J' & J & L \\
0 & 1 & -1
\end{array}
\right)}
\nonumber\\
&&\mbox{}\times
\left(\xi^+_{J'-l',J-l}I^{TL1}_{\sigma'\sigma}
-\xi^-_{J'-l',J-l}I^{TL2}_{\sigma'\sigma}
\right)
\\
\widetilde{W}^{TT}_{11}
&=&
-\frac{1}{K}
\sum_{{{\cal J}}'L}(-1)^{{{\cal J}}'+L}\tilde{h}^1_{1{{\cal J}}'L2}(\theta')
\sum_{\sigma'\sigma}P^+_{l+l'+{{\cal J}}'}\Phi_{\sigma'\sigma}
{\left( \begin{array}{ccc}
J' & J & L \\
1 & 1 & -2
\end{array}
\right)}
\nonumber\\
&&\mbox{}\times
\left(\xi^+_{J'-l',J-l}I^{TT1}_{\sigma'\sigma}
-\xi^-_{J'-l',J-l}I^{TT2}_{\sigma'\sigma}
\right)
\\
\widetilde{W}^{TL'}_{11}
&=&
-\frac{1}{K}
2\sqrt{2}\sum_{{{\cal J}}'L}(-1)^{{{\cal J}}'+L}\tilde{h}^1_{1{{\cal J}}'L1}(\theta')
\sum_{\sigma'\sigma}P^+_{l+l'+{{\cal J}}'}\Phi_{\sigma'\sigma}
{\left( \begin{array}{ccc}
J' & J & L \\
0 & 1 & -1
\end{array}
\right)}
\nonumber\\
&&\mbox{}\times
\left(\xi^+_{J'-l',J-l}R^{TL1}_{\sigma'\sigma}
-\xi^-_{J'-l',J-l}R^{TL2}_{\sigma'\sigma}
\right)
\\
{W}^{TL}_{1{{\cal M}}}
&=&
-\frac{1}{K}
2\sqrt{2}\sum_{{{\cal J}}'L}(-1)^{{{\cal J}}'+L}h^{{{\cal M}}}_{1{{\cal J}}'L1}(\theta')
\sum_{\sigma'\sigma}P^+_{l+l'+{{\cal J}}'}\Phi_{\sigma'\sigma}
{\left( \begin{array}{ccc}
J' & J & L \\
0 & 1 & -1
\end{array}
\right)}
\nonumber\\
&&\mbox{}\times
\left(\xi^+_{J'-l',J-l}I^{TL1}_{\sigma'\sigma}
-\xi^-_{J'-l',J-l}I^{TL2}_{\sigma'\sigma}
\right)
\\
{W}^{TT}_{1{{\cal M}}}
&=&
-\frac{1}{K}
\sum_{{{\cal J}}'L}(-1)^{{{\cal J}}'+L}{h}^{{{\cal M}}}_{1{{\cal J}}'L2}(\theta')
\sum_{\sigma'\sigma}P^+_{l+l'+{{\cal J}}'}\Phi_{\sigma'\sigma}
{\left( \begin{array}{ccc}
J' & J & L \\
1 & 1 & -2
\end{array}
\right)}
\nonumber\\
&&\mbox{}\times
\left(\xi^+_{J'-l',J-l}I^{TT1}_{\sigma'\sigma}
-\xi^-_{J'-l',J-l}I^{TT2}_{\sigma'\sigma}
\right)
\\
{W}^{T'}_{1{{\cal M}}}
&=&
-\frac{1}{K}
\sum_{{{\cal J}}'L} P^-_{{{\cal J}}'+L} {h}^{{{\cal M}}}_{1{{\cal J}}'L0}(\theta')
\sum_{\sigma'\sigma}P^+_{l+l'+{{\cal J}}'}\Phi_{\sigma'\sigma}
{\left( \begin{array}{ccc}
J' & J & L \\
1 & -1 & 0
\end{array}
\right)}
\nonumber\\
&&\mbox{}\times
\left(\xi^+_{J'-l',J-l}R^{T1}_{\sigma'\sigma}
+\xi^-_{J'-l',J-l}R^{T2}_{\sigma'\sigma}
\right)
\\
{W}^{TL'}_{1{{\cal M}}}
&=&
-\frac{1}{K}
2\sqrt{2}\sum_{{{\cal J}}'L}(-1)^{{{\cal J}}'+L}{h}^{{{\cal M}}}_{1{{\cal J}}'L1}(\theta')
\sum_{\sigma'\sigma}P^+_{l+l'+{{\cal J}}'}\Phi_{\sigma'\sigma}
{\left( \begin{array}{ccc}
J' & J & L \\
0 & 1 & -1
\end{array}
\right)}
\nonumber\\
&&\mbox{}\times
\left(\xi^+_{J'-l',J-l}R^{TL1}_{\sigma'\sigma}
-\xi^-_{J'-l',J-l}R^{TL2}_{\sigma'\sigma}
\right) \, ,
\label{WTL'1M}\end{aligned}$$ where we use the parity functions $$P^{\pm}_{n}=(1\pm (-1)^n)/2,
\kern 0.3cm
\xi^+_{J'J}\equiv (-1)^{(J'-J)/2}P^+_{J'+J},
\kern 0.3cm
\xi^-_{J'J}\equiv (-1)^{(J'-J+1)/2}P^-_{J'+J}$$ and the angular dependence of the above responses is determined by the functions $h^{{{\cal M}}}_{{{\cal J}}{{\cal J}}'LM}(\theta')$ and $\tilde{h}^{{{\cal M}}}_{{{\cal J}}{{\cal J}}'LM}(\theta')$, defined through the coupling of two spherical harmonics (see (\[Ycoupling\])). $$\begin{aligned}
{\rm Re}\left[ Y_{{{\cal J}}}({\mbox{\boldmath{$s$}}})Y_{{{\cal J}}'}({{\bf \hat{p}}}')\right]_{LM}
&=&
\cos M\phi'
\sum_{{{\cal M}}=0}^{{{\cal J}}}
h^{{{\cal M}}}_{{{\cal J}}{{\cal J}}'LM}(\theta')
P^{{{\cal M}}}_{{{\cal J}}}(\cos\theta_s)\cos({{\cal M}}\Delta\phi)
\nonumber\\
&&
\mbox{}+
\sin M\phi'
\sum_{{{\cal M}}=0}^{{{\cal J}}}
\tilde{h}^{{{\cal M}}}_{{{\cal J}}{{\cal J}}'LM}(\theta')
P^{{{\cal M}}}_{{{\cal J}}}(\cos\theta_s)\sin({{\cal M}}\Delta\phi) \, .\end{aligned}$$ Finally, the coefficients $\Phi_{\sigma'\sigma}$ are derived in the Appendix and are given by eq. (\[phi\]) selecting ${{\cal J}}=1$. Although the above expressions correspond formally to those denoted as $W^{K(-)}_{{{\cal J}}{{\cal M}}}$ and $\widetilde{W}^{K(-)}_{{{\cal J}}{{\cal M}}}$ in [@Ama98b] for polarized nuclei and ${{\cal J}}=1$, it is important to point out that the coefficients $\Phi_{\sigma'\sigma}$ contain the whole information on the polarization distribution of the particles. Hence the significance of $\Phi_{\sigma'\sigma}$ is clearly different when polarization degrees of freedom are considered for the ejected nucleon (present work) or the target nucleus [@Ama98b].
The nuclear structure information in (\[WL11\]-\[WTL’1M\]) is contained in the quadratic forms (\[ccL\]–\[emTT2\]) of the $C$, $E$, $M$ multipoles[^3]. Thus the present expansion can be applied to any nuclear model of the reaction as far as it provides multipole matrix elements (\[Csigma\],\[Esigma\],\[Msigma\]) for high enough angular momenta $\sigma=(l,j,J)$. Note that only the responses involving the real parts $R^K_{\sigma'\sigma}$ survive when FSI are neglected since in this case all the $C$, $E$, $M$ multipoles are strictly real functions. Therefore those responses which depend on the imaginary parts are expected to be particularly sensitive to the description of FSI.
Writing down explicitly the Legendre polynomials involved in the multipole expansion (\[Wexpansion\],\[Wtexpansion\]), and comparing with the general expression (\[Wgeneral\],\[Wtgeneral\]), we get the following relation between both sets of response functions: $$\begin{aligned}
W^K_n
&=&
-2\pi \widetilde{W}^K_{11}, \kern 5cm K=L,T,TL,TT,TL'
\label{WKn}
\\
W^{K}_l
&=&
2\pi\alpha_K\left(W^{K}_{11}\sin\theta'+W^{K}_{10}\cos\theta'\right),
\kern 1cm K=TL,TT,TL',T'
\label{WKl}
\\
W^{K}_t
&=&
2\pi\alpha_K\left(W^{K}_{11}\cos\theta'-W^{K}_{10}\sin\theta'\right),
\kern 1cm K=TL,TT,TL',T'
\label{WKt}\end{aligned}$$ with $\alpha_K$ as introduced in (\[Wtexpansion\]).
Electromagnetic operators and PWIA
----------------------------------
In this work we evaluate the exclusive polarized responses using a semi-relativistic (SR) model for describing the electromagnetic one-body (OB) and two-body MEC current operators. The OB current has been obtained by a direct Pauli reduction of the fully relativistic operator in powers only of the initial nucleon momentum over the nucleon mass ${{\bf p}}/m_N$. The dependence on the transfer and final momenta, which can be large [@Ama96a; @Ama96b; @Ama98a; @Ama02c], is treated exactly. The SR-OB current in momentum space can be written as $$\begin{aligned}
J^0({{\bf p}}',{{\bf p}})
&=&
\rho_c+ i \rho_{so}(\cos\phi\ \sigma_x-\sin\phi\ \sigma_y)\chi
\label{rho}
\\
J^x({{\bf p}}',{{\bf p}})
&=&
iJ_m\sigma_y + J_c\ \chi\cos\phi
\label{Jx}
\\
J^y({{\bf p}}',{{\bf p}})
&=&
-iJ_m\sigma_x + J_c\ \chi\sin\phi \, ,
\label{Jy}\end{aligned}$$ where $\chi=(p/m_N)\sin\theta$, and $(\theta,\phi)$ are the angles determining the direction of the initial momentum ${{\bf p}}$ in the $(x,y,z)$ coordinate system. The charge ($\rho_c$), spin-orbit ($\rho_{so}$), magnetization ($J_m$) and convection ($J_c$) terms shown above include relativistic corrections and are given by the following expressions $$\begin{aligned}
\rho_c = \frac{\kappa}{\sqrt{\tau}}G_E, &&
\rho_{so} = \kappa\frac{2G_M-G_E}{2\sqrt{1+\tau}}
\label{rho-factors}
\\
J_m = \sqrt{\tau}G_M ,
&&
J_c = \frac{\sqrt{\tau}}{\kappa}G_E \, ,
\label{J-factors}\end{aligned}$$ where $\kappa=q/2m_N$, $\tau=|Q^2|/4m_N^2$, and $G_E$, $G_M$ are the electric and magnetic nucleon form factors for which we take the Galster parameterization [@Gal71].
The two-body MEC operators of pionic (P), seagull or contact (S) and $\Delta$-isobar kinds, displayed in the Feynman diagrams of Fig. 2, have been also obtained by making use of a SR approach leading to simple prescriptions that include relativistic corrections through a multiplicative factor (see Refs. [@Ama03b; @Ama02c; @Ama03a] for details on the SR expansion method) $${{\bf J}}^{MEC}_{SR} = \frac{1}{\sqrt{1+\tau}}{{\bf J}}^{MEC}_{NR} \, ,$$ where ${{\bf J}}^{MEC}_{NR}$ is the traditional non-relativistic MEC operator.
\#1\#2[1.0\#1]{}
The expressions for the reduced matrix elements of the OB and MEC multipole operators (\[Csigma\],\[Esigma\],\[Msigma\]) in the shell model are given in Refs. [@Ama93; @Ama94; @Ama96b] except for the relativistic correction factors appearing within the SR operators. The somewhat complex structure displayed by these multipoles makes it not possible to predict the relative importance of each contribution separately without explicit numerical evaluation, even in the case of the OB current.
Although in this work we perform a DWIA analysis of the response functions, we may take advantage of the significant simplifications introduced within the plane wave impulse approximation (PWIA), where analytical expressions for the response functions can be obtained [@Mar02a; @Mar02b]. First, for intermediate to high values of $q$, the PWIA approach is expected to provide reasonable results, thus the analytical PWIA expressions allow us to estimate the contributions of the different pieces of the currents to the polarized response functions. Second, since the PWIA results should be recovered using the present multipole expansion in the limit of no FSI, the comparison between our calculation and the analytical PWIA responses makes it possible to fix the number of multipoles needed to get convergence.
Hence, within PWIA, the matrix element of the OB current is written as $$\langle {{\bf p}}'{\mbox{\boldmath{$s$}}},B|J^\mu({{\bf q}})|A\rangle
=\sum_{\beta'\beta}
{\cal D}^*_{\beta'\frac12}({\mbox{\boldmath{$s$}}})
J^{\mu}({{\bf p}}',{{\bf p}})_{\beta'\beta}
\langle B|a_{{{\bf p}},\beta}|A\rangle \, ,$$ where $a_{{{\bf p}},\beta}$ is the annihilation operator corresponding to a particle with momentum ${{\bf p}}$ and spin projection $\beta$ referred to the quantization axis. Inserting this expression into the hadronic tensor (\[hadronic-tensor\]), and following the procedure described in [@Ama96b], we obtain $$\label{factorized}
W^{\mu\nu} = \frac12 m_Np' w^{\mu\nu}({{\bf p}}',{{\bf p}},{\mbox{\boldmath{$s$}}})M^S({{\bf p}}) \, ,$$ where we have defined the polarized single-nucleon tensor $$w^{\mu\nu}({{\bf p}}',{{\bf p}},{\mbox{\boldmath{$s$}}}) =
\sum_{\alpha\alpha'\beta'}
{\cal D}^*_{\beta'\frac12}({\mbox{\boldmath{$s$}}})
J^{\nu}({{\bf p}}',{{\bf p}})_{\beta'\alpha}
J^{\mu}({{\bf p}}',{{\bf p}})_{\alpha'\alpha}^*
{\cal D}_{\alpha'\frac12}({\mbox{\boldmath{$s$}}}) \, .$$ In the case of interest here, a closed-shell nucleus, the scalar momentum distribution $M^S({{\bf p}})$ for nucleon knock-out from a shell $nlj$ is given by $$M^S({{\bf p}})=\frac{2j+1}{4\pi}\widetilde{R}^2(p)$$ with $\widetilde{R}(p)$ the radial wave function of the hole in momentum space.
Using the current matrix elements (\[rho\],\[Jx\],\[Jy\]) one can compute in the factorized approximation (\[factorized\]) the response functions (\[rlrt\]–\[rt’rtl’\]). From these results the PWIA expressions for the reduced response functions can be identified. Expressions for the unpolarized responses in PWIA were given in [@Ama96b; @Ama98b]. In the case of the polarized responses, from the total of eighteen, only five survive in PWIA. These are given by $$\begin{aligned}
W^{T'}_{11}
&=& \frac{m_Np'}{4\pi} 2J_cJ_m\chi M^S({{\bf p}})
\label{WT'11-pw}
\\
W^{T'}_{10}
&=& \frac{m_Np'}{4\pi} 2J_m^2 M^S({{\bf p}})
\label{WT'10-pw}
\\
\widetilde{W}^{TL'}_{11}
&=& \frac{m_Np'}{4\pi} 2\sqrt{2} (\rho_cJ_m-\rho_{so}J_c\chi^2) M^S({{\bf p}})
\\
W^{TL'}_{11}
&=& \frac{m_Np'}{4\pi} 2\sqrt{2}\rho_cJ_m M^S({{\bf p}})
\\
W^{TL'}_{10}
&=& \frac{m_Np'}{4\pi} 2\sqrt{2}\rho_{so}J_m\chi M^S({{\bf p}}) \, ,
\label{WTL'10-pw}\end{aligned}$$ where we have used the factors introduced in (\[rho-factors\],\[J-factors\]). Notice that all the $L,T,TL$ and $TT$-type polarized responses are zero in this approximation.
Results
=======
In this section we present results for selected recoil nucleon polarization observables corresponding to proton knockout from the $p_{1/2}$ and $p_{3/2}$ shells in $^{16}$O. In particular, we restrict ourselves to the analysis of all the polarized response functions, including the fifth response $W^{TL'}_0$ that does not depend on the nucleon polarization and only enters when the initial electron beam is polarized, and the transferred polarization asymmetries $P'_{l,t,n}$. The study of cross sections and induced polarizations will be presented in a forthcoming publication [@Kaz03]. Two different kinematical situations corresponding to $(q,\omega)$-constant kinematics (also referred as quasiperpendicular kinematics) have been selected: i) $q=460$ MeV/c, $\omega=100$ MeV, and ii) $q=1$ GeV/c, $\omega=450$ MeV. In both cases the value of the transfer energy $\omega$ corresponds almost to the quasielastic peak.
In this work our main interest is focused on the role of the two-body MEC operators upon the recoil nucleon polarization observables, trying to identify kinematical conditions for which these effects can be important; however, a brief excursion on the FSI effects is also presented. All the calculations have been done within the formalism described in the previous section, i.e., semirelativistic expressions for the one- and two-body current operators and a multipole expansion method have been used. The number of multipoles needed has been fixed by comparing the DWIA results, in the particular case of no FSI, with the exact factorized PWIA responses (\[WT’11-pw\]–\[WTL’10-pw\]). Convergence in the multipole analysis is obtained with $J_{max}=30$ for $q=460$ MeV/c, and $J_{max}=35$ for $q=1000$ MeV/c. Finally, in all of the results which follow, the kinematics of the ejected nucleon is treated exactly.
Polarized response functions
----------------------------
Here we analyze the thirteen responses defined in (\[WKn\]–\[WKt\]) which arise from the ejected nucleon polarization, plus the “fifth” response function $W^{TL'}_o$. Results are displayed in Figs. 3–12. A similar analysis for the unpolarized responses $L$, $T$, $TL$ and $TT$ has been performed recently in [@Ama03b].
\#1\#2[0.9\#1]{}
\#1\#2[0.9\#1]{}
### Effects of FSI
We start our discussion with the effects of FSI on the polarized responses. A study of the dependence of the unpolarized responses on the particular FSI model was already presented in [@Ama99]. In Fig. 3 we show the eight induced polarized responses for proton knock-out from the $1p_{1/2}$ shell in $^{16}$O as function of the missing momentum $p$. Kinematics corresponds to $q=460$ MeV/c and $\omega=100$ MeV. The five transferred polarized responses plus the fifth one ($T'$ and $TL'$ types) are displayed in Fig. 4. Similar results are obtained for the $1p_{3/2}$ shell and thus they are not shown here. In all of these results we use bound wave functions obtained as solutions of the Shrödinger equation with a Woods-Saxon potential, with parameters taken from ref. [@Ama96a]. For the final states we use solutions for two different optical potentials. Solid lines correspond to calculations performed with the Comfort and Karp potential [@Com80], which was originally fitted to elastic proton scattering from $^{12}$C for energies below 183 MeV. We have extended it to $^{16}$O by introducing a dependence $A^{1/3}$ in the radius parameters. The results shown with dashed lines have been computed with the Schwandt potential [@Sch82], which also has been extrapolated here for $^{16}$O since it was originally fitted to higher mass nuclei.
The induced polarized $L,T,TL,TT$ and the fifth response functions, which are zero in absence of FSI, are expected to be highly sensitive to the details of the particular optical potential considered, and in particular, to the spin-orbit term in the potential. In this sense notice the significant difference introduced by both potentials in the case of the polarized $TL$ and $TT$ responses (fig. \[fig-v1\]), while the FSI discrepancy gets smaller for the fifth response function (fig. \[fig-v2\]) and is considerably reduced for $W^L_n$ and $W^T_n$.
The five transferred polarized responses which survive in PWIA (\[WT’11-pw\]–\[WTL’10-pw\]), depend less on the details of the potential, being mostly affected by the central imaginary part of it. As known, these responses enter in the case in which also the initial electron is polarized, and they contribute to the transferred nucleon polarization asymmetry. We observe (fig. \[fig-v2\]) that both potentials lead to close results, differing by less than 10% for the dominant responses $W^{TL'}_n$, $W^{T'}_l$ and $W^{TL'}_l$, while the largest differences are shown for $W^{T'}_t$, which is however very small. Similar results are found for the $p_{3/2}$ shell. The sensitivity shown by some polarized responses to the details of the potential, makes these observables of special interest to disentangle between the different models of FSI that can fit reasonably well the unpolarized cross sections.
\#1\#2[0.9\#1]{}
\#1\#2[0.9\#1]{}
### Effects of MEC
The impact of MEC on the recoil nucleon polarized responses is shown in Figs. \[fig-mec1\]–\[fig-mec8\]. In each panel we compare the distorted wave responses evaluated by using the OB current only (dotted line) with the results obtained when including also the two-body MEC operators considered in fig. 2, namely, the seagull or contact (OB+S) current (dashed lines), the contact and pion-in-flight (OB+S+P) currents (dot-dashed lines), and finally, including also the $\Delta$ current (solid lines), denoted as (OB+MEC). Results in figs. 5–8 correspond to kinematics $q=460$ MeV/c, $\omega=100$ MeV (kinematics I), whereas in figs. 9–12 we present the responses evaluated at $q=1$ GeV/c, $\omega=450$ MeV (kinematics II). For both kinematics proton knock-out from the $p_{1/2}$ (figs. 5,6 and 9,10) and $p_{3/2}$ (figs. 7,8 and 11,12) have been considered.
Let us discuss first the results for kinematics I (figs. 5-8). Here we observe that the global sign of the polarized $T$, $TL$ and $TT$ responses changes when comparing the $p_{1/2}$ (fig. \[fig-mec1\]) and $p_{3/2}$ (fig. \[fig-mec3\]) shells. The same occurs for kinematics II. Concerning MEC effects, the various polarized responses display different sensitivities to the two-body component of the nuclear current. Apart from the pure longitudinal response $W^L_n$, which shows no dependence on MEC because the “semi-relativistic” MEC expressions only include the leading transverse components, the role of MEC on $W^T_n$ is shown to be similar to the one found for the unpolarized $T$-response in [@Ama03b]: the enhancement (in absolute value) produced by the S current is partially cancelled by the reduction introduced by the P current; the $\Delta$ current gives rise to an additional reduction, leading to a global decrease of the $W^T_n$ response of the order of $\sim 10\%$ at the maximum. This effect being similar for both shells (figs. \[fig-mec1\] and \[fig-mec3\]).
Larger MEC effects are found for some of the induced polarized $TL$ responses, particularly for $W^{TL}_t$ where the $\Delta$ current produces a very significant modification of the response, changing even its shape in the region close to $p\sim 100$ MeV/c. Note that, although the global effect introduced by the $\Delta$ in this response is similar for both shells, in the case of the $p_{1/2}$ there is a large increase, whereas for $p_{3/2}$ the response is significantly reduced in absolute value. It is also interesting to point out that the $\Delta$ current plays also the most important role for the $W^{TL}_n$ response, this being clearly shown in the case of the $p_{3/2}$ shell.
The role of MEC on the three polarized $TT$ responses shows a very different behaviour for the two shells considered. In the case of the $p_{1/2}$ (fig. \[fig-mec1\]), the global effect of MEC is a very significant reduction of the responses, particularly for $W^{TT}_t$ ($\sim 20\%$) and $W^{TT}_n$ ($\sim 30\%$), being the separate contributions of the S, P and $\Delta$ currents of rather similar importance. Note that the contributions introduced by the S and P currents have opposite signs for the $p_{1/2}$ and $p_{3/2}$ shells. As a consequence, for the $p_{3/2}$ shell (fig.\[fig-mec3\]) the large enhancement (in absolute value) produced by the S current is almost cancelled exactly by the contributions of the P and $\Delta$ currents, so the net MEC effect is almost negligible for the three $TT$ responses.
\#1\#2[0.9\#1]{}
\#1\#2[0.9\#1]{}
The transferred polarized responses ($T'$ and $TL'$-type responses) are shown in Figs. \[fig-mec2\] and \[fig-mec4\]. From these results we find in general a small effect of MEC, less than $\sim
5\%$. An exception is $W^{T'}_t$ where the role of $\Delta$ gives rise to an important reduction of the response; however notice that $W^{T'}_t$ is very small, of the order of $\sim 10\%$ compared with $W^{T'}_l$ and $W^{TL'}_t$, and hence difficult to measure. The anomalous smallness of the response $W^{T'}_t$ was already discussed in detail in [@Mar02b] within the context of the PWIA and different non-relativistic reduction schemes. This result can be also understood within the multipole analysis performed in this work by taking into account the general relations given in eqs. (\[WKl\],\[WKt\]) and the explicit expressions obtained for the multipole functions in PWIA (\[WT’11-pw\],\[WT’10-pw\]). Since we are close to the quasielastic-peak, the angle $\theta'$ is close to zero for moderate missing momentum, so the biggest contribution comes from the factor multiplied by $\cos\theta'$ in eqs. (\[WKl\],\[WKt\]). This factor is $W^K_{10}=O(1)$ in the case of the $l$-responses, and $W^K_{11}=O(\chi)$ for the $t$-response. Precise values of the nucleon form factors and kinematical variables can be introduced in these equations to verify the exact relation between the $l$ and $t$ components in PWIA, which is not very different from the distorted-wave results of figs. \[fig-mec2\] and \[fig-mec4\].
\#1\#2[0.9\#1]{}
\#1\#2[0.9\#1]{}
Results for higher momentum and energy transfer, $q=1000$ MeV/c and $\omega=450$ MeV (kinematics II), are shown in Figs. \[fig-mec5\]–\[fig-mec8\] for the two $p$ shells in $^{16}$O. This kinematics corresponds to the experimental setting of [@Gao00; @Mal00] where $Q^2=-0.8$ (GeV/c)$^2$. Obviously in this case relativity is expected to play a more important role and in fact, studies within the relativistic distorted wave impulse approximation (RDWIA) [@Udi99] have proved the importance of these effects. The present SR model, although lacking some of the relativistic ingredients inherent in the RDWIA, incorporates exact relativistic kinematics for the ejected nucleon, a SR expansion of the current which can be used for high $q$ values, and finally, the use of the Schrödinger-equivalent form of the S-V Dirac global optical potential of [@Coo93], including the Darwing term in the wave function. The validity of the expansion procedure used in the SR model was tested in [@Udi99] where unpolarized observables evaluated within the SR approach were compared with a RDWIA calculation for this kinematics.
\#1\#2[0.9\#1]{}
\#1\#2[0.9\#1]{}
The discussion of the results presented in figs. \[fig-mec5\]-\[fig-mec8\] follows similar trends to the ones already presented for kinematics I, so here we simply summarize those aspects which can be of more relevance. As shown in figs. \[fig-mec5\]-\[fig-mec8\], the general effect introduced by MEC is a global reduction of the responses (in absolute value) whose magnitude depends on the specific response, being of the order of a few percent for $W^{TL}_{l,t,n}$ and $W^{TL'}_{0,l,t,n}$, larger for $W^{T}_{n}$, $W^{TT}_{t,n}$ and $W^{T'}_{l,t}$ (particularly because of the $\Delta$-contribution) and the largest for $W^{TT}_l$, where the reduction (basically due to $\Delta$) is about $\sim 20-25\%$. Note however that the response $W^{TT}_l$ is the smallest one and so hardly measurable.
The dependence of MEC effects on the momentum transfer shown in the results of figs. 5–12 is consistent with the findings of ref. [@Ama03b] for the unpolarized responses. In general the importance of MEC decreases with $q$. This is in accord with the results for the $T$ response in the $1p-1h$ channel in the case of quasielastic inclusive $(e,e')$ reactions [@Ama02c; @Ama03a]. This behaviour can be roughly understood from the relativistic expressions for the particle-hole transverse current matrix elements ${{{\bf J}}}_T({{\bf p}}',{{\bf p}})$ in Fermi gas [@Ama02c; @Ama03a], and also from the traditional non relativistic expressions. At the non relativistic level, the OB current is dominated by the magnetization contribution which goes as $\sim q$. On the contrary, MEC present a much more complex dependence on $q$ and on the momenta of the two holes involved: the missing momentum ${{\bf p}}$ and an intermediate momentum ${{\bf k}}$ which should be integrated. Moreover, MEC also contain pion propagators involving inverse squared pion momenta. For high $q$, a crude estimation of the (transverse) seagull and pion in flight currents is shown to behave as $\sim
q/(q^2+m_\pi^2)$, while the $\Delta$ current goes as $\sim
q^3/(q^2+m_\pi^2)$, hence the latter clearly dominates, which is in accord with the results shown here. Once the $\pi N$ form factor, which becomes smaller when high momenta are probed, is added to the two-body currents, we find the OB contribution to dominate over the MEC. At the relativistic level the above dependences on $q$ change. In [@Ama98a] it was demonstrated that if the form factors are neglected, then the OB, seagull and pionic currents grow asymptotically as $\sqrt{q}$. Thus the inclusion of $\pi N$ form factors is essential for the dominance of the OB current This conclusion however applies to the response functions only for low missing momentum, since for other observables such as the $A_{TL}$ asymmetry [@Ama03b] and the polarization asymmetries (see below) larger effects are found for high values of $q$ and missing momentum.
Transferred polarization asymmetries
------------------------------------
Apart from the response functions, other observables of special interest are the nucleon polarization asymmetries introduced in eqs. (18-21). These observables are given as ratios between polarized and unpolarized responses, where one hopes to gain different insight into the underlying physics from what is revealed through the responses themselves. As already mentioned in the introduction and in order to clarify the discussion, here we restrict ourselves to the analysis of the transferred polarization asymmetries $P'_{l,t,n}$, which only enter with polarized incident electrons and persist in PWIA. Induced polarization ratios $P_{l,t,n}$ —that do not depend on the polarization of the incident electron and are zero within the plane wave approach—, and total cross sections will be analyzed in a forthcoming publication [@Kaz03].
Following the discussion presented for the responses, here we first study the effects introduced by FSI and later on we focus on the role of MEC.
### Effects of FSI
\#1\#2[0.8\#1]{}
In Figs. \[fig-pol0\]–\[fig-pol00\] we present the results obtained for the transferred polarization asymmetries corresponding to proton knockout from the $p_{1/2}$ and $p_{3/2}$ shells in $^{16}$O. Kinematics has been selected as (I), i.e., $q=460$ MeV/c and $\omega=100$ MeV. Results for kinematics (II) follow the same general trends although FSI effects are in general less important because of the higher momentum transfer involved. The longitudinal $P'_l$ and transverse (sideways) $P'_t$ components are shown in fig. \[fig-pol0\] for electron scattering angle fixed to $\theta_e=30^{\rm o}$ (forward scattering) and three values of the proton azimuthal angle $\phi=0,90^{\rm o}$ and 180$^{\rm o}$, while the normal polarization $P'_n$ is displayed in fig. \[fig-pol00\] for $\phi=90^{\rm o}$ (notice that $P'_n$ is zero for co-planar kinematics). Although not shown here for brevity, we have also explored the behaviour of the transferred polarization ratios at backward scattering angle ($\theta_e=150^o$). As known, the purely transverse responses dominate at backward angles, whereas all of the kinematical factors that enter in the description of $(\vec{e},e'\vec{p})$ reaction are of similar order at forward angles. In [@Mar02a] forward scattering angles were proved to enhance significantly the sensitivity to dynamical relativistic effects. Concerning FSI and MEC, the discussion of the results for $\theta_e=150^{\rm o}$ follow similar trends to the ones presented here for $\theta_e=30^{\rm o}$.
The PWIA calculation (dotted line) is compared with DWIA results using the two optical potentials already presented in the previous section, i.e., Comfort & Karp (solid lines) and Schwandt (dashed lines). First, notice the difference between PWIA and DWIA results. Within the plane wave approach, the responses factorize and hence the polarization ratios depend only on the single-nucleon responses, being cancelled the whole dependence with the momentum distribution. This means that PWIA results are identical for the two $p$-shells considered. Moreover, polarization ratios in PWIA may be written in the general form $$P'_i = \frac{a_i+b_i\chi+O(\chi^2)}{c_i+d_i\chi +O(\chi^2)},
\kern 1cm
\mbox{for $i=l,t,n$} \, ,$$ where $\chi=p/m_N \sin\theta$, already introduced in (\[rho\]-\[Jy\]), is the parameter in the SR expansion of the nuclear current. For low missing momentum, the above fraction has a linear dependence on $\chi$ plus a small correction of order $\chi^2$ which breaks linearity for higher $p$.
For low missing momentum values $p\leq 200$ MeV/c, the effects introduced by FSI are small, being almost negligible at the maximum of the momentum distribution ($p\approx 100$ MeV/c). This result is expected because of the global reduction of the polarized response functions produced by FSI: of the order of $\sim30\%$ (fig. 4). This is somewhat similar to the behaviour shown by the unpolarized responses [@Ama99]. Hence, although not exact because of the slightly different sensitivities to FSI shown by the various responses, a kind of cancellation of FSI between the numerator and denominator in the polarization ratios occurs for low $p$. From results in fig. 13, one also observes that FSI effects are slightly bigger in the case of the $p_{3/2}$ shell, particularly for $P'_l$ and $\phi=180^{\rm o}$. The reason of this is connected to the much less reduction that FSI cause upon the unpolarized $TL$ response for $p_{3/2}$ (see [@Ama99] for details).
For high missing momentum the DWIA polarizations deviate significantly from the PWIA results, showing a very pronounced oscillatory behavior which may even give rise to a change of sign in the polarizations. This is a clear indication that for high momentum the effects of FSI are not simply a global reduction of the responses due to the imaginary part of the potential, but on the contrary, each response turns out to present a peculiar sensitivity to the interaction. As shown in figs. 3 and 4, this is hardly visible in the separate response functions because of the smallness of the momentum distribution for high $p$. It is important to point out that the oscillatory behaviour presented by the polarization ratios is a direct consequence of the breaking of factorization property. This issue was already studied at the level of the plane wave approach taking care of the dynamical relativistic effects introduced by the lower components of the bound Dirac spinors [@Mar02a]. A general analysis of factorization within the context of the RDWIA and different non-relativistic approximations is presently in progress [@Cris03].
Focusing on the results presented in fig. 13, we observe that the shape and magnitude of both polarization asymmetries, $P'_l$ and $P'_t$, are similar for the two $p$-shells. In the particular case of $\phi=90^{\rm o}$ (out-of-plane kinematics) the ratio $P'_t$ is very small, almost negligible for low missing momentum. This is expected since only the response $W^{T'}_t$, which is very small, contributes to $P'_t$ in that situation. For co-planar kinematics a large discrepancy between the results obtained at $\phi=0^o$ and $\phi=180^{\rm o}$ exists. As shown by eqs. (\[RT’\]–\[Wtgeneral\]), the numerator in the ratios $P'_i$, $i=l,t$, is given through the linear combination $v_{T'}W^{T'}_i+v_{TL'}W^{TL'}_i\cos\phi$, with the kinematical factors being $v_{T'}=0.27$ and $v_{TL'}=-0.18$ for the kinematics considered here (I). Hence, from the transferred polarization asymmetries measured at $\phi=0^o$ and $\phi=180^{\rm o}$, the separate responses $W^{T'}_i$ and $W^{TL'}_i$ could be extracted.
\#1\#2[0.8\#1]{}
Comparing the solid and dashed lines in fig. \[fig-pol0\] we conclude that the uncertainties introduced by the optical potentials selected are rather small. For low momentum transfer these differences are negligible in contrast to fig. 4 where some responses are shown to be affected appreciably by the optical potential. This again is an outcome of the fact that the differences between the responses computed with these potentials are of the same size in numerator and denominator and they tend to cancel when taking the quotient to compute the polarizations. Both sets of results start to differ for $p\geq 300$ MeV/c. Note however that for high $p$-values other relativistic effects coming from the dynamical enhancement of the lower components in the wave functions, not included in the present model, may also contribute significantly to the oscillatory behaviour of the polarizations [@Mar02a; @Cris03].
The case of the normal polarization transfer $P'_n$ (fig. \[fig-pol00\]), present some peculiarities not observed for $P'_{l,t}$. First the difference between PWIA and DWIA results is rather constant for the two shells in the whole range of missing momentum. In particular, the distorted wave approach leads to results which are very similar to the ones obtained within PWIA in the case of the $p_{3/2}$. In addition, the strong oscillatory behaviour due to FSI and shown for $P'_{l,t}$ (fig. 13) does not appear here, being the differences introduced by both optical potentials small. These results could promote this observable $P'_n$, that can be obtained in out-of-plane experiments, as a good candidate in order to study properties of the reaction without being much affected by FSI.
### Effects of MEC
In Figs. \[fig-pol1\]–\[fig-pol3\] we present the effects introduced by MEC upon the transferred proton polarization components for the two $p$ shells in $^{16}$O, and for the two kinematics considered above. In the case of kinematics (I), i.e., $q=460$ MeV/c (fig. \[fig-pol1\]), where we use the optical potential parameterized by Comfort & Karp, MEC effects are shown to be small in the whole missing momentum range and similar for both shells. Note also that the role played by MEC is of the same order of magnitude or even smaller than the uncertainty introduced by the use of the two optical potentials (fig. \[fig-pol0\]). The smallness of MEC effects on the polarization asymmetries comes from their effective cancellation when taking ratios of response functions.
Results for higher momentum transfer, $q=1000$ MeV/c (kinematics II), are shown in Fig. \[fig-pol2\] for the Schrödinger-equivalent form of the S-V Dirac global optical potential of ref. [@Coo93]. As in the previous case, MEC effects are small for low missing momentum; however they tend to increase significantly for higher $p$-values due to the $\Delta$ exchange current, inducing a softening of the transferred polarization asymmetry, which makes its oscillatory behaviour to appear at slightly lower momenta (see for instance the important MEC effects observed for $P'_l$ at $\phi=0$, particularly in the region close to the minimum $p\sim 300$ MeV/c). The present results indicate that the response functions entering into the polarization ratios are importantly affected by MEC, mainly due to the $\Delta$ current, for high momentum transfer. Large effects of this kind have also been found recently for the $A_{TL}$ asymmetry obtained from the analysis of unpolarized (e,e’p) reactions corresponding to the same kinematics II [@Ama03b].
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\#1\#2[0.8\#1]{}
The normal polarization $P'_n$ is shown in Fig. \[fig-pol3\] for the two kinematics and both shells. MEC effects follow the same trends as those observed for $P'_l$ and $P'_t$: they increase significantly for high momentum transfer ($q=1000$ MeV/c) and high missing momentum. In contrast with the $P'_l$ and $P'_t$ cases, here the relative contributions of the separate MEC currents depend on the specific kinematics and $p$-shell selected, playing the seagull and pion-in-flight currents an important role.
\#1\#2[1.0\#1]{}
To finish we present in Fig. \[fig-pol4\] the asymmetries $P'_l$ and $P'_t$ evaluated for $q=1000$ MeV/c, $\omega=450$ MeV and electron incident energy $\epsilon_e=2450$ MeV. This kinematics corresponds to a recent experiment performed at TJlab. We compare our calculations with the experimental data presented in [@Mal00]. The azimuthal angle in this experiment was $\phi=180^{\rm o}$. Note the change of sign of $P'_t$ with respect to the results of Fig. \[fig-pol2\], due to the opposite definitions of the normal plane (and hence of the $t$ component) for $\phi=180^{\rm o}$ (in our case the normal plane for $\phi=180^{\rm o}$ would point down in Fig. 1, while in ref. [@Mal00] it was chosen along the up direction). Results for the $p$ and $s$ shells in $^{16}$O are shown from left to right. Although being aware of the possible modifications that the “dynamical” relativistic ingredients [@Udi99; @Kel99; @Mal00] may introduce in the present calculations, we are rather confident that the results in fig. \[fig-pol4\] give us a clear indication of how much the DWIA calculation is expected to be modified after including the two-body (MEC) contributions (compare dotted with solid lines). As noted, whereas the contribution of MEC over $P'_t$ is negligible, they give rise to a slight reduction of $P'_l$, which is well inside the experimental error except for the $s_{1/2}$ shell. Comparing the results for the two $p$-shells we observe that our model describes better the case of the $p_{3/2}$. This is in agreement with the findings in [@Udi99; @Ama03b] concerning the $A_{TL}$ asymmetry. The particular case of $s_{1/2}$ shows that the experimental data for $P'_t$ are well reproduced by the calculations, which however understimate very significantly the data for $P'_l$.
In order to clarify the importance of FSI, in fig. \[fig-pol4\] we also show with dot-dashed lines the results corresponding to the global OB+MEC calculation but without including the spin-orbit term of the optical potential, i.e., using a phenomenological optical potential consisting only of a central part. As shown, the corresponding polarizations present some kind of “linearity”, being similar to the PWIA results. This is expected since the spin-orbit interaction is the main responsible of the oscillatory behaviour of the polarization ratios, apart from the “dynamical” relativistic effects.
The ratio $P'_t/P'_l$, shown in the bottom panels of fig. \[fig-pol4\], has been proposed as a suitable observable for getting information on nucleon properties inside the nuclear medium [@Arn81]. From inspection of fig. \[fig-pol4\] we find that MEC produce a tiny reduction of this observable, particularly for low missing momentum, being larger as $p$ goes up.
\#1\#2[0.9\#1]{}
All of the above results have been computed using the Galster parameterization for the nucleon form factor. It is of interest to know the dependence of our results with the nucleon structure, hence we have also calculated the OB+MEC polarization asymmetries assuming the Gari-Krumplemann (GK) form factor parameterization [@Gar85]. The results are shown with dashed lines. The GK parameterization was used by Udias et al. [@Udi03] within the context of the relativistic calculations presented in ref. [@Mal00]. The $P'_l$ computed with GK form factors is increased with respect to the solid lines, being a little bit closer to the experimental data. Let us remind that the effects of MEC lead to a global reduction of all of these polarization observables, hence the OB calculation using the GK form factors would be clearly located above the corresponding results including MEC (dashed lines). This makes our present results to come closer to the relativistic ones of [@Mal00]. Note also that the uncertainty introduced by the nucleon form factor parameterization shows up in $P'_l$, being negligible for $P'_t$.
To finish the discussion, it is also interesting to point out that the behaviour shown by the $P'_t$ data, growing with $p$ for $p_{1/2}$ and the reverse for $p_{3/2}$, does not agree with the theoretical results which increase with $p$ for both shells. This is in accordance with other relativistic calculations [@Mal00]. For $p=140$ MeV/c our predictions for $P'_t$ in the case of the $p_{1/2}$ shell clearly underestimate the data; as already mentioned, other relativistic effects coming from the lower components of the Dirac wave functions, not considered here, may also play a significant role.
Comparison with previous works
------------------------------
Concerning previous calculations of MEC in $(e,e'p)$ reactions, in [@Ama99; @Ama03b] comparisons for unpolarized observables obtained with the present model with those of Refs. [@Bof91; @Ryc01; @Slu94; @Giu02] were presented. Next we summarize the differences of MEC effects on recoil polarization observables between the present work and Refs. [@Bof90; @Ryc99].
1. Boffi and collaborators [@Bof90] find for intermediate $q$ values large MEC effects on the separate polarized responses (reduction of the order of 15-30% or even larger), being the $\Delta$ current the main contribution. We get in general smaller and qualitatively different effects for this kinematics, the seagull and pion in flight currents being in our case as important as the $\Delta$. Concerning the transfer polarization ratios, they find $P'_l$ to be the most sensitive one, with a 20% decrease due to the $\Delta$ for low missing momentum $p<200$ MeV/c. In our calculation MEC effects are clearly less important for these missing momentum values.
2. Ryckebusch [*et al.*]{} [@Ryc99] do not present the separate $l,t,n$ response functions. In general they find small MEC effects, as we do, in the transferred polarizations for low $p<300$ MeV/c. These effects being larger as $q$ and $p$ increase. Comparing specifically our results to theirs for kinematics II, we observe that the OB results clearly differ due to the different treatment of FSI, while somewhat larger and qualitatively different MEC effects are found in this work.
Be it as it may, since the different treatment of FSI and of the current operators in [@Bof90; @Ryc99] and in the present work produces discrepancies already at the impulse approximation, it is hard to draw general conclusions on MEC effects beyond the fact that in [@Bof90] MEC lead to excesively large contributions compared with us, while their small size in [@Ryc99] is in accord with our calculation.
Summary and conclusions
=======================
In this paper we have presented a distorted wave model of $(\vec{e},e'\vec{p})$ reactions which goes beyond the impulse approximation with the inclusion of two-body meson exchange currents. Relativistic kinematics to relate the energy and momentum of the ejected proton is used and the currents are derived through an expansion in powers of the missing momentum over the nucleon mass. Explicit expressions of the polarized response functions in a general multipole expansion method are given. Results for the responses and transferred polarization asymmetries have been obtained for proton knock-out from the different shells in $^{16}$O for quasiperpendicular kinematics with the transfer momentum fixed to $q=460$ and 1000 MeV/c. FSI have been considered in each case by using different optical potentials.
One of our primary goals has been to estimate, within our present model, the validity of the impulse approximation by analyzing the effect of MEC on the different recoil nucleon polarized observables. Thus we compare the standard DWIA results, obtained using only the OB current, with the “full” calculation which includes the MEC. We have also explored the role played by the particular description of the FSI, hence we compare the results obtained by using different optical potentials which have been widely considered in the literature: Schwandt [@Sch82] and Comfort & Karp [@Com80] parameterizations. For higher energy we have used instead the Schrödinger-equivalent form of a Dirac optical potential.
From our present studies we may summarize and conclude the following:
1. The induced $T$, $TL$ and $TT$ polarized responses are particularly sensitive to the details of the optical potential, allowing them, specially the $TT$ ones, to constrain the theoretical model for FSI. The transferred polarized responses ($T'$ and $TL'$), which survive in PWIA, show a much less sensitivity to the interaction.
2. In general, MEC effects over the transferred $T'$, $TL'$ polarized responses for $q=460$ MeV/c and moderate missing momentum ($p<300$ MeV/c) are rather small ($< 5\%$), and tend to increase as $q$ goes higher, being of the order of a $\sim 10\%$ reduction (due mainly to the $\Delta$ current) in the particular case of $W^{T'}_l$ and $q=1$ GeV/c.
The role of MEC gets clearly more important for the induced $T$, $TL$ and $TT$ polarized responses. Emphasis should be placed on $W^{TT}_t$ and $W^{TT}_n$ which are reduced at the maximum by $\sim 20\%$ and $\sim 30\%$, respectively, for $q=460$ MeV/c and for the $p_{1/2}$ shell; notice however that these effects are negligible in the case of $p_{3/2}$. For $q=1$ GeV/c the role of MEC diminishes.
3. FSI give rise to an important deviation of the transferred polarization asymmetries $P'_l$ and $P'_t$ with respect to the PWIA results, showing a very pronounced oscillatory behaviour that starts for $p\geq 200$ MeV/c. This behaviour does not appear in the component $P'_n$. The uncertainties introduced by the optical potentials are rather small for the missing momentum region analyzed.
4. MEC effects on $P'_t$ and $P'_l$ are negligible for $q=460$ MeV/c and increase for $q=1$ GeV/c, especially for $p>200$ MeV/c. The role of MEC on $P'_n$ is clearly more important.
Finally we are confident that the significant sensitivity shown by some polarized observables to MEC, particularly to the $\Delta$ current, will be maintained within the scheme of a “fully” relativistic calculation which takes care of relativistic ingredients, such as the dynamical enhancement of lower components, not included in the present model. Work along this line is in progress.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was partially supported by funds provided by DGI (Spain) and FEDER funds, under Contracts Nos BFM2002-03218, BFM2002-03315 and FPA2002-04181-C04-04 and by the Junta de Andalucía.
Sum over third components and reduced response functions
========================================================
In this appendix we perform the sum over third components in the multipole expansion of response functions and give their explicit expressions in terms of the reduced matrix elements of the multipole operators.
First we write the response functions (\[rlrt\]–\[rt’rtl’\]) in terms of the spherical components of the current matrix elements $J_{\pm1}\equiv \langle{{\bf p}}' {\mbox{\boldmath{$s$}}}, B|\hat{J}_{\pm1}|A\rangle$ using the hadronic tensor (\[hadronic-tensor\]) $$\begin{aligned}
R^L &=& \frac{1}{K}
\sum \rho^*\rho \\
R^T &=& \frac{1}{K} \sum \left(|J_{-1}|^2+|J_{+1}|^2\right)\\
R^{TL} &=& -2\frac{1}{K} {\rm Re} \sum \rho^*\left(J_{+1}-J_{-1}\right)\\
R^{TT} &=& \frac{1}{K} \sum \left(J_{-1}^*J_{+1}+J_{+1}^*J_{-1}\right)\\
R^{TL'} &=& -2 \frac{1}{K} {\rm Re} \sum \rho^*\left(J_{+1}+J_{-1}\right)\\
R^{T'} &=& \frac{1}{K} \sum \left(|J_{+1}|^2-|J_{-1}|^2\right)\, .\\\end{aligned}$$ Inserting the multipole expansion for the charge and current components as given in (\[m-rho\],\[m-J\]) we find that each response can be written as a sum of terms of the type $$B^{m'm}_{J'J}
\equiv \frac{1}{K}
\sum_{M_B}
\langle {{\bf p}}'{\mbox{\boldmath{$s$}}},J_BM_B| \hat{T'}_{J'm'}({{\bf q}})|A\rangle^*
\langle {{\bf p}}'{\mbox{\boldmath{$s$}}},J_BM_B| \hat{T}_{Jm}({{\bf q}})|A\rangle \, ,$$ where $\hat{T}'_{J'm'}$ and $\hat{T}_{Jm}$ represent in general the Coulomb, electric or magnetic multipole operators. Introducing now the multipole expansion (\[final-states\]) corresponding to the final state, which is polarized along an arbitrary direction ${\mbox{\boldmath{$s$}}}$ (\[rotation\]), we get $$\begin{aligned}
B^{m'm}_{J'J}
&=& \frac{1}{K}
\sum_{M_B}\sum_{\nu'\nu}
{\cal D}^{(1/2)}_{\nu'\frac12}({\mbox{\boldmath{$s$}}})
{\cal D}^{(1/2)}_{\nu\frac12}({\mbox{\boldmath{$s$}}})^*
\nonumber\\
&&
\kern -1cm \times
\sum_{l'M'j'm'_p}
i^{l'} Y^*_{l'M'}({{\bf \hat{p}}}')
\langle \frac12\nu' l' M'| j'm'_p\rangle
\langle j' m'_p J_BM_B|J'm'\rangle
\langle (l'j')J_B,J'm'|T'_{J'm'}|A\rangle^*
\nonumber\\
&&
\kern -1cm \times
\sum_{lMjm_p}
i^{-l} Y_{lM}({{\bf \hat{p}}}')
\langle \frac12\nu l M| jm_p\rangle
\langle j m_p J_BM_B|Jm\rangle
\langle (lj)J_B,Jm|T_{Jm}|A\rangle \, ,\end{aligned}$$ where we have used $J_f=J$ and $M_f=m$, since the initial nucleus has total angular momentum equal zero.
Using the Wigner-Eckart theorem for the matrix elements of tensor operators between states of definite angular momenta $$\langle (lj)J_B,Jm|T_{Jm}|0\rangle
=
\frac{1}{[J]}\langle (lj)J_B,J\|T_{J}\|0\rangle$$ and reducing the products of two rotation matrices and two spherical harmonics to linear combinations of spherical harmonics $$\begin{aligned}
{\cal D}^{(1/2)}_{\nu'\frac12}({\mbox{\boldmath{$\hat{s}$}}})
{\cal D}^{(1/2)}_{\nu\frac12}({\mbox{\boldmath{$\hat{s}$}}})^*
&=&
\sqrt{4\pi}\sum_{{{\cal J}}{{\cal M}}}(-1)^{1/2+\nu+{{\cal J}}}f^{(1/2)}_{{{\cal J}}}
{\left( \begin{array}{ccc}
\frac12 & \frac12 & {{\cal J}}\\
-\nu & \nu' & {{\cal M}}\end{array}
\right)}
Y_{{{\cal J}}{{\cal M}}}({\mbox{\boldmath{$\hat{s}$}}})
\\
Y^*_{l'M'}({{\bf \hat{p}}}')Y_{lM}({{\bf \hat{p}}}')
&=&
\nonumber\\
&&
\kern -1.5cm
\sum_{{{\cal J}}'{{\cal M}}'}(-1)^M\frac{[l][l'][{{\cal J}}']}{\sqrt{4\pi}}
{\left( \begin{array}{ccc}
l & l' & {{\cal J}}' \\
-M & M' & {{\cal M}}'
\end{array}
\right)}
{\left( \begin{array}{ccc}
l & l' & {{\cal J}}' \\
0 & 0 & 0
\end{array}
\right)}
Y_{{{\cal J}}'{{\cal M}}'}({{\bf \hat{p}}}'),
\nonumber\\\end{aligned}$$ with $f^{(1/2)}_{{{\cal J}}}=\frac{1}{\sqrt{2}}$ the Fano tensor for spin-1/2 polarization, we obtain $$\begin{aligned}
B^{m'm}_{J'J}
&=&
\sum_{M_B}\sum_{\nu\nu'}\sum_{{{\cal J}}{{\cal M}}}
\sum_{lMjm_p}\sum_{l'M'j'm'_p}\sum_{{{\cal J}}'{{\cal M}}'}
i^{l'-l}(-1)^{\frac12+\nu+{{\cal J}}}f^{(\frac12)}_{{{\cal J}}}
{\left( \begin{array}{ccc}
\frac12 & \frac12 & {{\cal J}}\\
-\nu & \nu' & {{\cal M}}\end{array}
\right)}
Y_{{{\cal J}}{{\cal M}}}({\mbox{\boldmath{$s$}}})
\nonumber\\
&&
\mbox{}\times
(-1)^M[l][l'][{{\cal J}}']
{\left( \begin{array}{ccc}
l & l' & {{\cal J}}' \\
-M & M' & {{\cal M}}'
\end{array}
\right)}
{\left( \begin{array}{ccc}
l & l' & {{\cal J}}' \\
0 & 0 & 0
\end{array}
\right)}
Y_{{{\cal J}}'{{\cal M}}'}({{\bf \hat{p}}}')
\nonumber\\
&&
\mbox{}\times
(-1)^{l'-\frac12-m'_p}[j']{\left( \begin{array}{ccc}
\frac12 & l' & j' \\
\nu' & M' & -m'_p
\end{array}
\right)}
(-1)^{J_B-j'-m'}{\left( \begin{array}{ccc}
j' & J_B & J' \\
m'_p & M_B & -m'
\end{array}
\right)}
\nonumber\\
&&
\mbox{}\times
(-1)^{l-\frac12-m_p}[j]{\left( \begin{array}{ccc}
\frac12 & l & j \\
\nu & M & -m_p
\end{array}
\right)}
(-1)^{J_B-j-m}{\left( \begin{array}{ccc}
j & J_B & J \\
m_p & M_B & -m
\end{array}
\right)}
\nonumber\\
&&
\mbox{}\times
\langle (l'j')J_B,J'\|T'_{J'}\|0\rangle^*
\langle (lj)J_B,J\|T_{J}\|0\rangle \, ,
\label{B}\end{aligned}$$ where we have transformed the Clebsch-Jordan to three-J coefficients. Next we perform the sums over third components of angular momenta in the above expression. Note that the total phase inside the sum can be simplified to $$\mbox{phase}= (-1)^{\frac12+m_p}(-1)^{{{\cal J}}+l+l'}(-1)^{j-j'}.$$ Therefore the following coefficient appears $$\begin{aligned}
S &\equiv&
\sum_{M_B}\sum_{\nu\nu'}\sum_{MM'}\sum_{m_pm'_p}
(-1)^{\frac12+m_p+{{\cal J}}+l+l'+j-j'}
{\left( \begin{array}{ccc}
\frac12 & \frac12 & {{\cal J}}\\
-\nu & \nu' & {{\cal M}}\end{array}
\right)}
{\left( \begin{array}{ccc}
l & l' & {{\cal J}}' \\
-M & M' & {{\cal M}}'
\end{array}
\right)}
\nonumber\\
&&
\kern -1.5cm
\mbox{}\times
{\left( \begin{array}{ccc}
\frac12 & l' & j' \\
\nu' & M' & -m'_p
\end{array}
\right)}
{\left( \begin{array}{ccc}
j' & J_B & J' \\
m'_p & M_B & -m'
\end{array}
\right)}
{\left( \begin{array}{ccc}
\frac12 & l & j \\
\nu & M & -m_p
\end{array}
\right)}
{\left( \begin{array}{ccc}
j & J_B & J \\
m_p & M_B & -m
\end{array}
\right)} \, .
\nonumber\\
\label{S}\end{aligned}$$ We first perform the sum over $\nu,\nu',M,M'$ by using a 9-j coefficient $$\begin{aligned}
\lefteqn{
\sum_{\nu\nu'}\sum_{MM'}
{\left( \begin{array}{ccc}
\frac12 & \frac12 & {{\cal J}}\\
-\nu & \nu' & {{\cal M}}\end{array}
\right)}
{\left( \begin{array}{ccc}
l & l' & {{\cal J}}' \\
-M & M' & {{\cal M}}'
\end{array}
\right)}
{\left( \begin{array}{ccc}
\frac12 & l & j \\
\nu & M & -m_p
\end{array}
\right)}
{\left( \begin{array}{ccc}
\frac12 & l' & j' \\
\nu' & M' & -m'_p
\end{array}
\right)} =}
\nonumber\\
&&
(-1)^{\frac12+l+j}\sum_{LM}[L]^2
{\left( \begin{array}{ccc}
{{\cal J}}& {{\cal J}}' & L \\
{{\cal M}}& {{\cal M}}' & M
\end{array}
\right)}
{\left( \begin{array}{ccc}
L & j & j' \\
M & m_p & -m'_p
\end{array}
\right)}
{\left\{ \begin{array}{ccc}
{{\cal J}}& {{\cal J}}' & L \\
\frac12 & l & j \\
\frac12 & l' & j'
\end{array}
\right\} } \, .\end{aligned}$$ Next we compute the sum over $m_p,m'_p,M_B$ using a 6-j coefficient $$\begin{aligned}
\lefteqn{\sum_{m_pm'_pM_B}
(-1)^{m_p+\frac12}
{\left( \begin{array}{ccc}
L & j & j' \\
M & m_p & -m'_p
\end{array}
\right)}
{\left( \begin{array}{ccc}
j & J_B & J \\
m_p & M_B & -m
\end{array}
\right)}
{\left( \begin{array}{ccc}
j' & J_B & J' \\
m'_p & M_B & -m'
\end{array}
\right)} =}
\nonumber\\
&&
(-1)^{\frac12+m'+j+j'+J_B}
{\left( \begin{array}{ccc}
L & J' & J \\
M & -m' & m
\end{array}
\right)}
{\left\{ \begin{array}{ccc}
L & J' & J \\
J_B & j & j'
\end{array}
\right\} }\, .\end{aligned}$$ Then the $S$-coefficient (\[S\]) results $$\begin{aligned}
S &=&
\sum_{LM}[L]^2(-1)^{{{\cal J}}+l'+j+J_B+m'}
{\left( \begin{array}{ccc}
L & J' & J \\
M & -m' & m
\end{array}
\right)}
{\left( \begin{array}{ccc}
{{\cal J}}& {{\cal J}}' & L \\
{{\cal M}}& {{\cal M}}' & M
\end{array}
\right)}
\nonumber\\
&&
\mbox{}\times
{\left\{ \begin{array}{ccc}
{{\cal J}}& {{\cal J}}' & L \\
\frac12 & l & j \\
\frac12 & l' & j'
\end{array}
\right\} }
{\left\{ \begin{array}{ccc}
L & J' & J \\
J_B & j & j'
\end{array}
\right\} }\, .
\label{S-final}\end{aligned}$$ To finish we insert the result (\[S-final\]) into (\[B\]), and define indices $\sigma,\sigma'$ corresponding to the quantum numbers of the final states $$\sigma = (l,j,J), \kern 1cm
\sigma' = (l',j',J')$$ and a coupling coefficient $$\begin{aligned}
\Phi_{\sigma'\sigma}({{\cal J}},{{\cal J}}',L)
&=&
\sqrt{2}[l][l'][j][j'][J][J'][{{\cal J}}'][L]
(-1)^{l+j+J_B+L+J+J'}
\nonumber\\
&&
\mbox{}\times
{\left( \begin{array}{ccc}
l & l' & {{\cal J}}' \\
0 & 0 & 0
\end{array}
\right)}
{\left\{ \begin{array}{ccc}
L & J' & J \\
J_B & j & j'
\end{array}
\right\} }
{\left\{ \begin{array}{ccc}
{{\cal J}}& {{\cal J}}' & L \\
\frac12 & l & j \\
\frac12 & l' & j'
\end{array}
\right\} }\, .
\label{phi}\end{aligned}$$ The final expression for $B$ is $$\begin{aligned}
B^{m'm}_{J'J}
&=&
\frac12\sum_{lj}\sum_{l'j'}\sum_{{{\cal J}}{{\cal J}}'LM}
i^{l'-l}
(-1)^m
{\left( \begin{array}{ccc}
J & J' & L \\
m & -m' & M
\end{array}
\right)}
\Phi_{\sigma'\sigma}({{\cal J}},{{\cal J}}',L)
\nonumber\\
&&
\mbox{}\times
\left[ Y_{{{\cal J}}}({\mbox{\boldmath{$\hat{s}$}}}) Y_{{{\cal J}}'}({{\bf \hat{p}}}')\right]_{L,-M}
\langle (l'j')J_B,J'\|T'_{J'}\|0\rangle^*
\langle (lj)J_B,J\|T_{J}\|0\rangle \, ,
\label{Bfinal}\end{aligned}$$ where the coupling between two spherical harmonics has been used $$\begin{aligned}
\lefteqn{\left[Y_{{{\cal J}}}({\mbox{\boldmath{$\hat{s}$}}})Y_{{{\cal J}}'}({{\bf \hat{p}}}')\right]_{LM}=}
\nonumber\\
&=& (-1)^{{{\cal J}}-{{\cal J}}'+M}\sum_{{{\cal M}}{{\cal M}}'}[L]
{\left( \begin{array}{ccc}
{{\cal J}}& {{\cal J}}' & L \\
{{\cal M}}& {{\cal M}}' & -M
\end{array}
\right)}
Y_{{{\cal J}}{{\cal M}}}({\mbox{\boldmath{$\hat{s}$}}})Y_{{{\cal J}}'{{\cal M}}'}({{\bf \hat{p}}}')\, .
\label{Ycoupling}\end{aligned}$$
Although the result given in eq. (\[Bfinal\]) is formally identical, with the exception of the factor $1/2$, to the one obtained in [@Ama98b] for the case of polarized nuclei, there exists a basic difference concerning the polarization coefficient $\Phi_{\sigma'\sigma}({{\cal J}},{{\cal J}}',L)$, which contains all the information on the polarization properties of the particles in the initial and/or final state. Note that in the present case (spin-1/2 polarized particles), the angular momentum in the expansion of the rotation matrices, ${{\cal J}}$, only takes the values 0,1. The case ${{\cal J}}=0$ is the only one surviving when the final nucleon is not polarized, i.e., when summing the cross sections for $\pm s$ values. In this case the present formalism reduces simply to the standard unpolarized one of ref. [@Maz02]. In fact, for ${{\cal J}}=0$ we have ${{\cal J}}'=L$ and the reader can prove after some Racah algebra, that $\Phi_{\sigma'\sigma}(0,L,L)$ reduces to the expression given in eq. (A11) of Ref. [@Maz02] for the unpolarized case.
Moreover, using the properties of the 9-j symbol, the following important symmetry property is found for the polarization coefficient under exchange of the indices $$\Phi_{\sigma'\sigma}({{\cal J}},{{\cal J}}',L)= (-1)^{{{\cal J}}+{{\cal J}}'+L}
\Phi_{\sigma\sigma'}({{\cal J}},{{\cal J}}',L).$$ This property coincides with the one already presented in [@Ama98b] in the case of polarized targets. Since the multipole expansion of response functions performed in [@Ama98b] was based on this symmetry, then the same formalism can be applied to the present case. In this way one arrives to eqs. (\[WL11\]–\[WTL’1M\]) (see [@Ama98b] for more details on the expansion).
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[^1]: In this work we consider the kinematical factors similar to those expressions presented in [@Ras89]. Notice that these factors differ from the ones of ref. [@Pic89] in a global sign for $v_{TL}$, $v_{TL'}$ and $v_{TT}$, and an additional $1/\sqrt{2}$ factor in the case of the interference $TL$ coefficients
[^2]: Notice that this notation does not coincide with Refs. [@Mar02a; @Mar02b] where the ${\mbox{\boldmath{$t$}}}$ direction is denoted as ${\mbox{\boldmath{$s$}}}$ (sideways)
[^3]: Note that there is a typo in eqs. (40–51) of ref. [@Ama98b]: the order of $J$ and $J'$ in the three-j should be reversed. This error has been corrected in eqs. (37–45) in the present paper.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We study the phenomenon of internal avalanching within the context of recently introduced lattice models of granular media. The avalanche is produced by pulling out a grain at the base of the packing and studying how many grains have to rearrange before the packing is once more stable. We find that the avalanches are long-ranged, decaying as a power-law. We study the distriution of avalanches as a function of the density of the packing and find that the avalanche distribution is a very sensitive structural probe of the system.'
address: |
$a)$ P.M.M.H. Ecole Supérieure de Physique et Chimie Industrielles,\
10, rue Vauquelin, 75231 Paris CEDEX 05 France\
$b)$ ICA1, Univ. Stuttgart, Germany\
$c)$ Dipartimento di Fisica, Universitá di Napoli “Federico II”, Unità INFM and INFN Napoli\
Mostra d’Oltremare, Pad. 19, 80125 Napoli, Italy,\
$d)$ Laboratoire Surface du Verre et Interfaces, Unité Mixte de Recherche CNRS/St-Gobain,\
39, Quai Lucien Lefranc, F-93303 Aubervilliers Cedex, France.
author:
- 'Supriya Krishnamurthy$^a$, Hans Herrmann$^{a,b}$ Vittorio Loreto$^a$, Mario Nicodemi$^c$ and Stephane Roux $^d$'
title: Internal avalanches in models of granular media
---
Introduction
============
The internal structure and geometry of granular packings are very different from those of other systems such as liquids or solids and a lot of work has been devoted to understand them [@grain]. In particular, surface avalanches occurring in granular packings have been extensively studied [@grain]. In this paper we look at another sort of avalanching phenomenon also indicative of the internal structure - the phenomenon of internal avalanching occurring under small perturbations.
Recently simple lattice gas models have been proposed to describe slow dynamical processes in granular media, models whose basic ingredient is the geometric frustration in particle motion [@conihermann; @nicohans1; @tetris]. These models reproduce experimentally observed phenomena such as slow relaxation in compaction, segregation[@segtet], experimental irreversible-reversible cycles as well as the presence of “aging” and glassy behavior[@nicodemi].
Within the context of these lattice gas models, we study in the present paper the features of internal avalanching. In particular we observe how the internal structure of the packing is reorganized after a small perturbation, such as pulling out a grain from the base. We find that surprisingly, the packing can undergo large rearrangements even under such a minimal perturbation. The size distribution of the produced events is very broad, being a power law over several orders of magnitude. This shows the strong sensitivity of packings, in the above models, to small perturbations which may trigger huge events up to the scale of the system itself. We study the avalanche distribution as a function of the density of the packing and find that the distribution shows a strong dependence on density, indicating that it is a very sensitive structural probe of the system.
The outline of the paper is as follows. In the following section, we define the model which we have studied numerically. In section $III$ we make precise the definition of an internal avalanche and define the quantity studied. Our results are described in section $IV$ and the conclusions and discussion follow in section $V$.
[*Tetris*]{}-like models
========================
In this section we briefly review the definitions and some basic properties of the [*Tetris*]{}-like lattice models [@tetris] used in our simulations. The choice of the name “Tetris” captures the original idea of the computer game, of difficult parking problems in a packing of objects of different shapes. Frustration arises in granular packings owing to the interlocking of grains having different shapes. Different shaped particles have different sorts of excluded volume effects which leads to frustration in the packing. This geometrical feature is captured in this class of models. Hence, in this model, the complexity of the problem lies in the complexity of the particle arrangements in the packing. In a very general way we can define the model with a complex interaction matrix which tells us, for each particle, what are the constraints on the combinations of particles around it. For the sake of simplicity, in what follows, we define and study the model in its simplest version with two kinds of elongated particles.
In order to describe various experimentally observed properties of granular media, another class of frustrated lattice gas models with quenched disorder has also been introduced [@conihermann; @nicohans1]. Although in this paper we just focus on the [*Tetris*]{}-like models, it would be interesting to check the same phenomenology in this other class of frustrated models as well.
The simplest version of the [*Tetris*]{}-like model can be defined by considering a system of particles which occupy the sites of a square lattice tilted by $45^0$ (see Fig. \[figmodel\]), with periodic boundary conditions in the horizontal direction (cylindrical geometry) and a rigid wall at the bottom. Particles cannot overlap and this condition produces very strong constraints (frustration) on their relative positions. For instance in the simplest case of two kinds of elongated particles pointing in two (orthogonal) directions, the frustration implies that two identical particles (pointing in the same direction) cannot occupy neighboring sites in this direction (Fig. \[figmodel\]). There is no other form of interaction between particles, and in this sense the model is purely geometrical. The system is initialized by filling the container, by inserting the grains at the top of the system, one at a time, and letting them fall under gravity. The particles perform an oriented random walk on the lattice until they reach a stable position defined as a position in which they cannot fall any further because of other particles below them.
The density reached by this filling procedure is $\sim 0.747$ and is the lowest density a random packing can reach in this model with two kinds of elongated particles (the density of a packing is measured by averaging over the densities of each row in the lower half of the system). Higher densities are reached by “shaking” the system, a procedure described below. This procedure has been studied earlier in relation to the experimentally observed slow density increase which occurs in these models [@tetris] as the shaking continues. In this paper however we concern ourselves with probing the changes in the structure of the packing caused by the shaking procedure, by studying internal avalanching as a function of the density.
The effect of vibrations is introduced by the possibility of the particles moving up with a probability $p_{up}$ and of moving down with a probability $p_{down}=1-p_{up}$ The quantity $1/\ln \frac{1}{x}$, with $x=p_{up}/p_{down}$, plays the role of an effective temperature and can be related to the tap intensity amplitude. The shaking procedure we use can be divided into two alternating steps. First, in a [*heating*]{} process (tapping) the system is perturbed by putting $p_{up}\ne 0$ and performing a fixed number $N$ of attempts of movements per particle. At the end of this step, due to a nonzero $p_{up}$, the system is in an unstable state with many particles in positions unsupported by particles below. We now allow the system to relax under gravity setting $p_{up}=0$. The relaxation process ([*Cooling*]{}) is terminated when all the particles once more acquire stable positions. The system is now in a static state and the process of heating and relaxing the system is repeated a specified number of times. The density the system reaches depends, on average, on this number. The basic features of our model are robust with respect to variations in the exact Monte-Carlo procedure used.
This simple version of the Tetris model presents a trivial “antiferromagnetic” ground state. That is, the highest density packing that the system can reach with the above tapping procedure always has $ \rho =1$ and is ordered in one of two possible orderings : Even(odd) rows consist of rods with $+45^{0}$($-45^{0}$) orientations. A state with only one kind of ordering is hence called a “single-domain” state. The existing of only two distinct orderings is potentially a drawback since a real granular system contains much more disorder due to a wider shape distribution and the absence of a lattice. In order to incorporate this effect, the Tetris model can be modified by considering particles with more complicated shapes[@tetris; @segtet]. This prevents the occurrence of an ordered ground state. In this paper however we study exclusively the simpler model described, with rods of two different orientations, with a brief discussion of the more complicated case in section $V$.
Internal avalanches
===================
We study the effects of small perturbations on packings as a function of density, by studying in detail the phenomenon of internal avalanching within the scope of the models described above. Specifically we focus our attention on the rearrangements of grains generated in a static assembly by the extraction of a grain at the base. The creation of a void in the lattice destabilizes the neighboring grains above it. One of these may then fall down to fill the void, if the geometry of the packing allows the motion (i.e., if the orientation of the grain fits the local conformation). In this case, the net effect is that the void propagates one lattice step upwards destabilizing its neighbors in the layer above and so on. How effective this process is in causing the restructuring of the configuration depends on the precise structure of the packing. That large scale restructuring events are indeed possible is reflected in the fact that there are certain local configurations in which the motion of a single particle results in the motion of two particles above it, thus creating a second moving void. As a result of this “birth” process, moving voids can not only propagate up or get trapped but also multiply and hence lead to large avalanches.
We begin by preparing the system in the loose packed state by the procedure described in the previous section. As mentioned earlier, in this way the system attains in average a density of about $\rho=0.747$ ($\rho_{ld}$). The avalanche distribution in this state is studied by the following procedure. An initial state at the loose packed density is produced and a particle is randomly removed from the base. The total number of grains that move as a result of this removal is then calculated by checking the system row by row and letting all unstable particles settle under gravity. We have also checked that choosing unstable particles completely stochastically rather than row by row does not change the results quantitatively. By invoking the rules of stability introduced in this model, it is easy to see that when a particle is removed from the packing, it at most destabilizes all the particles within a cone with its apex at the removed particle (Fig. \[figmodel\]). Statistics for the avalanches is obtained by repeating this process of counting the number of unstable grains for various other initial states at the loose packed density. Thus the avalanche distribution obtained from this ensemble averaged procedure is indicative of this particular density. To obtain the distribution at higher densities the same ensemble averaging procedure is followed where now each member of the ensemble is generated by starting from an initial state at the loose packed density followed by a specified number of shakes. For each member of the ensemble, the magnitude of the internal avalanche is studied by counting as before the total number of particles rearranged as a result of removing one particle from the last row. The density that parametrises this avalanche distribution is just the average density of the ensemble. In everything that follows unless otherwise mentioned, this is the procedure we used to generate avalanche distributions at different densities, by averaging over statistics obtained for $100,000$ different initial conditions. A figure of an actual avalanche is shown in Fig \[figaval\]. The shaded dots represent the original positions of particles which moved as a result of the avalanche. The total number of these dots is then the quantity we measure as the size of the avalanche.
Results
=======
In studying the avalanche distribution as a function of the tapping density, we used two different values of $p_{up}$ ( $p_{up}=0.1$ and $p_{up}=0.5$) in order to test the sensitivity of the results to different procedures. The avalanche size distribution for the loose-packed density is shown in Figure \[figloose\] as a function of the size of the system. As can be seen, it follows a power-law $ P(s) \sim s^{-\tau}$ with an avalanche exponent $\tau$ close to $1.5$. The distribution for the higher densities, according to the tapping procedure with $p_{up}=0.1$ and $p_{up}=0.5$, are shown in Figures \[fighigh01\] and \[fighigh05\] for a single system size ($L_x=200,L_y=200$).
The density dependence of the avalanches is highly non-trivial. Since the loose packed density is the lowest that the packing can reach, shaking, as mentioned earlier, results in a monotonic increase of the density. It seems evident from Figure \[figloose\] and Figures \[fighigh01\] and \[fighigh05\] that the loose-packed density and other densities close to this value seem to exhibit a power-law behavior for the avalanche size distribution. At very high densities, when the structure is ordered (due to the ground state being a completely ordered one) the avalanche distribution is exponentially distributed (as will be clear later). Though this information is insufficient to infer the behaviour of intermediate densities, a possible hypothesis is that there exists a second order critical point located at some density $\rho_c$. Then one would expect that for densities larger than the critical one the system develops a characteristic size for the avalanches that acts as a cut-off for the avalanche size distribution. From this point of view one would then expect to be able to rescale all the avalanche size distributions obtained at different densities in one single scaling function such as $$P(s,\rho)= s^{-\tau}F(s{(\rho-\rho_c)}^{1/\sigma})
\label{eq:scaling}$$ where $\rho_c$ represents the location of the critical point and $P(s,\rho)$ is the probability for avalanches propagating in a medium of density $\rho$.
If we make this hypothesis of a single critical density even for the data, then the results for scaling the avalanche data in Figures (\[fighigh01\]) and (\[fighigh05\]) are shown in Figures \[figcoll\_tap01\] and \[figcoll\_tap05\] respectively. In each case, it is only the last three curves of the avalanche data that are scaled since the value of the critical density lies in between the lowest and the highest we have measured. As indicated the best value of the exponents $\tau$ and $\sigma$ seem to match for the two sets of data within the error bars. As for the values of $\rho_c$, though they seem to depend on the particular procedure used to generate the avalanches, we cannot rule out the possibility that they coincide within the error bar of our numerics.
In order to test this hypothesis of a single critical density in a simpler situation as well as elucidate the possible meaning of a critical density, we have also looked at a toy model which is a simple limiting case of the more general situation. As already mentioned, the [*Tetris*]{} model with rods of two orientations has a very simple ground state (highest density state) - the completely antiferromagnetic one. We take advantage of this fact by constructing the toy model in the following way. We begin with a $\rho =1$ completely antiferromagnetic state and generate lower density states by randomly removing particles. After each removal, the system is allowed to re-settle into a stable state via the avalanche dynamics already described. As a further simplification, we consider periodic boundary conditions in both $X$ and $Y$ directions. This allows us to eliminate system size effects as well as edge effects on the avalanche statistics. We call this example the Fully Periodic Single Domain (FPSD) model.
On this simplified version of the model, we perform the same set of measurements described earlier, in order to measure the avalanche distribution as a function of the density. A given density here is accessed by the removal of a certain number of particles instead of by shaking. Once a given density is reached, the avalanche distribution is measured by randomly removing a particle from the system and counting the number of particles destabilized as a result. A distribution is obtained by the same procedure of averaging over a ensemble of systems each originating from a $\rho =1$ state from which a given number of particles (corresponding to the density we want to access) is removed. The numerical results for avalanche size distribution as a function of the density is shown in Fig. \[figsingdom\]. In this case the scaling hypothesis is satisfied and all the curves collapse for a critical density $\rho_c^{SD} \sim 0.76 \pm 0.01$, with an avalanche distribution decaying as a power with the exponent $\tau^{\prime} = 1.45 \pm .05$ and $1/\sigma = 1.5 \pm 0.1$ (see Fig. (\[figcoll\_singdom\])). As can be seen, the values of the scaling exponents are similar to those obtained for the original data.
The rationale for the existence of a critical point can be understood by mapping the avalanche to a problem of a branching process. The avalanche seed represents the insertion of a vacancy in the system and the subsequent evolution of the avalanche is just the propagation of this vacancy. There is an effective probability $p_1 (\rho)$ for the vacancy to move through the medium, a probability $p_0 (\rho)$ for stopping and a probability $p_2 (\rho)$ for branching, [*i.e.*]{} meeting some other frozen vacancy and freeing it. In this mean field description, the critical point lies at the density which satisfies the condition $p_2(\rho)= p_0(\rho)$. In the FPSD case considered this mapping can be made quantitative by identifying the configurations leading to branching and death. It is possible to have in this way a mean-field estimate of the critical density, whose value is in good agreement with the numerical data presented in Fig. (\[figcoll\_singdom\]).
The FPSD does not fully reflect the complexity of the original problem since we basically probe an ordered state here while the configurations ensuing from shaking the loose-packed density (used to generate Figures \[fighigh01\] and \[fighigh05\]) are disordered. The value of the critical density in the FPSD signifies the ordered loosest packing that a single domain state with periodic boundary conditions can achieve. The significance of a critical density in the “shaking” case is not so clear. The data seems to suggest a density larger than the loose packed value. However as mentioned earlier, we are unable to conclusively pick out a critical density in the latter case due to the large error bars.
It is obvious from Fig. (\[figcoll\_singdom\]) that the quality of the collapse is much better than in the case with the open boundaries. The error bars on the various parameters of the collapse are also reduced. This seems to imply that boundary conditions and surface effects play a dominant role in affecting avalanches. One reason that they might do this is the following. In the FPSD case, by virtue of the boundary conditions, the system is homogeneous and more or less has the same density everywhere. Whereas in the curves generated by shaking, the system develops a density profile with an interfacial region towards the top where the density decays to zero. This affects avalanche statistics by enhancing avalanches over a certain size. Avalanches which reach the low density region towards the top can very quickly move right to the surface. The width of the density profile can be reduced by tapping very gently, [*i.e.*]{} shaking with a very small amplitude. This is possibly an important point to be considered and taken into account when doing real life experiments in the laboratory to study internal avalanching.
Conclusions and Discussion
==========================
Our main conclusions are the following. The avalanche distribution is very sensitive to the internal structure of the packing and clearly appears to be long-ranged and decaying as a power law for a range of densities. As obvious in Figs. ( \[fighigh01\]) and ( \[fighigh05\]) a slight change in the density changes the avalanche distribution considerably. However as evident from the figures, the avalanche size cut-off is dominated by the system size over a range of densities, indicating that they are all effectively critical for the system sizes measured. Scaling behavior is thus visible only for the largest densities we have studied. The cleanest case, the case of the FPSD toy model seems to indicate that there is a critical density about which other densities obey the scaling relation Eq. \[eq:scaling\] (Figs. \[figsingdom\] and \[figcoll\_singdom\]). This might indicate one of two possibilities. The first is that there is a unique critical density $ \rho_c \geq \rho_{ld} $ even in the case of Figs. \[fighigh01\] and \[fighigh05\] and we have to go to systems large enough that the density dependent cut off appears clearly in the avalanche distribution. A second possibility is that the avalanche distribution and hence the critical density crucially depends on the initial state begun with as well as the procedure under which the system reaches higher densities (as for example the case considered here, the case of different values of $ p_{up}$). Our numerical data thus far is unable to distinguish conclusively between these two possibilities.
In order to understand the role that the dynamics under which the system evolves plays in determining the internal structure of the medium (and hence the avalanche distribution) we have also studied the following dynamics as an alternative to shaking the system. We begin as usual with the loose packed density and take out a particle from the bottom row, hence initiating an avalanche. After the avalanche terminates, we add the particle back to the system at a random position on the top. This dual procedure of removing a particle and adding it back randomly to the top is continued till the system reaches a steady state in which the avalanche distribution is studied. We find that in this case, the steady state the system reaches is different from any reached in the packing procedure in that the avalanche distribution decays with a different power $ \tau =1.75 \pm 0.05$. Further details of this procedure as well as a description of the steady state reached are reported elsewhere [@sketal].
There are several possibilities which remain to be investigated. Amongst the most important of these is how much of this scenario holds for similar models with particles of more complicated shapes. An example is the [*Tetris*]{}-like model with “T”-shaped particles [@tetris; @segtet]. As mentioned earlier, this model has a highly degenerate ground state unlike the simpler model we have studied in this paper. It is of interest to investigate whether this property affects the scenario discussed above for the avalanche distribution as a function of density. While preliminary results indicate that the avalanche distribution is again a power-law in this case, we have yet to investigate in detail its density or dynamics dependence.
Acknowledgements {#acknowledgements .unnumbered}
================
We would like to thank S. Zapperi for useful comments. We are grateful to CEFIPRA for their support. In particular SK would like to acknowledge financial support under project no 1508-3/192. VL acknowledges financial support under project ERBFMBICT961220. This work has been partially supported from the European Network-Fractals under contract No.FMRXCT980183.
[90]{}
for a general introduction to the overall phenomenology of granular matter one can address the following papers: H.M. Jaeger and S.R. Nagel, Science [**255**]{}, 1523 (1992); H.M. Jaeger, S.R. Nagel and R.P. Behringer, Rev. Mod. Phys. [**68**]{}, 1259 (1996). D. Bideau and A. Hansen, eds. [*Disorder and Granular Media*]{}, (North-Holland, Amsterdam, 1993). A. Mehta, ed., [*Granular Matter: an interdisciplinary approach*]{}, (Springer-Verlag, New York, 1994). See also Proceedings of the NATO Advanced Study Institute on [*Physics of Dry Granular Media*]{}, Eds. H. J. Herrmann [*et al*]{}, Kluwer Academic Publishers, Netherlands (1998).
A. Coniglio and H.J. Herrmann, [*Physica A*]{}, [**225**]{}, 1 (1996).
M. Nicodemi, A. Coniglio and H.J. Herrmann, [*Phys. Rev. E*]{} [**55**]{}, 3962 (1997); [*J. Phys. A*]{} [**30**]{}, L379 (1997); [*Physica A*]{} [**240**]{}, 405 (1997).
E. Caglioti, V. Loreto, H.J. Herrmann and M. Nicodemi, [*Phys. Rev. Lett.*]{} [**79**]{}, 1575 (1997) and in preparation (1998).
E. Caglioti, A. Coniglio, H.J. Herrmann, V. Loreto and M. Nicodemi, [*Europhys. Lett.*]{} [**43**]{}, 591 (1998).
M. Nicodemi and A. Coniglio, preprint (1998), cond-mat/9803148.
S. Krishnamurthy, H. J. Herrmann, V. Loreto and S. Roux, in preparation (1998).
Figure Captions
[**Fig. 1**]{} An example of a stable configuration on which internal avalanche measurements can be made. If the circled particle is removed, from the lower most layer, it at most destabilizes all the particles within the cone shown. The boundary conditions are periodic in the horizontal direction ($X$).\
\
[**Fig. 2**]{} A picture of an avalanche. The shaded dots represent all the particles in the initial configuration which moved as a result of the removal of a particle from the lower most layer.\
\
[**Fig. 3**]{} Avalanche size probability distribution $P(s)$ for the loose-packed density for different system sizes. In increasing order, the system sizes are $Lx=100,Ly=100$, $Lx=200,Ly=200$, $Lx=200,Ly=400$ and $Lx=100,Ly=900$. Average over $100,000$ realizations.\
\
[**Fig. 4**]{} $P(s)$ vs. $s$ for the densities mentioned for $p_{up}=0.1$.\
\
[**Fig. 5**]{} The scaling plot $f(s^{*})$ vs. $s^{*}$ of the data shown in Fig. 4 where $s^{*} = s (\rho -
\rho_c)^{1/{\sigma}}$ and $ f(s^{*}) = (s^{*})^{-\tau} F(s^{*})$. The last three densities are scaled with parameters $\tau= 1.5 \pm 0.1$, $1/{\sigma} = 1.5 \pm 0.1$ and $\rho_c=0.77 \pm 0.1$\
\
[**Fig. 6**]{} $P(s)$ vs. $s$ for the densities $\rho =0.76-0.83$ for $p_{up}=0.5$.\
\
[**Fig. 7**]{} The scaling plot $f(s^{*})$ vs. $s^{*}$ of the data shown in Fig. 6 where $s^{*} = s (\rho -
\rho_c)^{1/{\sigma}}$ and $ f(s^{*}) = (s^{*})^{-\tau} F(s^{*})$. The densities scaled are $\rho =0.81-0.83$ of the data shown in Figure 6. The scaling parameters are $\tau= 1.5 \pm 0.1$, $1/{\sigma} = 1.5 \pm 0.1$ and $\rho_c=0.79 \pm 0.1$\
\
[**Fig. 8**]{} $P(s)$ vs. $s$ for the densities mentioned for the FPSD model.\
\
[**Fig. 9**]{} The scaling plot for the data in Fig. 8. with the scaling parameters $ \tau =1.45 \pm 0.05$, $1/{\sigma} = 1.5 \pm 0.1$ and $\rho_c
= 0.76 \pm 0.01$\
\
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Scheduling of jobs on multiprocessing systems has been studied extensively since last five decades in two well defined algorithmic frameworks such as offline and online. In offline setting, all the information on the input jobs are known at the outset. Whereas in online setting, jobs are available one by one and each job must be scheduled irrevocably before the availability of the next job. Semi-online is an intermediate framework to address the practicability of online and offline frameworks. Semi-online scheduling is a relaxed variant of online scheduling, where an additional memory in terms of buffer or an *Extra Piece of Information(EPI)* is provided along with input data. The *EPI* may include one or more of the parameter(s) such as size of the largest job, total size of all jobs, arrival sequence of the jobs, optimum makespan value or range of job’s processing time. A semi-online scheduling algorithm was first introduced in 1997 by Kellerer et al. They envisioned semi-online scheduling as a practically significant model and obtained improved results for $2$-identical machine setting. This paper surveys scholarly contributions in the design of semi-online scheduling algorithms in various parallel machine models such as identical and uniformly related by considering job’s processing formats such as preemptive and non-preemptive with the optimality criteria such as *Min-Max* and *Max-Min*. The main focus is to present state of the art competitive analysis results of well-known semi-online scheduling algorithms in a chronological overview. The survey first introduces the online and semi-online algorithmic frameworks for the multi-processor scheduling problem with important applications and research motivation, outlines a general taxonomy for semi-online scheduling. Fifteen well-known semi-online scheduling algorithms are stated. Important competitive analysis results are presented in a chronological way by highlighting the critical ideas and intuition behind the results. An evolution time-line of semi-online scheduling setups and a classification of the references based on *EPI* are outlined. Finally, the survey concludes with the exploration of some of the interesting research challenges and open problems.'
author:
- Debasis Dwibedy and Rakesh Mohanty
title: 'Semi-online Scheduling: A Survey'
---
Introduction {#sec:Introduction}
============
Scheduling deals with allocation of resources to jobs in some order with application specific objectives and constraints. The concept of scheduling was introduced to address the following research question \[1\]: *Given a list of $n$ jobs and $m$($\geq 2$)machines, what can be a sequence of executing the jobs on the machines such that all jobs are finished by latest time possible?* Scheduling has now become ubiquitous in the sense that it inherently appears in all facets of daily life. Everyday, we involve ourselves in essential activities such as scheduling of meetings, setting of deadlines for projects, scheduling the maintenance periods of various tools, planning and management of events, allocating lecture halls to various courses, organizing vacations, work periods and academic curriculum etc. Scheduling finds practical applications in broad domains of computers, operations research, production, manufacturing, medical, transport and industries \[17\]. Widespread applicability has made scheduling an exciting area of investigation across all domains.\
Scheduling of jobs on multiprocessing systems has been studied extensively over the years in well defined algorithmic frameworks of offline and online scheduling \[6, 16, 17, 41, 49\]. A common consideration in *offline scheduling* is that all information about the input jobs are known at the outset. However, in most of the current practical applications, jobs are given incrementally one by one. An irrevocable scheduling decision must be made upon receiving a job with no prior information on successive jobs \[2, 12\]. Scheduling in such applications is known as *online scheduling*. In this survey, we study a relaxed variant of online scheduling, known as *semi-online scheduling*, where some *extra piece of information* about the future jobs are known at the outset. We present the structure and organization of our survey in Figure 1.
![Organization of our Survey[]{data-label="fig:structureofoursurvey.png"}](structureofoursurvey.PNG)
Algorithmic Frameworks {#subsec:Algorithmic Frameworks}
----------------------
We present three algorithmic frameworks such as offline, online and semi-online based on availability of input information in processing of a computational problem as shown in Figure \[fig:processingframeworks.png\].
![Algorithmic Frameworks (a) Offline (b) Online (c) Semi-online[]{data-label="fig:processingframeworks.png"}](algorithmframeworks.PNG)
- In **Offline framework**, complete input information is known at the outset. Let us consider a set $I$=$\{i_1,i_2,...,i_n\}$ representing all inputs of a computational problem $X$. In offline framework, $I$ is known prior to construct a solution for $X$. The algorithm designed for computation of $I$ in the offline framework is known as *offline algorithm*. An offline algorithm processes all inputs $I$ simultaneously to produce the final output $o$.
- In **Online framework**, the inputs are given one by one in order. Each available input must be processed immediately with no information on the successive inputs. In online framework, at the time step $t$, the input sequence $I_t$:$<i_1,i_2,...,i_{t-1},i_t>$ is known and must be processed irrevocably with no information on future input sequence $<i_{t+1},...,i_{n-1},i_n>$, where $t\geq 1$. The algorithm designed for computation of $I$ in the online framework is known as *online algorithm*. An online algorithm produces a partial output $o_t$ for each input $I_t$ on the fly, where, $1\leq t\leq n-1$ before producing the final output $o_n$.
- In **Semi-Online framework** inputs are given one by one like online framework along with some *Extra Piece of Information(EPI)* on future inputs. At any time step $t$, a semi-online algorithm receives input sequence $I_t$ with an *EPI* and processes them irrevocably to obtain a partial output $o_t$ on the fly, where $1\leq t\leq n-1$ before producing the final output $0_n$.
Semi-online is an intermediate framework to address the practicability and limitations of online and offline frameworks. In most of the current practical scenarios, neither all the inputs are available at the beginning nor the inputs occur exclusively in online fashion, but may occur one by one with additional information on the successive inputs. For example, an online *video on demand* application receives requests for downloading video files on the fly, however, it knows the highly requested video file and the largest video file among all video files before processing the current request \[79\]. A related model to semi-online framework is the *advice model*, where the *EPI* has been referred to as *bits of advice*. A comprehensive survey on advice models can be found in \[92\].
Semi-online Scheduling Problem {#subsec:Semi-online Scheduling Problem}
------------------------------
Semi-online scheduling \[13\] is a variant of online scheduling with an *EPI* on future jobs or with additional algorithmic extensions by allowing two parallel policies to operate on each incoming job. It may also include a *buffer* of finite length for pre-processing of a newly arrived job before the actual assignment. We now formally define the semi-online scheduling problem by presenting inputs, constraints and output as follows.
- Inputs:
- A sequence $J:<J_1, J_2...,J_n>$ of $n$ jobs with corresponding processing time of $p_i$, where $1\leq i\leq n$ and $p_i> 0$ are revealed one by one for processing on a list $M$=$(M_1, M_2,...,M_m)$ of $m$ parallel machines, where $m\geq 2$ and $n>>>m$.
- An *EPI* such as arrival order of the jobs or largest processing time or upper and lower bounds on the processing time of the incoming jobs is given a priori.
- Constraints:
- Each incoming job $J_i$ must be assigned irrevocably to one of the machines $M_j$ as soon as $J_i$ is given.
- Jobs are non-preemptive, however the preemptive variant of the problem supports job splitting to execute distinct pieces of a job at non-overlapping time spans on the same or different machines.
- Output: Generation of a schedule, representing assignment of all jobs over $m$ machines.(we shall discuss about the output parameters and objectives in section 2.3).
Practical Applications {#subsec: Practical Applications}
----------------------
Here, we discuss some of the important applications, where semi-online scheduling serves as a major algorithmic framework.
- **Resource Management in Operating System** \[2\]: In a multi-user, time-shared operating system, it is not known at the outset the sequence of jobs or the number of jobs that would be submitted to the system. Here, jobs are given to the scheduler over time. However, it is the inherent property of the scheduler to make an educated guess about the *maximum and minimum time required to complete a resident job*. The objective is to irrevocably assign the required computer resources such as memory, processors immediately upon the availability of a job to attain a minimum completion time.
- **Distributed Data Management** \[15\]: Distributed and parallel systems often confronted to store files of varying sizes on limited capacity remote servers. It is evident that files are submitted from a known source on the fly and each received file must be assigned immediately to one of the remote servers. The central scheduler of the system is handicapped about the successive submissions prior to make an irrevocable assignment. However, it is known for an instance that the submitting source stored the files in $k$ unit capacity servers, which provides a hint for the *total size of files* to be received. The challenge is to store the files on the remote servers with minimum storage requirement.
- **Server Request Management or Web Caching** \[40\]: In a client-server model, it is not known in advance the number of requests that would be submitted to the remote servers nor the time required to process the requests. However, the hierarchical organizations of servers can serve as an extra piece of information for scheduler to cater different level of services to the requests with a broader objective of processing all requests as latest as possible.
- **Production and Manufacturing:** Orders from clients arrive on the fly to a production system. The resources such as human beings, machinery equipment(s) and manufacturing unit(s) must be allocated immediately upon receiving each client order with no knowledge on the future orders. However, one could estimate the *minimum or maximum time* required to complete the order. Online arrival of the orders have high impact on the renting and purchasing of the high cost machines in the manufacturing units.
- **Maintenance and upgrade of industrial tools \[52\]:** Scheduling of various maintenance and operational activities for modular gas turbine aircraft engines. The goal is to distribute different activities to the machines in such a way that the loads of the the machines will be balanced. The common practice is to maximize load of the least loaded machine.
Performance Measure for Semi-online Scheduling {#subsec:Performance Measure of Semi-online Scheduling}
----------------------------------------------
Traditional techniques \[17\] for analyzing the performance of offline scheduling algorithms are largely relied on the entire job sequence, therefore are insignificant in the performance evaluation of semi-online algorithms, which operate on single incoming input at any given time step with minimal knowledge on the future arrivals.\
**Competitive analysis** method \[8\] measures the worst-case performance of a semi-online algorithm $ALG$ designed either for a cost minimization or maximization problem by evaluating *competitive ratio(CR)*. For a cost minimization problem, *CR* is defined as the smallest positive integer $k(\geq 1)$, such that for all valid sequences of inputs in the set $I$= $\{i_1, i_2,...,i_n\}$, we have $C_{ALG}\leq k\cdot C_{OPT}$, where $C_{ALG}$ is the cost obtained by semi-online algorithm $ALG$ for any sequence of $I$ and $C_{OPT}$ is the optimum cost incurred by the optimal offline algorithm $OPT$ for $I$. The *Upper Bound*(UB) on the *CR* obtained by $ALG$ guarantees the maximum value of *CR* for all legal sequences of $I$. The *Lower Bound*(LB) on the *CR* of a *semi-online problem $X$* ensures that there exists an instance of $I$ such that any semi-online algorithm $ALG$ must incur a cost $C_{ALG}\geq b \cdot C_{OPT}$, where $b$ is referred to as *LB* for $X$. The performance of $ALG$ is considered to be *tight*, when $ALG$ ensures no gap between achieved *LB* and *UB* for the problem considered. Sometimes, the performance of $ALG$ is referred to as *tight* if $C_{ALG}$=$k\cdot C_{OPT}$. For a cost maximization problem, *CR* is defined as the infimum $k$ such that for any valid input sequence of $I$, we have $k\cdot C_{ALG}\geq C_{OPT}$. The objective of a semi-online algorithm is to obtain a *CR* as closer as possible to $1$(strictly greater than or equal to $1$).
Research Motivation {#subsec: Research Motivation}
-------------------
Research in semi-online scheduling has been pioneered by the following non-trivial issues.
- The offline $m$-machine($m\geq2$) scheduling problem with makespan minimization objective has been proved to be *NP-complete* by a polynomial time reduction from well-known *Partition* problem \[7\]. Let us consider an instance of scheduling $n$ jobs on $m$ parallel machines, where $n>>>m$. There are $m^n$ possible assignments of jobs. An optimum schedule can be obtained in worst case with probability $\frac{1}{m^n}$. Further, unavailability of prior information about the whole set of jobs poses a non-trivial challenge in the design of efficient algorithms for semi-online scheduling problems.
- Given an online scheduling problem considered in the semi-online framework, the non-trivial question raised is:\
*What can be an additional realistic information on successive jobs that is necessary and sufficient to achieve $1$-competitiveness or to beat the best known bounds on the CR?*
- A semi-online algorithm is equivalent to an *online algorithm with advice* in the sense that an *EPI* considered in semi-online model can be encoded into bits of advice. The quantification of information into bits will help in analyzing the advice complexity of a semi-online algorithm. Any advancement in semi-online scheduling may lead to significant improvements in the best known bounds obtained by the advice models.
- Semi-online model is practically significant than the advice model as it considers feasible information on future inputs unlike bits of advice, which sometimes may constitute an unrealistic information.
Scope and Uniqueness of Our Survey {#subsec:Scope and Uniqueness of Our Survey }
-----------------------------------
**Scope.** This paper surveys scholarly contributions in the design of semi-online scheduling algorithms in various parallel machine models such as identical and uniformly related by considering job’s processing formats such as preemptive and non-preemptive with the optimality criteria such as *Min-Max* and *Max-Min*. The aim of the paper is to record important competitive analysis results with the exploration of novel intuitions and critical ideas in a historical chronological overview.\
**Uniqueness.** This is a comprehensive survey article on semi-online scheduling, which describes the motivation towards semi-online scheduling research, outlines a general taxonomy, states fifteen well-known semi-online scheduling algorithms, presents state of the art contributions, explains critical ideas, overviews important results in a chronological manner, organized by *EPI* considered in various semi-online scheduling setups. Several non-trivial research challenges and open problems are explored for future research work. Important references are grouped together in a single article to develop basic understanding, systematic study and to update the literature on semi-online scheduling for future investigation.
Taxonomy of Semi-Online Scheduling {#sec: Taxonomy of Semi-Online Scheduling}
===================================
The basic terminologies, notations and definitions related to semi-online scheduling are presented in Table \[tab:Basic Terminologies Notations and Definition\].
[ccp[8.0cm]{}]{} **Terms** & **Notations** &**Definitions/Descriptions/Formula**\
Job\[1\] & $J_i$ & Program under execution, which consists of a finite number of instructions. A job is also referred to as a collection of at least one smallest indivisible sub task called *thread*. Unless specified explicitly, we assume that a job consists of single thread only. Here, we use terms job and task in the same sense.\
Machine\[1\] & $M_j$ & An automated system capable of processing some jobs by following a set of rules. Machine can be a router, web server, robot, industrial tool, processing unit or processor, which is capable of processing the jobs. Here, we use terms machine and processor in the same sense.\
Processing Time\[1-3\] & $p_{ij}$ & Total time of execution of a job $J_i$ on machine $M_j$. For identical machines $p_{ij}$=$p_i$.\
Largest Processing Time & $p_{max}$ & $\max\{p_i|1\leq i\leq n\}$.\
Release Time\[2, 25\] & $r_i$ & The time at which any job $J_i$ becomes available or ready for processing.\
Completion Time\[3, 25\] & $c_i$ & The time at which any job $J_i$ finishes its execution\
Deadline\[3, 25\] & $d_i$ & Latest time by which $J_i$ must be finished.\
Load \[11\] & $l_j$ & Sum of processing times of the jobs that have been assigned to machine $M_j$.\
Speed \[2, 3\] & $S_j$ & The number of instructions processed by machine $M_j$ in unit time\
Speed Ratio & $s$ & The ratio between the speeds of two machines. For $2$-machines with speeds $1$ and $\frac{1}{S}$ respectively. We have speed ratio $s$ = $\frac{1}{\frac{1}{S}} = S$\
Idle Time \[1, 5\] & $\varphi$ & The duration of time at which a machine is not processing any task. During the idle time a machine is called idle.\
Optimal Makespan \[2\] & $C_{OPT}$ & $C_{OPT}$= $\max\{p_{max},\hspace*{0.2cm} \frac{1}{m}\cdot \sum_{i=1}^{n}{p_i}\}$\
\[tab:Basic Terminologies Notations and Definition\]
\
Based on the literature study, a general taxonomy of semi-online scheduling is outlined using the three parameters($\alpha| \beta| \gamma$) based framework of Graham et al. \[6\] in Figure \[fig:semionlinescheduling.png\]. Here, $\alpha$ represents parallel machine models, $\beta$ specifies different job characteristics and $\gamma$ represents optimality criteria.
![A General Taxonomy of Semi-Online Scheduling[]{data-label="fig:semionlinescheduling.png"}](semionlinescheduling.png)
Parallel Machine Models($\alpha$) {#subsec:Machine Models}
---------------------------------
Parallel machine models support simultaneous execution of multiple threads of a single job or a number of jobs on $m$ machines, where $m\geq 2$. Semi-online scheduling problem has been studied in parallel machine models such as identical, uniformly related(or related machines in short) and unbounded batch machines. One model differs from another based on its processing power defined in the literature \[6\] as follows.
- **Identical Machines(P)**: Here, all machines have equal speeds of processing any job $J_i$. We have $p_{ij}=p_i$, $\forall M_j, 1\leq j \leq m$.
- **Related Machines (Q)**: Here, the machines operate at different speeds. For a machine $M_j$ with speed $S_j$, execution time of job $J_i$ on $M_j$ is $p_{ij}=\frac{p_i}{S_j}$.
- **Unbounded Batch Machine (U-batch)**: A batch machine receives jobs in batches, where a batch($U(t)$) is formed by considering all jobs that are received at time $t$. The jobs in $U(t)$ are processed at the same time in the sense that the completion time($U(c_t)$) of all jobs in a batch are same. The processing time of $U(t)$ is $U(p_t)$=$\max \{p_1, p_2,...,p_k\}$ and the completion time $U(c_t)$=$t+U(p_t)$, where $k$ is the size of $U(t)$ i.e. the number of jobs in a batch. When the size of the batches are not bounded with any positive integers, then it is called *unbounded batch machine* with $k$=$\infty$.
Job Characteristics($\beta$) {#subsec:Job Characteristics}
----------------------------
Job characteristics describe the nature of the jobs and related characteristics to job scheduling \[6, 49\]. All jobs of any scheduling problem must possess at least one of the characteristics specified in set $\beta=\{\beta_1, \beta_2, \beta_3, \beta_4, \beta_5\}$. In semi-online scheduling, a new job characteristic $\beta_6$ is introduced to represent *extra piece of information(EPI)* on future jobs. The job characteristics are presented as follows: $\beta_1$ specifies whether *preemption* or job splitting is allowed, $\beta_2$ specifies precedence relations or dependencies among the jobs, $\beta_3$ specifies release time for each job, $\beta_4$ specifies restrictions related to processing times of the jobs, for example, if $\beta_4$=$1$, that means $ p_{i}=1$, $\forall J_i$, $\beta_5$ specifies deadline($d_i$) for each job $j_i$, indicating the execution of each $j_i$ must be finished by time $d_i$, otherwise an extra penalty may incur due to deadline over run. Let us throw more clarity on some of the important job characteristics as follows.\
**Preemption(pmtn)** allows splitting of a job into pieces, where each piece is executed on the same or different machines in non-overlapping time spans.\
**Non-preemption(N-pmtn)** ensures that once a job $J_i$ with processing time $p_i$ begins to execute on machine $M_j$ at time $t$, then $J_i$ continues the execution on $M_j$ until time $t+p_i$ with no interruption in between.\
**Precedence Relation** defines dependencies among the jobs by the partial order ’$\prec$’ rule on the set of jobs \[5\]. A partial order can be defined on two jobs $J_i$ and $J_k$ as $J_i$ $\prec$ $J_k$, which indicates execution of $J_k$ never starts before the completion of $J_i$. The dependencies among different jobs can be illustrated with a precedence graph $G(p, \prec)$, where each vertex represents a job $J_i$ and labeled with its processing time $p_i$. A directed arc between two vertices in $G(p, <)$ i.e $J_i$ $\rightarrow$ $J_k$ represents $J_i$ $\prec$ $J_k$, where $J_i$ is referred to as *predecessor* of $J_k$. If there exists a cycle in the precedence graph, then scheduling is not possible for the jobs. When there is no precedence relation defined on the jobs, then they are said to be *independent*. We represent precedence relation among the jobs through precedence graphs by considering three jobs $J_1$, $J_2$, $J_3$ and their dependency relations as shown in Figure \[fig:precedencegraph.png\].
![(a) Cyclic dependencies among jobs. (b) $J_2$ $\prec$ $J_3$ and $J_1$ is independent of $J_2 $, $J_3$. (c) All jobs are independent.[]{data-label="fig:precedencegraph.png"}](precedencegraph2.PNG)
\
**Extra Piece of Information(EPI)** is the additional information given to an online scheduling algorithm about the future jobs. Motivated by the interactive applications, a number of *EPI*s have been considered in the literature(see the recent surveys \[92\], \[108\]) to gain a significant performance improvement over the pure online scheduling policies \[92\]. We now present the definitions and notations of some well studied *EPI*s as follows.
- **Sum(*T*).** $\sum_{i=1}^{n}{p_i}$ Total size of all jobs \[13\].
- **Max($p_{max}$).** $\max \{p_i|1\leq i\leq n\}$ Largest processing time or largest size job \[20\].
- **Optimum Makespan(*Opt*).** Value of the optimum makespan is often represented by the following two general bounds \[15\].\
$C_{OPT}\geq \frac{1}{m}\cdot\sum_{i=1}^{n}{p_{ij}}$ and\
$C_{OPT}\geq p_{max}$.
- **Tightly Grouped Processing time (*TGRP*).** Lower and upper bounds on the processing times of all jobs \[20\]. Some authors \[22, 31, 45\] considered either lower bound *TGRP(lb)* or upper bound *TGRP(ub*) on the processing times of the jobs.
- **Arrival Order of Jobs.** $p_{i+1} \leq p_i$, for $1\leq i\leq n$ Jobs arrive, in order of non-incresing sizes (*Decr*) \[21\] or $r_{i+1} \geq r_i$ in order of non-decreasing release times (*Incr-r*) \[70,87\].
- **Buffer(B(k)).** A buffer (B(k)) is a storage unit of finite length $k$($\geq 1$), capable of storing at most $k$ jobs \[13\]. The weight of B(k) is $w(B(k))\leq \sum_ {i=1}^{k}{p_i}$. Availability of buffer allows an online scheduling algorithm either to keep an incoming job temporarily in the buffer or to irrevocably assign a job to a machine in case the buffer is full \[13\]. Therefore, information about $k+1$ future jobs is always known prior to make an efficient scheduling decision. The following variations in the buffer length and usage of buffer have been explored in the literature: buffer with length $k$($\geq 1$) i.e *B(k)* \[13\], buffer with length 1 i.e *B(1)* \[13, 14\] and re-ordering of buffer presented as *re B(k)* \[56\].
- **Information on Last Job.** It is known in advance that last job has the largest processing time i.e. $p_n$=$p_{max}$, this *EPI* is denoted by *LL* in \[26\]. In \[28\], it is considered that several jobs arrive at the same as last job and this *EPI* is denoted by *Sugg*.
- **Inexact Partial Information.** Inexact partial information is also referred to as *disturbed partial information*, which deals with the scenario, where the extra piece of information available to the online algorithm is not exact. For example, the algorithm knows a nearest value of the actual *Sum* but not the exact value. This *EPI* is represented as *$dis Sum$* in \[53\].
- **Reassignment of Jobs($reasgn$).** Once all jobs are assigned to the machines, again they can be reallocated to different machines with some pre-defined conditions. Several conditions on reassignment policies have been proposed in the literature \[57, 62\] such as reassign the last $k$ jobs, we represent as $reasgn(last(k))$, reassign arbitrary $k$ jobs i.e. $reasgn(k^*)$, reassign only the last job of all machines i.e. $reasgn{(last)}^*$, reassign last job of any one of the machines, represented by $reasgn({(last)}^1)$.
- **Machine availability ($mchavl$).** All machines may not available initially. Machines are available on demand and the release time ($r_j$) of machine $M_j$ is known in advance \[64\]. Some authors have also considered the scenario where one machine is available for all jobs and other machine is available for few designated jobs \[82\].
- **Grade of Service (GOS) or Machine Hierarchy.** It is known a priori that machines are arranged in a hierarchical fashion to cater different levels of services to the jobs with some defined *GOS* \[36, 46\]. For example, if a *GOS* of $g_2$ is defined for any job $J_i$, then $J_i$ can only be assigned to machine $M_2$ and if $J_i$ has *GOS* of $g_1$, then it can be scheduled on any of the machines.
Optimality Criteria($\gamma$) {#subsec:Optimality Criteria}
-----------------------------
Several optimality criteria or output parameters have been investigated in the offline and online settings \[17, 49\]. However, in semi-online scheduling the following output parameters have been considered mostly: *makespan* and *load balancing*.
- **Makespan($C_{max}$)** represents completion time($c_i$) of the job that finishes last in the schedule, $C_{max}$=$\max \{c_i|1\leq i\leq n\}$ or $C_{max}$=$\min\{l_j|1\leq j\leq m\}$. The objective is to *minimize $C_{max}$*, otherwise termed as minimization of the load of highest loaded machine*(min-max)*.
- **Load Balancing** describes the objective to *maximize the minimum machine load(max-min)* or *machine cover*. The scheduler assigns certain number of jobs to each machine for the processing of $n$ jobs on $m$ machines. Each job $J_i$ adds $p_i$ amount of *load* to the assigned machine $M_j$. The goal is to maximize the minimum load occurs on any of the machines so as to keep a balance in the incurred loads among all machines. We refer $C_{min}$ to represent *max-min* objective. As an example, Figure \[fig:loadprofiles.png\] shows the loads of machines during the processing of a specified number of jobs.
![Timing Diagram of a Sample Schedule Showing Loads of Machines[]{data-label="fig:loadprofiles.png"}](loadprofiles.png)
***Examples:*** We present various semi-online scheduling setups based on the three fields ($\alpha| \beta| \gamma$) classification format as shown in Table \[tab: Representation of Semi-online Problems \].\
**Setup($\alpha| \beta| \gamma$)** **Descriptions**
------------------------------------ --------------------------------------------------------------------------------------------------------------------------
$P_2| Sum | C_{max}$ 2-identical machines $|$ no preemption, total processing time $|$ min-max.
$P_2 | Sum, Max | C_{max}$ 2-identical machines $|$ no preemption, total processing time and maximum size job $|$ min-max.
$P_m | B(k) | C_{max}$ $m$-identical machines $|$ no preemption, given a buffer of length $k$ $|$ min-max
$Q_2| Decr | C_{max}$ 2-uniform related machines $|$ no preemption, jobs arrive in non-increasing order of their processing times $|$ min-max.
$Q_2| pmtn, TGRP | C_{min}$ 2-related machine $|$ preemption, lower and upper bounds on the processing times of the jobs $|$ max-min.
$U-batch| Max | C_{max}$ Unbounded batch machine $|$ no preemption, maximum size job $|$ min-max.
: A Sample Format For Representing Semi-online Scheduling Setups
\[tab: Representation of Semi-online Problems \]
\
Well-Known Semi-Online Scheduling Algorithms {#sec:Well-Known Semi-Online Scheduling Algorithms}
============================================
For developing a basic understanding on semi-online policies, we represent fifteen well-known semi-online scheduling algorithms as follows.
- Algorithm **$H_1$** was proposed by Kellerer et al. \[13\] for $P_2|B(k)|C_{max}$. Algorithm $H_1$ assigns first $k$ incoming jobs to the buffer $B(k)$, where $k \geq 1$. When $({k+1})^{th}$ job arrives, then a job $J_i$ is selected from the buffer, where $J_i \in \{J_1, J_2,.......J_k, J_{k+1}\}$ and is scheduled on machine $M_1$ such that $l_1 + p_i \leq \frac{2}{3} (l_1 + l_2 + w(B))$. If such a $J_i$ does not exist, then any arbitrary job is picked up from the buffer and is assigned to machine $M_2$. Here, $l_1, l_2$ are the loads of machines $M_1, M_2$ respectively before assigning $J_i$ and $w(B)= \sum_{i=1}^{k+1}{p_i}$.
- **Algorithm $H_3$** was proposed by Kellerer et al. \[13\] for $P_2|Sum|C_{max}$. Algorithm **$H_3$** schedules each available job $J_i$ on machine $M_1$ as long as $l_1+p_i\leq (\frac{1}{3})\cdot T$, where $T$=$\sum_{i=1}^{n}{p_i}$ and $l_1$ is the load of $M_1$ before the assignment of $J_i$. If $l_1+p_i\leq (\frac{2}{3})\cdot T$, then algorithm $H_3$ schedules $J_i$ on $M_1$ and the remaining jobs $J_t$ are scheduled on machine $M_2$, where $(i+1)\leq t\leq n$.
- Algorithm **Premeditated List Scheduling(PLS)** is due to He and Zhang \[20\] for $P_2|Max|C_{max}$. Algorithm *PLS* assigns each incoming job $J_i$ to machine $M_1$ as long as $p_i\neq p_{max}$ and $l_1 + p_i \leq 2\cdot (p_{max})$, otherwise, $J_i$ is scheduled on machine $M_2$. Thereafter, each incoming job is scheduled on machine $M_j$ for which $l_j$=$\min\{l_1, l_2\}$.
- **Algorithm $H$** is due to Angelelli \[22\] for $P_2|Sum|C_{max}$. Algorithm $H$ assigns an incoming job $J_i$ to machine $M_j$ for which $l_j$=$\max\{l_1, l_2\}$ if $V\geq \max\{|l_1-l_2|, p_i\}$, else schedules $J_i$ on $M_j$ for which $l_j$=$\min\{l_1, l_2\}$, where $V$=$T-(l_1+l_2+p_i)$.
- Algorithm **Ordinal** is due to Tan and He \[23\] for the setup $Q_2|Decr|C_{max}$. Algorithm *Ordinal* schedules all jobs on machine $M_2$ for speed ratio $s\geq (1+\sqrt{3})$. For $s\in [s(k-1), s(k))$, $k\geq 1$, the sub set of jobs $\{J_{ki}, J_{3i+3}|i\geq 0\}$ is scheduled on machine $M_1$ and the sub set of jobs $\{J_1\}\cup \{J_{ki+3}, J_{ki+4},...,J_{ki+k+1}|i\geq 0\}$ is scheduled on machine $M_2$, where $s(k)$=$1$ for $k=0$; $s(k)$=$\frac{1+\sqrt{3}}{4}$ for $k$=$1$ and $s(k)$=$\frac{k^2-1+\sqrt{(k^2-1)^{2}+2k^3(k+1)}}{k(k+1)}$ for $k\geq 2$.
- Algorithm **Highest Loaded Machine(HLM)** was proposed by Angelelli et al. \[35\] and was originally named as *algorithm $H$* for the setup $P_m|Sum|C_{max}$. By observing the behavior of the algorithm, we rename it to *HLM*. Algorithm *HLM* schedules a newly arrive job either on the highest loaded machine in the set of heavily loaded machines or on the highest loaded machine in the set of lightly loaded machine.
- Algorithm **Extended Longest Processing Time(ELPT)** was proposed by Epstein and Favrholdt \[42\] for the setup $Q_2|Decr|C_{max}$. Algorithm *ELPT* assigns each incoming job $J_i$ to the fastest machine $M_j\in \{M_1, M_2\}$ for which $l_j + \frac{p_i}{S_j}$ is minimum, where $S_1$=$1$ and $S_2$=$\frac{1}{s}$ for $s\geq 1$.
- Algorithm **Slow LPT** was proposed by Epstein and Favrholdt \[42\] for $Q_2|Decr|C_{max}$. It schedules the first available job $J_1$ to the slowest machine $M_2$ and the next job $J_2$ is scheduled on the fastest machine $M_1$. It assigns the next incoming job $J_3$ to $M_2$ if $s\cdot (p_1 + p_3) \leq c(s)\cdot (p_2 + p_3)$, otherwise $J_3$ is assigned to machine $M_1$. Next incoming jobs are assigned to the machine $M_j$ for which $l_j + \frac{p_i}{S_j}$ is minimum, where $S_1$=$1$ and $S_2$=$\frac{1}{s}$ for $s\geq 1$. ($c(s)$ is a function of the speed ratio interval $s$)
- Algorithm **Grade of Service Eligibility(GSE)** is due to Park et al. \[46\] for the setup $P_2|GOS,Sum|C_{max}$. It states that upon the arrival of any job $J_i$ with $GOS=1$, assign $J_i$ to machine $M_1$. When $J_i$ arrives with $GOS=2$, then $J_i$ is assigned to machine $M_2$ if and only if $l_2 + p_i \leq \frac{3}{2}\cdot T$, otherwise, $J_i$ is scheduled on machine $M_1$.
- Algorithm **Fastest Last(FL)** was proposed by Epstein and Ye \[51\] for $P_2|LL|C_{min}$. Algorithm *FL* schedules an incoming job $J_i$ on the slowest machine $M_1$ if and only if $l_2 + p_i > \alpha(S) (l_1 + S\cdot p_i)$, else $J_i$ is scheduled on the fastest machine $M_2$. If $J_i$ is the last job, then it is scheduled exclusively on the fastest machine $M_2$. ($\alpha(S)$ is a function of $S$ and $0 < \alpha(S) < \frac{1}{S}$)
- Algorithm **Fractional Semi-online Assignment(FSA)** was proposed by Chassid and Epstein \[59\] for $Q_2|pmtn,GOS,Sum|C_{min}$. Algorithm *FSA* assigns a newly arrive job $J_i$ with $g_i$=$1$ to machine $M_1$. If $J_i$ has $g_i = 2$, then if $l_2$=$\frac{1}{b+1}$, then $J_i$ is scheduled on $M_1$; else if $l_2+p_i\leq \frac{1}{b+1}$, then $J_i$ is assigned to machine $M_2$; else $\frac{1}{b+1}-l_2$ portion of $J_i$ is assigned to machine $M_2$ and the remaining part of $J_i$ is scheduled on $M_1$. (Note that: $S_1$=$1$, $S_2$=$b$ and $Sum$=$1$, where $b\geq 1$)
- Algorithm **RatioStretch** was developed by Ebenlendr and Sgall \[61\] for $Q_m|pmtn,Decr|C_{max}$. Algorithm *RatioStretch* first estmates for each incoming job $J_i$ the completion time $c_i$=$r\cdot C_{OPT}(i)$, where $r$ is the required approximation ratio and $C_{OPT}(i)$ is the least value of estimated makespan for processing of jobs $J_1, J_2,...,J_i$. Then, two consecutive fastest machines $M_j, M_{j+1}$ are chosen along with time $t_j$ such that if $J_i$ is scheduled on $M_{j+1}$ in the interval $(0, t_j]$ and on $M_j$ from time $t_j$ on wards, then $J_i$ finishes by time $c_i$.
- Algorithm **High Speed Machine Priority(HSMP)** was given by Cai and Yang \[97\] for the setup $Q_2|Max|C_{max}$. Algorithm *HSMP* schedules each incoming job $J_i$ on machine $M_2$ if $p_i$=$p_{max}$; thereafter schedules each incoming $J_{i+1}$ on the machine that will finish $J_{i+1}$ at the earliest; otherwise, schedules $J_i$ on machine $M_1$ if $p_i< p_{max}$ and if $l^{i}_{1}+p_i< l^{i}_{2}+\frac{p_i+p{max}}{s}$, where $l^{j}_{i}$ is the load of machine $M_j$ just before the scheduling of $J_i$ and $1.414\leq s\leq 2.732$; otherwise, schedules $J_i$ on machine $M_2$.
- Algorithm **OM** was proposed by Cao et al. \[74\] for $P_2|Opt,Max|C_{max}$. It is known at the outset that the first incoming job $J_1$ has the largest processing time $p_{max}$. Algorithm *OM* assigns $J_1$ to machine $M_2$. Thereafter, each incoming job $J_i$, where $2\leq i\leq n$ is scheduled on $M_2$ if and only if $l_2+p_i \leq (\frac{6}{5})\cdot Opt$; otherwise $J_i$ is assigned to machine $M_1$.
- Algorithm **Light Load** was proposed by Albers and Hellwig \[75\] for $P_m|Sum|C_{max}$. It assigns a new job $J_i$ to the $\lceil \frac{m}{2} \rceil ^{th}$ highest loaded machine $M_j$ if and only if $l_m > 0.25 (\frac{T}{m})$ and $l_j + p_i \leq 1.75 (\frac{T}{m})$; otherwise, $J_i$ is scheduled on the least loaded machine $M_m$.
Historical Overview of Semi-online Scheduling: Important Results {#sec:Historical Overview of Semi-online Scheduling: Important Results}
================================================================
In 1960’s, the curiosity to explore computational advantages of multi-processor systems resulted in a number of scheduling models. Online scheduling is one among such models. Graham \[2\] initiated the study of online scheduling of a list of $n$ jobs on $m(\geq 2)$ identical parallel machines and proposed the famous *List Scheduling(LS)* algorithm. Algorithm *LS* selects the first unscheduled job $J_i$ from the list such that all its *predecessors* ($J_k\prec J_i$) have been completed and schedules $J_i$ on the most lightly loaded machine. Algorithm *LS* achieves performance ratios of $1.5$ for $m$=$2$ and $2-\frac{1}{m}$ for all $m$ by considering $C_{OPT}\geq \frac{\sum_{i=1}^{n}{p_i}}{m}$. In \[3\], Graham considered the offline setting of $m$-machine scheduling problem and proposed the seminal algorithm *Largest Processing Time(LPT)*. Algorithm *LPT* first sorts the jobs in the list by non-increasing *sizes* and assigns them one by one to a machine that incurs smallest load after each assignment. Algorithm *LPT* achieves a worst-case performance ratio of $1.16$ for $m=2$ and $1.33-\frac{1}{3m}$ for $m\geq 2$ with the time complexity of $O(n\log n)$. These two seminal contributions of Graham served as a motivation for further investigations to address research challenges in online scheduling. Initial three decades(1966-1996) of the online scheduling research were concentrated on the improvements of the *LB* and *UB* on the *CR* to achieve optimal competitiveness(please, see \[16-17\] for a comprehensive survey on the seminal contributions). However, no significant attention has been paid for exploring the practicability of the online scheduling model of Graham.\
Motivated by real world applications, Kellerer et al. \[13\] proposed a novel variant of the online scheduling model by considering *EPIs* for pre-processing of online arriving jobs and named the variant as semi-online scheduling. They conjectured that additional information on future jobs would immensely help in improving the best competitive bounds in various online scheduling setups. Following the conjecture of Keller et al., ocean of literature have been produced since last two decades in pursuance of achieving optimum competitiveness with the exploration of practically significant new *EPIs*. We now survey the critical ideas and important results given for semi-online scheduling in a historical chronological manner by classifying the results based on the *EPI* as follows.
Early Works in Semi-online Scheduling (1997-2000) {#subsec:Early Works on Semi-online Scheduling(1997-2000)}
-------------------------------------------------
**Buffer, Sum.** Kellerer et al. \[13\] envisioned semi-online scheduling as a theoretically significant and practically well performed online scheduling model. They initiated the study on semi-online scheduling by considering *Sum* as the known *EPI* and proposed algorithm $H_3$ for the setup $P_2|Sum|C_{max}$. Algorithm $H_3$ outperforms algorithm *LS* and achieves a *tight* bound $1.33$ for $m=2$. To show the *LB* $1.33$ of algorithm $H_3$, let us consider an instance of $P_2|Sum|C_{max}$, where $Sum$=$2$. Algorithm $H_3$ schedules each incoming job $J_i$ to machine $M_1$ until $l_1+p_i\leq \frac{2}{3}\cdot$(*Sum*) and assigns the remaining jobs to machine $M_2$. If we consider $Sum$=$2$, then irrespective of the input instances, the final loads $l_1$, $l_2$ would be $\frac{4}{3}$, $\frac{2}{3}$ respectively and $C_{OPT}$ would be $1$. This implies, $C_{H_3} \geq 1.33\cdot (C_{OPT})$. The semi-online strategy of Kellerer et al. unveils that advance knowledge of *Sum* helps any online algorithm $A$ to schedule incoming jobs to a particular machine until its load reaches upto a judiciously chosen fraction of the *Sum* and assigns the remaining jobs to the other machine such that the ratio between $C_A$ and $C_{OPT}$ results in the improved competitive bound. They also studied semi-online scheduling with a buffer($B$) of length $k$ and proposed algorithm $H_1$ for the setup $P_2|B(k)|C_{max}$. They proved that any online scheduling algorithm with $B(k)$ achieves a *CR* of at least $1.33$. The *LB* can be shown by considering an online sequence $J:<J_1/1, J_2/1, J_3/1, J_4/3>$ of four jobs with specified processing times and $k$=$1$. Algorithm $H_1$ keeps the first available job $J_1$ in the buffer. Thereafter, each incoming $J_{i+1}$, where $1\leq i\leq 3$, is either kept in the buffer or any $J_x\in \{J_i, J_{i+1}\}$ is scheduled on machine $M_1$ if $l_1+p_x\leq \frac{2}{3}\cdot (l_1+l_2+w(B))$, else any $J_x$ is assigned to machine $M_2$ (Note that $w(B)$=$p_i+p_{i+1}$). We now have a schedule due to algorithm $H_1$ with the sequence of assignments of $J$ on the machines as follows: $J_1/1$ on $M_1$, $J_2/1$ on $M_2$, $J_4/3$ on $M_1$ and $J_3/1$ on $M_2$ such that $C_{H_1}\geq 4$, where $C_{OPT}\geq 3$. Therefore, $C_{H_1}\geq 1.33\cdot (C_{OPT})$. A matching *UB* was shown to achieve a *tight* bound $1.33$ for algorithm $H_1$. They also studied a semi-online variant, where two parallel processors are given to virtually schedule a sequence of incoming jobs over $2$-identical machines by two distinct procedures independently. Finally, the jobs are scheduled by the procedure that has incurred minimum $C_{max}$ for the entire job sequence. They obtained a *tight* bound $1.33$ for the semi-online variant $P_2|2$-$Proc|C_{max}$. Zhang \[14\] studied the setup $P_2|B(1)|C_{max}$ and obtained the *tight* bound $1.33$ with an alternate policy. The policy keeps the first job $J_1$ in the buffer and if no further jobs arrive, then $J_1$ is scheduled on machine $M_2$, else for next incoming job $J_{i+1}$, where $1\leq i\leq n-1$, the job $J_x$ is chosen from $\{J_i, J_{i+1}\}$ such that $p_x$ is minimum (let us denote the other job as $J_y$). Now, $J_x$ is assigned to machine $M_1$ if $l_1+p_x\leq 2\cdot(l_2+p_y)$, else $J_x$ is scheduled on machine $M_2$. If there is no jobs to arrive further, then the last job in the buffer is assigned to machine $M_2$. The aim of the policy is to keep a larger load difference between machines $M_1$ and $M_2$ by assigning smaller jobs to $M_2$ such that at any time step, the availability and assignment of an unexpected larger job would not incur a makespan beyond $1.33\cdot (C_{OPT})$. Angelelli \[22\] proposed an alternative to algorithm $H_3$ \[13\] as *algorithm H* for $P_2|Sum|C_{max}$ and obtained *tight* bound $1.33$. He analyzed the performance of *algorithm H* by considering various ranges of lower bounds on the processing times of the jobs. Girlich et al. \[18\] obtained an *UB* $1.66$ for $P_m|Sum|C_{max}$.\
**TGRP, Max.** He and Zhang \[20\] initiated study on the setup $P_2|TGRP|C_{max}$ by assuming that all jobs have processing times within the interval of $p$ and $rp$, where $p>0$ and $r\geq 1$. They proved that any online algorithm $A$ must have $C_A\geq \frac{r+1}{2}$ for $r\leq 2$ and $C_A\geq 1.5$ for $r>2$. They analyzed algorithm *LS* for $P_m|TGRP|C_{max}$ and showed that $C_{LS}\leq (1+\frac{(m-1)(r-1)}{m})\cdot C_{OPT}$. They obtained *LB* $1.33$ for the setup $P_2|Max|C_{max}$ by considering the online availability of the job sequence $J:<J_1/1, J_2/1, J_3/2, J_4/2>$, where $p_{max}$=$2$ is known a priori. Following the optimum policy \[2\] of keeping a machine free for the largest job and assigning the sequence of comparatively shorter jobs to the remaining machines, any online algorithm $A$ assigns $J_1/1$ and $J_2/1$ on $M_1$ followed by the assignments of $J_3/2$ on $M_2$ and $J_4/2$ on either $M_1$ or $M_2$ to incur $C_A\geq 4$, where $C_{OPT}\geq 3$. This implies $C_A\geq 1.33\cdot (C_{OPT})$. They proposed algorithm *PLS* to achieve a *tight* bound $1.33$. Algorithm *PLS* always maintains a load difference maximum of upto $p_{max}$ between machines $M_1$ and $M_2$ such that scheduling of the largest job on the smallest loaded machine almost equalizes the loads of both the machines.\
**Opt.** Azar and Regev \[15\] introduced a variant of the classical bin-stretching problem, where items are available one by one in order and each available item must be packed into one of the $m$ bins before the availability of the next item. It is known apriori that all items can be placed into $m$ unit sized bins. The goal is to stretch the bins as minimum as possible so as to fit all items into the bins. Therefore, the bin stretching problem considered by Azar and Regev is analogous to the setup $P_m|Opt|C_{max}$. They achieved an *UB* $1.625$ for large $m$. The key idea is to first define the threshold values $\alpha$ and $2\alpha-1$ based on the value of known *Opt*. Then, arrange $m$ machines in at least two distinct sets based on their loads with respect to $\alpha$ and $2\alpha-1$. A new job is now assigned to the selected machine belongs to the chosen set. Improved rules can be defined for the selections of set and machine. Here, a non-trivial challenge is to define and characterize the threshold values.\
**Decr and Preemptive Semi-online Scheduling.** Seiden et al. \[21\] analyzed algorithm *LPT* \[3\] with known *Decr* and achieved a *tight* bound $1.166$ for $m$=$2$ and *LB* $1.18$ for $m$=$3$. They initiated the study on $P_m|pmtn, Decr|C_{max}$ and obtained *tight* bound $1.366$. They assumed preemption as *job splitting* for scheduling distinct pieces of an incoming job in the non-overlapping time slots. To understand the notion of job splitting, let us consider a sequence of jobs of unit sizes. Suppose, the required *CR* to be obtained is $k$. Now, the initial $m$ incoming jobs are splitted into at most two pieces each such that all pieces of a job execute in distinct time slots and all jobs are finished by time $k$. Let $r$=$\lfloor\frac{m}{k}\rfloor$, $i\leq r$, each incoming job $J_{m+i}$ is splitted and assigned to the first $i$ machines such that each machine gets $\frac{k}{m}$ fraction of the processing time of job $J_{m+i}$ and its remaining fraction is assigned prior to time $k$. We now have $i$ highest loads of the machines represented as: $k(1+\frac{i}{m})$, $k(1+\frac{i-1}{m})$,...,$k(1+\frac{1}{m})$, which ensures non-overlapping time slots in the subsequent rounds. Similarly, the next jobs followed by the $({m+r})^{th}$ job are scheduled only in the time slots after $k$ on at most $r+1$ machines. A non-trivial challenge is to rightly choose the values of $r$ and $k$ such that the load to be scheduled prior to time $k$ is at most $k\cdot m$. The authors conjectured that the achieved *tight* bound $1.366$ with known *Decr* can possibly be achieved with only known $p_{max}$. Further, they showed that randomization in scheduling decision making does not lead to improved the *CR* for the setup $P_m|pmtn, Decr|C_{max}$. We now present the main results obtained for semi-online scheduling in identical machines for the years 1997-2000 in table \[tab: Important Results: 1997-2000\].
**Author(s), Year** **Setup($\alpha| \beta| \gamma$)** **Competitiveness Results**
----------------------------- ------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------
Kellerer et al. 1997 \[13\] $P_2 | Sum | C_{max}$ $P_2 | B(k) | C_{max}$ $P_2|2-Proc|C_{max}$ $1.33$ Tight for each setup
Zhang 1997 \[14\] $P_2 | B(1) | C_{max}$ $1.33$ Tight
Azar and Regev 1998 \[15\] $P_m | Opt | C_{max}$ $1.625$ UB
Girlich et al. 1998 \[18\] $P_m | Sum | C_{max}$ $1.66$ UB
He and Zhang 1999 \[20\] $P_2 | Max | C_{max}$ $P_2 | TGRP[r, rp] | C_{max}$ $P_m|TGRP[r, rp]|C_{max}$ $1.33$ Tight with Max, $1.5$ LB for $P_2$ and $r>2$ with TGRP, ($1+\frac{(m-1)(r-1)}{m}$) UB for $P_m$ with TGRP.
Seiden et al. 2000 \[21\] $P_m | pmtn, Decr | C_{max}$ $P_{2,3} | Decr | C_{max}$ $1.366$ Tight for $m \rightarrow \infty$, $1.166$ Tight for $m=2$, $1.18$ LB for $m=3$
Angelelli 2000 \[22\] $P_2 | Sum, TGRP(lb) | C_{max}$ $1.33$ Tight
: Main Results for Identical Machines: 1997-2000
\[tab: Important Results: 1997-2000\]
Well-known Results in Semi-online Scheduling (2001-2005)
--------------------------------------------------------
During the years 2001-2005, semi-online scheduling was studied not only for identical machines but for uniform related machines as well. Both preemptive and non-preemptive processing formats were investigated. The concept of combined *EPI* and a new *EPI* on the last job were introduced. We present the state of the art contributions in semi-online scheduling for related machines and identical machines as follows.\
\
**Related Machines:**\
**Decr.** Tan and He \[23\] proposed algorithm *Ordinal* for non-preemptive semi-online scheduling with *ordinal data* \[11\] and known *Decr* for $2$-uniformly related machines, where $S_1$=$1$ and $S_2\geq 1$. They analyzed and proved competitiveness of the algorithm as an interval wise function of machines’ speed ratio $s$. They proved the *tightness* of the algorithm in most of the intervals of $s \in [1, \infty)$. As a main result, they produced *UB* $\frac{s+1}{s}$ for $s\geq 2.732$ and *LB* $\frac{s+1}{s}$ for $s\in [2.732, \infty)$. However, the *LB* of algorithm *Ordinal* does not match with its *UB* when the total length of the speed ratio interval reduces to $0.7784$, where the largest gap between the intervals is at most $0.0521$. Epstein and Favrholdt \[30\] initiated study on the setup $Q_2|pmtn, Decr|C_{max}$ and achieved competitive ratios of $\frac{3(s+1)}{3s + 2}$ for $ 1 \leq s \leq 3$ and $\frac{2s + (s+1)}{2s^2 + s + 1}$ for $ s \geq 3$. In \[42\], they investigated the setup $Q_2|Decr|C_{max}$, where $S_1$=$1$ and $S_2$=$\frac{1}{s}$. They expressed competitive ratios as a function of $15$ speed ratio intervals. They proposed algorithm *ELPT* and achieved *tight* bound $1.28$ for $s$=$1.28$. They proposed algorithm *Slow-LPT* for $s\in [1, \frac{1}{6}(1+\sqrt{37})]$ and obtained a *tight* bound $1.28$. Here, the key idea is to initially use the slowest machine and keep the fastest machine free for incoming larger jobs. They proposed algorithms *Balanced-LPT* and *Opposite-LPT* for the remaining intervals, where algorithms *ELPT* and *Slow-LPT* do not obtain tight bounds. Algorithm *Balanced-LPT* schedules the first job $J_1$ on the fastest machine $M_1$. The second job $J_2$ is assigned to machine $M_2$ if $s>c(s)(p_1+p_2)$, else job $J_2$ is scheduled on $M_1$, where $c(s)$=$2.19$ for $s\in [2, 2.19]$ and $c(s)$=$2.57$ for $s\in [2.35, 2.57]$. Thereafter, remaining jobs are scheduled by algorithm *ELPT*. Algorithm *Opposite-LPT* also schedules job $J_1$ on machine $M_1$. The second job $J_2$ is scheduled on $M_1$ if $s\cdot p_2< (p_1+p_2)\leq c(s)\cdot s\cdot p_2$, else $J_2$ is scheduled on $M_2$, where $c(s)$=$2.35$ for $s\in [2.19, 2.35]$. Thereafter, the subsequent jobs are scheduled by the *ELPT* rule.\
**Opt.** Epstein \[33\] studied semi-online scheduling for the setup $Q_2|Opt|C_{max}$, where $S_1$=$1$ and $S_2$=$\frac{1}{s}$. He proposed algorithm *SLOW* by considering $C_{OPT}$=$1$ and $s\geq \sqrt{2}$. Algorithm *SLOW* schedules an incoming job $J_i$ to machine $M_1$ if $l_2\geq \frac{1}{s^2+s}$; else if $l_2+p_j\geq \frac{C_{SL}(s)}{s}$, then job $J_i$ is assigned to machine $M_2$; else $J_i$ is scheduled on machine $M_1$, where $C_{SL}(s)$=$\frac{s+2}{s+1}$. Algorithm *SLOW* performs better in the scenario, where the slowest machine $M_2$ is relatively very slow and initial jobs needs to be assigned to it for keeping the high speed machine $M_1$ relatively free for future larger jobs. For $s\leq \sqrt{2}$, Epstein proposed algorithm *FAST* by considering $C_{OPT}$=$1$. Algorithm *FAST* assigns an incoming job $J_i$ to machine $M_2$ if a job $J_x$ was earlier assigned to $M_2$ due to $l_1<(1+\frac{1}{s}-\frac{C_{FA}(s)}{s})$ and $(l_1+p_x)> C_{FA}(s)$; else if $l_1\geq (1+\frac{1}{s}-\frac{C_{FA}(s)}{s})$, then $J_i$ is scheduled on $M_2$; else if $(l_1+p_j)\leq C_{FA}(s)$, then $J_i$ is scheduled on $M_1$; else $J_i$ is assigned to machine $M_2$, where $C_{FA}(s)$=$\frac{2s+2}{2s+1}$ for $1\leq s\leq \frac{1+\sqrt{17}}{4}$ and $C_{FA}(s)$=$s$ for $\frac{1+\sqrt{17}}{4}\leq s\leq \sqrt{2}$. Algorithm *FAST* performs better in the cases, where the slowest machine $M_2$ is considerably fast, thus allowing initial jobs to be scheduled on the fastest machine $M_1$. He achieved lower bounds in terms of function of defined speed ratio intervals and obtained overall *CR* of $1.414$ and *LB* of $1.366$.\
**TGRP, Max.** He and Jiang \[34\] studied the setup $Q_2|pmtn, TGRP(p, xp)|C_{max}$ by considering $S_1$=$1$ and $S_2\geq 1$, where $p>0$ and job size ratio $x\geq 1$. They initiated analysis of algorithm with respect to speed ratio intervals ($s \geq 1$) and job size ratios. They achieved a *tight* bound $\frac{s^2+s}{s^2+1}$ for $s\geq 1$ and $x<2s$. For $s\geq 1$ and $x\geq 2s$, the *tight* bound $\frac{1+2s+s^2}{1+s+s^2}$ was obtained. Further, they investigated the setup $Q_2|pmtn, Max|C_{max}$ by considering known $p_{max}$=$s\geq 1$. They achieved a *CR* of $\frac{2s^2+3s+1}{2s^2+2s+1}$ for $s\geq 1$. They explored that information on *Max* is weaker than known *Decr* for preemptive semi-online scheduling on $2$-related machine. We now present the main results obtained for semi-online scheduling on related machines for the years 2001-2005 in table \[tab:Important Results for Uniform Machines: 2001-2005\].
**Author(s), Year** **setup($\alpha| \beta| \gamma$)** **Competitiveness Results**
----------------------------------- ----------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------
Tan and He 2001 \[23\] $Q_2 | Decr | C_{max}$ $\frac{s+1}{s}$ Tight
Epstein and Favrholdt 2002 \[30\] $Q_2 | pmtn, Decr | C_{max}$ $\frac{3(s+1)}{3s + 2}$ Tight for $ 1 \leq s \leq 3$, $\frac{2s + (s+1)}{2s^2 + s + 1}$ Tight for $ s \geq 3$
Epstein 2003 \[33\] $Q_2 | Opt | C_{max}$ $1.414$ UB, $1.366$ LB
He and Jiang 2004 \[34\] $Q_2 | pmtn, Max | C_{max}$ $Q_2 | pmtn, TGRP(p, xp) | C_{max}$ $\frac{2s^2 + 3s + 1}{2s^2 + 2s + 1}$ Tight with *Max*, $\frac{s^2 + s}{s^2 + 1}$ Tight with TGRP.
Epstein and Favrholdt 2005 \[42\] $Q_2 | Decr | C_{max}$ $1.28$ Tight
: Main Results for Related Machines: 2001-2005
\[tab:Important Results for Uniform Machines: 2001-2005\]
\
**Identical Machines:**\
**Information on Last job(LL).** Zhang and Ye \[26\] studied a semi-online variant, where it is known apriori that the last job $J_n$ is the largest one i.e. $p_{n}$=$p_{max}$. Upon availability of a job $J_i$, it is also revealed that whether $J_i$ is the last job. They proposed algorithm $A_1$ for the setup $P_2|LL|C_{max}$ and achieved a *tight* bound $1.414$. Algorithm $A_1$ schedules an incoming job $J_i$ on machine $M_2$ if $J_i$=$J_n$; else if $(l_2+p_i)>(0.414)\cdot (l_1+p_i)$, then $J_i$ is assigned to machine $M_1$; else $J_i$ is scheduled on $M_2$. The key idea is to reserve a machine for $J_n$ to obtain relatively minimum makespan irrespective of the size of $J_n$. They proposed algorithm *List Scheduling with a waiting machine(LSw)* for the setup $P_3|LL|C_{max}$ by keeping machine $M_3$ free for $J_n$. Algorithm *LSw* schedules an incoming job $J_i$ on $M_3$ if $J_i$=$J_n$; else assigns job $J_i$ to machine $M_j\in \{M_1, M_2\}$ for which $l_j$=$\min\{l_1, l_2\}$. They proved a *tight* bound $1.5$ for algorithm *LSw*. However, it would be interesting to analyze the cases, where the value of $p_n$=$p_{max}$ is relatively smaller or there are multiple jobs with $p_i$=$p_{max}$.\
**Combined Information.** Tan and He \[28\] exploited the limitation of prior knowledge of $LL$ \[26\] by considering the following sequences $J:<J_1/1, J_2/1, J_3/2>$ and $J':<J_1/1, J_2/1, J_3/\epsilon>$, where $\epsilon> 0$. They studied the semi-online variants, where two *EPIs* are known at the outset. They proposed the $1.2$ competitive algorithm *SM* for the setup $P_2|Sum, Max|C_{max}$. Algorithm *SM* is designed based on the ratio between known *Sum(T)* and *Max($p_{max}$)*. If $p_{max}\in [\frac{2T}{5}, T]$, then the first job $J_i$ is assigned to machine $M_2$ for which $p_i$=$p_{max}$(such a job is denoted as $J^{1}_{max}$) and other jobs are scheduled on machine $M_1$. If $p_{max}\in (0, \frac{T}{5}]$, then all incoming jobs are scheduled by algorithm *LS*. If $p_{max}\in (\frac{T}{5}, \frac{2T}{5})$, then an incoming job $J_i$ is assigned to machine $M_j\in\{M_1, M_2\}$ such that $(l_j+p_i)\in [\frac{2T}{5}, \frac{3T}{5}]$ and the successive jobs are scheduled on machine $M_{3-j}$. If $(l_j+p_i)\in [\frac{2T}{5}$-$p_{max}, \frac{3T}{5}$-$p_{max}]$ and if $J^{1}_{max}$ has not been revealed yet, then both $J_i$ and $J^{1}_{max}$ are assigned to $M_j$ and other jobs are scheduled on machine $M_{3-j}$. If $p_i\leq \frac{T}{5}$ or $J_i$=$J^{1}_{max}$, then $J_i$ is scheduled on $M_1$; else if $\frac{T}{5}<p_i\leq p_{max}$, then $J_i$ is scheduled on $M_2$; if two jobs have already been scheduled on $M_2$ such that $l_2\geq \frac{2T}{5}$, then successive jobs are scheduled on $M_1$. Further, they proposed a $1.11$ competitive algorithm for the setup $P_2|Sum, Decr|C_{max}$. They also showed that if *Sum* is given, then information on *LL* is useless and if *Decr* is known, then knowledge of *Max* dos not substantiate to improve the best competitive bound of $1.16$ \[21\] for $P_2|Decr|C_{max}$. Epstein \[33\] followed the work of \[15\] and achieved a *tight* bound $1.11$ for the setup $P_2|Decr, Opt|C_{max}$. He proved the *LB* by considering $C_{OPT}$=$1$ and six jobs, where the jobs $J_1$ and $J_2$ are of *size* $\frac{4}{9}$ each and jobs $J_3, J_4, J_5, J_6$ are of *size* $\frac{5}{18}$ each. If any semi-online algorithm *A* schedules $J_1$ and $J_2$ either on machine $M_1$ or on $M_2$ and schedules the remaining jobs to the other vacant machine, then we have $C_A$=$\frac{10}{9}$. If $J_1$ and $J_2$ are scheduled on two different machines, then by considering the *size* of next three jobs($J'_3, J'_4$, $J'_5$) as $\frac{1}{3}$ each, we have $C_A$=$\frac{10}{9}$ (as any two jobs from $J'_3, J'_4, J'_5$ must be scheduled on a single machine). However, algorithm $OPT$ schedules $J_1$ and $J_2$ on one machine and assigns the remaining three jobs to the other machine to incur $C_{OPT}$=$1$. Therefore, we have $\frac{C_A}{C_{OPT}}$=$\frac{10}{9}$=$1.11$. He proposed the algorithm *SIZES*, which schedules an incoming job $J_i$ to any $M_j\in \{M_1, M_2\}$ such that $\frac{8}{9}\leq (l_j+p_i)\leq \frac{10}{9}$, the remaining jobs are scheduled on the other machine. If $p_i\leq \frac{2}{9}$, then $J_i$ and all remaining jobs are assigned to machine $M_j$ which incurs $l_j$=$\min\{l_1, l_2\}$ after each assignment.\
**List Model.** Yong and Shengyi \[27\] studied the *list model* of \[19\], which is a variant of Graham’s list scheduling model \[2\], where it is considered that the machines are not available at the outset. Upon availability of a job $J_i$, an algorithm may purchase a machine by incurring an unit cost. A machine $M_j$ is purchased such that the existing $j$ machines satisfies the following inequality: $l_j\leq T_i<l_{j+1}$, where $T_i$=$\sum_{j=1}^{i}{l_j}$(total work load incurred by initial $i$ jobs). The aim is to optimize the *sum of makespan($C_{max}$) and total machine cost($m$)*. They showed that with known *Max*, an algorithm makes decisions on purchasing of a machine and scheduling of an incoming job $J_i$ by comparing the values of $p_i$, loads of the existing machines or total machine cost with some judiciously chosen bounds on the known $p_{max}$. They obtained an *UB* $1.5309$ and a *LB* $1.33$ with known *Max*. They achieved an *UB* $1.414$ and *LB* $1.161$ with known *Sum*. Further, *List model* can be studied to improve the existing competitive bounds by considering other well-known *EPI*s. We may obtain a natural variant of the *list model* by considering non-identical machines with well defined characteristics, which may influence the choice of an algorithm in purchasing of a machine.\
**Max.** Cai \[29\] extended the work of \[20\] to obtain a *tight* bound ($\frac{m-2 + \sqrt{(m-2)^2 + 8m^2}}{2m}$) for $P_m|Max|C_{max}$, where $3\leq m\leq 17$. Further, he achieved a *tight* bound $1.414$ for $m\rightarrow \infty$.\
**TGRP.** Angelelli et al. \[31\] considered $TGRP(ub)<1$ and $Sum$=$2$ are known in advance. For $2$-identical machine setup, they obtained lower bounds for various ranges of $ub$. They showed that algorithm *LS* is optimal for smaller $ub$ and proposed optimal algorithms for $0.5\leq ub\leq 0.6$; $ub$=$0.75$ and $0.9\leq ub < 1$. He and Dosa \[43\] investigated the $3$-identical machine setting by considering $TGRP(p, xp)$, where $p > 0$ and job size ratio $x \geq 1$. They proved that algorithm *LS* is optimal for different intervals of $x \in [1,1.5], [1.73, 2], [6, +\infty]$. They designed algorithms for various ranges of *x* with improved bounds for which the gap between the competitive ratio and the lower bounds is at most $0.01417$.\
**Sum, Buffer.** Angelleli et al. \[35\] extended their previous work \[22,31\] and obtained an *UB* $1.725$ for $m$-identical machine with known *Sum*. Cheng et al. \[44\] investigated the setup $p_m|Sum|C_{max}$ by considering *Sum*=$m$. They followed the work of \[15, 35\] and obtained *UB* $1.6$ and improved *LB* $1.5$ for $m \geq 6$. Dosa et al. \[37\] studied the setup $P_2|B(1), Sum|C_{max}$ and obtained a *tight* bound $1.25$. They showed that considering a $B(k)$, where $k>1$ does not help to improve the $1.25$ competitiveness. They explored that when *Sum* is known at the outset, then the *knowledge of the sizes of $k(>1)$ future jobs($k$-look ahead)* does not help in improving the competitive bound. Further, they studied the setup $P_2|2$-$Proc|C_{max}$ with known *Sum* and improved the *tight* bound $1.33$ obtained in \[13\] to $1.2$. We now present the main results obtained for semi-online scheduling on identical machines for the years 2001-2005 in table \[tab: Important Results on Identical Machines: 2001-2005\].
**Author(s), Year** **setup($\alpha| \beta| \gamma$)** **Competitiveness Results**
------------------------------ ------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------
Zhang and Ye 2002 \[26\] $P_2 | LL | C_{max}$ $P_3 | LL | C_{max}$ $1.414$ Tight for $m=2$, $1.5$ Tight for $m=3$.
Yong and shengyi 2002 \[27\] $P_m | Max | C_{max}$+$m$ $P_m | Sum | C_{max}$+$m$ $1.53$ UB and $1.33$ LB with *Max*, $1.414$ UB and $1.161$ LB with *Sum*
Tan and He 2002 \[28\] $P_2 | Sum, Max | C_{max}$ $P_2 | Sum, Decr | C_{max}$ $1.2$ Tight with *Sum* and *Max* , $1.11$ Tight with *Sum* and *Decr*
Cai 2002 \[29\] $P_m | Max | C_{max}$ ($\frac{m-2 + \sqrt{(m-2)^2 + 8m^2}}{2m}$) Tight for $3 \leq m \leq 17$, $1.414$ Tight for $m\rightarrow \infty$.
Angelelli et al. 2003 \[31\] $P_2 | Sum, TGRP(ub) | C_{max}$ $1.2$ Tight for $ub \in [0.5, 0.6]$ , ($1 + (\frac{ub}{3}$)) Tight for $ub \in [0.75, 1]$, ($0.666(1+ ub)$) Tight for $ub \in [0.94, 1]$ .
Epstein 2003 \[33\] $P_2 | Decr, Opt | C_{max}$ $1.11$ Tight.
Angelelli et al. 2004 \[35\] $P_m | Sum | C_{max}$ $1.725$ UB, $1.565$ LB for $m \rightarrow \infty$
Dosa et al. 2004 \[37\] $P_2 | B(1), Sum | C_{max}$ $P_2 | 2$-$Proc, Sum | C_{max}$ ($1.25$, $1.2$) Tight for respective setups
He and Dosa 2005 \[43\] $P_3 | TGRP | C_{max}$ $1.5$ Tight for $x \in (2, 2.5]$, ($\frac{4r +2}{2r + 3}$) Tight for $x \in (2.5, 3]$, ($1.66 - \frac{\delta}{18}$) Tight for $x \in (3, 6)$
Cheng et al. 2005 \[44\] $P_m | Sum | C_{max}$ ($1.6, 1.5$) UB and LB respectively for $m \geq 6$
: Important Results for Identical Machines: 2001-2005
\[tab: Important Results on Identical Machines: 2001-2005\]
Advancements in Semi-online Scheduling (2006-2010) {#sebsec:Advancements in Semi-online Scheduling(2006-2010)}
--------------------------------------------------
The initial decade in semi-online scheduling research was devoted to the traditional online scheduling setups with fundamental *EPI*s on the future jobs. Moreover, the significance of *EPI* was realized with the improvement in the competitive bounds for pure online scheduling setups. During the years 2006-2010, new semi-online scheduling setups such as *GOS* or machine hierarchy; a variant of *EPI* such as *inexact EPI* and new policies such as *job re-assignment* and *buffer re-ordering* have been introduced. We now discuss on the important results contributed during the years 2006-2010 for semi-online scheduling on identical and related machines as follows.\
**Identical Machines:**\
**Sum.** Angelelli et al. \[45\] studied the setup $P_2|Sum, TGRP(ub)|C_{max}$ and advanced their previous work \[31\] for the unexplored intervals of $ub$. They showed *LB*s for the interval, where $ub\in [\frac{1}{k}, \frac{1}{k-1}]$ and $k\geq 2$. For an instance, a *LB* of $(\frac{k-1}{3})\cdot ub$+$\frac{2}{3}\cdot(\frac{k+1}{k})$ was shown for $ub\in [\frac{2(k+1)}{k(2k+1)}, \frac{2k-1}{2k(k-1)}]$. The *LB* was proved by considering two job sequences $J'$ and $J''$, where $J'$=$\{J_1/x, J_2/x, J_3/y, J_4/y$ and $2(k-1)$ jobs of size $ub \}$, where $x\in [0, ub]$ such that $x+y+(k-1)\cdot ub$=$1$ and $y\leq x\leq \frac{ub}{2}$ and $J''$=$\{J_1/x, J_2/x, J_3/z$ and $2(k-1)$ jobs of size $\frac{1}{k} \}$, where $2x+z+(k-2)\cdot \frac{1}{k}$=$1$ and $x<z<\frac{1}{k}$. Any algorithm *A* has the option to schedule the fist two jobs $J_1$ and $J_2$ either on the same machine or on different machines. If algorithm *A* schedules $J_1$ and $J_2$ on the same machine, then for the sequence $J'$, we obtain $C_A\geq 2x+(k-1)\cdot ub$. If $J_1$ and $J_2$ are scheduled on different machines, then for $J''$ we have $C_A\geq x+(k-1)\cdot \frac{1}{k}+z$=$1-x+\frac{1}{k}$. Therefore, in both cases, we obtain $C_A\geq \min\{1-x+\frac{1}{k}, 2x+(k-1)\cdot ub\}$. We obtain $C_{OPT}$=$1$ by assigning $J_1$ and $J_2$ to different machines for $J'$ and by scheduling them on the same machine for $J''$. Therefore, we have $\frac{C_A}{C_{OPT}}$=$\min\{1-x+\frac{1}{k}, 2x+(k-1)\cdot ub\}$. By maximizing w.r.t $x$, we achieve $\frac{C_A}{C_{OPT}}\geq (\frac{k-1}{3})\cdot ub$+$\frac{2}{3}\cdot(\frac{k+1}{k})$. They proposed the optimal algorithm $H'$ for $ub\in [\frac{1}{k}, \frac{2(k+1)}{k(2k+1)}]$, which is $(k\cdot ub)$-competitive for $ub\in [\frac{2(k+1)}{k(2k+1)}, \frac{1+2k}{2k^2})$. Algorithm $H'$ schedules an incoming job $J_i$ on the machine $M_1$ if $l_1+p_i\leq 1+\frac{1}{2k+1}$; else if $l_2+p_i\leq 1+\frac{1}{2k+1}$, then $J_i$ is scheduled on the machine $M_2$; else $J_i$ is assigned to the machine $M_j\in \{M_1, M_2\}$ such that $l_j$=$\min\{l_1, l_2\}$. They also proposed a $(1+\frac{1}{2k})$-competitive algorithm for $ub\in [\frac{1+2k}{2k^2}, \frac{2k-1}{2k(k-1)}]$. In \[50\], they studied the setup $P_3|Sum|C_{max}$ and obtained the *LB* ($1+ \frac{\sqrt{129}-9}{6}) > 1.392$. An *UB* $1.421$ was shown by a pre-processing policy of the available jobs. Here, a non-trivial challenge is to tighten or diminish the gap between the obtained *LB* and *UB*.\
**Max.** Wu et al. \[58\] followed the work of \[29\] and obtained a *tight* bound $2-\frac{1}{m-1}$ for $m$=$3, 4$ with known *Max*. Sun and Huang \[64\] considered a variant, where all machines are not given at the outset. However, machine availability time $r_j$ is given at the outset for each machine. W.l.o.g, it is assumed that $r_m\geq r_{m-1}\geq....\geq r_1$. They obtained a *LB* $1.457$ for $m>6$. They proposed a $(2-\frac{1}{m-1})$-competitive algorithm, which assigns an incoming job $J_i$ by algorithm *LS* unless $p_i$=$p_{max}$ and $(r_{min}+l_{min}+p_i)> 2\cdot p_{max}$; otherwise $J_i$ is scheduled on machine $M_1$ and the successive jobs are scheduled by algorithm *LS*, where $r_{min}$ and $l_{min}$ are the release time and load of the most lightly loaded machine respectively.\
**Combined Information.** Hua et al. \[48\] advanced the work of \[28\] for $3$-identical machine setting with known *Sum* and *Max*. They obtained an *UB* $1.4$ and a *LB* $1.33$. Wu et al. \[54\] tighten the gap between the obtained *UB* and *LB* of \[48\] and obtained a *tight* bound $1.33$ for the setup $P_3|Sum, Max|C_{max}$.\
**GOS.** Park et al. \[46\] initiated the study on semi-online scheduling under *GOS* eligibility with known *Sum*. They considered that a job with $g_i$=$1$ can only be processed by machine $M_1$ and if $g_i$=$2$, then $J_i$ can be processed by any of the machines. They proposed a $1.5$-competitive semi-online algorithm for the setup $P_2|GOS, Sum|C_{max}$. The algorithm schedules an incoming job $J_i$ to machine $M_1$ if $g_i$=$1$; else if $g_i$=$2$ and $l_2+p_j\leq (\frac{3}{2})\cdot L$, then $J_i$ is scheduled on machine $M_2$; else $J_i$ is assigned to machine $M_1$, where $L$=$\frac{Sum}{2}$. For the same problem, Jiang et al. \[47\] studied the preemptive version with *GOS* and proposed a $1.5$ competitive algorithm. For the non-preemptive case with *GOS*, they improved the *UB* from $2$ obtained in \[10\] to $1.66$. Wu and Yang \[66\] studied $2$-identical machine case with *GOS*. They investigated the problem separately for known *Opt* and *Max*.\
**Inexact EPI.** Tan and He \[53\] studied semi-online settings, where the value of a known *EPI* is given in interval or in the inexact form unlike the exact value. For some $x>0$ and the *disturbance parameter* $y\geq 1$, the following *EPI*s were considered for the respective settings: for $P_2|disOpt|C_{max}$, it is given that $C_{OPT}\in [x, yx]$; for $P_2|disSum|C_{max}$, it is known that $Sum\in [x, yx]$ and for $P_2|disMax|C_{max}$, it is known that $p_{max}\in [x, yx]$. For $P_2|disOpt|C_{max}$, they achieved a *LB* $1.5$ for $y\geq 1.5$ and obtained *UB*s $\frac{7y+1}{4y+2}$ for $1\leq y\leq \frac{5+\sqrt{41}}{8}$ and $y$ for $\frac{5+\sqrt{41}}{8}\leq y< 1.5$. They proved *LB* $1.5$ for the setup $P_2|disSum|C_{max}$, where $y\geq 1.5$. For $P_2|disMax|C_{max}$, they proved *LB*s $\frac{2y+2}{y+2}$ for $y$=$1.23$ and $1.5$ for $y\geq 2$. Further, they proposed the algorithm *modified PLS(MPLS)* and achieved an *UB* $\frac{2y+2}{y+2}$ for $y\in [1, 2]$ and showed its tightness for $y\in [1, \sqrt{5}-1]$. Algorithm *MPLS* assigns each incoming job $J_i$ to machine $M_1$ until the arrival of any job $J_b$ for which $p_b\in [1, y]$ and $(l_1+p_b)>2$. Thereafter, $J_b$ and all successive jobs are scheduled by algorithm *LS*.\
**Job Reassignment.** Tan and Yu \[57\] studied a semi-online variant, where an algorithm is allowed to re-schedule some of the already assigned jobs under certain conditions. For the setup $P_2|reasgn(last(k))|C_{max}$, they proved *LB* $1.5$ and showed that algorithm *LS* is optimal with no re-assignments. For $P_2|reasgn(last)|C_{max}$, they proposed algorithm *RE* and obtained a *tight* bound $1.414$. Algorithm *RE* assigns an incoming job $J_i$ to the highest loaded machine $M_j$ if $l_1\leq (\sqrt{2}+1)\cdot (l_2+p_i)$ and $p_i\leq \sqrt{2}\cdot l_1$; otherwise, $J_i$ is scheduled on machine $M_{3-j}$. After the assignment of all jobs, algorithm *RE* checks for re-assignment. If all jobs have been scheduled on the same machine $M_j$, then the job $J_n$(last job) is re-scheduled on the machine $M_{3-j}$. Let $J^{1}_{n_1}$ and $J^{2}_{n_2}$ be the last two jobs scheduled on machines $M_1$ and $M_2$ respectively. Let us consider $p_x$=$\max\{p^{1}_{n_1}, p^{2}_{n_2}\}$ and $p_y$=$\min\{p^{1}_{n_1}, p^{2}_{n_2}\}$. Algorithm *RE* re-assigns $J_x$ followed by $J_y$ to the $M_j$, which can obtain minimum $c_x$ and $c_y$ respectively. For $P_2|reasgn(k^*)|C_{max}$, they proposed algorithm *RA* and achieved a *tight* bound $1.33$. Algorithm *RA* schedules jobs $J_1$ and $J_2$ on two different machines. Let $l_1$=$\max\{l_1, l_2\}$. Each incoming job $J_i$, where $3\leq i\leq n$, is scheduled on machine $M_1$ if $l_1+p_i\leq 2\cdot l_2$; otherwise, $J_i$ is scheduled on machine $M_2$. After the scheduling of all jobs, if $l_2> 2\cdot l_1$, then the job $J^{2}_{n_2-1}$ is re-scheduled on machine $M_1$. The following non-trivial questions remain open: What is the minimum number of re-assignments that is sufficient to improve the known competitive bounds? Is the re-assignment policy with *EPI* such as *Decr*, *Opt*, *Sum* or *Max* practically significant and helps in achieving optimal bounds on the *CR*?\
**Max-Min Objective.** Tan and Wu \[52\] studied non-preemptive semi-online scheduling on $m$-identical machine($m \geq 3$) with $C_{min}$ objective. They proposed a $(m-1)$-competitive algorithm for the setup $P_m|Sum|C_{min}$. The idea is to keep the loads of all machines under $\frac{Sum}{2m}$. The machine $M_m$ is reserved from starting to schedule a job $J_i$, if there exists no machine $M_j$, where $1\leq j\leq m-1$ for which $l_j$ is at most $\frac{Sum}{m}$ or $\frac{Sum}{2m}$ after the assignment of $J_i$. If there exists some machines with load at most $\frac{Sum}{m}$, then assignment of $J_i$ to $M_m$ makes $l_m> \frac{Sum}{2m}$ and if there are some machines with load at most $\frac{Sum}{2m}$, then $J_i$ and the remaining jobs are scheduled on $M_m$. They proposed a $(m-1)$-competitive algorithm for $P_m|Max|C_{min}$. Each incoming job is scheduled on any one of the $m-1$ machines by algorithm *LS* until the arrival of a job $J_i$ with $p_i$=$p_{max}$ or $p_i+\min\{l_1, l_2,...,l_{m-1}\}> 2\cdot (p_{max})$. Such a $J_i$ is scheduled on machine $M_m$ and the successive jobs are scheduled over $m$-machines by algorithm *LS*. The idea is to maintain a load of at most $2\cdot (p_{max})$ in each machine, where the machine $M_m$ is kept idle for the largest job $J_b$ with $p_b$=$p_{max}$. They obtained *tight* bounds $1.5$ and $m-2$ for $m$=$3$ and $m\geq 4$ respectively with combined information on *Sum* and *Max*. We now present the main results obtained for semi-online scheduling on identical machines for the years 2006-2010 in table \[tab: Important Results on Identical Machines: 2006-2010\].
**Author(s), Year** **setup($\alpha| \beta| \gamma$)** **Competitiveness Results**
------------------------------ ------------------------------------------------------------------------------------------------ ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Angelelli et al. 2006 \[45\] $P_2 | Sum, TGRP(ub) | C_{max}$ ($1+ \frac{1}{2n + 1}$) Tight for $ub \in [ \frac{1}{n}, \frac{2(n+1)}{n(2n+1)}]$, ($(\frac{n-1}{3}) ub + (0.666) (\frac{n+1}{n})$) Tight for $ub \in (\frac{2n-1}{2n(n-1)}, \frac{1}{n-1}]$
Park et al. 2006 \[46\] $P_2 | GOS, Sum | C_{max}$ $1.5$ Tight
Jiang et al. 2006 \[47\] $P_2 | GOS | C_{max}$ $P_2 | pmtn, GOS | C_{max}$ $1.66$ UB, $1.5$ Tight
Hua et al. 2006 \[48\] $P_3 | Sum, Max | C_{max}$ $1.4$ UB, $1.33$ LB.
Angelelli et al. 2007 \[50\] $P_3 | Sum | C_{max}$ $1.392$ LB, $1.421$ UB.
Tan and Wu 2007 \[52\] $P_m | Sum | C_{min}$ $P_m | Max | C_{min}$ $P_m | Sum, Max | C_{min}$ ($m-1$)-competitive for *Sum* or *Max*, $1.5$ Tight with *Sum* and *Max* for $m=3$, ($m-2$) Tight for $m \geq 4$.
Tan and He 2007 \[53\] $P_2 | dis Opt | C_{max}$ $P_2 | dis Sum | C_{max}$ $P_2 | dis Max | C_{max}$ $1.5$ Tight with *Opt* or *Sum* for $y \geq 1.5$, $1.5$ Tight with *Max* for $y \geq 2$
Wu et al. 2007 \[54\] $P_3 | Sum, Max | C_{max}$ $1.33$ Tight.
Tan and Yu 2008 \[57\] $P_2 | reasgn(last(k)) | C_{max}$ $P_2 | reasgn(k^*) | C_{max}$ $P_2 | reasgn(last) | C_{max}$ ($1.5$, $1.33$, $1.414$) LB for respective setups
Wu et al. 2008 \[58\] $P_m | Max | C_{max}$ ($2- \frac{1}{m-1}$ ) Tight
Sun and Huang 2010 \[64\] $P_m | r_j, Max | C_{max}$ $1.457$ LB for $m > 6 $, ($2- \frac{1}{m-1}$) Tight.
Wu and Yang 2010 \[66\] $P_2 | GOS, Max | C_{max}$ $P_2 | GOS, Opt | C_{max}$ ($1.618, 1.5$) Tight for respective setups.
: Important Results for Identical Machines: 2006-2010
\[tab: Important Results on Identical Machines: 2006-2010\]
####
**Related Machines:**\
**Last Job.** Epstein and Ye \[51\] followed the work of \[26\] and considered *LL* as the known *EPI* in their study of semi-online scheduling on $2$-related machines with *min-max* and *max-min* optimality criteria. They considered $S_1$=$\frac{1}{s}$ and $S_2$=$1$, where $s\geq 1$. They proposed in general an algorithm for both optimality criteria and analyzed its performance for various intervals of $s$. The algorithm schedules an incoming job $J_i$ on machine $M_1$ if $l_2+p_i> \alpha(s)\cdot (l_1+s\cdot p_i)$; otherwise job $J_i$ is scheduled on machine $M_2$, where $0< \alpha(s)< \frac{1}{s}$. If $J_i$=$J_n$, then $J_i$ is scheduled on $M_2$. The key idea is to keep the highest speed machine $M_2$ relatively light loaded to schedule $J_n$(*largest job*) on it. They obtained *tight* bound $2.618$ for the setup $Q_2|LL|C_{min}$. They achieved *UB* $1.5$ and *LB* $1.465$ for the setup $Q_2|LL|C_{max}$.\
**Sum.** Angelelli et al. \[55\] studied the setup $Q_2|Sum|C_{max}$. They considered speeds $S_1$=$x$, $S_2$=$1$ and $Sum$=$1+x$, where $x\geq 1$. They showed *LB* and *UB* as functions of $x$. They proposed algorithm $H'$ for $x\in [1, 1.28]$, which assigns an incoming job $J_i$ to machine $M_1$ if $l_1+p_i\leq x\cdot (1+\frac{1}{2x+1})$; otherwise $J_i$ is scheduled on machine $M_2$. They proved $(\frac{2+2x}{2x+1})$-competitiveness of algorithm $H'$. They developed algorithm $H''$ for $x\in[1.28, 1.41]$, which assigns an incoming job $J_i$ to machine $M_1$ if $l_1+p_i \leq x^2$; else $J_i$ is assigned to machine $M_2$. They showed that algorithm $H''$ is $x$-competitive. For $x\geq 1.41$, they designed algorithm $H'''$, which assigns an incoming job $J_i$ to machine $M_2$ if $l_2+p_i\leq 1+\frac{1}{x+1}$; else $J_i$ is scheduled on machine $M_1$. They proved ($\frac{x+2}{x+1}$)-competitiveness of algorithm $H'''$. Ng et al. \[60\] studied the setup $Q_2|Sum|C_{max}$ by considering $S_1$=$1$ and $S_2\geq 1$. They obtained competitive bounds as functions of intervals of $s\geq 1$, where the largest gap between the *LB* and *UB* is at most $0.01762$. They achieved a *LB* $\frac{s+2}{s+1}$ for $s\geq \sqrt(3)$ and overall *UB* $1.369$ for $s \in [1, \infty)$. Angelelli et al. \[63\] investigated for the setup stated in \[55, 60\] by introducing a *geometric representation* of the scheduling problem through a *planar model*. They considered $2$-related machine setup with speeds $S_1$=$1$, $S_2$=$b$ and $Sum$=$b+1$, where $b\geq 1$. They represented scheduling of jobs in planar model as a game between *constructor(K)* and *scheduler(H)*, where *K* submits jobs one by one and *H* schedules a job upon its availability on a machine by following an algorithm. They illustrated the game in a plane by representing each point($x, y$) as the situation, where $x$=$l_1$ and $y$=$l_2$. Here, a move of *K* corresponds to the arrival of a new job $J_i$ with $p_i>0$ and the move of *H* specifies, whether to move to the point($x+p_i, y$) or to the point ($x, y+p_i$) from the point ($x, y$) in the plane. The game ends after reaching the line $x+y$=$b+1$. Now, the current position of the point($x, y$) determines the makespan incurred by the scheduler $H$. They showed a *LB* $1.359$ for $b\in[1.366, 1.732]$, which they proved to be optimal for $b$=$1.5$.\
**Buffer.** Englert et al. \[56\] investigated both $m$-identical and $m$-related machines settings with a buffer of size $k\in \theta(m)$(where, $\theta(m)$ is a function on number of machines). They introduced the *re-ordering of buffer* policy which does not assign each incoming job immediately to any of the machines, rather stores the jobs in the buffer and re-order the stored input job sequence prior to construct the actual schedule so as to achieve minimum makespan. They obtained *LB* and *UB* $1.333$, $1.465$ respectively for $m$-identical machine which beats the previous best results obtained by non-reordering buffer strategies of \[13, 14, 37\]. For $m$-related machine setup, they obtained an *UB* $2$ with a buffer of size $m$.\
**Preemptive Semi-online Scheduling.** Chassid and Epstein \[59\] studied preemptive semi-online scheduling on $2$-related machine setup. They considered both *max-min* and *min-max* optimality criteria with known *GOS* and *Sum*. They considered that $S_1$=$1$, $S_2$=$b$, $Sum$=$1$, where $b\geq 1$. They assumed that a job $J_i$ with $g_i$=$1$ must be processed only on machine $M_1$ and with $g_i$=$2$ it can be processed on any $M_j\in\{M_1, M_2\}$. They proposed $1$-competitive algorithm *FSA* and proved its *tightness* for both optimality criterion with the key idea of keeping the load of machine $M_2$ at most $\frac{Sum}{b+1}$. The optimality of algorithm *FSA* was shown by analyzing the following two cases. Case 1: if $l_2$=$\frac{Sum}{b+1}$, then we have $l_1$=$\frac{Sum}{b+1}$ by considering $Sum$=$1$ and $b$=$1$, this implies $C_{FSA}$=$\frac{1}{2}$ followed by $\frac{C_{FSA}}{C_{OPT}}$=$1$, where $C_{OPT}\geq \frac{Sum}{2}$=$\frac{1}{2}$. Case 2: if $l_2<l_1$, means machine $M_2$ has been equipped with all $J_i$s’ with $g_i$=$2$, so the remaining $J_i$s’ with $g_i$=$1$ have been scheduled on machine $M_1$, which eventually balances the loads of $M_1$ and $M_2$. Therefore, we obtain optimal schedules in both the cases. Ebenlendr and sgall \[61\] proposed an unified algorithm *RatioStretch* for preemptive semi-online scheduling on $m$-uniform machines($m\geq 2$). They proved that the algorithm achieves optimum approximation ratio that holds for any values of $s$ with any known *EPI*. They computed the ratio by linear program, where machines speeds are considered as input parameters. They established relationships among well-known semi-online setups for uniform machines and obtained competitive bounds in each setup for large $m$.\
**Opt.** Ng et al. \[60\] improved the results of Epstein \[33\] for the speed ratio interval $s\in [1.366, 1.395]$ and obtained an *UB* $\frac{2s+1}{2s}$. They showed *tight* bound $1.366$ for overall $s\in [1, \infty)$.\
**Job Reassignment.** Liu et al. \[62\] studied the setup $Q_2|GOS|C_{max}$ by considering $S_1$=$1$ for higher *GOS* machine($M_1$) and $S_2$=$x$ for ordinary machine($M_2$). They obtained *LB*s $1+ \frac{2x}{x+2}$ for $0 < x\leq 1$ and $1+ \frac{x+1}{x(2x+1)}$ for $x > 1$ by considering different *GOS* levels. They proved *LB* $1+ \frac{1}{1+x}$ with *re-assignment of last $k$ jobs(reasgn(last(k)))* and *LB* $\frac{{(S+1)}^2}{S^2 + S + 1}$ for *re-assignment of one job from every machine(reasgn$(last)^*$)*. They proposed ($\frac{{(x+1)}^2}{x+2}$)-competitive algorithm *EX-RA* for both types of re-assignment policies by considering $S_1$=$x$ and $S_2$=$1$, where $1\leq x \leq 1.414$. Algorithm *EX-RA* schedules the jobs $J_1$ and $J_2$ on different machines such that $l_1$=$\max\{p_1, p_2\}$ and $l_2$=$\min\{p_1, p_2\}$. For each incoming job $J_i$($3\leq i\leq n$), if $l_j+\frac{p_i}{x}\leq (x+1)\cdot l_2$, then job $J_i$ is scheduled on machine $M_1$; otherwise $J_i$ is assigned to machine $M_2$. After the scheduling of job $J_n$, if $l_2\leq (x+1)\cdot l_1$, then we have $C_{EX-RA}$=$\max\{l_1, l_2\}$; otherwise the second last job of machine $M_2$ is re-scheduled on machine $M_1$ and $l_1$, $l_2$ is updated to obtain the final $C_{EX-RA}$=$\max\{l_1, l_2\}$. Cao and Liu \[65\] followed the re-assignment policies of \[57, 62\] for $2$-related machine setup. They considered re-assignment of last job of each machine and obtained overall competitive ratio of $min \{\sqrt{s+1}, \frac{s+1}{s}\}$ for different speed ratio($s$) intervals. We now present the main results obtained for semi-online scheduling on uniform related machines for the years 2006-2010 in table \[tab: Important Contributions for Uniform Machines: 2006-2010\].
**Author(s), Year** **setup($\alpha| \beta| \gamma$)** **Competitiveness Results**
--------------------------------- -------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Epstein and Ye 2007 \[51\] $Q_2 | LL | C_{min}$, $Q_2 | LL | C_{max}$ $1.5$ UB and $1.465$ LB for $C_{max}$, $2.618$ Tight for $C_{min}$.
Angelelli et al. 2008 \[55\] $Q_2 | Sum | C_{max}$ $1.33$ Tight for $x = 1$, $x$ Tight for $x \in (1.28, 1.366)$, $\frac{x+2}{x+1}$ Tight for $x \geq 1.732$
Englert et al. 2008 \[56\] $P_m | re B(k) | C_{max}$, $Q_m | re B(k) | C_{max}$ ($1.333, 1.465$) LB and UB respectively for $P_m$ with $k \in \theta(m)$, ($2- \frac{1}{m-k+1}$) Tight for $P_m$ with $k \in [1, \frac{m+1}{2}]$, $2$ Tight for $Q_m$ with $k \in m$
Chassid and Epstein 2008 \[59\] $Q_2 | pmtn, GOS, Sum | C_{min}$ $Q_2 | pmtn, GOS, Sum | C_{max}$ $1$ Tight for both setups
Ng et al. 2009 \[60\] $Q_2 | Opt | C_{max}$, $Q_2 | Sum | C_{max}$ ($1.366, 1.369$) Tight with *Opt* or *Sum* respectively.
Ebenlendr and Sgall 2009 \[61\] $Q_3 | pmtn, Sum | C_{max}$ $Q_3 | pmtn, Max | C_{max}$ $Q_3 | pmtn, Decr | C_{max}$ $1.138$ Tight with $S_1 = 1.414$, $S_2 = S_3 = 1$ and known *Sum*, $1.252$ Tight with $S_1 =2$, $S_2 = S_3 = 1.732$ and known *Max* , $1.52$ Tight with known *Decr*.
Liu et al. 2009 \[62\] $Q_2 | GOS | C_{max}$ $Q_2 | reasgn(last(k)) | C_{max}$ $Q_2 | reasgn(last)^{*} | C_{max}$ ($1+ \frac{x+1}{x(2x+1)}$) LB with *GOS* for $x>1$, ($1+\frac{1}{1+x}$) LB with reasgn(last(k)), ($\frac{(x+1)^2}{x+2}$) Tight with both re-assignment policies for $1\leq x\leq 1.414$ .
Angelelli et al. 2010 \[63\] $Q_2 | Sum | C_{max}$ $1.359$ LB with $b = 1.5$.
Cao and Liu 2010 \[65\] $Q_2 | reasgn(last) | C_{max}$ ($\sqrt{s}+1$) Tight for $1 \leq s < 1.618$, $\frac{s+1}{s}$ Tight for $s \geq 1.618$
: Main Results for Related Machines: 2006-2010
\[tab: Important Contributions for Uniform Machines: 2006-2010\]
Recent Works in Semi-online Scheduling {#subsec:Recent Works in Semi-online Scheduling}
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The recent era of semi-online scheduling has been dominated by non-preemptive scheduling in identical machines with multiple grades of service levels(*GOS*) or machine hierarchy. Semi-online scheduling in *unbounded batch machine* has been introduced. Several instances of related machines have been studied for various unexplored speed ratio intervals. Job rejection and reassignment policies have been introduced for various setups of related machines. We now present an overview of the state of the art in semi-online scheduling for unbounded batch machine, uniform related machines and identical machines as follows.
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**Unbounded Batch Machine:** Yuan et al. \[68\] introduced semi-online scheduling in single unbounded batch machine to improve the $1.618$ competitive bound obtained by pure online strategies in \[25,32\]. They considered that at any time step $t$, we are given with $p_t$ and $r_t$ of job $J_t$, where $J_t$ is the largest job that will arrive after time $t$. They obtained *tight* bound $1.382$ with known $p_t$ by considering at most two batches. With given $r_t$, they achieved *LB* $1.442$ and *UB* $1.5$ by constructing at most three batches. With known $r_t$, they proposed an algorithm, which constructs at most two batches. The algorithm resets the value of $r_{t_1}$=$\max\{r_{t_1}, \alpha\cdot(p_{t_1})\}$, then forms the first batch ($U(t_1)$) by considering all jobs that are available by time $r_{t_1}$ and schedules them irrevocably on the machine, where $r_{t_1}$ is the release time of the first largest job and $\alpha$=$0.618$. The second batch $U(t_2)$ is formed by considering all jobs that are received at the time step $t_2$=$r_{t_1}+p_{t_1}$, then the value of $r_{t_2}$ is reset to $\max\{r_{t_2}, \alpha\cdot(p_{t_2})\}$ prior to schedule all jobs of batch $U(t_2)$. They obtained a matching *UB* $1.618$ to that of pure online strategies. It is now a non-trivial challenge to beat the $1.618$ competitive bound by forming at most $2$ batches with known $r_t$.
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**Related Machines:**\
**Buffer.** Epstein et al. \[95\] investigated the setup $Q_m|re B(k)|C_{min}$ and proposed a $m$-competitive algorithm, where $m\geq 2$ and $k$=$m+1$. The algorithm keeps initial $m+1$ incoming jobs in the buffer. After arrival of the $(m+2)^{th}$ job until availability of the ${n}^{th}$ job, each time the smallest job $J_i$ is selected from $m+2$ available jobs and is scheduled by algorithm *LS*, while not considering the machine speeds. When there is no jobs to arrive and the buffer contains $m+1$ jobs such that $p_1\leq p_2\leq ...\leq p_m\leq p_{m+1}$, the algorithm schedules the jobs in any of the following rules.
1. Schedule $J_1$ by *LS* rule and schedule the jobs $J_i$, where $2\leq i\leq (m+1)$ to the corresponding machine $M_j$ respectively, where $1\leq j\leq m$.
2. Schedule the jobs by rule $1$, but migrate $J_i$ to the machine $M_m$ for some $2\leq i\leq m$.
3. Schedule $J_{i+1}$ to $M_{i}$ for $1\leq i\leq m$ and schedule $J_1$ to a machine $M_k$ such that $2\leq k\leq (m-1)$.
Interestingly, the algorithm ignores the machine speeds until the arrival of all jobs, and then the relative order of the machines speeds are considered for making scheduling decision. Further, they studied the setup $Q_2|B(1)|C_{min}$, where $S_1$=$1$ and $S_2\geq 1$ and proposed a $\frac{2s+1}{s+1}$ competitive algorithm. The algorithm keeps the first job $J_1$ in the buffer, thereafter on the arrival of each incoming $J_i$, $2\leq i\leq n$, it is assumed that $p_x$=$\min\{p_{i-1}, p_i\}$ and $p_y$=$\max\{p_{i-1}, p_i\}$ . Now, job $J_x$ is scheduled on machine $M_1$ if $\frac{l_2+p_y}{s}\geq \frac{l_1+p_x}{s+1}$; otherwise $J_x$ is assigned to machine $M_2$. The goal is to schedule the smaller jobs to the slowest machine and relatively larger jobs to the fastest machine so as to maximize the minimum work load incurred on a machine. Lan et al. \[76\] studied the setup $Q_m|B(k)|C_{max}$ and achieved a tight bound ($2-\frac{1}{m}+\epsilon$) with $k$=$m$ and $m\geq 2$, where $\epsilon > 0$.\
**Job Rejection.** Min et al. \[96\] initiated the study on semi-online scheduling in $2$-uniform machine with job rejection policy by considering $S_1$=$1$, $S_2\geq 1$. The rejection policy describes a scenario, where an incoming job $J_i$ can either be assigned to a machine or can be rejected permanently by incorporating a penalty of $x_i$. The objective of any semi-online algorithm is to incur a minimum value for the sum of makespan with sum of all penalties. The algorithm is given beforehand with two parallel processors for making scheduling policies, finally the best policy is opted for actual assignment of the jobs. Min et al. proposed a semi-online algorithm with the following rules for scheduling of each incoming job: Upon availability of a new job $J_i$, processor 1 rejects $J_i$, if $x_i\leq \alpha\cdot p_i$, where $\alpha$=$\frac{1}{s+1}$; else schedules $J_i$ by algorithm *LS*. On the other hand, processor 2 rejects $J_i$, if $x_i\leq \beta\cdot p_i$, where $\beta$=$\frac{2}{2s+1}$; else schedules $J_i$ by algorithm *LS*. After the assignments of all jobs, one of the policies that has yielded a minimum objective value is opted by the algorithm for actual scheduling of the jobs. The algorithm achieves tight bounds $\frac{2s+1}{s+1}$ for $1\leq s\leq 1.618$; and $\frac{s+1}{s}$ for $s> 1.618$.\
**Max.** Cai and Yang \[97\] investigated the setup $Q_2|Max|C_{max}$ by considering $S_1$=$1$ and $S_2\geq 1$. They proposed algorithm *Low Speed Machine Priority(LSMP)* and obtained tight bound $\max\{\frac{2s+2}{2s+1}, s\}$ for $s\in [1, 1.414]$. Algorithm *LSMP* schedules an incoming job $J_i$ to machine $M_1$ if $p_i$=$p_{max}$; thereafter the remaining jobs are scheduled by algorithm *LS*. If $p_i< p_{max}$, and if $l^{i}_{1}+p_{max}+p_i < l^{i}_{2}+\frac{p_i}{s}$, then schedules $J_i$ on $M_1$ (where, $l^{i}_{j}$ is the load of machine $M_j$ just before the scheduling of $J_i$); otherwise $J_i$ is scheduled on $M_2$. They proposed algorithm *HSMP* and obtained tight bound $\frac{2s+2}{s+2}$ for $1\leq s\leq 1.414$. The tight bounds achieved for $s\geq 1.414$ are expressed by an algebraic function $r(s)$ as follows. $$r(s)=
\begin{cases}
\frac{s+2}{s+1},& \text{for}\hspace*{0.2cm} 1.414\leq s\leq 2\\
\frac{3s+2}{2s+2},& \text{for}\hspace*{0.2cm} 2\leq s\leq 2.732\\
\frac{s+1}{s},& \text{for}\hspace*{0.2cm} s\geq 2.732\\
\end{cases}$$ The idea is to schedule the first largest job on machine $M_2$ and to schedule the remainning jobs by algorithm *LS*.\
**Opt.** Dosa et al. \[69\] followed the work of \[33, 55, 60\] for $Q_2|Opt|C_{max}$ by considering $S_1$=$1$, $S_2\geq 1$ and $s\in[1, 1.28]$. They obtained *LB* $\min \{1+\frac{1}{3s}, 1+\frac{3s}{5s+5}, 1+\frac{1}{2s+1}\}$. The *LB* was derived by constructing a *lower bound binary tree*, where each node represents a job $J_i$ along with its *size* $p_i$ and each arc specifies an assignment of $J_i$ on machine $M_j\in\{M_1, M_2\}$. The left branch of a node represents scheduling of $J_i$ on $M_1$ and right branch specifies scheduling of $J_i$ on $M_2$. The *size* of the next job $J_{i+1}$ is chosen based on its assignment to any of the $M_j$ in correspondence to the *size* and scheduling of $J_i$. By traversing the *lower bound binary tree* from root to the leaf nodes, one can obtain the instances, for which any semi-online algorithm achieves a *CR* of at least the defined *LB*. In \[89, 94\], the authors considered the setup studied in \[69\] and obtained lower bounds in terms of an algebraic function $r(s)$ for the following unexplored speed ratio intervals. $$r(s)=
\begin{cases}
\frac{6s+6}{4s+5},& \text{if}\hspace*{0.2cm} 1.3956\leq s\leq 1.443\\
\frac{12s+10}{9s+7},& \text{if}\hspace*{0.2cm} 1.66\leq s\leq 1.6934\\
\frac{18s+16}{16s+7},& \text{if}\hspace*{0.2cm} 1.6934\leq s\leq 1.6955\\
\frac{8s+7}{3s+10},& \text{if}\hspace*{0.2cm} 1.6955\leq s\leq 1.6963\\
\frac{12s+10}{9s+7},& \text{if}\hspace*{0.2cm} 1.6963\leq s\leq 1.7258\\
\end{cases}$$ In \[91\], they studied for the interval $s \in[1.710, 1.732]$ and achieved tight bounds of $\frac{2s+10}{9s+7}$ for $s$=$1.7258$ and $\frac{s+1}{2}$ for $1.725\leq s\leq 1.732$ respectively. The obtained results draw an insight that a single algebraic function can not formulate the tightness of the *LB*.\
**Sum.** Dosa et al. \[69\] investigated the setup $Q_2|Sum|C_{max}$ by considering $S_1$=$1$, $S_2\geq 1$ and *Sum*=$3s\cdot(1+s)$. They achieved *tight* bounds for the unexplored speed ratio interval $1\leq s< 1.2808$, which are presented by an algebraic function $r(s)$ as follows. $$r(s)=
\begin{cases}
1+ \frac{1}{3s},& \text{for}\hspace*{0.2cm} s\in [1, 1.071]\\
1+ \frac{3s}{4s+6},& \text{for}\hspace*{0.2cm} s\in [1.071, 1.0868]\\
1+\frac{1}{2s+1},& \text{for}\hspace*{0.2cm} s\in [1.0868, 1.2808]\\
\end{cases}$$ They proposed an algorithm by considering various time interval ranges as safe sets for scheduling decision making. The algorithm involves three subroutines as described below.\
*Subroutine 1 Master*
1. Upon the arrival of a new job $J_i$, run subroutine Slave.
2. If $J_i$=$J_1$, then run subroutine *CoalA* from starting; else continue *CoalA* from the breaking point of the last call for scheduling $J_{i-1}$.
3. If no more jobs to arrive, then stop; else move to step 1.
*Subroutine 2 Slave*
1. Schedule $J_i$ on machine $M_j$, if the value of $l_j+p_i$ is within the time interval range $[2s, 3s+1]$ for $j$=$1$ or $[3s^2-1, 3s^2+s] $ for $j$=$2$; thereafter, remaining jobs are scheduled on machine $M_{3-j}$ and stop.
2. Schedule $J_i$ on $M_j$, if $l_j\leq T_j$ and $l_j+p_j> T_{0j}$, where $T_1$=$3s+1-3s^2$, $T_2$=$s$, $T_{01}$=$3s+1$ and $T_{02}$=$3s^2+s$; thereafter, schedule the remaining jobs to machine $M_{3-j}$ and stop.
3. Schedule $J_i$ on $M_j$ if the value of $l_j+p_i$ is within the time interval range $[s-1, 3s+1-3s^2]$ for $j$=$1$ or the value within the range $[3s^2-s-2, s]$ for $j$=$2$; thereafter, schedule the remaining jobs on $M_{3-j}$ until $l_{3-j}< 2s$ for $j$=$2$ or $l_{3-j}< 3s^2-1$ for $j$=$1$; otherwise run the subroutine *Slave* once more.
*Subroutine 3 CoalA*
1. Schedule $J_i$ on $M_j$ until $l_j+p_i< s-1$.
2. Schedule $J_i$ on machine $M_1$, if $p_i< 3s^2-s-2$ and $l_1< 3s^2-2$.
3. Schedule $J_i$ on machine $M_2$, if $l_2+p_i< 3s^2-s-2$.
4. If $J_i$ is assigned to $M_2$, then the remaining jobs are scheduled on $M_2$ as long as $l_2\leq 3s^2-1$; thereafter, schedule the next job on $M_1$ and the remaining jobs on $M_2$; Stop.
The algorithm was shown to be *tight* with *CR* of $1+\frac{1}{3s}$ for $s\in [1, 1.071]$. A similar algorithm with slight modification in the time interval ranges of the safe sets was proposed for $s\in [1.071, 1.08]$.\
**GOS.** Hou and Kang \[98\] invesigated semi-online hierarchical scheduling on $m$-uniform machine setup with *max-min* and *min-max* objectives. They considered $m$ machines in two hierarchies, where in hierarchy $1$, we have $k$ machines, each with speed $S_1\geq 1$ and $g_j$=$1$, capable of executing all jobs. In hierarchy $2$, we have rest $m-k$ machines, each with speed $S_2$=$1$ and $g_j$=$2$, capable of executing the jobs having $g_i$=$2$. For $Q_m|GOS|C_{min}$, they proved that no online algorithm can be possible with a bounded *CR*. They investigated the setup $Q_m|pmtn, GOS|C_{min}$ and obtained *UB* of $\frac{2ks+m-k}{ks+m-k}$ for $0< s< \infty$ by applying the fractional assignment policy, where each incoming job $J_i$ can be splitted arbitrarily among the machines. For $Q_m|pmtn, GOS|C_{max}$, they achieved *UB* of $\frac{(ks+m-k)^2}{k^2s^2+ks(m-k)+(m-k)^2}$ for $0< s< \infty$. For $Q_m|pmtn, GOS, Sum|C_{max}$, they proposed a $1$-competitive algorithm. The idea is to schedule the jobs having $g_i$=$1$ evenly on the machines having $g_j$=$1$ and schedule the jobs having $g_i$=$2$ on the machines with $g_j$=$2$ as long as the loads of the machines are under a given threshold value. Lu and Liu \[81\] studied three variants of $Q_2|GOS|C_{max}$ with known Opt or Sum or Max by considering $S_1$=$1$, $S_2$=$s>0$, $g_i$=$1$(for processing job $J_i$ exclusively on machine $M_1$) and $g_i$=$2$(for making $J_i$ eligible for processing in any one of the $M_j\in\{M_1, M_2\}$). They proposed algorithm *Gos-OPT* for $Q_2|GOS, Opt|C_{max}$ and obtained *UB* of $\min\{\frac{1+2s}{1+s}, \frac{1+s}{s}\}$. Algorithm *Gos-OPT* schedules each incoming job $J_i$ on the machine $M_1$ if $g_i$=$1$; else if $g_i$=$2$ then $J_i$ is scheduled by the following policy: if $s\geq \frac{1+\sqrt{5}}{2}$, then $J_i$ is scheduled on machine $M_2$; if $0<s<\frac{1+\sqrt{5}}{2}$ and $\frac{l_2+p_i}{s}\leq (\frac{1+2s}{1+s})\cdot C_{OPT}$, then $J_i$ is scheduled on $M_2$; if $0<s<\frac{1+\sqrt{5}}{2}$ and $\frac{l_2+p_i}{s}> (\frac{1+2s}{1+s})\cdot C_{OPT}$, then $J_i$ is scheduled on $M_1$. They achieved a matching *UB* for $Q_2|GOS, Sum|C_{max}$ with an equivalent algorithm *Gos-SUM*, which is equivalent to algorithm *Gos-OPT*, just simply replacing $C_{OPT}$ by $\frac{Sum}{1+s}$ in the policy. They proposed algorithm *GoS-MAX* for $Q_2|GOS, Max|C_{max}$ and obtained *UB* $1+s$ for $0<s<\frac{\sqrt{5}-1}{2}$. Algorithm *GoS-MAX* schedules an incoming job $J_i$ on $M_1$ if $g_i$=$1$; else if $g_i$=$2$, then $J_i$ is scheduled by the following policy: if $0<s\leq \frac{\sqrt{5}-1}{2}$, then $J_i$ is scheduled on $M_1$; if $\frac{\sqrt{5}-1}{2}<s\leq 1$ and $\frac{l_2+p_i}{s}\leq \frac{1+\sqrt{5}}{2}\cdot (\max\{p_{max}, \frac{T_k}{1+s}, T^{k}_{1}\}$), then $J_i$ is scheduled on $M_1$; if $1<s<s^*$ and $\frac{l_2+p_j}{s}\leq (\frac{1+\sqrt{1+4s}}{2}\cdot (\max\{\frac{p_{max}}{s}, \frac{T_k}{s}, T^{k}_1\}))$, then $J_i$ is scheduled on $M_2$, else $J_i$ is assigned to machine $M_1$; and if $s\geq s^*$, then $J_i$ is scheduled on $M_2$. They achieved *UB* $\min\{1+s, \frac{1+\sqrt{5}}{2}\}$ for $0<s<1$ and *UB* $\min\{\frac{1+\sqrt{1+4s}}{2}, \frac{1+s}{s}\}$ for $s\geq 1$. (Note that: $T_k$ is referred to as sum of the sizes of the first $k$ jobs, $T^{1}_{k}$ represents the sum of the sizes of the set of jobs belongs to first $k$ jobs for which $g_i$=$1$ and $s^*\in (1.3247, 1.3248)$).\
**TGRP.** Luo and Xu \[88\] followed the work of Chassid and Epstein \[59\] for semi-online scheduling on $2$-parallel machines with *Max-Min* objective. They considered hierarchical scheduling to cater different levels of services to the input jobs. They investigated two semi-online variants, one with *TGRP*\[$1, b$\] and obtained lower bound of $1+b$ for $b \geq 1$. In the second case, they considered *TGRP*\[$1, b$\], *Sum* and achieved lower bound of $b$ for $1 \leq b < 2$. Cao and Liu \[99\] studied the setup $Q_2|TGRP|C_{max}$ by considering $S_1$=$1$, $S_1\leq S_2$=$s$ and $\forall_{i}$, $p_i\in [p, xp]$, where $p> 0$ and $x\geq 1$. They proved tight bound of $\min\{\frac{2s+1}{s+1}, \frac{s+1}{s}, x\}$ for algorithm *LS*, where $1\leq s\leq 1.618$ and $x\geq \frac{1+s}{s{\alpha}^2}$ with $\alpha$=$\frac{1+s-s^2}{s^2}$. They proposed a $s$-competitive algorithm, which is tight for $1.325\leq s\leq 1.618$ and $x\leq \frac{s^2-1}{1+s-s^2}$. The algorithm schedules an incoming job $J_i$ on machine $M_2$ if $l^{i}_{2}+\frac{p_i}{s}\leq s\cdot \frac{l^{i}_{1}+s\cdot l^{i}_{2}+p_i}{1+s}$; otherwise schedules $J_i$ on machine $M_1$. Further, they designed a new algorithm that achieves the following tight bounds for the unexplored speed ratio intervals, expressed in an algebraic function $r(s)$. $$r(s)=
\begin{cases}
s,& \text{for}\hspace*{0.2cm} s\in [1.206, 1.5]\hspace*{0.2cm} and \hspace*{0.2cm} s\leq x\leq \min\{2s-1, \frac{2s^2-2}{1+s-s^2}\}\\
\frac{1+x}{2},& \text{for}\hspace*{0.2cm} s\in [1, 1.28]\hspace*{0.2cm} and \hspace*{0.2cm}\max\{2s-1, \frac{-s+\sqrt{9s^2+8s}}{2s}\}\leq x\leq \frac{2}{s}\\
\end{cases}$$ The new algorithm schedules the initial four incoming jobs $J_1$, $J_2$, $J_3$ and $J_4$ on machines $M_2$, $M_1$, $M_2$ and $M_1$ respectively. Thereafter, each incoming $J_i$, where $i\geq 5$ is scheduled on machine $M_2$ if $l^{i}_{2}+\frac{p_i}{s}\leq \frac{k\cdot (l^{i}_{1}+sl^{i}_{2}+p_i)}{1+s}$, where $k$=$\max\{s, \frac{1+x}{2}\}$; else $J_i$ is scheduled on machine $M_1$. (Note that: $l^{i}_{j}$ is the load of machine $M_j$ just before the scheduling of job $J_i$.)\
**Job Reassignment.** Englert et al. \[93\] initiated the study on online scheduling in $m$-uniform machine with job reassignment policy, where $m\geq 2$. The aim is to explore, how far the reassignment of a small number of jobs helps to improve the *CR* of an algorithm designed for makespan minimization in pure online setup. They proposed an algorithm that achieves a *CR* between $1.33$ and $1.7992$ for different speed ratio intervals with at most $m$ reassignments. The algorithm functions in two phases, where in the first phase, online arriving jobs are scheduled on $m$ machines. And in the second phase, a specified number of jobs(at most $m$) are removed from the allocated machines and are re-scheduled on different machines as per a defined set of rules. We now present a summary of the main results obtained in recent years for semi-online scheduling on related machines in table \[tab: Important Results for Uniform Machines: 2011-2017\].
**Author(s), Year** **setup($\alpha| \beta| \gamma$)** **Competitiveness Results**
---------------------------- ------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Epstein et al. 2011 \[95\] $Q_2 | re B(k) | C_{min}$ $Q_m | re B(k)| C_{min}$ $\frac{2s+1}{s+1}$ Tight for $Q_2$ and $m$ Tight for $Q_m$
Cai and Yang 2011 \[97\] $Q_2 | Max | C_{max}$ $\frac{2s+2}{2s+1}$ for $s\in [1, 1.414]$; $\frac{s+1}{s}$ for $s\in [1.414, 2.732]$
Dosa et al. 2011 \[69\] $Q_2 | Opt | C_{max}$ $Q_2 | Sum | C_{max}$ min$\{1+\frac{1}{3s}, 1+\frac{3s}{5s+5}, 1+\frac{1}{2s+1}\}$ LB with *Opt*, ($1+\frac{1}{3s}$) Tight with *Sum*.
Hou and Kang 2011 \[98\] $Q_m|pmtn, GOS|C_{min}$ $Q_m|pmtn, GOS, Sum|C_{max}$ $\frac{2ks+m-k}{ks+m-k}$ UB for $C_{min}$, $1$ Tight for $C_{max}$
Lan et al. 2012 \[76\] $Q_m|B(k)|C_{max}$ ($2-\frac{1}{m}+\epsilon$) Tight with $k$=$m$
Lu and Liu 2013 \[81\] $Q_2 | GOS, Opt | C_{max}$ $Q_2 | GOS, Sum | C_{max}$ $Q_2 | GOS, Max | C_{max}$ $min \{\frac{1+2s}{1+s}, \frac{1+s}{s}\}$ Tight with *Opt* or *Sum*, $min \{\frac{1+ \sqrt{1+4s}}{2}, \frac{1+s}{s}\}$ Tight with *Max* for $s \geq 1$.
Luo and Xu 2015 \[88\] $Q_2 | GOS, TGRP | C_{min}$ $Q_2 | GOS, TGRP, Sum | C_{min}$ ($1+b$) LB with *TGRP*, ($b$) LB with *TGRP* and *Sum*.
Dosa et al. 2015 \[89\] $Q_2 | Opt | C_{max}$ ($\frac{6(s+1)}{4s+5}$) LB for $s \in [1.3956, 1.443]$, $min \{\frac{12s+10}{9s+7}, \frac{18s+16}{16s+7}, \frac{8s+7}{3s+10}, \frac{12s+10}{9s+7}\}$ LB for $s \in [1.666, 1.725]$.
Cao and Liu 2016 \[99\] $Q_2|TGRP|C_{max}$ $s$ Tight for $s\in[1.325, 1.618]$
Dosa et al. 2017 \[91\] $Q_2 | Opt | C_{max}$ ($\frac{12s+10}{9s+7}$) Tight for $s \approx 1.7258$, ($\frac{s+1}{2}$) Tight for $1.725 \leq s < 1.732$.
Englert et al. 2018 \[93\] $Q_m | reasgn | C_{max}$ $1.7992$ UB
: Summary of the Recent Contributions for Related Machines
\[tab: Important Results for Uniform Machines: 2011-2017\]
####
**Identical Machines:**\
**GOS.** Liu et al. \[73\] studied the setup $P_2|GOS, TGRP[a, ba]C_{max}$, where $a> 0$ and $b> 1$. The *GOS* model studied here considers two machines, where one machine can afford higher quality in service called *higher GOS machine* and the other one can cater normal quality in service called *lower GOS machine*. Each newly available job $J_i$ reveals its $p_i$ and $g_i$, where $g_i\in \{1, 2\}$. If $g_i$=$1$, then $J_i$ must be executed on machine $M_1$; if $g_i$=$2$, then $J_i$ can be executed on any of the machines. They proposed the algorithm *B-ONLINE* by following the the policy of Park et al. \[46\] and obtained the following *tight* bounds, expressed by an algebraic function $r(s)$. $$r(s)=
\begin{cases}
\frac{1+b}{2},& \text{for}\hspace*{0.2cm} 1.785\leq b\leq 2\\
1.5,& \text{for}\hspace*{0.2cm} 2\leq b\leq 5\\
\frac{4+b}{6},& \text{for}\hspace*{0.2cm} 5\leq b\leq 6\\
\end{cases}$$ Algorithm *B-ONLINE* works by the following policy. Initialize the parameters $l_1$=$0$, $l_2$=$0$, $P_{max}$=$0$, $X$=$0$ and $T$=$0$. Upon receiving a new job $J_i$, update $P_{max}$=$\max\{P_{max}, p_i\}$ and $X$=$X+\frac{p_i}{2}$. If $g_i$=$1$, then schedule $J_i$ on machine $M_1$ and update $T$=$T+p_i$. If $g_i$=$2$, then $J_i$ is scheduled on $M_2$ if $l_2+p_i\leq r(s)\cdot L$, where $L$=$\max\{X, T, P_{max}\}$; otherwise, $J_i$ is assigned to machine $M_1$. Further, they studied the setup $P_2|GOS, TGRP[a, ba], Sum|C_{max}$ and proposed the ($\frac{1+b}{2}$)-competitive optimal algorithm *B-SUM-ONLINE* for $Sum\geq (\frac{2b}{b-1})\cdot a$ and $1< b< 2$. Algorithm *B-SUM-ONLINE* schedules an incoming job $J_i$ on $M_1$ if $g_i$=$1$. If $g_i$=$2$ and $l_2+p_i\leq (\frac{1+b}{2})\cdot L$, then $J_i$ is scheduled on $M_2$; otherwise $J_i$ is scheduled on $M_1$. Wu et al. \[77\] followed the work of Liu et al. in the study of the setup $P_2|GOS, Opt|C_{max}$ and proposed a $1.5$ competitive optimal algorithm. The algorithm schedules an incoming job $J_i$ on machine $M_1$ if $g_i$=$1$. If $g_i$=$2$ and $l_2+p_i\leq (1.5)\cdot C_{OPT}$, then $J_i$ is scheduled on $M_2$; otherwise $J_i$ is assigned to machine $M_1$. The objective is to keep the loads of both machines under $(1.5)\cdot C_{OPT}$. Further, they considered *GOS* with known $p_{max}$ in $2$-identical machine setup and proposed $1.618$ competitive optimal algorithm *Gos-Max*. Algorithm *Gos-Max* works as follows: upon receiving a job $J_i$, update $X$=$X+\frac{p_i}{2}$, where $X$=$0$. If $g_i$=$1$, schedule $J_i$ on $M_1$ and update $T$=$T+p_i$, where, $T$=$0$. If $g_i$=$2$ and $l_2+p_i\leq (1.618)\cdot L$, then $J_i$ is scheduled on $M_2$, where, $L$=$\max\{p_{max}, X, T\}$; else schedule $J_i$ on machine $M_1$.\
Chen et al. \[80\] studied the setup $P_2|GOS, B(k)|C_{max}$ by considering known $g_i\in \{1, 2\}$ for each incoming job $J_i$. It is assumed that machine $M_1$ can execute all jobs, whereas machine $M_2$ can execute the jobs having $g_i$=$2$. They proposed a $1.5$ competitive optimal algorithm, which always tries to keep the largest job having $g_i$=$2$ in the buffer and schedule it at the end. The algorithm works in two phases, wherein the first phase, all jobs having $g_i$=$1$ are scheduled on $M_1$ and maximum possible jobs are assigned to $M_2$ as long as the desired *CR* holds. In the $2^{nd}$ phase, the largest job in the buffer is scheduled on the smallest loaded machine. Further, they studied the setup $P_2|GOS, reasgn(k)|C_{max}$ and proposed a $1.5$ competitive optimal algorithm with $k$=$1$. The idea is to schedule maximum number of jobs on a particular machine $M_j$ until $l_j$ reaches upto a defined threshold, then reassign the largest job scheduled on $M_j$ to the other machine $M_{3-j}$. Zhang et al. \[82\] improved the bounds obtained by Liu et al. in \[73\] with *GOS* and *TGRP*($a, ba$) for $1 \leq b < 3$. Further, they proved that use of preemption and idle time do not improve the competitiveness of the pure online setting of hierarchical scheduling in $2$-identical machines. Luo and Xu \[85\] improved the bounds given in \[46, 77\] for $2$-identical machines with known *Sum* and different *GOS* levels such as higher *GOS* and lower *GOS*.\
Chen et al. \[100\] extended their previous work \[80\] with similar idea and considered three different setups of online hierarchical scheduling in $2$-identical machines. They studied the setup, where $\sum_{}^{}{p_i}$ for the jobs with $g_i$=$1$ is known and proved a *tight* bound $1.5$ for algorithm *LS*. In another setup, they assumed known values of $T_1$=$\sum_{}^{}{p_i}$=$1$, $\forall J_i$ such that $g_i$=$1$ and the value of $T_2$=$\sum_{}^{}{p_i}$=$T> 0$, $\forall J_i$ such that $g_i$=$2$. They proposed the algorithm *CMF*, which achieves a *tight* bound of $1.33$. Algorithm *CMF* adopts the following rule: schedule an incoming job $J_i$ by its $g_i\in \{1, 2\}$ to respective $M_j\in \{M_1, M_2\}$ if $T\leq 2$. If $T> 2$, else if $g_i$=$1$, then $J_i$ is scheduled on machine $M_1$, else if $g_i$=$2$, then $J_i$ is assigned to $M_1$ if $l^{i}_{1}+1+p_j\leq \frac{1+T}{3}$, else let $x$=$i$, schedule $J_x$ and the remaining jobs by following rule: assign $J_x$ to $M_2$ and remaining jobs to $M_1$ if $l^{x}_{1}+1+p_x> \frac{2(1+T)}{3}$, else $J_x$ along with all future jobs for which $g_i$=$1$ are scheduled on $M_1$ and rest of the jobs are assigned to machine $M_2$. Further, they considered $B(k)$ in the first setup and obtained a *tight* bound of $1.33$ with $k$=$1$.\
Qi and Yuan \[101\] addressed the research challenge posed by Chen et al. in \[80\] regarding an unified approach for semi-online hierarchical scheduling with buffer or reassignment. However, they opened up a new direction by introducing *$L_p$-norm load balancing($C^{(p)}$)* as an optimality criterion for semi-online hierarchical scheduling in $2$-identical machine setup. Let us represent in a schedule the final loads of $M_1$ and $M_2$ by $l_1$ and $l_2$ respectively. The load vector is represented by $L$=$\{l_1, l_2\}$. The $L_p$-norm is denoted as $\|L_p\|$ and defined as follows: $$\|L_p\|=
\begin{cases}
(l^{p}_{1}+l^{p}_{2})^{\frac{1}{p}},& \text{for}\hspace*{0.2cm} 1\leq p< \infty\\
\max\{l_1, l_2\},& \text{for}\hspace*{0.2cm} p=\infty\\
\end{cases}$$ They argued that $L_p$-norm objective is practically more significant than makespan, as it captures the average machine loads instead of the largest load among the machines. They obtained *tight* bound $1.5$ by separately considering $B(k)$ and $reasgn(k)$ respectively with $k$=$1$ for $p$=$\infty$. Xiao et al. \[102\] followed the work of Chen et al. \[100\] and addressed $C_{min}$ objective in a setting, where *sum of the sizes of low hierarchy jobs($T_1$=$1$)* is known and $B(1)$ is given. They proposed the algorithm *BLS*, which achieves a *tight* bound $1.5$ for $C_{min}$. Algorithm *BLS* schedules an incoming job $J_i$ on $M_1$ if $g_i$=$1$. If $g_i$=$2$, put $J_i$ on the buffer, $\Big($let $B_{max}$=$\max\{p_i|jobs\hspace*{0.1cm} in \hspace*{0.1cm} the\hspace*{0.1cm} buffer\}$, $B_{min}$=$\min\{p_i|jobs\hspace*{0.1cm} in \hspace*{0.1cm} the\hspace*{0.1cm} buffer\}\Big)$ and if $l_2+B_{max}\geq \frac{l_1+T_1+B_{min}}{2}$, then $J_{min}$ is scheduled on $M_1$; else, $J_{min}$ is assigned to machine $M_2$. Further, they considered the setting, where $T_1$ is given and *$p_{max}$* for hierarchy 2 i.e. $p^{2}_{max}$ is known. They obtained a *tight* bound $1.5$ for $C_{min}$. Qi and Yuan \[103\] studied the setup $P_2|GOS, Sum|C^{(p)}$ and proposed an algorithm that achieves a *tight* bound $1.5$ for $p$=$\infty$. The algorithm schedules an incoming $J_i$ on machine $M_i$ if $g_i$=$1$. If $g_i$=$2$, and $l_2+p_i\leq \frac{3}{4}\cdot T$, then $J_i$ is assigned to machine $M_2$. If $g_i$=$2$ and $l_2+p_j> \frac{3}{4}\cdot T$, then schedules $J_i$ by the following rule: if $l_2< \frac{1}{4}\cdot T$ and $l_2\leq \frac{T-p_i}{2}$, then schedule $J_i$ on $M_2$ and all future jobs on $M_1$, If $l_2< \frac{1}{4}\cdot T$ and $l_2> \frac{T-p_i}{2}$, then schedule $J_i$ on $M_1$ and all future jobs with $g_i$=$2$ on $M_2$ and jobs with $g_i$=1 on $M_1$. If $l_2\geq \frac{1}{4}\cdot T$, then schedule $J_i$ along with all future jobs on machine $M_1$. The idea is to schedule larger number of jobs on $M_2$ as long as $l_2$ would not exceed $l_1$. Importantly, the algorithm handles the larger size jobs with known $T$. Further, they studied the setup $P_2|GOS, T_1, T_2|C^{(p)}$ and obtained a *tight* bound of $1.33$ for $p$=$\infty$. In future, some interesting consideration would be semi-online hierarchical scheduling for $L_p$-norm optimization with other unexplored *EPIs* such as *Max*, *TGRP*, *Opt*, *Decr* etc. The study remains open in $m$-identical machine for $m>2$ and related machine setups. We now present the summary of important results for semi-online scheduling in identical machines with *GOS* in table \[tab: Important Results for Identical Machines with GOS: 2011-2017\].
**Author(s), Year** **setup($\alpha| \beta| \gamma$)** **Competitiveness Results**
-------------------------- ------------------------------------------------------------------ ---------------------------------------------------------------------------------------------------------------------------------------------------
Liu et al. 2011 \[73\] $P_2 | GOS, TGRP | C_{max}$ $P_2 | GOS, TGRP, Sum | C_{max}$ max$\{\frac{1+b}{2}, 1.5, \frac{4+b}{6}\}$ Tight with GOS and TGRP for $1 < b < 6$, $\frac{1+b}{2}$ Tight with GOS, TGRP and Sum for $1 < b < 2$.
Wu et al. 2012 \[77\] $P_2 | GOS, Opt | C_{max}$ $P_2 | GOS, Max | C_{max}$ ($1.5, 1.618$) Tight for respective setups.
Chen et al. 2013 \[80\] $P_2 | GOS, B(k) | C_{max}$, $P_2 | GOS, reasgn(k) | C_{max}$ $1.5$ Tight with $k=1$ for both setups
Zhang et al. 2013 \[82\] $P_2 | GOS, TGRP | C_{max}$ $P_2 | pmtn, GOS, TGRP | C_{max}$ $1.66$ Tight with $b \geq 3$ for N-pmtn, $1.5$ Tight with $b \geq 2$ for pmtn.
Luo and Xu 2014 \[85\] $P_2 | GOS, Sum | C_{max}$ ($1.5, 1.53, 1.33$) Tight for *Sum* with (higher GOS or lower GOS or both) respectively.
Chen et al. 2015 \[100\] $P_2|GOS, Sum|C_{max}$ $P_2|GOS, Sum, B(1)|C_{max}$ $1.33$ Tight for respective setups
Qi and Yuan 2016 \[101\] $P_2|GOS, B(1)|C^{(p)}$ $P_2|GOS, reasgn(1)|C^{(p)}$ $1.5$ Tight for both setups for $p$=$\infty$
Xiao et al. 2019 \[102\] $P_2|GOS, T_1, B(1)|C_{min}$ $P_2|GOS, T_1, p^{2}_{max}|C_{min}$ $1.5$ Tight for both setups
Qi and Yuan 2019 \[103\] $P_2|GOS, Sum|C^{(p)}$ $1.5$ Tight for $p$=$\infty$
: Summary of the Recent Works on Identical Machines with GOS
\[tab: Important Results for Identical Machines with GOS: 2011-2017\]
\
**TGRP.** Cao et al. \[72\] studied the setup $P_2|TGRP, Max|C_{max}$ by considering $TGRP[a, ba]$ and $p_{max}$=$ba$, where $a> 0$ and $b\geq 1$. They obtained $\frac{b+1}{2}$ *LB* for $1\leq b< 1.33$ and $1.33$ *LB* for $b\geq 2$. They proposed algorithm *PIJS*, which achieves a *tight* bound $\max\{\frac{4(b+1)}{3b+4}, \frac{2b}{b+1}\}$ for $1.33\leq b\leq 2$. Algorithm *PIJS* schedules an incoming job $J_i$ on machine $M_1$ if $l^{i}_{1}+p_i\leq k\cdot \max\{q^{1}_{i}+q^{2}_{i}+...+q^{\lceil\frac{i+1}{2}\rceil}, \frac{l^{i}_{1}+l^{i}_{2}+p_i+p_{max}}{2}\}$; otherwise $J_i$ is scheduled on $M_2$. And this continues until the arrival of the first largest job (let $J_{max}$). When $J_{max}$ arrives, it is scheduled on machine $M_2$. Thereafter, each incoming $J_i$ is scheduled on $M_1$, if $l^{i}_{1}+p_i\leq k\cdot \max\{q^{1}_{i}+q^{2}_{i}+...+q^{\lceil\frac{i}{2}\rceil}, \frac{l^{i}_{1}+l^{i}_{2}+p_i}{2}\}$; else $J_i$ is scheduled on machine $M_2$. (Note that: $k$=$\max\{\frac{4(b+1)}{3b+4}, \frac{2b}{b+1}\}$, $l^{i}_{j}$ is the load of machine $M_j$ just before the assignment of $J_i$ and $q^{r}_{i}$ is the $r^{th}$ smallest job at the arrival of $J_i$ i.e. $\{q^{1}_{i}, q^{2}_{i},...,q^{r}_{i}\}$ such that $p^{1}_{i}\leq p^{2}_{i}\leq...\leq p^{r}_{i}$.) The idea given in this study reveals that when $p_{max}$ is known in advance, it is better to assign $J_{max}$ at the outset. Cao and Wan \[84\] studied the setup $P_2|TGRP[1, b], Decr|C_{max}$. They showed that algorithm *LS* achieves a *tight* bound $1.16$ for $1\leq b\leq 1.5$. With only known $TGRP(ub)$, they obtained *LB* $1.16$, which matches the *UB* given by Seiden in \[21\], where, $ub\geq 1.5$.\
**Arrival Order of Jobs.** Li et al. \[70\] studied the scenario where an incoming job $J_i$ requests an order to the scheduler with its release time $r_i$ and processing time $p_i$. Then, the scheduler service the order by non-preemptively schedule the job with the objective to optimize the makespan. They considered that jobs are arriving by non-decreasing release times(*Incr-r*) and non-increasing sizes(*Decr*). They analyzed the performance of algorithm *LS* and obtained *UB* $\frac{3}{2}-\frac{1}{2m}$ for $m$-identical machine setup. Cheng et al. \[78\] studied the setup $P_m|Decr|C_{max}$. They analyzed algorithm *LPT* and proved *tight* bounds $1.18$ for $m$=3 and $1.25$ for $m>3$. Tang and Nai \[87\] refined the results of Li et al. \[70\] and derived a new proof for the *UB* of algorithm *LS*.\
**Job Reassignment.** Min et al. \[71\] considered the job’s assignment policy of Tan and Yu \[57\]. They obtained a tight bound of $1.41$ for semi-online scheduling on $2$-identical machine by allowing the reassignment of the last job of one machine only. Further, they considered known *Sum* besides the reassignment policy and improved the previous best *UB* of $1.33$ to $1.25$.\
**Combined Information.** Cao et al. \[74\] considered several semi-online variants for scheduling on $2$-identical machine with *min-max* objective. They proposed $1.2$ competitive optimal algorithm *OM* with known *Opt* and *Max*. Further, they considered combined information on (*B(k), Max*), (*B(1), Decr*), (B(1), TGRP(1,b)) and obtained *tight* bounds ($1.25, 1.16, 1.33$) respectively. Lee and Lim \[79\] studied the setup $P_m|Sum, Max| C_{max}$ and achieved *UB*s ($1.462$, $1.5$) for $m$=$4$, $5$ respectively.\
**Sum.** Albers and Hellwig \[75\] studied the setup $P_m|Sum|C_{max}$. They improved the *LB* $1.565$ \[35\] to $1.585$ for $m$-identical machine setup, where $m\rightarrow \infty$. They proposed algorithm *Light Load*, which is free from traditional job class policy considered in \[35, 44\] and achieves an *UB* $1.75$. Lee and Lim \[79\] investigated the setup $P_m|Sum|C_{max}$ for small number of machines. They obtained *LB*s of $1.442$, $1.482$ and $1.5$ for $m$=$4$, $5$ and $6$ respectively. An algorithm named *ForwardFit-BackwardFit-ListScheduling* was proposed by assuming $Sum$=$m$, which achieves *UB*s of $1.4$, $1.4615$ and $1.5$ for $m$=$3$, $4$ and $5$ respectively. The algorithm prefers two conditions, where in the first condition, a *load threshold* is set upto the defined competitive bound to keep the loads of the machines under the threshold value. Before scheduling an incoming job, the loads of each machine is checked against the load threshold value. If the first condition fails, then the incoming job is scheduled by algorithm *LS*. An obvious question raised here is: How to choose the threshold value, which always guarantees that the scheduling of all jobs would yield the defined competitive bound for any $m$?\
Kellerer et al. \[90\] obtained an *UB* $1.585$ by considering $Sum$=$m$, which matches the *LB* achieved by Albers and Hellwig in \[75\] for $m$-identical machine setup. They adopted the job class policy and classified the incoming jobs into four classes such as tiny, small, medium and large depending on their sizes, defined by the time intervals $(0, \frac{\alpha}{2}]$ for tiny, $(0, \alpha]$ for small, $(\alpha, \frac{1}{2\alpha}]$ for medium, $[>\frac{1}{2\alpha}]$ for large. Similarly, $m$ machines were classified as tiny, small, medium, big and huge depending on their loads defined by the time intervals $(0, \frac{\alpha}{2}]$ for tiny, $(0, \alpha]$ for small, $(\alpha, \frac{1}{2\alpha}]$ for medium, $(\frac{1}{2\alpha}, 1]$ for big and $[> 1]$ for huge, where $\alpha$=$0.585$. They proposed an algorithm, which executes in 2 phases, where in the $1^{st}$ phase, jobs are scheduled on the machines depending on the classes of jobs and machines. The $2^{nd}$ phase of the algorithm, emerges from the $1^{st}$ phase and runs two policies with respect to the current machines loads after the $1^{st}$ phase. Here, the classification of jobs and machines helps in improving the tightness in the competitive bound to $1+\alpha$. A natural question pops out from here is: Can an algorithm be possible for $P_m|Sum|C_{max}$ with job class policy and $\alpha< 0.585$?\
**Buffer.** Lan et al. \[76\] studied the general cases for identical machines by considering buffer as additional feature. For $m$-identical machines they obtained a tight bound $1.5$ with a buffer of size $1.5m$.\
**Opt.** Kellerer and Kotov \[104\] studied the setup considered by Azar and Regev in \[15\]. They improved the *UB* from $1.625$ to $1.571$ by considering the job class policy, where the incoming jobs and available machines are classified by their sizes and current loads respectively, defined by the specified time interval ranges. It was proved that a two phase algorithm with job class policy always guarantees the loads of each machine to be under $1.571$ of the known $Opt$. Gabay et al. \[105\] further improved the *UB* to $1.5294$. The current best known *UB* $1.5$ for the setup $P_m|Opt|C_{max}$ is due to Bohm et al. \[106\]. Further, they obtained UB $1.375$ for $m$=$3$. Gabay et al. \[107\] improved the *LB* $1.33$ to $1.357$ for the setup $P_m|Opt|C_{max}$. Thus, minimizing the gap between the current best *LB* and *UB* in $m$-identical machine semi-online scheduling with known *Opt*. We now present the summary of important results obtained for semi-online scheduling in identical machines other than known *GOS* in table \[tab: Important Results for Identical Machines: 2011-2017\].
**Author(s), Year** **setup($\alpha| \beta| \gamma$)** **Competitiveness Results**
--------------------------------- ------------------------------------------------------------------------------------------------------------------ --------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Cao et al. 2011 \[72\] $P_2 | TGRP, Max | C_{max}$ $\frac{b+1}{2}$ LB for $1 \leq b < 1.33$, $1.33$ LB for $b \geq 2$, $\max\{\frac{4(b+1)}{3b+4}, \frac{2b}{b+1}\}$ Tight for $1.33 \leq b < 2$.
Li et al. 2011 \[70\] $P_m | Incr-r, Decr | C_{max}$ ($\frac{3}{2} - \frac{1}{2m}$) Tight.
Min et al. 2011 \[71\] $P_2 | reasgn(last(1)^*) | C_{max}$ $P_2 | reasgn(last(1)^*), Sum | C_{max}$ ($1.41, 1.25$) Tight for respective setups .
Cao et al. 2012 \[74\] $P_2 | Opt, Max | C_{max}$ $P_2 | B(k), Max | C_{max}$ $P_2 | B(1), Decr | C_{max}$ $P_2 | B(1), TGRP | C_{max}$ ($1.2, 1.25, 1.16, 1.33$) Tight for respective setups.
Albers and Hellwig 2012 \[75\] $P_m | Sum | C_{max}$ $1.585$ LB, $1.75$ UB
Lan et al. 2012 \[76\] $P_m | B(1.5m) | C_{max}$ $1.5$ Tight
Cheng et al. 2012 \[78\] $P_m | Decr | C_{max}$ $1.18$ Tight for $m=3$, $1.25$ Tight for $m > 3$.
Lee and Lim 2013 \[79\] $P_m | Sum | C_{max}$ $P_m | Max | C_{max}$ $P_m | Sum, Max | C_{max}$ ($1.4$, $1.4615$, $1.5$) UB for $m$=$3$,$4$,$5$ with *Sum*, ($1.618$, $1.667$) Tight for $m$=$4$,$5$ with *Max*, ($1.462, 1.5$) UB for $m$=$4$,$5$ with *Sum* and *Max*.
Kellerer and Kotov 2013 \[104\] $P_m|Opt|C_{max}$ $1.571$ UB.
Cao and Wan 2014 \[84\] $P_2 | TGRP[1, b], Decr | C_{max}$ $P_2 | TGRP[ub], Decr | C_{max}$ $1.16$ Tight for $TGRP[1, b]$ and $1 \leq b < 1.5$, $1.166$ LB for $TGRP[ub]$ and $ub \geq 1.5$.
Tang and Nie 2015 \[87\] $P_m | Incr-r, Decr | C_{max}$ ($\frac{3}{2} - \frac{1}{2m}$) UB.
Kellerer et al. 2015 \[90\] $P_m | Sum | C_{max}$ $1.585$ Tight for $m\rightarrow \infty$.
Gabay et al. 2015 \[105\] $P_m|Opt|C_{max}$ $1.5294$ UB.
Bohm et al. 2017 \[106\] $P_m|Opt|C_{max}$ $P_3|Opt|C_{max}$ $1.5$ UB for $P_m$, $1.375$ UB for $P_3$.
Gabay et al. 2017 \[107\] $P_m|Opt|C_{max}$ $1.357$ LB.
: Summary of the Recent Works on Identical Machines
\[tab: Important Results for Identical Machines: 2011-2017\]
Emergence of Semi-online Scheduling Setups and Classification of the Related Works {#sec:Emergence of Semi-online Scheduling Setups and Classification of the Related Works}
==================================================================================
**Evolution Time-line for Semi-online Scheduling Setups.** After making a comprehensive literature survey, we understand and explore various problem setups, research directions and research trends in semi-online scheduling. In this section, we sketch a time-line to represent the emergence of various semi-online scheduling setups as shown in figure \[fig:researchtrends.png\].\
![Evolution Timeline of Semi-online Scheduling Setups[]{data-label="fig:researchtrends.png"}](researchtrends.png)
**Classification of Related Works based on EPI.** Though many researchers have exhaustively studied semi-online scheduling based on either a single *EPI* or more than one *EPI*s, there is hardly any attempt to develop a taxonomy to classify the literature and related works based on *EPI*. Here we attempt to classify the whole literature on semi-online scheduling based on *EPI*s for identifying related works for various setups. We present our classification in figure \[fig:Relatedpapers.png\]. We consider the setups for identical($P$) and uniform($Q$) machines in our classification. The other parameters for various problem setups are processing formats such as non-preemptive($N-pmtn$) and preemptive($pmtn$) and optimality criteria such as makespan($C_{max}$) and *Max-Min*($C_{min}$). We also provide links to references of related works for each problem setup. For each *EPI*, we have mentioned the various setups which are studied in the literature along with their references. Our classification may help the researchers to focus on related works and explore specific research directions for future work. The future research work can also be carried out based on specific *EPI* by choosing a particular set up from our classification.
![Classification of Related Works based on EPI[]{data-label="fig:Relatedpapers.png"}](relatedpapers.png)
Research Challenges and Open Problems {#sec:Research Challenges and Open Problems}
=====================================
Semi-online scheduling has been extensively studied in various setups for the last two decades. Still, there are many research issues, which can lead to further investigations in this area. We conclude our survey on the important results and critical ideas for semi-online scheduling by exploring some of the non-trivial research challenges and open problems as follows.
Research Challenges {#subsec:Research Challenges}
-------------------
- Exploration of practically significant new *EPI*s that can help in improving the *CR* of the existing online scheduling algorithms.
- Generation, characterization and classification of the input job sequences that can resemble the real world inputs in various semi-online scheduling setups.
- Minimize or diminish the gap between *LB* $1.585$ and *UB* $1.6$ for the setup **$P_m|Sum|C_{max}$**.
- Reduce the *CR* $1.366$ for preemptive **$P_m|C_{max}$** setting.
- Improvement of $1.5$-competitive strategy for **$P_2|C_{max}$** problem with inexact partial information. The solution for **$P_m|C_{max}$** problem is unknown in this case.
- Close or remove the gap \[$1.442, 1.5$\] between lower and upper bound for semi-online scheduling on unbounded parallel batch machine.
- Design of optimal semi-online scheduling algorithm with at most $1.5$-competitiveness for scheduling on $m$-identical machines ($m\geq 2$) under *GOS* and known *Opt*.
- Exploration of optimal semi-online algorithms for **$P_m|C_{max}$** and **$Q_m|C_{max}$** setups with reassignment of job policy.
Open Problems {#subsec:Open Problems}
-------------
- Can *EPI* be used to develop a new complexity class for evaluating the performance of online algorithms?
- Does there exist an optimal semi-online algorithm with *CR* less than $1.33$ for the setup $P_m|B(k)|C_{max}$ with $k$=$1$?
- Can the *CR* $1.2$ be improved for the setup $P_2|Sum|C_{max}$? Can a matching bound be possible for $P_m|Sum|C_{max}$? How far preemption can help in improving the results in this setting?
- How can we establish a relationship between semi-online scheduling and online scheduling with look ahead? Which one is practically significant? For an instance, is it possible to obtain a *CR* less than or equal to $1.33$ for $P_m|Sum|C_{max}$, where $T$ is known for $k$ future jobs and $1\leq k < n$.
- Can a tight bound be possible for $P_3|C_{min}$, which is independent of number of machines?
- Does there exist an optimal semi-online algorithm for the setup $Q_m|C_{min}$? Already, a $1$-competitive algorithm is known for $Q_2|C_{min}$ due to \[59\].
- What can be an optimal semi-online policy for $P_m|C_{max}$? Can a $1$-competitive semi-online algorithm be possible for this setting with known *Decr* \[108\]?
- What can be a tight bound for semi-online scheduling on uniform machines with overall speed ratio interval of $[1, \infty)$?
- Does there exist an optimal semi-online strategy for multiple unbounded parallel batch processors?
- Can *EPI* be helpful in improving the best competitive bounds obtained for online scheduling in various setups of unrelated parallel machines?
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
An additional event near the upper kinematic limit for $K^+ \to \pi^+ \nu \bar\nu$ has been observed by Experiment E949 at Brookhaven National Laboratory. Combining previously reported and new data, the branching ratio is ${\cal B}$($K^+ \to \pi^+ \nu
\bar\nu$)$={\mbox {$({{\mbox {$1.47$}}}^{+1.30}_{-0.89}) \times 10^{-10}$}}$ based on three events observed in the pion momentum region $211<P<229$ MeV/$c$. At the measured central value of the branching ratio, the additional event had a signal-to-background ratio of .
author:
- 'V.V. Anisimovsky'
- 'A.V. Artamonov'
- 'B. Bassalleck'
- 'B. Bhuyan'
- 'E.W. Blackmore'
- 'D.A. Bryman'
- 'S. Chen'
- 'I-H. Chiang'
- 'I.-A. Christidi'
- 'P.S. Cooper'
- 'M.V. Diwan'
- 'J.S. Frank'
- 'T. Fujiwara'
- 'J. Hu'
- 'A.P. Ivashkin'
- 'D.E. Jaffe'
- 'S. Kabe'
- 'S.H. Kettell'
- 'M.M. Khabibullin'
- 'A.N. Khotjantsev'
- 'P. Kitching'
- 'M. Kobayashi'
- 'T.K. Komatsubara'
- 'A. Konaka'
- 'A.P. Kozhevnikov'
- 'Yu.G. Kudenko'
- 'A. Kushnirenko'
- 'L.G. Landsberg'
- 'B. Lewis'
- 'K.K. Li'
- 'L.S. Littenberg'
- 'J.A. Macdonald'
- 'J. Mildenberger'
- 'O.V. Mineev'
- 'M. Miyajima'
- 'K. Mizouchi'
- 'V.A. Mukhin'
- 'N. Muramatsu'
- 'T. Nakano'
- 'M. Nomachi'
- 'T. Nomura'
- 'T. Numao'
- 'V.F. Obraztsov'
- 'K. Omata'
- 'D.I. Patalakha'
- 'S.V. Petrenko'
- 'R. Poutissou'
- 'E.J. Ramberg'
- 'G. Redlinger'
- 'T. Sato'
- 'T. Sekiguchi'
- 'T. Shinkawa'
- 'R.C. Strand'
- 'S. Sugimoto'
- 'Y. Tamagawa'
- 'R. Tschirhart'
- 'T. Tsunemi'
- 'D.V. Vavilov'
- 'B. Viren'
- 'N.V. Yershov'
- 'Y. Yoshimura'
- 'T. Yoshioka'
title: 'Improved Measurement of the $K^+ \to \pi^+ \nu \bar\nu$ Branching Ratio'
---
In the standard model (SM), the decay $K^+ \to \pi^+ \nu \bar\nu$ is sensitive to the couplings of top quarks which dominate the internal processes involved in this flavor changing neutral current reaction. A reliable SM prediction for the branching ratio ${\cal B}(K^+ \to
\pi^+ \nu \bar\nu$)= [@BB; @newbb] can be made due to knowledge of the hadronic transition matrix element from similar processes, and minimal complications from hadronic effects. ${\cal B}(K^+ \to \pi^+ \nu \bar\nu)$ is a sensitive probe of new physics, since, for example, the apparent couplings between top and down quarks may also be determined by measurements in the B meson system resulting in a possible discrepancy[@B2004; @BSM]. In earlier studies, two events consistent with the decay $K^+ \to \pi^+ \nu \bar\nu$ were reported giving ${\cal B}$($K^+ \to \pi^+ \nu
\bar\nu$)= by Experiment E787 at the Alternating Gradient Synchrotron (AGS) of Brookhaven National Laboratory [@pnn2002]. In this letter, the first results from Experiment E949 [@det.E949] at the AGS are presented.
Measurement of $K^+ \! \rightarrow \! \pi^+ \nu \overline{\nu}$ decay from kaons at rest involved observation of the $\pi^+$ in the momentum region $211< P
<229$ MeV/$c$ in the absence of other coincident activity. Pions were identified by comparing momentum ($P$), range ($R$), and energy ($E$) measurements, and by observation of the $\pi^+ \! \rightarrow \! \mu^+ \! \rightarrow \! e^+$ decay sequence. Primary background sources were pions from the two-body decay $K^+\to\pi^+\pi^0$ ($K_{\pi 2}$), muons from $K^+\to\mu^+\nu$ ($K_{\mu 2}$) and other $K^+$ decays, pions scattered from the beam, and $K^+$ charge exchange reactions followed by $K_L^0\to\pi^+ l^- \overline{\nu}_l$, where $l=e$ or $\mu$.
The new data were acquired in 2002 using beams, apparatus, and procedures similar to those of Experiment E787 [@pnn2002; @det.TD; @det; @det.collar; @newl0]. The number of kaons stopped in the scintillating fiber target was $N_K=1.8 \times 10^{12}$. Measurements of charged decay products were made in a 1T magnetic field using the target, a central drift chamber, and a cylindrical range stack (RS) of scintillator detectors. Photons were detected in a $4\pi$ sr calorimeter consisting of a lead/scintillator sandwich barrel veto detector (upgraded for E949) surrounding the RS, endcaps of undoped CsI crystals, and other detectors. The upgraded apparatus also included replacement of one third of the RS, and an improved trigger system[@newl0]. Although the instantaneous detector rates were twice those in E787, detector upgrades and the use of improved pattern recognition software enabled comparable acceptance to be obtained.
Each background source was suppressed by two groups of complementary but independent selection criteria (cuts), and the desired level of background rejection was obtained by adjusting the severity of the cuts. For example, the cut pair for $K_{\pi 2}$ background involved kinematic measurements of the $\pi^+$, and photon detection in $\pi^0\to\gamma\gamma$ decay. The photon detection criteria, for instance, could be varied by changing the energy threshold and timing coincidence interval (relative to the $\pi^+$ signal) of the photon detectors. The effectiveness of each cut at rejecting background was determined using data selected by inverting the criteria of the complementary cut. Unbiased estimates of the effectiveness of the cuts were obtained using a uniformly-sampled 1/3 portion of the data for cut development and the remaining 2/3 portion for background measurement. Examination of the pre-determined signal region was avoided throughout the procedure. The level of signal acceptance as a function of cut severity was determined using data and simulations. This procedure enabled estimates of the expected background and signal rates inside and outside the signal region at different levels of background rejection and signal acceptance.
As a check of the method, the observed background levels near but outside the signal region were compared to the predicted background rates when both cuts for each background type were applied. The results are summarized in Table \[tab:otb\] for the two-body backgrounds, $K_{\pi 2}$ and $K_{\mu 2}$, and the multi-body background ($K_{\mu m}$) with contributions from $K^+\to\mu^+\nu\gamma$, $K^+\to\mu^+\pi^0\nu$, and $K_{\pi 2}$ with $\pi^+\to\mu^+\nu$ decay-in-flight. Five cases were considered corresponding to increasing background levels outside the signal region. For example, for the $K_{\mu 2}$ component, the region nearest to (farthest from) the signal region was chosen to have 7 (400) times the background expected in the signal region. The five ratios of the observed to predicted backgrounds were fitted to a constant $c$ for each background type. The consistency of $c$ with unity and the acceptable probability of $\chi^2$ of each fit confirmed both the independence of the pairs of cuts and the reliability of the background estimates. The measured uncertainties in the constants c were used to estimate the systematic uncertainties in the predicted background rates in the signal region.
Background $\chi^2$ Probability Events
------------- -------- ------------------------- ---------------------- ------ -----------------
$K_{\pi 2}$ $0.85$ ${}^{+0.12}_{-0.11}$ ${}^{+0.15}_{-0.11}$ 0.17 $0.216\pm0.023$
$K_{\mu 2}$ $1.15$ ${}^{+0.25}_{-0.21}$ ${}^{+0.16}_{-0.12}$ 0.67 $0.044\pm0.005$
$K_{\mu m}$ $1.06$ $ {}^{+0.35 }_{-0.29 }$ ${}^{+0.93}_{-0.34}$ 0.40 $0.024\pm0.010$
: \[tab:otb\] The fitted constants $c$ of the ratios of the observed to the predicted numbers of background events and the probability of $\chi^2$ of the fits for the $K_{\pi 2}$, $K_{\mu 2}$ and $K_{\mu m}$ backgrounds near but outside the signal region. The first uncertainty in $c$ was due to the statistics of the observed events and the second was due to the uncertainty in the predicted rate. The predicted numbers of background events within the signal region and their statistical uncertainties are also tabulated in the fourth column. Other backgrounds contributed an additional $0.014\pm0.003$ events resulting in a total number of background events expected in the signal region of ${\mbox {$0.30$}}\pm 0.03$.
To estimate ${\cal B}(K^+ \to \pi^+ \nu\bar\nu)$, the parameter space of observables in the signal region was subdivided into 3781 bins corresponding to different ranges of cut severity and each observed event could be assigned to the bin corresponding to its measured quantities. Bin $i$ was characterized by the value of $S_i/b_i$, the relative probability of an event in the bin to originate from $K^+ \to
\pi^+ \nu\bar\nu$ ($S_i$) or background ($b_i$) [@pnn2002]. The signal rate of a bin was $S_i\equiv{\cal B}(K^+ \to \pi^+
\nu\bar\nu)A_iN_K$ where $A_i$ was the acceptance of the $i^{\rm th}$ bin. Each observed event could also be described by a weight $W\equiv
S/(S+b)$ that represented its effective contribution to ${\cal B}(K^+
\to \pi^+ \nu\bar\nu)$. ${\cal B}(K^+ \to \pi^+ \nu\bar\nu)$ was obtained by a likelihood ratio technique [@Junk] that determined the confidence level (C.L.) of a given branching ratio based on the observed events. For the 2002 data set, the candidate selection requirements were similar to those used previously. The pre-determined signal region was enlarged, resulting in 10% more acceptance but also allowing more background. Estimated background levels dominated by $K_{\pi 2}$ and $K_{\mu 2}$ are listed in Table \[tab:otb\].
Examination of the signal region for the new data set yielded one event with $P=227.3 \pm 2.7$ MeV/$c$, $R=39.2 \pm 1.2$ cm (in equivalent cm of scintillator), and $E=128.9 \pm 3.6$ MeV. The event (2002A) has all the characteristics of a signal event although its high momentum and low apparent time of $\pi\to\mu$ decay (6.2 ns) indicate a higher probability than the two previously observed candidate events that it was due to background, particularly $K_{\mu 2}$ decay.
The combined result for the E949 and E787 data is shown in Figure \[rve\] with the range and kinetic energy of the events surviving all other cuts. The result obtained from the likelihood method described above was ${\cal B}(K^+
\to \pi^+ \nu \overline{\nu}) ={\mbox {$({{\mbox {$1.47$}}}^{+1.30}_{-0.89}) \times 10^{-10}$}}\ $ incorporating the three observed events and their associated weights $W$ given in Table \[Nbg\]. For event 2002A the weight was $W= {\mbox {$0.48$}}$ ($S/b={\mbox {$0.9$}}$). The estimated probability that the background alone gave rise to this or any more signal-like event was . Table \[Nbg\] also shows the estimated probability that the background alone gave rise to each event (or any more signal-like event), the acceptances [@pnn2002], $N_K$, and the total expected background levels. This result is consistent with the SM expectation [@BB; @newbb]. The quoted 68% C.L. interval includes statistical and estimated systematic uncertainties. The 80% and 90% C.L. intervals for ${\cal
B}(K^+\to\pi^+\nu\bar\nu)$ were and , respectively[@cl]. The estimated systematic uncertainties do not significantly affect the confidence levels. The estimated probability that background alone gave rise to the three observed events (or to any more signal-like configuration) was $0.001$[@E787redux0].
![ Range (R) vs. energy (E) distribution of events passing all other cuts of the final sample. The circles represent E787 data and the triangles E949 data. The group of events around $E=108$ MeV was due to the $K_{\pi 2}$ background. The simulated distribution of events from $K^+ \! \rightarrow \! \pi^+ \nu\overline{\nu}$ decay is indicated by dots. The solid-line (dashed-line) box represents the signal region for E949 (E787).[]{data-label="rve"}](figure1.eps){width="\linewidth"}
E949
------------------ ------- ---------------------------- -------
$N_{K} $ $1.8\times 10^{12}$
Total Acceptance $ 0.0022\pm 0.0002 $
Total Background ${\mbox {$0.30$}}\pm 0.03$
Candidate 1995A 1998C 2002A
$S/b$
$ W$
Background Prob.
: \[Nbg\] Numbers of kaons stopped in the target $N_{K}$, total acceptance, total numbers of estimated background events for the E949 and E787 data samples[@E787redux], $S/b$ and $W$ for each observed candidate event calculated from the likelihood analysis described in the text, and the estimated probability that the background alone gave rise to each event (or any more signal-like event).
The E787 and E949 data were also used to set a limit on the branching ratio for $K^+ \!\rightarrow \! \pi^+ X^0$, where $X^0$ is a neutral weakly interacting massless particle [@x0]. Previous E787 data produced a limit of ${\cal B}(K^+ \! \rightarrow \!\pi^+ X^0) < 0.59 \times 10^{-10}$[@pnn2002]. The new result was ${\cal B}(K^+ \! \rightarrow \!\pi^+ X^0) < {\mbox {$ 0.73 \times 10^{-10}$}}\ $ (90% C.L.), based on the inclusion of event 2002A which was observed within two standard deviations of the expected pion momentum.
We acknowledge the dedicated effort of the technical staff supporting E949, the Brookhaven C-A Department, and the contributions made by colleagues who participated in E787. This research was supported in part by the U.S. Department of Energy, the Ministry of Education, Culture, Sports, Science and Technology of Japan through the Japan-U.S. Cooperative Research Program in High Energy Physics and under Grant-in-Aids for Scientific Research, the Natural Sciences and Engineering Research Council and the National Research Council of Canada, the Russian Federation State Scientific Center Institute for High Energy Physics, and the Ministry of Industry, Science and New Technologies of the Russian Federation.
[ (a) ]{}
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A corrected estimate of the number of background events expected in the E787 data sample due to a reevaluation of the beam background is presented here.
Additional information on the confidence levels are available at http://www.phy.bnl.gov/e949/E949Archive/.
The value (0.02%) quoted in ref. [@pnn2002] for the probability that the observation of two events was due entirely to background was incorrect due to insufficient precision in the calculation.
F. Wilczek, Phys. Rev. Lett. [**49**]{}, 1549 (1982).
|
{
"pile_set_name": "ArXiv"
}
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---
abstract: 'Distributed compression is the task of compressing correlated data by several parties, each one possessing one piece of data and acting separately. The classical Slepian-Wolf theorem [@sle-wol:j:distribcompression] shows that if data is generated by independent draws from a joint distribution, that is by a memoryless stochastic process, then distributed compression can achieve the same compression rates as centralized compression when the parties act together. Recently, the author [@zim:c:kolmslepianwolf] has obtained an analogue version of the Slepian-Wolf theorem in the framework of Algorithmic Information Theory (also known as Kolmogorov complexity). The advantage over the classical theorem, is that the AIT version works for individual strings, without any assumption regarding the generative process. The only requirement is that the parties know the complexity profile of the input strings, which is a simple quantitative measure of the data correlation. The goal of this paper is to present in an accessible form that omits some technical details the main ideas from the reference [@zim:c:kolmslepianwolf].'
author:
- '[Marius Zimand]{} [^1]'
title: 'Distributed compression through the lens of algorithmic information theory: a primer'
---
[CRVW02]{}
Bruno Bauwens and Marius Zimand. Linear list-approximation for short programs (or the power of a few random bits). In [*[IEEE]{} 29th Conference on Computational Complexity, [CCC]{} 2014, Vancouver, BC, Canada, June 11-13, 2014*]{}, pages 241–247. [IEEE]{}, 2014.
G. Chaitin. On the length of programs for computing finite binary sequences. , 13:547–569, 1966.
Thomas M. Cover. A proof of the data compression theorem of [S]{}lepian and [W]{}olf for ergodic sources (corresp.). , 21(2):226–228, 1975.
M. R. Capalbo, O. Reingold, S. P. Vadhan, and A. Wigderson. Randomness conductors and constant-degree lossless expanders. In John H. Reif, editor, [*STOC*]{}, pages 659–668. ACM, 2002.
G. Dueck and L. Wolters. The [S]{}lepian-[W]{}olf theorem for individual sequences. , 14:437–450, 1985.
A.N. Kolmogorov. Three approaches to the quantitative definition of information. , 1(1):1–7, 1965.
S. Kuzuoka. lepian-[W]{}olf coding of individual sequences based on ensembles of linear functions. , E92-A(10):2393–2401, 2009.
A. Lempel and J. Ziv. On the complexity of finite sequences. , IT-22:75–81, 1976.
Andrei A. Muchnik. Conditional complexity and codes. , 271(1-2):97–109, 2002.
A. Romashchenko. Complexity interpretation for the fork network coding. , 5(1):20–28, 2005. In Russian. Available in English as [@rom:t:slepwolf].
Andrei Romashchenko. Coding in the fork network in the framework of [K]{}olmogorov complexity. , abs/1602.02648, 2016.
Ran Raz and Omer Reingold. On recycling the randomness of states in space bounded computation. In Jeffrey Scott Vitter, Lawrence L. Larmore, and Frank Thomson Leighton, editors, [*STOC*]{}, pages 159–168. ACM, 1999.
R. Raz, O. Reingold, and S. Vadhan. Extracting all the randomness and reducing the error in [T]{}revisan’s extractor. In [*Proceedings of the 30th ACM Symposium on Theory of Computing*]{}, pages 149–158. ACM Press, May 1999.
R. Solomonoff. A formal theory of inductive inference. , 7:224–254, 1964.
D. Slepian and J.K. Wolf. Noiseless coding of correlated information sources. , 19(4):471–480, 1973.
A.D. Wyner and J. Ziv. The sliding window [L]{}empel-[Z]{}iv is asymptotically optimal. , 2(6):872–877, 1994.
M. Zimand. olmogorov complexity version of [S]{}lepian-[W]{}olf coding. In [*STOC 2017*]{}, pages 22–32. ACM, June 2017.
J. Ziv. Coding theorems for individual sequences. , IT-24:405–412, 1978.
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[^1]: Department of Computer and Information Sciences, Towson University, Baltimore, MD. http://triton.towson.edu/ mzimand
|
{
"pile_set_name": "ArXiv"
}
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---
abstract: |
Non-thermal desorption from icy grains containing H$_2$CO has been invoked to explain the observed H$_2$CO gas phase abundances in ProtoPlanetary Disks (PPDs) and Photon Dominated Regions (PDRs). Photodesorption is thought to play a key role, however no absolute measurement of the photodesorption from H$_2$CO ices were performed up to now, so that a default value is used in the current astrophysical models. As photodesorption yields differ from one molecule to the other, it is crucial to experimentally investigate photodesorption from H$_2$CO ices.
We measured absolute wavelength-resolved photodesorption yields from pure H$_2$CO ices, H$_2$CO on top of a CO ice (H$_2$CO/CO), and H$_2$CO mixed with CO ice (H$_2$CO:CO) irradiated in the Vacuum UltraViolet (VUV) range (7-13.6 eV). Photodesorption from a pure H$_2$CO ice releases H$_2$CO in the gas phase, but also fragments, such as CO and H$_2$. Energy-resolved photodesorption spectra, coupled with InfraRed (IR) and Temperature Programmed Desorption (TPD) diagnostics, showed the important role played by photodissociation and allowed to discuss photodesorption mechanisms. For the release of H$_2$CO in the gas phase, they include Desorption Induced by Electronic Transitions (DIET), indirect DIET through CO-induced desorption of H$_2$CO and photochemical desorption.
We found that H$_2$CO photodesorbs with an average efficiency of $\sim 4-10 \times 10^{-4}$ molecule/photon, in various astrophysical environments. H$_2$CO and CO photodesorption yields and photodesorption mechanisms, involving photofragmentation of H$_2$CO, can be implemented in astrochemical codes. The effects of photodesorption on gas/solid abundances of H$_2$CO and all linked species from CO to Complex Organic Molecules (COMs), and on the H$_2$CO snowline location, are now on the verge of being unravelled.
author:
- Géraldine Féraud
- Mathieu Bertin
- Claire Romanzin
- Rémi Dupuy
- Franck Le Petit
- Evelyne Roueff
- Laurent Philippe
- Xavier Michaut
- Pascal Jeseck
- 'Jean-Hugues Fillion'
bibliography:
- 'H2CO.bib'
title: 'Vacuum ultraviolet photodesorption and photofragmentation of formaldehyde-containing ices'
---
**Accepted May 3, 2019 in ACS Earth and Space Chemistry, Special Issue: Complex Organic Molecules (COMs) in Star-Forming Regions**
Introduction
============
Formaldehyde (H$_2$CO) is an organic molecule that was detected in the interstellar medium in 1969.[@snyder1969] It has been observed in several environments, such as Young Stellar Objects,[@roueff2006; @araya2007] PDRs (e.g. Refs ), protostellar cores,[@maret2004] protoplanetary disks (e.g. Ref. ) and comets (e.g. Refs ). It has been studied as a potential probe of planet formation in protoplanetary disks (PPDs),[@cleeves2015] and it is even used in cosmology as an extinction-free tracer of star formation across the epoch of Galaxy evolution.[@darling2012] Its protonated form, H$_2$COH$^+$, detected for the first time in Sgr B2 and several hot cores,[@ohishi1996] has been recently observed in a cold prestellar core.[@bacmann2016a] In the coma of several comets, its abundance has been found at the percent level relative to H$_2$O (e.g. Refs ), and its polymerized form, polyoxymethylene (POM), has also been detected.[@huebner1987; @mitchell1987] In the solid phase, H$_2$CO is a likely identified species towards protostars, with the observation of a very weak IR absorption feature at 3.47 $\mu$m (C-H stretch),[@schutte1996] and of a stronger one at 5.83 $\mu$m (C-O stretch band), which is blended with a water band.[@ehrenfreund1997; @keane2001; @pontoppidan2004; @boogert2015] The derived abundances are $\approx2-7\%$ relative to H$_2$O ice, which gives H$_2$CO/CH$_3$OH ice ratios of 0.09 to 0.51 in high-mass young stellar objects.[@ehrenfreund1997; @dartois1999; @keane2001; @gibb2004; @boogert2015] Towards the massive star W33, H$_2$CO has been detected both in the solid phase and in the gas phase, with a gas/ice ratio of 3 %.[@roueff2006]
The origin of interstellar H$_2$CO appears to be manifold as it can be formed both on grains[@soma2018; @guzman2011; @guzman2018] and in the gas phase. This is different from CH$_3$OH, whose formation in the gas phase is considered inefficient,[@garrod2006; @geppert2006; @stoecklin2018] even if it could be at play in stellar outflows or in shocked circumstellar regions.[@dartois1999] On grains, H$_2$CO formation depends on the competition between H addition and desorption (e.g. Refs ). The location of H$_2$CO in the icy mantles of the interstellar grains changes with many parameters such as time, temperature of the grain, n$_H$/n$_{CO}$ gas phase abundance, grain porosity, [@cuppen2009; @taquet2012] and as a consequence it evolves with the processing of the interstellar ice. It could be in the upper layers of the ice, together with CO, CO$_2$ and CH$_3$OH, or it could also be buried deeper in the ice, possibly with H$_2$O (Refs and references therein).
Laboratory studies have given a wealth of information regarding the formation of solid H$_2$CO from various ices, using different processes: (i) by the hydrogenation of CO ices [@watanabe2002; @fuchs2009; @pirim2011] which has been recently revisited [@minissale2016a] (ii) by processing of CH$_3$OH ices either by ion irradiation,[@deBarros2011] by UV photons [@oberg2009b] or by soft X-ray photons [@ciaravella2010] (iii) by processing of H$_2$O:CO ice mixtures, either by UV photons [@schutte1996], by protons [@hudson1999] or by electrons.[@yamamoto2004] Some experiments also unravelled the evolution of solid phase H$_2$CO under specific conditions, for example by reactions with oxygen atoms (leading to CO$_2$),[@minissale2015] or by VUV irradiation.[@gerakines1996; @butscher2016] VUV-lamp irradiation of pure H$_2$CO ices or of H$_2$CO-containing ices showed the formation of many products within the ice, showing the richness and the complexity of photochemistry in the solid phase,[@gerakines1996; @butscher2016] even for a small organic molecule such as H$_2$CO.
It has been proposed that hydrogenation of H$_2$CO forms CH$_3$OH, so that H$_2$CO and CH$_3$OH are both of interest as precursors of larger organic molecules, the so-called Complex Organic Molecules (COMs) (e.g. Ref. ). Understanding the formation of COMs is the objective of several experiments [@theule2013; @butscher2016; @chuang2017] and of several models, including grain surface reactions or possibly gas-phase reactions,[@woods2012; @vasyunin2013; @balucani2015; @ruaud2015; @kalvans2015a] but the recent observation of COMs in the gas phase in pre-stellar cores, which are cold environments, was quite puzzling.[@bacmann2012; @vastel2014; @jimenez-serra2016] Their presence could be explained by non-thermal desorption, however the nature of the desorption mechanisms still has to be unravelled, with a joint experimental and modelling effort.
Non-thermal desorption from icy grain surfaces includes chemical desorption (also called reactive desorption, that is a chemical reaction at the surface of the ice forming a product that could desorb if exothermicity is sufficient)[@minissale2016b; @minissale2016a], desorption by cosmic rays,[@dartois2015] desorption by shock sputtering,[@gusdorf2008] photodesorption by UV (6-13.6 eV) primary photons or by secondary photons from cosmic rays and photochemical desorption (where photoproducts at the surface of the ice react and desorb).[@fillion2014; @martin-domenech2016] We can also add photodesorption by soft X-ray photons which has recently been experimentally explored and suggested to be important in PPDs.[@dupuy2018] UV photodesorption from ices has been proposed to play a key role in the gas-to-ice balance in various environments such as PDR, PPDs and protostellar envelopes (eg Ref. ), and also in the determination of snowlines.[@ligterink2018]
If one considers that H$_2$CO is a precursor of COMs, it is of prime importance to understand its photochemistry and the gas-grain exchanges it is involved in, as it could impact the formation and evolution of COMs. Even for a small molecule like H$_2$CO, gas-grain interactions are complex, and the gas-to-ice ratio results from the competition between thermal desorption, non-thermal desorption and freeze-out/processing of dust grains. It is crucial to understand in particular to which extent H$_2$CO can photodesorb as intact or dissociated in photofragments.
UV photodesorption of H$_2$CO has been called upon to explain molecular abundances and spatial distribution in PDR[@guzman2011; @guzman2013] and in PPDs.[@loomis2015; @carney2017; @oberg2017] There is a large number of astrochemical models that include H$_2$CO photodesorption, [@guzman2011; @guzman2013; @walsh2014; @agundez2018; @esplugues2016; @esplugues2017; @koumpia2017] but an arbitrary H$_2$CO photodesorption yield of 10$^{-3}$ molecules/photon is often considered, due to the absence of experimental data. However knowledge of photodesorption yields is crucial, as its variations, for the formaldehyde molecule, could increase gas phase abundances by up to several orders of magnitude in some environments, [@guzman2011] together with changes in the solid-phase chemistry.[@esplugues2016; @esplugues2017] It is clear that photodesorption yields from pure ices do vary on orders of magnitude from one molecule to the other, for example it is $\sim 10^{-2}$ molecules/photon for CO [@fayolle2011] and $\sim 10^{-5}$ molecules/photon for CH$_3$OH. [@bertin2016; @cruz-diaz2016] Photodesorption has been well studied for diatomics (CO,[@fayolle2011; @dupuy2017a] N$_2$,[@fayolle2013a; @bertin2013] NO),[@dupuy2017] but is much less known for larger molecules, and especially for organic molecules. Besides, photodesorption of a given molecule also varies with ice composition.
Whereas several experimental works focus on the photochemistry of H$_2$CO-containing ices, which is bulk sensitive, despite a clear need none studied the interaction between H$_2$CO ices and gas, which essentially involves the surface of the ice. To our knowledge, measurements of the photodesorption of H$_2$CO in the gas phase were performed from ices containing H$_2$CO as a photoproduct of the irradiation of pure methanol ices,[@bertin2016; @cruz-diaz2016] ethanol ices, or H$_2$O:CH$_4$ ice mixtures,[@martin-domenech2016] but never from ices grown from H$_2$CO, so that the amount of H$_2$CO in these ices is not well controlled.
We present wavelength-resolved and quantified data on the photodesorption of H$_2$CO and of CO and H$_2$ fragments from model ices : pure H$_2$CO ice, H$_2$CO on top of a CO ice (H$_2$CO/CO), and H$_2$CO mixed with CO ice (H$_2$CO:CO). Absolute photodesorption yields for each desorbing species, H$_2$CO, CO and H$_2$, are derived for each ice. Based on the results obtained from the pure H$_2$CO ice and the ones containing H$_2$CO and CO, we explore possible photodesorption mechanisms. Average photodesorption yields and branching ratios are deduced in various astrophysical environments (InterStellar Radiation Field ISRF, PDR at different extinctions, dense cores and PPDs). We discuss the possible effects of the photodesorption of H$_2$CO and of CO fragments in dense cores, PDR and PPDs.
Experiment
==========
The SPICES (Surface Processes & ICES) set-up, described in @doronin2015, was used for these experiments. It consists of an ultra-high vacuum (UHV) chamber with a base pressure of typically $10^{-10}$ mbar, within which a polycrystalline gold surface is mounted on a rotatable cold head that can be cooled down to $\sim$ 10 K using a closed cycle helium cryostat. Several diagnostics are possible in this set-up : detection of photodesorbing neutral molecules in the gas phase through mass-spectrometry with a quadrupole mass spectrometer (QMS, from Balzers), IR spectroscopy of the ice with Reflection Absorption InfraRed Spectroscopy (RAIRS), and temperature programmed desorption (TPD) of the ice with the same QMS as used for photodesorption measurements.[@doronin2015]
Ices are dosed by exposing the cold surface (10 K) to a partial pressure of gas using a tube positioned a few millimeters in front of the surface, allowing rapid growth without increasing the chamber pressure to more than a few 10$^{-9}$ mBar. In this set of experiments, different ices were grown : pure H$_2$CO ices, layered ices where H$_2$CO is deposited on a CO ice (H$_2$CO/CO)[^1] and mixed H$_2$CO:CO ices, where H$_2$CO and CO gases are mixed prior to deposition. Ice thicknesses are based on those measured from CO (Air liquide, >99.9% purity) ices, that are controlled with a precision better than 1 monolayer (ML) via a calibration using TPD, as detailed in @doronin2015. Condensing a pure H$_2$CO ice is not easy. Several tests have been performed in order to obtain the purest possible ice, and the following was performed (note that other techniques exist). First, solid paraformaldehyde, a (OCH$_2$)$_n$ multimer of formaldehyde, was pumped with primary pumps at room temperature for several hours. Then, its temperature was gradually increased up to 60 $^{\circ}$C with a water bath. Additional turbomolecular pumping is performed before the introduction of gas phase H$_2$CO to ensure the elimination of remaining water. This purified H$_2$CO in the gas phase was then released in the injection circuit and condensed on the cold substrate. The purity was checked with mass spectrometry and with IR spectroscopy of the pristine ice, with the RAIRS technique (a gold surface has been chosen for its good reflectance properties in the IR, so that RAIRS spectroscopy could be performed). A typical IR spectrum is shown in Fig.\[fig\_IR\_attribution\]. It confirms that H$_2$CO is the main component of the ice, with no H$_2$O or polymer contribution. Different H$_2$CO vibrational bands are labelled in the figure.[@bouilloud2015] For each new ice deposition, fresh H$_2$CO in the gas phase was expanded in the injection circuit.
The chamber was coupled to the undulator-based DESIRS beamline [@nahon2012] at the SOLEIL synchrotron facility, which provides monochromatic, tunable VUV light for irradiation of our ice samples. The coupling is window-free to prevent cut-off of the higher energy photons. The size of the VUV beam on the gold surface is $\sim0.7$ cm$^2$.[@fayolle2011] To acquire photodesorption spectra, the narrow bandwith ($\sim25$ meV) output of a grating monochromator is continuously scanned between 7 and 13.6 eV. Higher harmonics of the undulator are suppressed using a Kr gas filter. The experimental procedure is the following : we deposit an ice, record an IR spectrum to check its purity, irradiate it with VUV from 7 to 13.6 eV and record the photodesorption signal as a function of energy. The photodesorption of molecules in the gas phase following VUV irradiation of the ices is monitored by means of the QMS. Each 25 meV photon energy step lasts about 5 s, which is sufficiently higher than the dwell time of the QMS (0.5 s). Typical photon fluxes, as measured with a calibrated AXUV photodiode, depend on the photon energy and vary between $1.3 \times 10^{13}$ photons cm$^{-2}$ s$^{-1}$ at 7 eV and $5 \times 10^{12}$ photons cm$^{-2}$ s$^{-1}$ at 10.5 eV. A typical energy scan from 7 to 13.6 eV thus lasts around 20 minutes, which corresponds to a fluence of $\sim$ 10$^{16}$ photons cm$^{-2}$. In the following, a ’fresh’ ice refers to an ice that has just been deposited. We also performed several VUV irradiations on the same ice, further qualified as ’aged’. IR spectra after VUV irradiations are recorded and at the end, the ice and its photoproducts are released in the gas phase through TPD.
The conversion from the QMS signal to the absolute photodesorption efficiency, in molecule per incident photon, has been described in detail in @dupuy2017a. It is based on the knowledge of the absolute VUV photon flux, the apparatus function of the QMS, the 70 eV electron-impact ionization cross sections and comparative measurements to well-known photodesorption yields measured in the same experimental conditions. Due to an uncertainty in the photon flux measurement between 9.5 and 10.5 eV, the shape of the photodesorption spectrum has to be taken with caution in this region (see Supporting Information). In the electron-impact ionization process, H$_2$CO gives H$_2$CO$^+$, but also HCO$^+$ and CO$^+$. So any signal measured on the HCO$^+$ and on the CO$^+$ channels is corrected from this cracking. For all the detected species, electron-impact ionization cross-sections[@vacher2009; @tian1998a; @straub1996] were used to obtain absolute photodesorption yields. We should note that in the particular case of H$_2$, the apparatus function of the QMS is not known well and the partial electron-impact ionization cross-section of H$_2^+$ from H$_2$CO$^+$ seems unknown, so that photodesorption yields of H$_2$ from H$_2$CO are indicative values, from which not too many conclusions can be drawn.
![RAIRS spectrum of a 15 ML thick H$_2$CO ice (baseline subtracted). Vibrational modes are assigned according to @truong1993. $\nu(CH)$ are stretching vibrational modes (either symetric ($s$) or asymetric ($a$)). $\omega (CH_2)$, $\rho (CH_2)$ and $\delta (CH_2)$ are wagging, rocking and scissoring modes of the CH$_2$ group respectively. []{data-label="fig_IR_attribution"}](IR_attributions.pdf){width="8.26cm"}
Results
=======
Pure H$_2$CO ices
-----------------
### Photodesorption spectra of fresh H$_2$CO ices {#section_fresh}
![ Absolute photodesorption spectra of a 15 ML thick fresh H$_2$CO ice on gold at 10 K between 7 and 13.6 eV, recorded on the H$_2$CO mass (m/z 30, *upper panel*) and the CO mass (m/z 28, *middle panel*). The photodesorption spectrum recorded on the H$_2$ mass (m/z 2, *lower panel*) gives indicative yields (see Experiment). Note that the vertical scale on the upper panel is ten times lower than that in the other two panels. Spectra are averaged on three energy scans. Smoothed data (adjacent-averaging on 40 points) are represented by a bold black line. []{data-label="fig_pure_H2CO"}](pure_H2CO.pdf){width="8.26cm"}
![ Comparison between the H$_2$CO photodesorption spectra of a 15 ML thick H$_2$CO ice (upper panel; same as Fig. \[fig\_pure\_H2CO\]) and H$_2$CO gas phase absorption spectrum from @gentieu1970 (lower panel). The first ionization potential (1st IP) of gaseous H$_2$CO is indicated with a vertical line (10.88 eV [@kimura1981]).[]{data-label="fig_H2CO_gentieu"}](H2CO_gentieu.pdf){width="8.26cm"}
First of all, we recorded photodesorption spectra of an as-deposited H$_2$CO ice (Fig.\[fig\_pure\_H2CO\]). Several molecules were detected in the gas phase following VUV irradiation : the parent molecule, H$_2$CO, and fragments, CO and H$_2$. No signal of HCO photodesorption from H$_2$CO ices was detected. Indeed, whereas a HCO signal was measured, once corrected from the cracking pattern of H$_2$CO by the electron impact ionization, the signal on the HCO channel is dominated by noise. We estimated an upper limit of HCO photodesorption at $\sim 5 \times 10^{-4}$ molecule/photon, assuming the ionization cross section of HCO to be the same as that of H$_2$CO. The fact that H$_2$CO molecules are photodesorbing is noticeable. The photodesorption yield of H$_2$CO is around $5 \times 10^{-4}$ molecule/photon. Besides, the detection of CO and H$_2$ fragments is an important result and shows that VUV photons photodissociate a part of H$_2$CO ice in CO and in H$_2$, and that the CO fragments can desorb efficiently. The CO photodesorption intensities, from $10^{-3}$ to $10^{-2}$ molecule/photon, are indeed higher than those measured on the H$_2$CO channel.
The H$_2$CO photodesorption spectrum presents two broad structures around 7.2 and 9.2 eV, and a continuous increase above 10 eV. Comparing the parent photodesorption spectrum with solid phase absorption usually gives useful insights for the interpretation of the photodesorption spectrum, as the absorption is the first step eventually leading to photodesorption. Due to the lack of VUV absorption spectrum of solid H$_2$CO, the H$_2$CO photodesorption spectrum can only be discussed in the light of the H$_2$CO gas phase absorption. Figure \[fig\_H2CO\_gentieu\] presents the VUV absorption spectrum of H$_2$CO in the gas phase obtained with the monochromatized output of Hydrogen or Helium lamps (resolution 1Å) from @gentieu1970, and the reproduction of H$_2$CO photodesorption spectrum from Fig. \[fig\_pure\_H2CO\] for comparison. In the gas phase, there is a first electronic state (n,$\pi\star$) that shows a very weak UV absorption between 240 and 360 nm (3.4-5.2 eV, not shown). The highly-structured H$_2$CO spectrum between 115 and 160 nm (7.7-10.8 eV) consists of predissociated Rydberg series (eg Ref. ). These series are formed via excitation from the (2b$_2$) nonbonding orbital, and are accompanied by little vibrational excitation of the ionic core. On the contrary, Rydberg states in the 12-14 eV region enhance the production of vibrationally excited ions i.e. there could be autoionization from these Rydberg states,[@praet1968; @brint1985; @holland1994] and a strong and large resonance is seen in the gas phase spectrum (Fig. \[fig\_H2CO\_gentieu\]). Gas phase studies in the VUV show that when exciting H$_2$CO at 13 eV, the production of ions is favored relatively to that of neutrals and H$_2$CO$^+$ is the major ion detected, HCO$^+$ being much smaller.[@tanaka2017]
The fact that the electronic states of H$_2$CO in the gas phase are predissociative, the large width of the features observed in the photodesorption spectrum (Figure \[fig\_pure\_H2CO\]), and the observation of photodesorbing fragments point to the dissociative nature of the electronic states of solid H$_2$CO. Another finding is that photodesorption yields lie in the $4-8 \times 10^{-4}$ molecule/photon in the 7-13.6 eV range, and increase slightly and continuuously above 10 eV, which is expected to correspond more or less to the ionization energy (the ionization energy of solid H$_2$CO is not known, but is expected about 1 eV below the gas phase value of 10.88 eV). [@kimura1981] Besides, it seems that no sign of an autoionized state (which is at around 13 eV in the gas phase) is observed in the photodesorption spectrum, so that autoionization is not correlated to the photodesorption of neutral molecules. Possible causes include a very short lifetime in the solid phase and/or an electronic transition that does not lead to desorption of neutral H$_2$CO or of CO fragments. Regarding the possible photodesorption of ions, one has to keep in mind that ions in the solid phase are more bound than neutrals for polar ices, and that the electron takes the major part of the energy in the ionization process, so that the photodesorption of ions requires more energy than the ionization energy.[@philippe1997] Yet, desorption mechanisms when energies are higher than the ionization energy are poorly known. The detection of photodesorbing ions is not optimized for this version of the SPICES set-up, and no attempt to detect ions was performed.
CO and H$_2$ photodesorption spectra from fresh H$_2$CO ice show a different shape than the H$_2$CO photodesorption spectrum (Fig. \[fig\_pure\_H2CO\]). The H$_2$ spectrum presents an increase up to 9.5 eV, and then it remains at a constant yield of $4 \times 10^{-3}$ molecule/photon. However, there is a strong uncertainty on the absolute intensity and on the shape of H$_2$ photodesorption spectrum. This is due to the correction of the H$_2$ photodesorption measurement by a strong baseline coming from the residual vacuum. Finally, the CO photodesorption spectrum from pure H$_2$CO ices does not look like the photodesorption spectrum from pure CO ices. Indeed, no structures are seen, but only a strong increase with the photon energy, from $1 \times 10^{-3}$ molecule/photon at 7 eV to $9 \times 10^{-3}$ molecule/photon at 13.6 eV (Fig. \[fig\_pure\_H2CO\]) whereas the CO photodesorption spectrum from pure CO ices is null at 7 eV and between 9.5 and 10.5 eV, and show characteristic peaks between 8 and 9 eV (it is reproduced from @dupuy2017a in Figure \[fig\_H2CO\_CO\]). CO desorption here thus proceeds through H$_2$CO excitation. Possible photodesorption mechanisms of H$_2$CO and of CO will be further developped in the discussion.
### Ice modifications with VUV fluence (ice ageing): bulk and surface diagnostics through Infrared spectoscopy, TPD, and photodesorption spectra {#section_IR_TPD}
In this subsection, the results obtained when irradiating an ice that was already irradiated (ie an ’aged’ ice) are described. The study of ice modifications could be performed both by probing the bulk of the ice, with IR spectroscopy and TPD, and the surface of the ice through repeated photodesorption spectra on the same ice.
{width="\textwidth"}
IR spectra and Temperature Programmed Desorption give interesting results on the formation of photoproducts in the bulk of the ice. The comparison between IR spectra of a pristine ice and of an ice which received $3 \times 10^{16}$ photons cm$^{-2}$ is presented in Figure \[fig\_IR\_irradiation\]. The diminution of H$_2$CO is clearly seen in the 1400–1900 cm$^{-1}$ and in the 2800-3100 cm$^{-1}$ region. As the IR light illuminates the whole ice area (1 cm$^2$), but the VUV irradiates only 70%, this has to be taken into account in the analysis of IR spectra. We can estimate that approximately 2.4 ML of H$_2$CO was removed assuming a constant oscillator strength with the fluence. Given the initial thickness of 15 ML, this gives $\sim$ 20 % of removal of H$_2$CO at this fluence. The formation of CO and CO$_2$ in the ice was also evidenced with IR spectroscopy, by the appearance of characteristic bands at 2138 cm$^{-1}$ and 2341 cm$^{-1}$, respectively (Figure \[fig\_IR\_irradiation\]). The detection of HCO fragments in the ice with IR spectroscopy is not conclusive in our experiments (signal to noise ratio too low).
Figure \[fig\_IR\_irradiation\] also shows the TPD of the same irradiated ice, which received a total fluence of $3 \times 10^{16}$ photons cm$^{-2}$. Together with the thermal desorption of H$_2$CO, desorption of the photoproducts H$_2$ and CO was observed. The signal measured on CO is corrected from the cracking of H$_2$CO. H$_2$CO desorbs around 105 K, and the co-desorption of CO and H$_2$ together with H$_2$CO at this temperature is also noticed. CO also desorbs at lower temperature, at $\sim$ 40 K, which is larger than its desorption temperature from pure CO ices. That could be due to its diffusion in the H$_2$CO ice, or its presence in H$_2$CO pores. TPD is very useful as a quantitative technique but it may be artificially altered by thermal activation of chemical reactions. The amount of CO fragments in the bulk of the irradiated H$_2$CO ice is estimated at $\sim 20\%$ in abundance from TPD measurements. The absence of higher masses rules out POM presence.
In the end, the IR spectra and TPD give compatible amounts for the photodissociation of H$_2$CO in CO, $\sim 20\%$. The carbon and oxygen budget is balanced in the irradiation process. H$_2$CO disappearance in the bulk is due to the photodissociation of H$_2$CO, which mainly leads to CO formation (see Discussion). The amount of CO and H$_2$CO ejected in the gas phase is negligeable as compared with the amount of material within the bulk, as it will be shown below.
![ Same as Fig. 2, but for an ice pre-irradiated (fluence from $2 \times 10^{16}$ photons cm$^{-2}$ at 7 eV to $3 \times 10^{16}$ photons cm$^{-2}$ at 13.6 eV.) Smoothed data from the ’fresh’ ice of Fig. 2 are reproduced in dashed lines. []{data-label="fig_pure_H2CO_aged"}](pure_H2CO_aged.pdf){width="8.26cm"}
As mentioned earlier, ice modifications due to VUV irradiation are noticed not only in bulk diagnostics, but also in the signatures of the surface processes, through photodesorption. Figure \[fig\_pure\_H2CO\_aged\] presents photodesorption spectra for an ice which was already irradiated (at a fluence of $2 \times 10^{16}$ photons cm$^{-2}$) so that while recording the spectrum, the fluence varied between 2 and 3 $\times 10^{16}$ photons cm$^{-2}$. We qualify such an ice as ’aged’, contrary to ’fresh’ ices of Fig. \[fig\_pure\_H2CO\]. From this figure, it is clear that there are changes in the intensity and shape of the spectra, showing that the VUV irradiation photoprocessed the ice. The VUV flux used here is thus high enough to process the ice within one energy scan. More precisely, it should be noted that photodesorption spectra at the intermediary fluence (1 to 2 $\times 10^{16}$ photons cm$^{-2}$ i.e. just between those of Figure \[fig\_pure\_H2CO\] and \[fig\_pure\_H2CO\_aged\]) look very much like the spectrum of the Figure \[fig\_pure\_H2CO\_aged\] (see Supporting Information). Here a steady state may be reached certainly during the second energy scan. The fact that a steady state was not reached during the first energy scan in the ’fresh’ ice shows that the shape of the H$_2$CO spectrum during the first energy scan is certainly distorted, especially at the end of the scan. However, due to the weak H$_2$CO signal, it was not possible to perform measurements at lower flux.
The spectral shape of CO changes as a function of fluence (Figures \[fig\_pure\_H2CO\] and \[fig\_pure\_H2CO\_aged\]). The bump in the 8–9 eV region (Figure \[fig\_pure\_H2CO\_aged\]) could be the signature of accumulated solid CO electronic A-X transition. This is consistent with the well-known photodesorption signatures of CO from CO ices (see for example Ref. and Figure \[fig\_H2CO\_CO\]) and with the fact that CO is present in the ice following VUV dissociation of H$_2$CO, as shown by IR and TPD measurements. This is another signature of the photoprocessing of the ice. Interestingly, whereas for fresh ices (Figure \[fig\_pure\_H2CO\]), H$_2$CO, CO and H$_2$ photodesorption spectra look different, for aged ices their spectral shape looks much alike (Figure \[fig\_pure\_H2CO\_aged\]).
From the VUV irradiation dose and the average photodesorption yield, it is possible to estimate the amount of desorbed species during one energy scan: $4 \times 10^{-3} $ ML of H$_2$CO desorbed, with $4 \times 10^{-2} $ ML of CO and with $3 \times 10^{-2} $ ML of H$_2$. This corresponds to a total of only 7% of a ML desorbing during one energy scan, which is very small compared to the ice thickness, accounting for only 0.5%. It is also very small compared with the disappearance of H$_2$CO in the bulk, estimated at 20% from IR spectra. The amount of photodesorbing molecules is thus very small compared with the amount of photoprocessed ones; the overall ice thickness can thus be considered as constant. This implies that the decrease of the photodesorption signal between a fresh and a photoprocessed ice (Figures \[fig\_pure\_H2CO\] and \[fig\_pure\_H2CO\_aged\]) cannot be related to the evolution of the ice thickness. The loss of solid H$_2$CO through photodissociation followed by chemistry, together with a possible modification of the ice surface may thus be responsible for the decrease of the H$_2$CO photodesorption signal.
H$_2$CO on top of CO ice (H$_2$CO/CO)
-------------------------------------
We also studied layered H$_2$CO/CO ices, where H$_2$CO was deposited on top of a CO ice. Photodesorption spectra of H$_2$CO/CO ices, recorded on H$_2$CO (a) and CO (b) masses are presented in Fig. \[fig\_H2CO\_CO\], for different quantities of H$_2$CO on CO, increasing from 0.4 ML to 2.6 ML, from top to bottom. For comparison, the H$_2$CO spectrum from a pure H$_2$CO ice (like that of Figure \[fig\_pure\_H2CO\]) and the CO spectrum from a pure CO ice [@dupuy2017a] are reproduced at the bottom of the figure.
First, let us comment on the photodesorption spectra recorded on the H$_2$CO channel. H$_2$CO photodesorption spectra show different spectral shape and different yields for each ice (Fig. \[fig\_H2CO\_CO\] a), from top to bottom). If only a thin layer of H$_2$CO (0.4 or 1.1 ML) is deposited on top of CO, it seems despite the low signal-to-noise ratio that there is a bump in the 8-9 eV region (see Supporting Information). This could be the signature of CO electronic excitation, transferred to surface H$_2$CO that can desorb through a CO-induced desorption i.e. via an indirect DIET mechanism (see Discussion). Besides, the H$_2$CO signal is not null at 7 eV nor between 9.5 and 10.5 eV, where CO photodesorption is inefficient, indicating that there is a small contribution of H$_2$CO photodesorption from H$_2$CO direct excitation. In the intermediate thickness case (middle panel of Fig. \[fig\_H2CO\_CO\]), there are both CO and H$_2$CO contributions, the H$_2$CO contribution being larger than in the 0.4 ML case, and that of CO smaller. When 2.6 ML of H$_2$CO lie on CO, H$_2$CO desorption signal resembles the one from pure H$_2$CO. Thus no strong effect of H$_2$CO thickness on the photodesorption signal is observed, for thicknesses larger than 3 ML. For thicknesses smaller than 3 ML, we see that the H$_2$CO photodesorption yield depends on the ice thickness and on the ice composition (effects of the underlying CO molecules).
Regarding CO photodesorption spectra, they also change in intensity and in shape with the amount of covering H$_2$CO. When 0.4 ML of H$_2$CO is deposited, CO signatures are clearly seen in the 8–9 eV region, and the spectrum looks rather like that from pure CO ices. It is interesting to see how much CO yields decrease as soon as it is covered with more than 1 ML of H$_2$CO. In this case, CO desorption looks like that from pure H$_2$CO, that is CO comes from photodissociated H$_2$CO and at least does not result from direct excitation of CO.
Previous studies have shown that photodesorption is a surface process (eg Ref. for a CO ice). All the results on layered H$_2$CO/CO ices in this study also confirm that photodesorption is a surface process, mostly depending on the first 3 monolayers composition, in this case.
{width="\textwidth"}
H$_2$CO and CO mixed ice (H$_2$CO:CO (1:3))
-------------------------------------------
Photodesorption spectra of a mixed H$_2$CO:CO (1:3) ice are presented in Fig. \[fig\_H2CO\_CO\_mix\]. Assuming a homogeneous ideal mixing in the solid phase, the surface of this ice is thus composed of $\sim 25\%$ of H$_2$CO and of $\sim 75\%$ of CO. Due to signal/noise limitation, it is not possible to comment on the shape of H$_2$CO photodesorption spectrum. It is however possible to estimate if there is a CO-induced effect in the H$_2$CO photodesorption signal in the mixed ice. H$_2$CO photodesorption efficiencies between 8 and 9 eV are smaller for the mixed ice than for the pure ice, as there is less H$_2$CO available at the surface. After correcting the H$_2$CO photodesorption efficiency by the dilution factor, it is two times larger in the mixed ice than in the pure ice, showing that there is certainly a CO-induced desorption of H$_2$CO in the mixed ice.
CO photodesorption spectrum from H$_2$CO:CO was also recorded. It looks like that from pure CO, except it is less intense but this is consistent with the fact that there is less CO at the surface of the mixed ice than in the thick pure ice of @dupuy2017a. As for the layered ice, these results illustrate how photodesorption depends on the ice composition and the thickness.
![ Photodesorption spectra of a 24 ML thick H$_2$CO:CO (1:3) mixed ice at 10 K between 7 and 13.6 eV, recorded on the H$_2$CO mass (m/z 30,*upper panel*) and the CO mass (m/z 28, *lower panel*). Smoothed data are represented by a thick black line in the upper panel.[]{data-label="fig_H2CO_CO_mix"}](figure_mix.pdf){width="8.26cm"}
Discussion
==========
Photodesorption mechanisms {#section_discussion_psd}
--------------------------
### Mechanisms involved in the photodesorption of H$_2$CO and of CO (pure H$_2$CO ice)
First of all, we develop some aspects of H$_2$CO photochemistry, that are essential to the discussion of possible mechanisms involved in the photodesorption of H$_2$CO and of CO, for which the excitation of H$_2$CO in dissociative electronic states is certainly the very first step.
Studies of the photolysis of condensed H$_2$CO in the VUV range have shown the existence of these two channels:[@thomas1973; @glicker1976]\
H$_2$CO $\xrightarrow{h\nu}$ HCO + H (radical channel) *(i)*\
H$_2$CO $\xrightarrow{h\nu}$ CO + H$_2$ (molecular channel) *(ii)*\
As it requires at least 3.5 eV to dissociate H$_2$CO in (i) and (ii) (Refs and references therein), VUV irradiations provide enough energy to access these channels.
Regarding the present VUV irradiation of solid H$_2$CO, photodesorption, TPD and IR measurements show that CO and H$_2$ are photoproducts present in the ice. This reveals that channel (ii) is a major channel, directly enriching the ice with CO and H$_2$. Besides, the unclear detection of HCO formation in the bulk of our ices could be due to a lack of IR sensitivity or could mean that its destruction pathways are favored over its formation pathways. The HCO radical can be formed from H$_2$CO dissociation through channel (i) or from the reaction between an electronically excited CO (CO\*) and H$_2$[@chuang2018] (CO\* + H$_2$ $\rightarrow$ HCO + H), but it could further evolve to give back H$_2$CO and/or CO (through HCO + H (addition) $\rightarrow$ H$_2$CO which could be barrier-less,[@fuchs2009] through HCO + H (abstraction) $\rightarrow$ CO + H$_2$[@minissale2016a], through HCO + HCO $\rightarrow$ H$_2$CO + CO which is barrier-less,[@butscher2017] or through HCO $\xrightarrow{h\nu}$ CO + H). In the end, the final products of the reactions involving HCO are CO and H$_2$CO, the ones that we observe in the bulk in our experimental conditions. CO$_2$ was also observed in the ice (see Results), and it could be formed from the following reaction : CO\* + CO $\rightarrow$ CO$_2$ + C.[@loeffler2005] It is important to keep in mind that the nature and the amount of photoproducts present in the ice depend on the fluence. For example, CH$_3$OH is produced in the work of @gerakines1996, for fluences higher than 10$^{17}$ photons/cm$^2$. In our experiment, we did not see any CH$_3$OH band, certainly due to a congestion in the IR spectrum, but that does not preclude its presence, even if certainly weak.
The formation of these CO and H$_2$ photoproducts was also detected in the H$_2$-lamp irradiation of H$_2$CO ices. [@gerakines1996; @butscher2016] In addition, more complex molecules (glycolaldehyde, ethylene glycol and the formaldehyde polymer, polyoxymethylene (POM)) were also observed in @butscher2016’s experiments where a higher fluence than ours was used.
{width="\textwidth"}
There is a competition between the reaction of photoproducts and their desorption, if they have enough kinetic energy and if they are located near the surface. Based on all the experimental results and on the possible photodissociation paths, the photodesorption of H$_2$CO and of CO fragments could occur through different mechanisms, that are sketched in figure \[fig\_mechanisms\]:
\(a) the electronic excitation of an H$_2$CO molecule could lead to its direct desorption (DIET), if it is located at the surface and if it has not photodissociated.
\(b) the excitation of a H$_2$CO molecule or of a CO fragment in the first three upper layers of the ice could be transferred to a surface H$_2$CO molecule that could consequently desorb. This process has been named indirect DIET, [@bertin2012] and it could contribute to H$_2$CO photodesorption from pure ices, once it is photoprocessed and contains some CO fragments. Indeed, experiments on H$_2$CO/CO layered ices and on H$2$CO:CO mixed ices have shown that CO-induced desorption of H$_2$CO is at play, so that it could also contribute to H$_2$CO photodesorption efficiency from pure H$_2$CO ices, if enough CO photofragments are present in the ice. As approximately 20% of the ice is CO at a fluence of $3 \times 10^{16}$ photons cm$^{-2}$, this mechanism is possible.
\(c) the photodesorption of CO and H$_2$ products was also observed and it is an important finding. It could happen for CO and H$_2$ if they are located near the surface and if they have enough energy right after H$_2$CO dissociation, or (d) result from surface reaction processes such as reactions between HCO photoproducts[@butscher2017] or H atom abstraction of an HCO photoproduct,[@minissale2016a] or be induced by the excitation of CO molecules, as the electronic excitation of CO is probably seen in Fig. \[fig\_pure\_H2CO\_aged\].
\(d) photochemical desorption of H$_2$CO could occur, through recombination of HCO + H photofragments (HCO directly coming from H$_2$CO dissociation, or from CO\*+H$_2$). Indeed the reaction is exothermic (by 3.9 eV), so that it releases an energy much larger than the H$_2$CO binding energy (0.324 eV),[@noble2012] and it could possibly lead to a photodesorption event if it occurs near the surface. Recombination of two HCO radicals could also produce H$_2$CO[@butscher2017] that could be released in the gas phase, together with CO fragments.
\(e) an H atom could kick-out a H$_2$CO molecule present at the surface. Such a kick-out mechanism by H atoms has been predicted for H$_2$O.[@andersson2008] It is expected that the efficiency of this mechanism decreases with the mass of the target molecule, so as the mass of H$_2$CO is larger than the mass of H$_2$O, this process will be less favorable for H$_2$CO ice than for H$_2$O ice.
Therefore this set of data shows that desorption driven by electronic transitions either in a direct or indirect way contribute to the photodesorption of H$_2$CO, possibly together with photochemical desorption. For CO$_2$ ice, the photochemical desorption mechanism, through recombination of CO and O, was quantitatively measured to be 10% of the total photodesorption efficiency.[@fillion2014] On the contrary, for CH$_3$OH, the photochemical recombination CH$_3$O + H is proposed as a possible desorption pathway.[@bertin2016] Through the present study of the photodesorption of H$_2$CO, we have hints of the presence of photochemical recombination through the dissociative nature of electronic states and the presence of photoproducts. However it does not allow one to estimate the relative contribution of photochemical recombination with respect to DIET or indirect DIET mechanisms, all these mechanisms being certainly at play.
H$_2$CO photodesorption efficiency is larger than CH$_3$OH:[@bertin2016; @cruz-diaz2016] H$_2$CO photodesorption yields from pure H$_2$CO ices are $\sim 30-40$ times higher than CH$_3$OH from pure CH$_3$OH ices (Table \[table\_yields\]). When discussing photodesorption efficiencies from one molecule to the other, it is convenient to compare yields in molecule per absorbed photon, and not in molecule per incident photon (as derived here). For H$_2$CO, it is not possible to obtain yields in molecule per absorbed photon, as the absorption cross section in the solid phase is missing. As absorption cross sections of different ices differ only by a factor 2 or 3,[@cruz-diaz2014b] the large photodesorption efficiency of H$_2$CO cannot be explained by differences in absorption cross sections and another parameter has to be invoked. It cannot be the mass of H$_2$CO and of CH$_3$OH, as they are very similar, but it could be differences in binding energies (larger for CH$_3$OH than for H$_2$CO), in vibrational degrees of freedom (12 for CH$_3$OH versus 6 for H$_2$CO), or in the fragmentation of these two molecules.
In addition to the photodesorption of intact H$_2$CO, the fact that CO photofragments desorb is an important finding, and we proposed several mechanisms leading to its photodesorption. It seems that in our experiments, CO desorption decreases with fluence (compare Fig. \[fig\_pure\_H2CO\] and Fig. \[fig\_pure\_H2CO\_aged\]), that could be explained by the consumption of CO to give HCO and H$_2$CO products, or by the modification of the ice surface, so that in the end less molecular fragments photodesorb. While the deposited H$_2$CO ice is amorphous,[@hiraoka2005] modifications of the ice structure can happen due to VUV irradiation. Possible modifications include an amorphisation or a compactification of the H$_2$CO ice (the latter is observed in the irradiation of water ice).[@palumbo2010; @dartois2018]
### CO-induced desorption of H$_2$CO (H$_2$CO/CO layered ices and H$_2$CO:CO mixed ices)
To unravel the efficiency of CO-induced desorption, several layered ices have been studied.[@bertin2012; @bertin2013; @dupuy2017a] The characteristic signatures of CO vibronic excitation were observed in the photodesorption pattern of the molecule X above CO. This indicates a transfer of energy from CO molecules to the surface molecules, the indirect DIET process.[@bertin2012] The maximum photodesorption yields of X species in the 8–9 eV region for X/CO ices are : $2.5 \times 10^{-2}$ molecule/photon for X=N$_2$,[@bertin2013] $2 \times 10^{-3}$ molecule/photon for X=CH$_4$,[@dupuy2017a] and $3 \times 10^{-4}$ molecule/photon for X=H$_2$CO (this work). These yields vary for the different X molecules. The efficiency of the indirect DIET mechanism depends on several parameters: the inter-molecular energy transfer, the intra or inter-molecular energy relaxation once it has been transferred, and the binding energy of the molecule.
The comparison between N$_2$ and CH$_4$ yielded interesting findings and pointed out the important role of intra- or inter-molecular energy relaxation in this system (the binding energy of the two molecules on CO should be approximately the same).[@dupuy2017a] Comparison between N$_2$ and H$_2$CO indirect desorption also gives interesting conclusions. The mass of N$_2$ and H$_2$CO is approximately the same, so a simple kinetic momentum transfer between CO and N$_2$ or H$_2$CO cannot explain the different efficiencies. The binding energy of H$_2$CO on CO (not known) is expected to be larger than the binding energy of N$_2$ on CO (because H$_2$CO has a permanent dipole moment whereas N$_2$ has none). A higher binding energy of H$_2$CO on CO could thus play a role in the quenching of the CO-induced efficiency. We also expect H$_2$CO to relax the vibrational excess energy more efficiently than N$_2$, because of the degrees of freedom in H$_2$CO with respect to diatomics. However, the vibrational degrees of freedom of H$_2$CO (6) are smaller than for CH$_4$ (9), whereas H$_2$CO desorbs less efficiently than CH$_4$ through indirect DIET. All these results on different systems show that both energy relaxation and large binding energies play a role on quenching the CO-induced photodesorption.
Mixed ices containing CO and N$_2$ or CH$_3$OH were also studied in @bertin2013 [@bertin2016]. In the CO:N$_2$ binary ice,[@bertin2013] the photodesorption spectrum is a linear combination of that of CO and N$_2$. In the H$_2$CO:CO mix (Fig. \[fig\_H2CO\_CO\_mix\]) or in the CH$_3$OH:CO mix,[@bertin2016] this is not the case, as the H$_2$CO or CH$_3$OH photodesorption spectrum is influenced by photochemical processes and surface modification. As already shown by H$_2$CO/CO experiments, CO-induced desorption of H$_2$CO is less efficient than N$_2$ indirect desorption. This shows once more that photodesorption depends on the ice composition and on the considered molecule, the main difference between N$_2$ on one side and H$_2$CO and CH$_3$OH being that the last two photodissociate and have a large binding energy.
Photodesorption yields in various astrophysical media {#discussion_table}
-----------------------------------------------------
-------------------- --------------- ---------------- ---------------- ---------------- -------------------- ---------------------
Ice Photodesorbed ISRF^*a*^ PDR^*b*^ PDR^*b*^ Secondary UV ^*c*^ Protoplanetary disk
species $i$ A$_V$=1 A$_V$=5 TW Hya^*d*^
Pure H$_2$CO H$_2$CO 5 4 4 4 4
CO 45 28 22 45 40
H$_2$CO:CO (1:3) H$_2$CO 8^*$\dagger$*^ 6^*$\dagger$*^ 6^*$\dagger$*^ 6^*$\dagger$*^ 10^*$\dagger$*^
CO 61 40 38 42 26
Pure CH$_3$OH^*e*^ CH$_3$OH 0.12 0.15
H$_2$CO 0.07 0.12
-------------------- --------------- ---------------- ---------------- ---------------- -------------------- ---------------------
UV fields between 7 and 13.6 eV are taken from : ^*a*^ @mathis1983; ^*b*^ PDR Meudon Code @lePetit2006 (see text for details); ^*c*^ @gredel1987; ^*d*^ @heays2017 (FUV observation of TW-Hydra from @france2014, extrapolated to a broader spectral range); ^*e*^ @bertin2016
^*$\dagger$*^ This average photodesorption yield is normalized to the fraction of the surface of the ice containing H$_2$CO $f_s$ (see text for details)
\[table\_yields\]
Average photodesorption yields $Y_i$ are reported in Table \[table\_yields\], for $i=$ H$_2$CO and CO, from the pure H$_2$CO ice and from the mixed H$_2$CO:CO ice, for different UV radiation fields. They are derived from experimental photodesorption spectra (Fig. \[fig\_pure\_H2CO\]) and radiation fields in different interstellar environments, as described in Ref. . In the particular case of the mixed H$_2$CO:CO ice, average photodesorption yields $Y_{H_2CO}$ are obtained through the following :
$Y_{H_2CO} = \frac{Y_{H_2CO}^{measured}}{f_s},$
with $f_s$ the fraction of the surface of the ice containing H$_2$CO, which is 0.25 for the H$_2$CO:CO (1:3) mixed ice. These photodesorption yields are given in molecule per incident photon, so that they can be directly implemented in astrochemical models, wihout any correction by the number of monolayers included in the surface. For any molecule in the upper three monolayer, photodesorption yields of Table \[table\_yields\] can be added to models.
Average H$_2$CO photodesorption yields are in the same range for the different regions explored : the Interstellar Radiation Field, PDR radiation fields, secondary UV photons or a typical PPD spectrum. This is due to the shape of the photodesorption spectrum (Fig. \[fig\_pure\_H2CO\]), which do not vary significantly with the photon energy. However, there is an effect of the ice composition, as H$_2$CO photodesorption yields vary if pure or mixed ices are considered (Table \[table\_yields\]). They are slightly larger when H$_2$CO is mixed with CO, which is an enhancement certainly due to the CO-induced photodesorption of H$_2$CO. Thus to model an ice whose surface contains H$_2$CO and CO, it is more appropriate to use the average photodesorption yield from the CO-containing ice than from the pure ice.
Photodesorption yields of Table \[table\_yields\] are valid for H$_2$CO present at the surface (that is in the first 3 ML) of an ice whose total thickness is larger than 3 ML. If one considers astrophysical environments or time periods where ices start to form on interstellar grains, only thin ices ($\lesssim 1$ ML) cover the grain substrate, and photodesorption yields could be different. Indeed, the underlying substrate could change the photodesorption, as a function of wavelength and in intensity, depending on its nature. If we consider Fe-silicates, they absorb in the VUV range (see eg Ref. ), so an energy transfer to the thin ice above could be possible and change the photodesorption yields. Besides, for these thin ices, the presence of pores and of inhomogeneities of the grain substrate can change the adsorption of molecules and their arrangement, which could result in different photodesorption mechanisms and efficiencies. This was observed for CO deposited on porous-amorphous H$_2$O, where pores in the H$_2$O ice reduce the CO photodesorption yield.[@bertin2012] It is thus probable that the nature of the surface of interstellar grains and their morphology affect photodesorption if thin H$_2$CO-containing layers of ices are involved, and laboratory experiments are needed to characterize and quantify this.
It is possible to compare the H$_2$CO photodesorption yields from pure ices with H$_2$CO photodesorption yields from previously studied ices. H$_2$CO was also observed as a photodesorbing fragment from CH$_3$OH ice.[@bertin2016; @cruz-diaz2016] However, yields are around $10^{-5}$ molecule/photon (values from @bertin2016 are reproduced in Table \[table\_yields\]), which is much smaller than H$_2$CO from pure H$_2$CO ices, $ > 4 \times 10^{-4}$ molecule/photon (Fig. \[fig\_pure\_H2CO\_aged\] and Table \[table\_yields\]) i.e., H$_2$CO photodesorption from pure H$_2$CO ices is 50 times more efficient than from pure CH$_3$OH ices. In addition to CH$_3$OH ices, H$_2$CO desorption has been detected from ethanol or H$_2$O:CH$_4$ ice.[@martin-domenech2016] In this experiment, H$_2$CO photodesorption yields of $6 \times 10^{-4}$ molecule/photon and $4.4 \times 10^{-4}$ molecule/photon were found, respectively. These are the same values as found in the present study, meaning that H$_2$CO photodesorption has to be taken into account both from H$_2$CO and from the recombination of photofragments at the surface of interstellar ices. In the end, all desorption channels have to be taken into account, when possible.[@guzman2013]
If we consider pure H$_2$CO or pure CH$_3$OH ices, CO fragment is the major species that desorbs. CO desorption from H$_2$CO ices is $4.5 \times 10^{-3}$ molecule/photon (Table \[table\_yields\]). This yield is directly derived from our measurements, so that it can be added in models, without any photodissociation ratio correction. We also found that CO photodesorption from H$_2$CO ices is only 2 times less than CO desorption from pure CO ices .[@fayolle2011] CO ice abundance (20% relative to H$_2$O ice, cf Ref. ) is larger than H$_2$CO ice abundance (a few percent relative to H$_2$O ice), but CO photodesorption from H$_2$CO and more generally from organic molecules could be taken into account in models.
Pure H$_2$CO ices constitute model ices, however astrophysical ices are much more complex, and H$_2$CO could be mixed with CO, on a CO ice, below CO ice, or with H$_2$O or CO$_2$. The mixture of H$_2$CO in a H$_2$O-dominated ice could for example give different photodesorption yields and mechanisms, because of the dangling O-H bonds of surface H$_2$O who rapidly and efficiently evacuate excess vibrational energy[@Zhang2011]. Besides, the phase of interstellar H$_2$O, if porous/compact amorphous[@dartois2015a] or crystalline,[@mcclure2015] could also influence photodesorption. Precisely taking into account properly the effects of the ice composition on the photodesorption of H$_2$CO necessitates a dedicated study. Still, from the H$_2$CO:CO mixed and H$_2$CO/CO layered ices studied here, we see that the abundance of H$_2$CO in the first three monolayers has to be taken into account, together with CO-induced desorption effects. Photodesorption yields adapted to the modelled interstellar ice have to be considered.
### Prestellar cores and PDR
H$_2$CO in the gas phase has been observed in dense cores and in low or high-UV flux PDR (e.g. Refs ), and UV photodesorption has often been called upon to explain the observations.
In dense cores, H$_2$CO in the gas phase could come from different processes : it could be formed directly in the gas, or desorb from grains through chemical desorption, cosmic-ray sputtering, or UV photodesorption. Chemical desorption is often considered in astrochemical models, [@garrod2006] however little experimental data exist, so that the efficiency taken in models is highly uncertain, especially when considering reactions occurring on icy grains. Indeed, experimental results of @minissale2016b [@minissale2016a] have shown that the chemical desorption process is much less efficient on icy grains than on bare grains, and sometimes too weak to be detected. Whereas H$_2$CO chemical desorption from *bare* grains has important effects on gas phase abundances, those changes were much smaller when considering chemical desorption from water *icy* grains.[@cazaux2016] The competition between UV photodesorption efficiency of H$_2$CO and chemical desorption efficiency on icy grains can now be modelled.
During the formation of interstellar ices in the early phases of core contraction, UV photodesorption plays an important role, as shown by @kalvans2015. At this early phase, the ISRF can still penetrate the cloud, so that photodesorption mostly governs the onset of ice accumulation onto grains. Later on, whereas the ISRF is shielded, there are nonetheless UV photons emitted by H$_2$ excited by cosmic rays (called secondary UV photons). H$_2$CO average photodesorption yields from secondary UV photons in the dense cores are the same as from the ISRF (Table \[table\_yields\]). The secondary UV flux in dense cores is 10$^4$ photons cm$^{-2}$ s$^{-1}$.[@Shen2004] If we consider the beginning of H$_2$CO ice formation at 1000 years[@cuppen2009], and a dense cloud lifetime of 10$^6$ years, this gives a fluence of $3 \times 10^{14}$ to $3 \times 10^{17}$ photons cm$^{-2}$. In our experiments the fluence was at most $3 \times 10^{16}$ photons cm$^{-2}$, which is thus representative of that in dense clouds, except in the last 10$^5$-10$^6$ years of the core. These yields could thus be implemented in models, and their effects on dense cores at various time explored.
In low-UV illuminated PDR, nonthermal desorption from H$_2$CO-containing icy grains is necessary to reproduce the observed abundances,[@guzman2011; @guzman2013] and photodesorption was considered to be a very good candidate. As the effects of VUV radiation on the photodesorption of H$_2$CO ices were not known, these authors considered two different pathways: the first one is the release of H$_2$CO in the gas phase with a yield of $5 \times 10^{-4}$ molecule per photon, and the second one is the photodissociation and release of HCO and H fragments, with a yield of $5 \times 10^{-4}$ molecule per photon. Our experimental work reveals that different yields and pathways have to be considered (Table \[table\_yields\]): the H$_2$CO photodesorption yield is $4-6 \times 10^{-4}$ molecule per photon, the HCO + H path was not observed; whereas the CO + H$_2$ was the most efficient path giving $22-28 \times 10^{-4}$ CO molecule per photon. Taking these findings into account could result in different gas phase abundances of H$_2$CO and related molecules.
In order to study the effect of the radiation field on the photodesorption efficiency, radiation fields at different extinctions were generated with the Meudon PDR code,[@lePetit2006] at A$_V$=1 and A$_V$=5. Low-flux PDR conditions corresponding to the Horsehead PDR were considered, with an incident field of 80 times that of @mathis1983, and a density profile as described in @guzman2011. A$_V$=1 is the closest to the star, whereas A$_V$=5 is further in the PDR. We assume that H$_2$CO ice could start to form at A$_V$=1, like CH$_3$OH ice.[@esplugues2016] When going further in the PDR, the high-energy part of the spectrum is strongly attenuated. Despite the spectral differences at A$_V$=1 and A$_V$=5, average photodesorption yields are approximately the same from an extinction to the other mainly because photodesorption spectra do not vary significantly with the photon energy. As a consequence, for H$_2$CO, a single photodesorption yield can be used at all extinctions. H$_2$CO photodesorption rates should thus match the diminishing photon flux in the cloud. One should keep in mind that in PDR, UV photon fluxes vary strongly: if we consider a flux of $2 \times 10^{10}$ photons cm$^{-2}$ s$^{-1}$ at the edge of the PDR, it will decrease down to $2 \times 10^{2}$ photons cm$^{-2}$ s$^{-1}$ at A$_V$=5 (considering only dust extinction). The fluence over the whole PDR lifetime ($5 \times 10^5$ years)[@Pound2003] thus varies from $3 \times 10^{23}$ photons cm$^{-2}$ at the edge of the PDR to $3 \times 10^{15}$ photons cm$^{-2}$ at A$_V$=5 and at 12 eV. As a consequence, ices in PDR regions experience various irradiation conditions, from those similar to the present study, to others where photoproducts could appear and the ice composition change, possibly affecting the photodesorption yield of H$_2$CO and of CO, but also introducing other photodesorbing species.
The Orion bar case is different from the Horsehead PDR. It is a high-UV flux PDR, so dust temperatures are around 55–70 K, which is higher than in the Horsehead PDR (around 20-30 K, @guzman2011). So there is a competition between thermal and nonthermal desorption, contrary to the Horsehead case. But keep in mind that the adsorption energy of pure H$_2$CO is 3770 K and the adsorption energy of H$_2$CO bound to H$_2$O is 3259 K,[@noble2012] the latter being much larger than the one usually considered from @garrod2006b, 2050 K. Implementing an accurate adsorption energy is necessary as variations in adsorption energies induce variations in ice abundances and ice composition (see eg @penteado2017). This could mean that H$_2$CO thermal desorption is overestimated in some models and in some environments such as high UV flux PDR, and that non-thermal desorption could play a role. We can estimate that below $\sim$ 76 K[^2], UV photodesorption dominates over thermal desorption. As dust temperatures are lower than 76 K in the Orion Bar, UV photodesorption is thought to impact the gas-grain ratio in this high-flux region. This was actually (indirectly) explored in the work of @esplugues2016 [@esplugues2017]. As soon as CO and H$_2$CO photodesorptions are taken into account in high-flux PDR, there are changes of several orders of magnitude in the gas phase abundances of CO and H$_2$CO, so that UV photodesorption clearly plays a role.
### Protoplanetary disks (PPDs)
Several spatially-resolved observations of H$_2$CO in the gas phase have been performed on disks around a T Tauri (eg Ref. ) or a Herbig star.[@carney2017] These observations, together with models, showed that there are several H$_2$CO components in these disks. Whereas close to the star, warm H$_2$CO should form in the gas phase, further from the star, grain surface formation (which is active in extended areas of disks [@aikawa1999; @walsh2014]) and non-thermal desorption are both needed to explain the observations. More precisely, the detection of gas phase H$_2$CO in the outer parts of the disk, where CO is frozen on grains, favors the role of surface processes and of nonthermal processes such as photodesorption by UV or X-ray photons.[@loomis2015; @oberg2017; @carney2017; @podio2019] Complex models of PPDs have demonstrated that in the midplane, non-thermal desorption processes such as cosmic-ray-induced thermal desorption, X-ray spot heating and photodesorption by internal and external UV photons are necessary to produce H$_2$CO in the gas phase.[@walsh2014] Now that experimental data on the photodesorption of H$_2$CO ices are available, a precise study could be performed, with the determination of the relative efficiency of each non-thermal process.
In protoplanetary disks such as TW Hydra, the UV flux varies from 10$^3$ to 10$^9$ photons cm$^{-2}$ s$^{-1}$ at different locations in the disk, approximately where ices exist. With time spanning from 1 to 10 million years, this gives fluences of $3 \times 10^{16} - 3 \times 10^{17}$ photons cm$^{-2}$ for the areas which receive less photons, to $3 \times 10^{22} - 3 \times 10^{23}$ photons cm$^{-2}$ for the areas which are strongly illuminated. Our experimental fluence reproduces the low-illuminated regions of disks, whereas in the strongly-illuminated regions, photodesorption yields could differ if the ice composition is strongly altered.
As already mentioned, H$_2$CO is connected with CH$_3$OH, which has been recently detected in PPDs.[@walsh2016] Given all the experimental and modelling work performed on CH$_3$OH, discussing some findings on CH$_3$OH photodesorption can directly inspire what could be done with H$_2$CO:
\(i) Varying the CH$_3$OH photodesorption yield [@oberg2009b; @bertin2016; @cruz-diaz2016] has drastic consequences on the abundance of both solid and gas phase CH$_3$OH by orders of magnitude, in some parts of the disk. [@walsh2017; @ligterink2018] It should be noted that the experimental CH$_3$OH photodesorption yield of $10^{-5}$ molecule/photon [@bertin2016; @cruz-diaz2016] is low, but it is *not negligible* as it is high enough to have an effect on the gas replenishment.[@walsh2017; @ligterink2018]
\(ii) The comparison between UV photodesorption and chemical desorption of CH$_3$OH showed that photodesorption largely dominates chemical desorption in the majority of the disk.[@ligterink2018]
\(iii) Photodesorption do shape snowlines differently than chemical desorption, as it gives CH$_3$OH snowlines deeper in the disk. Therefore photodesorption is important in setting the location of snowlines, especially for non-volatile species such as H$_2$O, CH$_3$OH [@ligterink2018; @agundez2018] and probably H$_2$CO (the H$_2$CO snowline is located closer to the star than the CO snowline,[@qi2013] as its formation on grains only requires that the CO molecule spends some time on the surface of grains [@aikawa1999; @walsh2014] and as its binding energy is larger than the CO one).
\(iv) Interestingly, taking into account the desorption of CO fragments from CH$_3$OH ice has diverse impacts: it changes the abundance of gas phase CO in some parts of the disk and the CH$_3$OH snowline location.[@walsh2017; @ligterink2018] Indeed, when no fragmentation is considered, CH$_3$OH re-adsorption competes with photodesorption, whereas when CH$_3$OH photofragmentation is added, CH$_3$OH has to reform on the grains, and that changes the location of the snowline.
\(v) Gas phase CH$_3$OH shows a spatial coincidence with H$_2$CO [@walsh2016; @oberg2017] in the T Tauri disk TW-Hydra. For this T Tauri, the methanol-to-formaldehyde ratio is 1.27,[@walsh2016] but it is much smaller (0.24) for a PPD around a Herbig star.[@carney2019] One possible explanation for these unequal ratios is that the photodesorption of CH$_3$OH and H$_2$CO is different in these two disks, as the spectral distribution and intensity of the stellar radiation differ between T Tauri and Herbig stars.[@carney2019] With wavelength-resolved photodesorption spectra now available, detailed chemical modelling of these two disks can be performed, and the effects of photodesorption characterized.
In the light of all the experimental, observational and modelling work that has been done on CH$_3$OH, experimental data on H$_2$CO ice could be implemented or updated. The implementation of H$_2$CO photodesorption yields and of the branching ratio, giving CO fragments, both larger than in the CH$_3$OH ice case (Table \[table\_yields\] and Ref. ) could give useful information: (i) on gas/solid abundances of H$_2$CO and on the gas abundance of CO; (ii) on the solid phase abundance of H$_2$CO as it was shown that it drops drastically when UV photodesorption is included in the upper layers of the disks;[@walsh2014] besides, any change in the abundance of the ’building block’ H$_2$CO will certainly have consequences on other solid phase abundances, such as that of CH$_3$OH and of COMs (iii) on the H$_2$CO snowline location, as photodesorption is important for the location of snowlines especially in the outer disk for species with large binding energies.
Conclusions
===========
A quantitative study of the photodesorption from pure and H$_2$CO-containing ice was performed. H$_2$CO photodesorption spectrum presents dissociative electronic states and the photodesorption of CO and H$_2$ fragments was also measured. Photodesorption mechanisms were constrained, including (indirect) desorption induced by electronic transitions and photochemical desorption.
From the energy-resolved photodesorption yields, we could derive H$_2$CO and CO average photodesorption yields in several astrophysical environments such as the ISRF, PDR, dense cloud or PPD, and found that these yields slightly vary from one environment to the other. These yields can be directly added to astrochemical models, without any branching ratio correction.
This study also confirms that photodesorption yields strongly differ from one molecule to the other, for example from CO to H$_2$CO and to CH$_3$OH and also that they depend on the ice composition. Laboratory experiments on ice analogs are a necessary step, as it is not possible yet to predict photodesorption efficiencies. Elements are gathered to investigate further the H$_2$CO non-thermal desorption in dense cores, PDR and disks, to tune more finely gas phase and solid phase abundances, and possibly snowline locations. Its impact on the gas-to-ice balance has to continue to be explored, for H$_2$CO and all related species, spanning from CO to COMs.
The authors thank SOLEIL for provision of synchrotron radiation facilities under the project 20150760 and also Laurent Nahon and the DESIRS beamline for their help. This work was supported by the Programme National ’Physique et Chimie du Milieu Interstellaire’ (PCMI) of CNRS/INSU with INC/INP co-funded by CEA and CNES. Financial support from LabEx MiChem, part of the French state funds managed by the ANR within the investissements d’avenir programme under reference ANR-11-10EX-0004-02, and by the Ile-de-France region DIM ACAV programme, is gratefully acknowledged.
Available free of charge on the ACS Publications website at DOI: 10.1021/acsearthspacechem.9b00057
Flux measurement correction\
Ageing effects\
Extrapolation of the photodesorption spectra to lower energies\
Illustration of the CO contribution in the H$_2$CO/CO layered ice\
[^1]: We have grown three different ices, in the following order : (1) 0.4 ML of H$_2$CO on CO, (2) 1.1 ML of H$_2$CO on CO, and (3) 2.6 ML of H$_2$CO on CO; to obtain the second ice, we added 0.7 ML of H$_2$CO on the first one, and to obtain the third one, we added 1.5 ML to the second one
[^2]: For this estimation, we equalled the photodesorption rate with the thermal desorption rate. For UV photodesorption, we considered a UV flux of $10^{10}$ photons cm$^{-2}$ s$^{-1}$. For thermal desorption, we used a first order kinetics with a prefactor of $10^{28}$ mol. cm$^{-2}$ s$^{-1}$ and an adsorption energy of 3765 K from @noble2012.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We consider an environmentally dependent violation of Lorentz invariance in scalar-tensor models of modified gravity where General Relativity is retrieved locally thanks to a screening mechanism. We find that fermions have a modified dispersion relation and would go faster than light in an anisotropic and space-dependent way along the scalar field lines of force. Phenomenologically, these models are tightly restricted by the amount of Cerenkov radiation emitted by the superluminal particles, a constraint which is only satisfied by chameleons. Measuring the speed of neutrinos emitted radially from the surface of the earth and observed on the other side of the earth would probe the scalar field profile of modified gravity models in dense environments. We argue that the test of the equivalence principle provided by the Lunar ranging experiment implies that a deviation from the speed of light, for natural values of the coupling scale between the scalar field and fermions, would be below detectable levels, unless gravity is modified by camouflaged chameleons where the field normalisation is environmentally dependent.'
author:
- Philippe Brax
title: Lorentz Invariance Violation in Modified Gravity
---
The discovery of the acceleration of the Universe has led to a flurry of scenarios involving scalar fields and leading to different types of modified gravity models [@Clifton:2011jh]. All of them allow for large deviations from General Relativity on astrophysical scales while preserving Newton’s law locally in the solar system and in laboratories on earth. This is achieved thanks to screening features such as the Vainshtein mechanism for theories with higher order derivative self interactions[@vain], or the chameleon[@chameleon], symmetron[@symmetron] or Damour-Polyakov properties[@dp1994] for theories with non-linear effective potentials in the presence of pressure-less matter. Recently and after the claim of super-luminal propagation of neutrinos by the OPERA experiment[@opera], it has been suggested that fermions may travel faster than the speed of light in dense environments where the presence of matter offers a breaking of Lorentz invariance[@dvali]. This was further pursued in [@Kehagias:2011cb; @Saridakis:2011eq; @Hebecker:2011yh] and then in [@Ciu] where Galileons were used to describe the OPERA claims although failing to respect the tight bounds on the deviation of the electron speed from the speed of light. In this work, we will describe a general mechanism which leads to anomalously fast fermions and where the breaking of Lorentz invariance by a two-tensor appears naturally in models of modified gravity with screening properties. Chameleon, symmetron and Galileon models can reproduce phenomena of the type indicated by the preliminary OPERA publication. Only models with a thin shell screening mechanism, i.e. chameleons, are compatible with the constraints on the speed of charged leptons. Unfortunately for chameleons, Galileons and symmetrons and for natural values of the coupling between the scalar field and fermions we find that the deviation of the fermion speed from the speed of light would be unobservably small. More generally and model independently, testing the properties of neutrinos in matter probes the scalar field profile of modified gravity models and gives us an unprecedented opportunity to analyse the properties of modified gravity in dense environments. We argue that the tests of the equivalence principle such as the Lunar Ranging experiment imply that no deviation from the speed of light due to modified gravity should be observable in the neutrino sector of the standard model unless the scalar field is a camouflaged chameleon. Camouflaged chameleons are such the field normalisation is environmentally dependent and the coupling between the scalar field and fermions becomes smaller in dense media than in vacuum.
We consider the action governing the dynamics of a scalar field $\phi$ in a scalar-tensor theory of the general form $$\begin{aligned}
S &=& \int d^4x\sqrt{-g}\left\{\frac{m_{\rm Pl}^2}{2}{
R}-F(\phi,\partial_\mu\phi,\partial_\mu\partial_\nu \phi)- V(\phi)\right\}\nonumber\\
&& + \sum_i S_m^i (\psi_m^{(i,j)},\tilde g^i_{\mu\nu})\,, \label{action}\end{aligned}$$ where $$S_m^i (\psi_m^{(i,j)},\tilde g^i_{\mu\nu})=\int d^4x \sqrt{-\tilde g_i} {\cal
L}^i_m(\psi_m^{(i,j)},\tilde g^i_{\mu\nu})$$ is the action in the ith sector of the model, $g$ is the determinant of the metric $g_{\mu\nu}$, ${ R}$ is the Ricci scalar and $\psi_m^{(i,j)}$ are various matter fields labeled by $j$ interacting with the metric $\tilde g_{\mu\nu}^i$ in the Lagrangian ${\cal L}^i_m$. When $F(\phi,\partial_\mu\phi,\partial_\mu\partial_\nu \phi)=\frac{1}{2}(\partial\phi)^2$, the field is canonically normalised. More complex functions $F(\partial_\mu\phi,\partial_\mu\partial_\nu \phi)$ appear in the Galileon scenario for instance [@galileon]. A key ingredient of the model is the coupling of $\phi$ with matter particles. More precisely, the excitations of each matter field $\psi_m^{(i,j)}$ couple to a metric $\tilde g_{\mu\nu}^i$ which is related to the Einstein-frame metric $g_{\mu\nu}$ by $$\tilde g_{\mu\nu}^i=A^2(\phi)g_{\mu\nu}^i$$ where $$g_{\mu\nu}^i=g_{\mu\nu}+\frac{2\partial_\mu \phi\partial_\nu\phi}{M_i^4}$$ depends on each sector of the theory. For instance, the fermion kinetic terms may couple to a different metric from the gauge kinetic terms.
In the following, we will assume that the bi-metric term $\frac{\partial_\mu \phi \partial_\nu\phi }{M_i^4}$ is a small correction to $g_{\mu\nu}$. In the context of extra dimensional models where matter lives on a brane, the scalar field can be seen as parameterising the normal to the brane, the coupling function $A(\phi)$ arises from the warping of the bulk metric while the bilinear term reflects the coupling of matter to the induced metric on the brane. Defining the energy momentum tensor in each sector as $
T^{\mu\nu}_i=\frac{2}{\sqrt{-g}} \frac{\delta S_m^i}{\delta g_{\mu\nu}}
$ and expanding the action to linear order, we find that the scalar field couples derivatively to matter $$\begin{aligned}
&&S = \int d^4x\sqrt{-g}\left\{\frac{m_{\rm Pl}^2}{2}{
R}-F(\partial_\mu\phi,\partial_\mu\partial_\nu\phi)- V(\phi) \right\}\nonumber\\
&& + \sum_i \int d^4x\sqrt{-g} \frac{\partial_\mu \phi\partial_\nu \phi}{ M_i^4} T_i^{\mu\nu}\nonumber \\ && + \sum_i \int d^4x \sqrt{- g}A^4(\phi) {\cal
L}_m(\psi_m^{(i,j)},A^2 (\phi) g_{\mu\nu})\,, \label{action}\end{aligned}$$ As soon as $\theta_{\mu\nu}=\partial_\mu\phi\partial_\nu\phi$ does not vanish due to the presence of matter, we find that Lorentz invariance is broken and a new Lorentz invariance violating coupling to the matter energy momentum tensors is present in the model[@Kehagias:2011cb; @Hebecker:2011yh].
Massless fermions with the action $
S_F=- \int d^4x \sqrt{-g} \frac{i}{2}(\bar \psi \gamma^\mu D_\mu \psi - (D_\mu \bar \psi) \gamma^\mu \psi)
$ have the energy momentum tensor $
T^F_{\mu\nu}= -\frac{i}{2} (\bar \psi \gamma_{(\mu} D_{\nu)} \psi - (D_{(\mu }\bar \psi) \gamma_{\nu)} \psi)
$ symmetrised over the indices. This induces the following interaction terms with the scalar field $$-\frac{i}{2}\int d^4 x \sqrt{-g} \frac{\partial^\mu \phi \partial^\nu\phi}{M_\psi^4} (\bar \psi \gamma_{(\mu} D_{\nu)} \psi - (D_{(\mu }\bar \psi) \gamma_{\nu)} \psi)$$ where the mass scale $M_\psi$ is the suppression scale for the fermion species $\psi$. A similar coupling was first considered in [@gauthier].
Let us focus on a typical situation where the metric is Minkowskian to a good approximation (this is the case on earth where Newton’s potential is $\Phi_\oplus \sim 10^{-9}$) and the scalar field varies on scales much larger than the Compton wave length of the fermions. Moreover, let us assume that the scalar field is static. Then the interaction term reduces to $$-\frac{i}{2}\int d^4 x \sqrt{-g} \ d^i d^j (\bar \psi \gamma_{i} \partial_j \psi- \partial_i \bar \psi \gamma_j \psi)$$ where $
d^i= \frac{\partial^i\phi}{M_\psi^2}
$ is a slowly varying function of space only. Hence a static configuration of the scalar field yields a Lorentz violating interaction in the Fermion Lagrangian. The resulting Dirac equation becomes $$i(-\gamma^0\partial_0 +\gamma^i \partial_i +d^i d^j \gamma_i \partial_j)\psi=0.$$ The dispersion relation is obtained by squaring the modified Dirac operator to obtain $$p_0^2 = (c^2)^{ia}p_i p_a.$$ This becomes the dispersion relation in an anisotropic medium with a square velocity tensor $$(c^2)^{ia}=(\delta^{ij}+ d^i d^j)(\delta^{aj} + d^a d^j).$$ The eigenmodes of the velocity tensor are $d^i$ and two vectors $e^i_\lambda, \lambda=1,2$ orthogonal to $d^i$. The eigenspeeds are $c_d=(1+\vert d\vert^2)$ and twice $c_\lambda=1$. Hence, fermions go faster than light in the direction of the gradient $\partial^i \phi$, i.e. along the scalar lines of force, with $$\Delta c\equiv c_d-1= \vert d\vert^2.$$ An interesting application concerns fermions, typically neutrinos with no electromagnetic interactions, produced at the surface of a spherical body, traversing the sphere on a straight line parameterised by an angle $\theta$ varying between $-\theta_{\rm max}$ and $\theta_{\rm max}$. We assume that $d^i$ is radial. Along this line, the fermion speed is $
v(\theta)=1 + \vert d\vert^2 \theta^2
$ implying that fermions would be observed earlier compared to a propagation with the speed of light by $$\frac{\Delta t}{t}= \frac{1}{3} \vert d\vert^2 \theta_{\rm max}^2.$$ A time difference of $\frac{\Delta t}{t}=2.5\ 10^{-5}$, as claimed by the OPERA experiment where $\theta_{\rm max}\sim 0.06$, would require $\vert d\vert \sim 0.15$. In the following we will use such a value as a template and unravel its physical consequences. If the OPERA claims turned out to be spurious, the mechanisms described here would still be valid and lower bounds on the scale $M_\psi$ would ensue. These bounds would be easily extracted from the formalism discussed here.
When the scalar field is canonically normalised, the Klein Gordon equation is modified due to the coupling of the scalar field $\phi$ to matter: $$\Box \phi- \sum_i \frac{2}{M_i^4} D_\mu (\partial_\nu \phi T_i^{\mu\nu})= -\beta \sum_i T_i + \frac{dV}{d\phi},$$ where $T_i$ is the trace of the energy momentum tensor $T_i^{\mu\nu}$ and the coupling of $\phi$ to matter is defined by $
\beta\equiv m_{\rm Pl}\frac{d\ln A}{d \phi}.
$ In static situations where matter is pressure-less and space-time is assumed to be Minkowskian, the Klein-Gordon equation reduces to $$\Delta \phi = -\beta T + \frac{dV}{d\phi}$$ corresponding to the case with no derivative coupling in the Lagrangian.
In a dense environment, the scalar field acquires a non-trivial profile due to the matter dependent source term. We will be interested in spherical situations corresponding to dense astronomical or astrophysical objets such as the earth or the sun. In the superluminal context, this was first considered in [@Kehagias:2011cb] where the role of scalar fields coupled to fermions was emphasized. In the absence of a potential term and assuming that the coupling to matter is also determined by the scale $M_\psi$, i.e. we have $\beta=\frac{m_{\rm Pl}}{M_\psi}$, a time difference of order $\frac{\Delta t}{t}=2.5\ 10^{-5}$ can be explained if $M_\psi\approx 1\ {\rm TeV}$. This scale is large enough to argue that the coupling of the scalar field to matter may come from integrating out heavy fields at a scale which lies beyond the standard model of particle physics. Unfortunately, it leads to a very large value of the coupling to matter $\beta$ which would be ruled out by experimental tests of Newton’s law. This can be avoided in modified gravity models where the effects of the scalar field are screened in local tests of gravity. This is the type of models we investigate in this paper. We will first focus on a large class of known models, i.e. chameleons, Galileons and symmetrons. We will then deal with more model independent considerations and eventually introduce camouflaged chameleons where chameleon fields have an environmentally dependent normalisation.
Large deviations from Newton’s law are prevented in chameleon-like theories[@chameleon] where the effective $
V_{\rm eff}(\phi) = V(\phi) +\rho_m A(\phi)
$ acquires an environment dependent minimum where the mass is large enough to Yukawa screen the fifth force deep in the body, leaving only a thin shell of size $\Delta R$ over which the field varies significantly. Here we have defined $\rho_m$ as the conserved matter density $T=-A(\phi)\rho_m$. In summary, for radial distances $r\le R_s$ where $R_s$ is the radius of the shell, the solution is constant $
\phi=\phi_c, \ \ r\le R_s
$ where $\phi_c$ is the minimum of the effective potential inside the dense body and we assume $A(\phi)\approx 1$. In the thin shell, the field varies according to $
\phi=\phi_\infty -\frac{\beta \rho_m R^2}{2 m_{\rm Pl}} +\frac{\beta \rho_m R_s^3}{3 m_{\rm Pl} r} + \frac{\beta\rho_m r^2}{6m_{\rm Pl}}.
$ Outside the body, the fifth force is suppressed by a factor $
\frac{3\Delta R}{R} \equiv \frac{ \vert \phi_\infty -\phi_c\vert}{2\beta m_{\rm Pl}\Phi_N}
$ where $\Phi_N$ is the Newton potential generated by the body at its surface and $\Delta R= R-R_s$. A thin shell exists when $3 \Delta R / R \lesssim 1$. Solar system tests of gravity like the Lunar Ranging experiment require that the thin shell on earth should be such that $\beta \frac{\Delta R_\oplus}{R_\oplus}\lesssim 10^{-7}$ [@chameleon].
Inside a dense body, the scalar field has a non-vanishing gradient in a thin shell. More precisely we find that the gradient of the scalar field is radial $
\partial^i \phi =\frac{d\phi}{dr} \frac{x^i}{r}
$ where $
\frac{d\phi}{dr}= -\frac{\beta \rho_m R_s^3}{3 m_{\rm Pl} r^2} + \frac{\beta\rho_m r}{3m_{\rm Pl}}, R_s \le r\le R,
$ vanishes for $r\le R_s$ and can be approximated by $
\frac{d\phi}{dr}= \frac{\beta \rho_m }{ m_{\rm Pl} }(r-R_s), R_s\le r\le R,
$ which is maximal at the surface of the body and proportional to $\Delta R$. As an example, let us consider fermions starting from the surface of the dense body and traversing the body at a small angle $\theta$ from the horizontal which starts at $-\theta_{\rm max}$ and finishes at $\theta_{\rm max}$. Define by $\theta_{\rm min}$ the angle when the fermions leave the thin shell. We have $
\frac{d\phi}{dr}= \frac{\beta \rho_m R}{ 2 m_{\rm pl}} (\theta^2 -\theta_{\rm min}^2)
$ for small angles and $
\Delta \theta= \frac{\Delta R}{R\theta_{\rm max}}
$ with $\Delta \theta= \theta_{\rm max} -\theta_{\rm min}$. Along this trajectory the speed of the fermions is $
v(\theta)= 1+ \vert d\vert^2 \theta^2
$ corresponding to the difference of time of arrival between fermions and photons $
\Delta t= R\int_{\theta_{\rm min}}^{\theta_{\rm max}} (1-\frac{d\theta}{1+\vert d^2 \vert \theta^2}).
$ Denoting by $t= R\theta_{\rm max}$ the time photons take to go from $\theta_{\rm max}\sim \theta_{\rm min}$ to $\theta=0$, we find that $$\frac{\Delta t}{t}= \frac{12 \Phi_N^2 m_{\rm Pl}^2}{\beta R^2 M_\psi^4} (\beta \frac{\Delta R}{R})^3.
\label{tt}$$ As the thin shell is very small, this time difference can reproduce $\frac{\Delta t}{t}\sim 2.5 \ 10^{-5}$ with a low suppression scale $M_\psi \lesssim 10^{-1}$ eV. The smallness of this scale can be understood as resulting from the large suppression of the scalar field fifth force by the thin-shell mechanism. As it stands, such a low value of $M_\psi$ appears to be much too low to be justifiable in an effective field theory context where we understand the origin of the coupling between the scalar field and fermions as a result of integrating out heavy fields. This points out that a result of the OPERA type would not be explained in a natural way by chameleon models. We will introduce camouflaged chameleons to deal with this issue in a positive way.
Some models of modified gravity can be compatible with gravitational tests while describing scalar fields as being unscreened on earth. This is the case of the symmetron where the potential and the coupling functions are $
V(\phi)= -\frac{\mu^2}{2} \phi^2 +\frac{\lambda}{4}\phi^4,\ A(\phi)= 1 + \frac{\phi^2}{2M_G^2}.
$ For a low energy density, the field is stabilised at a value $\phi_\star=\frac{\mu}{\sqrt{\lambda}}$ whilst for a large energy density $\rho\ge \rho_\star=\mu^2 M_G^2$, the minimum of the effective potential is at the origin. The solution inside an unscreened body reads $
\phi(r)= A\frac{R}{r} \sinh \frac{\sqrt{\rho_m}}{M_G} r
$ where $A\sim \frac{M_G^3}{m_{\rm Pl}^2} \frac{1}{\sqrt{6\Phi_N}}$ with $M_G\le 10^{-3} m_{\rm Pl}$ to comply with solar system tests. In particular we have $
\frac{d\phi}{dr}\vert_{r=R}= {\cal O}(\frac{A}{R})
$. We will focus on the upper value of $M_G$ which corresponds to $\phi_\star \sim 10^{-6} m_{\rm Pl}$. A contribution of order $\vert d\vert \sim 0.15$ could be explained by these models provided $M_\psi^2 \sim 10 A/R$, or equivalently $M_\psi\sim 7\cdot 10^{-5}$ GeV. Again, such a low value appears to be too low to be justifiable from an effective field theory point of view.
Galileon models [@galileon] depend on four unknown parameters and involve three non-canonical contributions to the kinetic terms. We consider the case with $c_{4,5}=0$ for simplicity. Inside the Vainshtein radius expressed as $
R_\star= (4 c_3 \Phi_N \frac{R}{m_{\rm Pl}^2})^{1/3}
$ the Galileon profile due to a spherical mass $M$ reads $
\frac{d\phi}{dr}= \frac{M}{2m_{\rm Pl}r^2}(\frac{r}{R_\star})^{3/2}
$ where we have normalised the Galileon choosing $c_2=m_{\rm Pl}^2$. In this case, we find that the gradient of the scalar field is $
\frac{d\phi}{dr}\vert_R= m_{\rm Pl}^2 (\frac{\Phi_N}{4c_3})^{1/3}.
$ Taking $c_3 \sim 10^{120}$[@Burrage:2010rs; @brax2] to satisfy the Lunar Ranging tests, we find that $d\sim 0.15$ could be due to a galileon coupled to matter provided $
M_\psi\sim 1.3\ {\rm MeV}.
$ Although very low, such a value may be more justifiable in the Galileon context as non-renormalisation theorems exist [@Hui:2010dn]. Moreover, the Galileon normalisation varies with the environment and the coupling scale becomes much larger close to dense media where the coupling scales to matter, viewed as the cut-off of the theory, become of natural values.
If the coupling to $g^i_{\mu\nu}$ respects gauge invariance and if flavour effects are taken into account [@cohen; @strumia], all neutrinos of the standard model should couple with the same scale $M_\psi$. This would also entail that electrons and muons would have the same speed as the neutrinos at the classical level inside the earth. In the atmosphere there is a drastic difference between models with a thin shell effect like chameleons for which $\phi \sim \phi_{\rm atm}$ sits at the minimum of the effective potential [@chameleon] and induces no change in the speed of fermions at all and models with no thin shell like symmetrons and Galileons where $d$ is nearly constant in the atmosphere. The latter would strongly violate the bounds on the deviation from the speed of light for electrons as they are in the $10^{-15}$ range[@strumia]. One possibility for these models would be to have an environmentally dependent violation of gauge invariance with $M_\nu\ne M_f$ for $f=e,\mu,\tau$. A large value of $M_f\ge 10^4 M_\nu$ would be enough to satisfy the experimental bounds. In vacuum where the scalar field has no gradient, gauge invariance would be retrieved. Unfortunately, the resulting large difference between the neutrino and the electron speeds inside the earth would lead to too much Cerenkov radiation and is therefore excluded[@cohen].
The only viable models are the chameleon ones where $d$ vanishes in the atmosphere and in the vacuum pipes of particle experiments, implying no deviation between the fermion speed and the speed of light there. Moreover, the fact that neutrinos are obtained from pion decay in vacuum where the neutrino speed is the speed of light implies that no bound on the resulting neutrino energy applies [@Bi:2011nd]. Similarly, in matter such as inside the earth, as long as all fermions couple to the same metric with the same $M_\psi$, no $e^+e^-$ Cerenkov radiation is induced. For chameleon models, Cerenkov photon radiation $\nu\to \nu +\gamma$ happens in the thin shell only and leads to an energy loss - ()\^7 where $k=25/448$ which can be a tiny effect if the size of the thin shell on earth is extremely small.
So far we have taken $\frac{\Delta t}{t}\sim 2.5 \ 10^{-5}$. If the OPERA claims were not confirmed and $\frac{\Delta t}{t}$ happened to be smaller but still greater than the bound on the deviation of the speed of charged leptons from the speed of light, i.e. $\frac{\Delta t}{t} \gtrsim 10^{-15}$, modified gravity models with no thin shell such as Galileons and symmetrons would not provide an explanation for such a phenomenon. Indeed, the deviation from the speed of light for neutrinos and charged leptons would be the same both inside the earth and in the atmosphere, implying a direct violation of the bound on the deviation of the speed of charged leptons from the speed of light. On the other hand, chameleons would still be able to reproduce a value of $\frac{\Delta t}{t}\gtrsim 10^{-15}$ with no violation of the bound on the speed of charged leptons deduced from experiments in particle accelerator vacua and the earth’s atmosphere. Indeed, the gradient of the chameleon field vanishes inside these environments. As a result, this would simply imply that $M_\psi$ would be larger than a few eV’s. Eventually, if experiments concluded that no significant deviation from the speed of light for neutrinos and charged leptons could be detected, this could simply mean that $M_\psi$ is large enough to lead to no experimentally detectable effects. Moreover, it is to be expected that $M_\psi$ should be larger than the electroweak scale if the coupling between matter and the scalar field results from the dynamics of theories beyond the standard model. In this case, the time difference $\frac{\Delta t}{t}$ would be unobservably low. We will see that this negative conclusion can be avoided if the scalar field is a camouflaged chameleon.
In the chameleon, symmetron and Galileon cases, we have found that a compatibility with $\frac{\Delta t}{t}\sim 2.5 \ 10^{-5}$ could be reached for low values of $M_\psi$. As already argued, these scales are unnaturally low. In order to assess more quantitatively how unnatural these scales are, we now analyse the large effects they may have in particle physics experiments such as the ones at LEP, i.e. increasing the width of the Z boson or modifying the electroweak precision tests. To do so, we focus on the coupling of the scalar field to the gauge fields of the standard model and we assume a low scale $M_F\sim M_\psi$. Although a detailed study is beyond the present work, a simplified analysis can be carried out. One can evaluate the order of magnitude of such effects by reducing the bilinear coupling of the scalar field to W and Z bosons, to a linear coupling. First of all notice that the energy momentum of gauge fields involves $
T^F_{\mu\nu} \supset \frac{1}{4} g_{\mu\nu} F^2
$ leading to the effective coupling between the scalar field and the gauge bosons $
{\cal L}_I=-\frac{1}{4M_F^4} (\partial \phi)^2 F^2.
$ In the vacuum where particle physics experiments take place, in the chameleon and symmetron cases, and expanding $\phi=\phi_{\rm vac} + \delta\phi$, this leads to the operator, after one integration by parts, $
{\cal L}_I \supset \frac{\phi_{\rm vac}}{4M_F^4} \partial^2 \delta \phi F^2.
$ The gauge boson vacuum polarisation diagrams receive contributions from scalar loops. The effect of these loop is to induce potentially divergent contributions to the precision parameters $S$, $T$, etc …. Fortunately, at high momentum the electroweak symmetry breaking is irrelevant, implying a cancellation of the UV divergences. This is the essence of the screening theorem for scalars. As a result, only momenta up to the breaking scale $M_Z$ are relevant. This implies that we can replace the previous operator by $
{\cal L}_I \sim \frac{\phi_{\rm vac} M^2_Z}{4 M_F^4} \delta \phi F^2.
$ The effect of such a vertex was studied in [@part] where the suppressions scale $
\hat M_F = \frac{4M_F^4}{\phi_{\rm vac} M_Z^2}
$ was constrained to be $\hat M_F \ge 1 \ {\rm TeV}$. Here we find that $\hat M_F \sim 10^{-35} {\rm GeV}$ for $M_F\sim M_\psi$ in the symmetron case where $\phi_{\rm vac}=\phi_\star$. Of course this is a situation which is strongly excluded as the effects on the precision tests of the standard model would be dramatic. On the other hand, if we take into account the bound $\hat M_F \ge 1$ TeV we find that the precision test bound is satisfied provided $M_F \ge 14 \ {\rm TeV}$. Assuming that the couplings of the symmetron to both matter and gauge fields have a similar origin, this would certainly indicate that $M_\psi$ should be larger than a few TeV’s.
For chameleons[@chameleon], $\phi_{\rm vac}\le 10^{-28} m_{\rm Pl}$, implying that $ M_F \ge 0.15\ {\rm GeV}$. Hence a value of $M_F$ slightly larger than the electroweak scale would be compatible with the standard model precision tests. In the Galileon case, the Lorentz invariant breaking background leads to the operator $
{\cal L}_I= -\frac{dM_\psi^2}{4M_F^4}{\partial_r \delta \phi} F^2
$ where $d\sim 0.15$, which would lead to the same effect in precision tests as $
{\cal L}_I \sim -\frac{dM_Z M_\psi^2}{4 M_F^4} \delta \phi F^2
$ corresponding to a scale $
\hat M_F= \frac{4M^4_F}{d M_Z M_\psi^2}
$ which is $\hat M_F \sim 10^{-6} {\rm GeV}$ when $M_F\sim M_\psi$, a value which is much too small. The precision test bound is satisfied provided $M_F\ge 100\ {\rm MeV}$.
All in all, we have found that if the coupling to matter and to photons have similar origins and share similar scales $M_F\sim M_\psi$, the precision tests of the standard model exclude the low values of the coupling scale $M_\psi$ which would be necessary to reproduce a time difference $\frac{\Delta t}{t}=2.5\ 10^{-5}$ for chameleon models. On the other hand, nothing seems to exclude values of $M_\psi\sim M_F$ slightly beyond the standard model. The coupling to photons with such a strength would certainly lead to strong effects in astrophysics and the laboratory along the lines of [@Brax:2007hi]. Work on this topic is in progress[@us].
Having so far focused on known models of modified gravity, i.e. chameleons, Galileons and symmetrons, let us now come back to model independent features. If scalar fields modifying the laws of gravity exist, then their profile in the presence of matter breaks Lorentz invariance. This Lorentz invariance violation could then be transmitted to the neutrino sector of the standard model as we have seen in (\[action\]). In these models, as the variation of the neutrino speed follows the scalar lines of force, the deviation from the speed of light would be maximal for neutrinos emitted radially towards the centre of the earth and detected in a laboratory symmetrical from the emission point. In this case, the time difference would be = \_0\^R dr d\^2 (r). which depends on the entire profile of the scalar field inside the earth. For a generic modified gravity model, gravity tests apply outside dense objects where constraints on the scalar field gradient are tight. Inside a dense object, no such constraints are known. It is simply expected that the scalar field gradient will be screened compared to the unscreened case. In the unscreened case, we have essentially = r which gives an upper bound for the time difference \^2 \_N\^2 . For $\beta={\cal O}(1)$, and $M_\psi={\cal O}(1)$ TeV, this upper bounds implies that $\frac{\Delta t}{t}\lesssim 10^{-38}$. Hence we do not expect that any time difference may ever be detected if the bare coupling $\beta$ is of order one. A larger time difference may be obtained if the modified gravity models have a large bare coupling $\beta=\frac{m_{\rm Pl}}{M_\psi}$ corresponding to a universal suppression scale $M_\psi$ for both the derivative and non-derivative couplings of the scalar field $\phi$ to matter. In this case the upper bound becomes \_N\^2 which is of order $2\cdot 10^{-5}$ for the natural scale $M_\psi^2 \sim 10^5\ {\rm GeV}^2$ corresponding to the results in [@Kehagias:2011cb]. For this value of the coupling, a deviation from the speed of light greater than the one in the charged lepton sector, i.e. $\frac{\Delta t}{t}\gtrsim 10^{-15}$, can only be achieved provided the screening factor[^1] s(r) = is in the range $10^{-5}\lesssim s(r) \lesssim 1$. Hence we have found that deviations from the speed of light could be present for neutrinos going through the earth if gravity were modified in a dense environment and two conditions were met. The first one is that the coupling to matter $\beta$ must be large, a situation which was investigated for chameleon models in [@Mota:2006ed] for instance. Secondly, the screening of the scalar field profile inside the earth cannot be too large, certainly not at a level below $10^{-5}$.
Of course, a model of modified gravity satisfying these requirements would also have to explain the absence of modified gravity effects in the Lunar ranging experiment at the $\eta_\oplus= 10^{-13}$ level [@Williams:2012nc]. This requires to know the profile of the scalar field outside the dense object. Assuming that the scalar field behaves like a free field close to the body, we find 2 m\_[Pl]{} \_N s(R) . Notice that when the field is extremely screened inside the dense body $s(R)\approx 0$ implying that the field is nearly constant outside the body and therefore reducing the amount of Cerenkov radiation there. The coupling between dense objects and the scalar field is of order $\beta s(R)$ implying that the acceleration difference between the moon and the earth in the gravitational field of the sun is of order[@chameleon] \_(s(R))\^2. The Lunar Ranging experiment leads to $\beta s(R) \lesssim 10^{-7}$. For large values of $\beta$, this essentially rules out the possibility of having $10^{-5}\lesssim s(R)\lesssim 1$ and therefore the order of magnitude estimate \_N\^2 is unobservably small for natural values of $M_\psi$. Of course, we have assumed here that the field is almost free outside the body. This is not the case for models of the Galileon type where the field profile is more complex. Nonetheless it seems difficult to reconcile these models with the Cerenkov constraints. This negative result can be altered if the field normalisation becomes environmentally dependent. This is one of the features of Galileon models where the coupling scales to matter for the canonically normalised Galileon become larger around dense media than in vacuum. This mechanism can be transposed to the chameleon models, enabling one to tackle the naturality of the scale $M_\psi$.
Let us extend chameleon models by replacing the canonical kinetic terms by $-\frac{f(\phi)}{2} (\partial \phi)^2$ where $f(\phi)$ is a smooth function. The Klein-Gordon becomes = - ()\^2. Vacuum configurations of chameleon models in a medium of density $\rho_m$ are not modified by the change of normalisation of $\phi$ and are still minima of the effective potential $V_{\rm eff}$. Denoting by $\phi_c$ the minimum in a medium of density $\rho_c$, the Lagrangian can be linearised using $\phi= \phi_c +\delta \phi$, and the canonically normalised excitation is $\Phi= \sqrt{f(\phi_c)} \delta\phi$. The derivative coupling of this field to matter becomes \_i d\^4x T\_i\^ corresponding to the environmentally dependent coupling scale M\_i(\_c)= f\^[1/4]{}(\_c)M\_i while the non-derivative coupling becomes \_i(\_c)= Let us now study how this change of normalisation affects the thin-shell mechanism and the speed of fermions.
Consider a spherical object of radius $R$ and density $\rho_c$ embedded in a vacuum region of density $\rho_\infty$. Let us assume that this object has a thin shell. For $r\le R_s$, the field is still constant $\phi=\phi_c$. In the thin-shell, we have corresponding to $\frac{d\phi}{dr}\approx \frac{\beta \rho_m}{f(\phi_c) m_{\rm Pl}} (r-R_s)$ in the shell. Outside the body we have = \_+ implying that the size of the shell is = f(\_c) where a factor $f(\phi_c)$ has now appeared compared to the usual chameleon result.
Particles with no thin shell evolve in the background of a spherical object by following the trajectories defined by the modified Newtonian potential $\Psi = \frac{\Phi_N R}{r} + \beta \frac{\phi}{m_{\rm Pl}}$ where $ \frac{\Phi_N R}{r}$ is the newtonian potential at the distance $r$. This is a direct consequence of the fact that particles couple to the metric $A^2(\phi)g_{\mu\nu}$. If the spherical object has no thin shell and behaves like a point-like particle we have = (1 +2 \_\^2) where $\beta_\infty = \frac{\beta}{f^{1/2}(\phi_\infty)}$ is the coupling of the canonically normalised field $\Phi$ to matter outside the sphere. When the spherical object has a thin shell then we have = (1 + 2 \_\_[c,eff]{}) where the coupling of the dense object to the scalar field $\beta _{c,eff}$ is obtained to be \_[c,eff]{}= Notice that we have \_[c,eff]{}= ()\^4 3\_=()\^2 3\_c. As in [@chameleon], the Lunar Ranging constraint depends on $\eta_\oplus \approx \beta_{c,eff}^2$, implying that $\beta_{c,eff} \lesssim 10^{-7}$ in the solar system. It is then easy to deduce the time difference $\Delta t$ for fermions through the earth = ()\^3 which can be expressed as = ()\^4 \_\^[3/2]{}. A natural scale for the coupling scale $M_\psi (\rho_c)$ in matter can be obtained in the strong coupling regime like in [@Kehagias:2011cb] where $\beta_c=\frac{m_{\rm Pl}}{M_\psi (\rho_c)}$. Choosing a reasonable value for the thin shell $\frac{\Delta R}{R} \sim 10^{-1}$, we find that $\frac{\Delta t}{t} \sim 2\cdot 10^{-5}$ for a coupling scale $M_\psi (\rho_c)\sim 140 \ {\rm GeV}$. The Lunar Ranging constraint can be satisfied with $\eta_\oplus \sim 10^{-7}$ and a low vacuum coupling scale $M_\psi (\rho_\infty) \sim 20 \ {\rm eV}$. As expected, the effect of the camouflage mechanism is to enhance the coupling scale in matter. In vacuum, the coupling scale is still low. Viewed as an order of magnitude for the cut-off of the effective theory describing the modification of gravity, the coupling scale in matter needs to be large enough to incorporate effects of the standard model. As such a coupling scale like $M_\psi (\rho_c)\sim 140 \ {\rm GeV}$ satisfies this criterion. On the other hand, the effective field theory in vacuum captures effects on very large cosmological scales at energies below the electron mass for which a low cut-off in the ballpark of $M_\psi (\rho_\infty) \sim 20 \ {\rm eV}$ is not contradictory. Of course, more work should be devoted to these camouflaged chameleons. For instance, the variety of coupling scales in different environments require an appropriate choice of $f_c$ and $f_\infty$ which have to be very different. This could be achieved using power laws for $f(\phi)$ as $\phi_\infty \gg \phi_c$ for most chameleon models. The phenomenology of these models will be studied elsewhere.
We still need to check that the neutrinos emerging from the supernova SN1987A are not too much in advance compared to photons. Let us model the trajectory of the neutrinos as radial from earth, on the verge of the milky way at a distance $R_{\rm E}\sim 8\ {\rm kpc}$ from the galactic centre, to a distance $R_{\rm SN}\sim 60\ \rm {kpc}$. This is not the exact trajectory although it will give us the order of magnitude of the deviation. Along this trajectory, the neutrino speed varies as $c_d=1 +\vert d\vert^2$ where $d= M^{-2}_\psi\frac{d\phi}{dr}$. The time difference between neutrinos and photons can be easily evaluated in the camouflaged chameleon case, we find that $$\frac{\Delta t}{t}= \frac{4}{3}\beta_{G,eff}^2 \Phi_N^2 \frac{m_{\rm pl}^2 R_{\rm E}^2}{R_{\rm SN}M_\psi(\rho_\infty)^4}(\frac{1}{R_{\rm E}^3}-\frac{1}{R_{\rm SN}^3})$$ where $\Phi_N\sim 10^{-6}$ and $\beta_{G,eff}\sim 10^{-1}$ for the Milky Way. Here we obtain an upper bound on the time delay which is typically $\frac{\Delta t}{t} \lesssim 10^{-16}$. Therefore, camouflaged chameleons could provide an explanation for the non-observation of a deviation from the speed of light for neutrinos emerging from SN1987A. Finally we also have to check that the amount of Cerenkov radiation is small. In fact, this is given by - ()\^7 which is always small, of order $6\cdot 10^{-4}$.
In conclusion, we have shown that modified gravity models with screening properties can induce Lorentz violation effects in the fermionic sector of the standard model. Such violations are induced by the profile of a scalar field coupled to matter in the presence of pressure-less over densities. In particular, fermions have a space-dependent speed along the scalar lines of force. Gravitational and Cerenkov radiation constraints are too tight to expect an observable signal on earth for natural values of the coupling scale between the scalar field and fermions unless gravity is modified by a camouflaged chameleon. Exploring the possible coupling of standard model particles with such scalar fields modifying the laws of gravity is an exciting prospect, still in its infancy, and worth pursuing. New experimental results in the near future will certainly give indications on the possible existence of modifications of gravity by screened scalar fields.
I would like to thank Clare Burrage, Anne Davis and Alexander Vikman for very stimulating suggestions.
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J. G. Williams, S. G. Turyshev and D. Boggs, arXiv:1203.2150 \[gr-qc\].
[^1]: For chameleon models, the screening factor is at most of order $s(R)\sim \frac{\Delta R}{R}$ varying over a thin shell of width $\Delta R$ and vanishing otherwise. Here we envisage models where $s(r)$ varies smoothly across the body and is non-vanishing over a finite interval of the order of the size of the dense body.
|
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SU-ITP-95-26\
hep-th/9511115\
November 15, 1995\
[**SUPERSYMMETRIC BALANCE OF FORCES\
0.6 cm AND CONDENSATION OF BPS STATES**]{}\
[**Renata Kallosh**]{}[^1] [** and Andrei Linde**]{}[^2] 0.05cm Physics Department, Stanford University, Stanford CA 94305\
0.7 cm
1 cm
**ABSTRACT**
> Until now all known static multi black hole solutions described BPS states with charges of the same sign. Such solutions could not be related to flat directions in the space of BPS states: The total number of such states could not spontaneously increase because of the charge conservation.
>
> We show that there exist static BPS configurations which remain in equilibrium even if they consist of states with opposite electric (or magnetic) charges from vector multiplets. This is possible because of the exact cancellation between the Coulomb and scalar forces. In particular, in the theories with N=4 or N=2 supersymmetry there exist stable massless multi center configurations with vanishing total charge. Since such configurations have vanishing energy and charge independently of their number, they can be associated with flat directions in the space of all possible BPS states. For N=2 case this provides a realization of the idea that BPS condensates could relate to each other different vacua of the string theory.
Introduction
============
Recently it was found that there exist supersymmetric BPS states with vanishing ADM mass. They include the solutions with one half of the N=4 [@Klaus2; @K; @KL; @CY] or N=2 [@FKS] supersymmetry unbroken. As usual for supersymmetric configurations, the multi-center solutions are also available [@K]. Those solutions have various extraordinary features; in particular they do not represent an extreme limit of any known non-extreme black holes. When considered in the four-dimensional canonical frame, they are singular and the singularity has a universal repulsive character. This is why it is not appropriate to call them “black” holes, and we called them “white” holes, or repulsons [@KL]. In what follows we will try to avoid attributing any color to the massless supersymmetric configurations under study to minimize the abuse of terminology [@Town], but we will keep calling them holes, or massless BPS states.
The existence of such solutions may look surprising [@Town]. However one should take into account several considerations:
i\) The existence of such states in exact quantum theory is expected according to conjectures of dualities and enhanced symmetries [@HullT; @W; @Strom].
ii\) BPS states of extended supersymmetry, if they exist as solutions of classical theory, may remain exact in quantum theory, at least for N=4 case according to supersymmetric non-renormalization theorems.
iii\) If they would not be found at least one of the standard lores i) or ii) would be incorrect.
With all this in view one may take the following attitude. Supersymmetric black$\&$white holes may have naked singularities and a better description of physical states of supersymmetric gravity is desirable. However, so far they served well by explicitly realizing various mass formulas for BPS states and mass and charge relations. The present investigation will show again that the new type of mass-charge relations which are believed to remain exact in quantum theory can be discovered when new soliton configurations are found.
The purpose of this note is to exhibit some dramatic differences between new, massless supersymmetric configurations and the ones which are known as extreme massive multi black hole solutions. The difference is due to the fact that massless solutions exist in case that we have more than one supersymmetric multiplet involved in the solution, let us say $1+n$ multiplets. This leads to the existence of $1+n$ independent supersymmetric balance of forces conditions. The analysis of these conditions shows that massless BPS states may consist of the hole-anti-hole configurations with the vanishing net charge which are in equilibrium. We will indeed find the multi-center BPS configurations with the vanishing total mass and charge. The mass of every hole vanishes. However, the sign of the charges with respect to vector fields other than graviphoton may alternate from one hole to another. One may argue, therefore that the massless holes may form a condensate. The argument for this is simple: if there is a state with $s$ holes which form a massless neutral set and the other state with $s+t$ holes which are also massless and neutral, it will not cost any energy to produce them, and no charge conservation will be violated when the system changes from an $s$-hole state to $(s+t)$-hole state. In particular, the total number of massless neutral clusters of $s$ holes may take any value. The effective theory will have a non-trivial ground state with the massless mode corresponding to the condensation of the neutral massless BPS configurations.
We would like to explain here under which conditions the massless neutral solutions can be found. Four-dimensional theories with one gravitational multiplet do not have non-trivial massless BPS configurations. Indeed, with one multiplet of N=4 supersymmetry any solution is characterized by the mass, which due to supersymmetry is related to the graviphoton charge and to the dilaton charge. When this parameter vanishes the solution becomes trivial. With one gravitational multiplet of N=2 supersymmetry we have the relation between the mass and the graviphoton charge. When the mass goes to zero, the charge goes to zero.
In N=8 case only the gravitational multiplet exists. It is characterized by one parameter and when the mass goes to zero, only trivial solutions can be expected. Unless more than one half of N=8 supersymmetry is broken, one cannot expect any interesting massless solutions.
The situation changes for N=4 theory since one can consider the interaction of N=4 supergravity with any number of N=4 vector multiplets. In particular, one can consider the theory with 22 N=4 vector multiplets. Together with 6 graviphotons in the gravitational multiplet they form the original 28 vectors which previously were all in one gravitational multiplet of N=8 supergravity. However, when the symmetry between 6 and 22 is broken by placing them into different multiplets, we may expect that there exists a configuration with 6 graviphoton charges vanishing but with 22 non-vanishing vector multiplet charges. This is indeed the case [@Klaus2; @K; @KL; @CY].
If even more supersymmetries are broken and we are looking for configurations with one half of N=2 supersymmetry unbroken, we may consider one gravitational multiplet of N=2 supergravity interacting with some number of vector multiplets of N=2 supersymmetry. During this step of breaking supersymmetry, the dilaton which was in one multiplet with graviton is now in a different multiplet, in a vector one. The dilaton charge, as well as the charges of the vector fields in the vector multiplets, are not related to the graviphoton charge anymore. Therefore one may have expected that when supersymmetric solutions in this theory will be found, that they will remain non-trivial when the mass of the configuration will tend to zero, since the breaking of supersymmetry has made the charges of some vectors and scalars not related to that of the graviphoton anymore. This is indeed the case: there is a rich variety of massless solutions in N=2 theory [@FKS].
Supersymmetric balance of forces for massive extreme black holes: gravitational multiplet solutions
===================================================================================================
Consider an example of supersymmetric balance of forces between two $a=1$ stringy black holes of Gibbons-Maeda-Gurfinkel-Horowitz-Strominger. These solutions are supersymmetric when embedded into pure N=4 supergravity without vector multiplets. Any extreme solution in this class with one half of N=4 supersymmetry unbroken saturate the following supersymmetric bound [@US] (we consider the electric solutions first): $$M^2 +( \Sigma_{\rm dil})^2 - (Q_{\rm gr})^2 = 0 \ .$$ The mass $M$ is related to the graviphoton charge $Q_{\rm gr}$ and to the dilaton charge $ \Sigma_{\rm dil}$ as follows: $$\label{BPSEL}
M= \frac {|Q_{\rm gr}|}{\sqrt{2}} =- \Sigma_{\rm dil} \ .$$ If there are two supersymmetric black holes, each of them has to saturate the same type of bounds: $$\begin{aligned}
m_1^2 + \sigma_1^2 - q_1^2 &=& 0 \ ,\nonumber\\
\nonumber\\
m_2^2 + \sigma_2^2 - q_2^2 &=& 0 \ .\end{aligned}$$ For each multiplet we have the same set of relations between charges in every hole. $$\begin{aligned}
\label{multi}
m_1 &=& -\sigma_1 = \frac
{|q_1|}{\sqrt{2}} \ , \nonumber\\
\nonumber\\
m_2 &=& -\sigma_2 = \frac {|q_2|}{\sqrt{2}}\ .\end{aligned}$$ We can study the forces between two such black holes and find whether the holes are in equilibrium. Consider Newtonian, Coulomb and dilatonic forces between two distant objects of masses and charges $(m_1, q_1, \sigma_1)$ and $(m_2, q_2, \sigma_2)$: $$\label{bal}
F_{12} = - \frac{m_1 m_2}{r_{12}^2} +
\frac{q_1 q_2}{r_{12}^2} - \frac{\sigma_1 \sigma_2}{r_{12}^2} \ .$$ The dilatonic force is attractive for charges of the same sign and repulsive for charges of opposite sign. For our configurations [*the dilaton charge does not change the sign when the electric charge does*]{}. Therefore if the electric charges of the two black holes are of the same sign we get $$\label{bal1}
F_{12} = - \frac{m_1 m_2}{r_{12}^2} +
\frac{|q_1| |q_2|}{r_{12}^2} - \frac{\sigma_1 \sigma_2}{r_{12}^2} \ .$$ According to eq. (\[multi\]), $$\label{bal2}
F_{12} = - \frac{m_1 m_2}{r_{12}^2} (1 -2+1) =0
\ .$$ Thus we see that $F_{12}$ vanishes and we conclude that two holes with the same sign of the electric charge are in equilibrium with each other, since the attractive force between them due to gravity and dilaton force is cancelled by Coulomb repulsion of two equal sign charges.
Now consider the situation when the electric charges have opposite sign. All three forces are attractive this time: $$\label{bal3}
F_{12} = - \frac{m_1 m_2}{r_{12}^2} (1 +2+1) = \frac{4 m_1 m_2}{r_{12}^2}\neq 0
\ .$$ This configuration is unstable since there is no balance of forces. Instead of performing this balance of forces analysis one could simply look into explicit form of the two-hole solution [@US]. There is no static solution available for two holes with positive masses and with opposite electric charges.
For a magnetic BPS state with the mass $M$, the magnetic graviphoton charge $P_{\rm gr}$ and the dilaton charge $ \Sigma_{\rm dil}$ the relation between charges is similar to (\[BPSEL\]): $$M= \frac {|P_{\rm gr}|}{\sqrt{2}} = \Sigma_{\rm dil} \ .$$ The conclusions of the balance of force analysis would be the same: one can get two holes in equilibrium under the condition that they have magnetic charges of the same sign.
Thus one could have concluded that two supersymmetric black holes may be in equilibrium only when their charges have the same sign. Therefore the sum of all masses of BPS states in equilibrium and the absolute value of their graviphoton charges is always positive: $$\sum_{a=1}^{n} m_a > 0 \ , \qquad |\sum_{a=1}^{n} (q_{\rm gr}) _a| > 0 \ .$$
This is were the massless configurations brought a new surprise.
Supersymmetric balance of forces: gravitational and vector multiplet solutions
==============================================================================
As explained above we have explicit solutions with vanishing ADM mass only in case of few multiplets, one gravitational and $n$ vector multiplets. In case of N=4 supersymmetry our solutions saturate $1+n$ independent bounds, one for each multiplet. $$\begin{aligned}
&&M^2 +( \Sigma_{\rm dil})^2 - (Q_{\rm gr})^2 = 0 \ , \\
&& ( \Sigma^I_{\rm vec})^2 - (Q^I_{\rm vec})^2= 0 \ , \qquad I=1, \dots , n.\end{aligned}$$ The first bound relates three type of charges: the mass, the dilaton charge and the electric charge of the graviphoton in the gravitational multiplet. Every other bound relates the modulus charge $\Sigma_{\rm vec}$ with the vector charge $Q_{\rm vec}$ inside each vector multiplet.
The nature of the second type of supersymmetric bounds turned our to be very different from what may have been expected. The important property of the supersymmetric solutions [@Klaus2; @K; @KL; @CY; @FKS] is the following: the sign of the charge of the modulus field $\Sigma^I_{\rm vec}$ is correlated with the sign of the charge of the electric field of the same vector multiplet. $$\label{chargerel}
\Sigma^I_{\rm vec} = Q^I_{\rm vec} \ .$$ When the sign of the vector charge changes, the sign of the scalar charge also changes. Therefore it is possible to consider the situation when there are two massive black holes, saturating all bounds. The picture described above for one gravitational multiplet remains intact: the graviphoton charge must always be of the same sign for two holes. However, the vector multiplet charges do not have to be constrained that way. Each black hole has to saturate its own vector multiplet bound. $$( \sigma^I_{\rm vec})^2_{1} - (q^I_{\rm vec})^2_{1} = 0 \ , \qquad
( \sigma^I_{\rm vec})^2_{1} - (q^I_{\rm vec})^2_{2} = 0 \ .$$ Given that the vector charge of every hole equals its scalar charge, $$(\sigma^I_{\rm vec})_{1} = (q^I_{\rm vec})_{1} \ , \qquad (\sigma^I_{\rm
vec})_{2} = (q^I_{\rm vec})_{2} \ ,$$ all bounds are saturated and there is no restriction on the relative sign of vector charges of two holes. There is always the balance of forces: whether we have the Coulomb attraction for opposite sign electric charges or Coulomb repulsion for same sign electric charges, we get the same picture from modulus field force. Since the sign of the scalar charge is correlated with the sign of the vector charge the corresponding part of the force between two holes vanishes $$\label{bal4}
F_{12}^{\rm vec} = \frac{ (q^I_{\rm vec})_{1} (q^I_{\rm vec})_{2}}{r_{12}^2} -
\frac{(\sigma^I_{\rm vec})_{ 1} (\sigma^I_{\rm vec})_{ 2}}{r_{12}^2}=0 \ .$$ Thus we have found that for the massive extreme black holes with graviphoton charges as well as with vector multiplet charges there are two possible configurations in equilibrium: The sign of graviphoton charge for two holes is always the same, but the sign of the vector multiplet charge may or may not alternate between two holes. One may have either $$(q_{\rm gr})_1 (q_{\rm gr})_2 > 0 \ , \qquad (q^I_{\rm vec})_{1} (q^I_{\rm
vec})_{2} >0 \ ,$$ or $$(q_{\rm gr})_1 (q_{\rm gr})_2 > 0 \ , \qquad (q^I_{\rm vec})_{1} (q^I_{\rm
vec})_{2} <0 \ .$$
In N=2 case we have found extreme black hole solutions [@FKS] which saturate the gravitational bound as well as the vector multiplet bound: $$\begin{aligned}
&&M^2 - (Q_{\rm gr})^2 = 0 \ , \\
&& ( \Sigma^I_{\rm vec})^2 - (Q^I_{\rm vec})^2 = 0 \ , \qquad I=1, \dots , n.\end{aligned}$$ The difference from N=4 case is in the structure of the gravitational bound which does not include the dilaton charge anymore. It is saturated when $M=
|Q_{\rm gr}|$. The dilaton (in heterotic theory) is now placed in one of the vector multiplets and as such, behave as any other scalar of the vector multiplets when the vector charge changes the sign. Alternatively, from type II string theory point of view, the dilaton is in one of the hypermultiplets. Apart from this there is no difference in the picture of balance of forces, as discussed above: the graviphoton charge splits into the same sign charges whereas the vector multiplet charge may split into same sign or opposite sign charges.
The crucial feature of the configuration with the vanishing mass, dilaton charge and graviphoton charge is the fact that the gravitational supersymmetric bound is saturated trivially for each hole. There is no gravitational, dilaton and electric graviton force to be compensated, all of them vanish in order ${1\over r^2_{12}}$. Thus the conclusion from the balance of forces study is: the configurations with the vanishing total mass and all charges may exist. $$M= \sum_{a=1}^{n} m_a = 0 \ , \qquad
Q_{\rm gr}= \sum_{a=1}^{n} (q_{\rm gr}) _a=0 \ , \qquad
Q^I _{\rm vec }=\sum_{a=1}^{n}
(q_{\rm vec }^I) _a = 0 \ .$$ The massless solutions with the vanishing vector multiplet charge $Q^I _{\rm vec }=0$ in all gauge groups were not known to exist before. The mechanism of the black hole condensation proposed by Greene, Morrison and Strominger [@GMS] (GMS mechanism) is based on the assumption that such configurations exist. Our analysis of the balance of forces proves that they may exist. In what follows we will describe them.
Hole-anti-hole solution in N=4 theory
=====================================
We would like to consider first the two-hole and $2s$-hole supersymmetric solution which may have an interpretation of the particle-antiparticle state. The class of massive black holes with non-vanishing graviphoton charge will not allow us to find such, as explained above. However, if we use directly the massless solutions, those with two opposite charges can be found. Starting with the Maharana-Schwarz-Sen action for toroidally compactified heterotic string theory we can get a two-center supersymmetric massless solution by slightly generalizing the construction presented in [@BK].
The generic pure magnetic $(6,22)$ symmetric solution is described completely in terms of magnetic potentials $\vec \chi$: $$\vec{\chi}(x) = \left( \begin{array}{c} \vec{\chi}_{\rm vec} (x)\\
\vec{\chi}_{\rm gr}(x)
\end{array} \right)
\ , \qquad
\partial_i \partial_i \vec{\chi}(x) =0 \ .
\label{magn}$$ The 28-dimensional harmonic $O(6,22)$-vector $\vec \chi$ consists of the 22-dimensional vector $\vec{\chi}_{\rm vec}$ , or $\chi ^I,\, I=1,\dots 22$, describing the vector multiplets and of the 6-dimensional vector $\vec{\chi}_{\rm gr}(x)$, or $\chi^\alpha ,\, \alpha=1,\dots 6$, describing the gravitational multiplet. The metric, the dilaton, the moduli fields ${\cal M}$ and the magnetic field $ \vec{H}_i = {1\over 2}
\epsilon_{ijk} \vec {F}_{jk} $ are $$\label{monopole}
\begin{array}{ccc}
ds^2_{\rm can}= - e^{2 U} dt^2 + e^{-2 U} d\vec{x}^2\ , && e^{-4U} = 2 \,
\left ( (\chi^\alpha)^2 - (\chi^I)^2\right)
= e^{4\phi} \ , \\
\nonumber\\
\nonumber \\
{\cal M}= {\bf 1}_{28} + 4 e^{4U} \left( \begin{array} {cc} \chi^I \chi^J &
\chi^ I \chi^\beta
\\
\chi^\alpha \chi^J & \xi \; \chi^\alpha \chi^\beta \end{array}
\right) , & & \vec{H}_i = \partial_i \vec{\chi}\ ,
\end{array}
\label{multisol}$$ where $ \xi \equiv (\chi^I)^2 / (\chi^m)^2$.
Consider the simplest case of asymptotically flat geometry and vanishing at infinity scalar fields. The massless solution with $a=2s$ holes and vanishing net charge for each of the 28 gauge groups is given by[^3] $$\chi^I = \sum_{a=1}^{a=2s} \frac{q ^I{}_a }{|\vec x- \vec x_a|} \ , \qquad
\chi^\alpha = {1\over \sqrt 2} n^\alpha \ , \qquad (n^\alpha)^2 =1 \ ,$$ $$Q^I \equiv \sum_a q ^I{}_a = 0 , \qquad m_a=0 \ , \qquad M= \sum m_a =0 \ .$$ The graviphoton charge vanishes for each hole since each of them is massless. However, the vector multiplet charge of each individual hole does not vanish, its sign alternates and only the total sum over all holes vanishes. For example, the simplest monopole-anti-monopole solution has two holes of the opposite charge, $$\chi^I = \frac{q ^I }{|\vec x- \vec x_1|} - \frac{q ^I }{|\vec x-\vec
x_2|} \ , \qquad
\chi^\alpha = {1\over \sqrt 2} n^\alpha \ , \qquad (n^\alpha)^2 =1 \ ,$$ $$Q_I \equiv q ^I{}_1 + q ^I{}_2 = 0 , \qquad m_1=m_2 = 0 \ , \qquad M= m_1
+ m_2 =0 \ .$$ To confirm our balance of force condition analysis we must check that the moduli field charges indeed compensate the Coulomb forces between the monopole-anti-monopole pair. For this purpose we will write down the total solution, corresponding to the potential of the pair above. We have for the metric, the dilaton and the magnetic fields: $$e^{-4U} = \, 1 - 2 \Bigl( \frac{q ^I }{|\vec x-\vec x_1|} - \frac{q ^I
}{|\vec x-\vec x_2|}\Bigr)^2 = e^{4\phi} \ , \qquad H_i ^I = { (x-x_2)^i q^I
\over |\vec x- \vec x_2|^3}-
{ (x-x_1)^i q^I \over |\vec x-\vec x_1|^3} \ ,$$ and there are no graviphoton magnetic fields, $H_i^\alpha=0 $. The moduli fields are given in eq. (\[multisol\]). We are particularly interested here in the moduli field charges. Therefore we will write down explicitly only the terms required for defining the scalar field charges. Those are $${\cal M}= {\bf 1}_{28} + 2\sqrt2 \, \pmatrix{
0& n^\beta\left( \frac{q ^I }{|\vec x- \vec x_1|} - \frac{q ^I
}{|\vec x-\vec x_2|} \right) \cr
\cr
n^\alpha \left( \frac{q ^J }{|\vec x- \vec x_1|} - \frac{q ^J }{|\vec
x-\vec x_2|} \right) & 0 \cr
} + \dots$$ Here $\dots $ stays for terms which will not contribute to scalar charges of each monopole. To find the charges of the ${\cal M}^{I\beta }$ and ${\cal
M}^{\alpha J}$ we will take the following two limits. First, we choose $\vec
x_1=0$, i.e. we place the first hole in the center of coordinates. The distance between two holes is given by the vector $\vec l$. The scalar ${\cal M}^{\alpha J }$ becomes $${\cal M}^{\alpha J }= 2 \sqrt 2 n^\alpha \left( \frac{q ^J }{|\vec x|} -
\frac{q ^J }{|\vec x-\vec l |} \right) \ .$$ The scalar charge of the first hole is defined as follows. We remove the second hole far away, i.e. we consider $l \rightarrow \infty$. In this limit $${1\over 2 \sqrt 2} ({\cal M}^{\alpha J })_1 \rightarrow \frac{ n^\alpha q
^J }{|\vec x|} \equiv
{(\Sigma ^{\alpha J } )_1 \over | \vec x|} \ .$$ This defines the scalar charge of the first hole. In our case this leads to $$(\Sigma ^{\alpha J })_1 = n^\alpha q ^J = n^\alpha (q ^J)_1 \ .$$ To find the scalar charge of the second hole we place it in the beginning of coordinates, remove the first hole far away and consider the $(\Sigma
^{\alpha J } )_2 / | \vec x|$ term. We get $$(\Sigma ^{\alpha J })_2 =- \, n^\alpha q ^J = n^\alpha (q ^J)_2 \ .$$ Thus we have confirmed the balance of force analysis: the scalar charge of the vector multiplet is sensitive to the sign of the magnetic charge of the same multiplet. Therefore the supersymmetric positivity bound permits massless holes with opposite charges to be in equilibrium.
The reader familiar with Sen’s spherically symmetric extreme massive supersymmetric black holes [@Sen] may easily verify that the dilaton charge of the electric (magnetic) solution is not sensitive to the sign of the electric (magnetic) graviphoton charge $Q_R$, whereas the charge of the modulus ${\cal M}$ does change when the sign of the charge of the vector in the vector multiplet $Q_L$ changes. This may give an additional explanation of why the BPS solutions with half of unbroken supersymmetry include massless neutral configurations with arbitrary number of centers.
We should emphasize, that the solutions discussed above are obtained by solving exact nonlinear equations. The balance of force analysis, which describes asymptotic behavior of forces between the two holes, is necessary only to interpret our exact solutions and to explain why they are consistent despite describing configurations with opposite charges.
Black hole multiplets with unbroken N=4 supersymmetry form the vector multiplets of N=4 supersymmetry. It is likely that the dynamics of such multiplets will show the condensation of the massless states. Indeed, it does not cost any energy to create as many pairs of massless oppositely charged BPS states as one wishes, and it does not violate charge conservation since each of these pairs is electrically and magnetically neutral. We will consider below the case of N=2 theory which is simpler from the point of view of the effective theory and which also has the massless neutral BPS states.
Condensation of Massless Holes in N=2 theory
============================================
The GMS mechanism of black hole condensation in N=2 theory is the following [@GMS]. One starts with the system of 16 massless black holes of unbroken N=2 supersymmetry. The low energy theory should contain 15 $U(1)$ gauge groups. The total charge of the system of 16 holes in each of the 15 gauge groups has to vanish, $$Q^I \equiv \sum_{a=1}^{16} (q^I)_a =0 \ .
\label{net}$$ A specific example in [@GMS] was to have the first hole with the charge +1 in the first group, the second one with the charge +1 in the second group, the third one with the charge +1 in the third group, etc. The 16-th hole, however, had to be negatively charged in all 15 groups. The problem which was not clearly resolved there was whether massless black holes can actually exist and whether they can be in equilibrium despite the Coulomb attraction between objects with opposite charges.
The Calabi-Yau manifold in question was described in a way that it is difficult to find what kind of a Kahler manifold corresponds to it in the low energy four-dimensional theory. Therefore, we will not consider the study below as the one describing a particular Calabi-Yau manifold. However, we will give an example of N=2 black holes which have the massless limit and have the set of 16 holes with the charges satisfying the constraint (\[net\]) in equilibrium. The simplest case is to use the example of ${ SU(1,15 ) \over SU(15)} $ N=2 black holes [@FKS].
The prepotential and the Kahler potential are $$\label{36}
F(X^0,X^I) = (X^0)^2- (X^I)^2 \ , \qquad e^{- K(Z,\bar Z)} = 1-|Z^I|^2 \ .$$ We choose 15 harmonic scalars $Z^I={X^I\over X^0}$ to vanish at infinity and to describe a 16-hole massless configuration: $$\label{15}
Z^I = \sum _{a= 1}^{16} {q^I{}_a \over |\vec x - \vec x_a| }\, \ , \qquad
\sum _{a= 1}^{16} q^I{}_a=0 \ ,$$
$$\label{39}
ds^2 = (1-|Z|^2)^{-1}\, dt^2 -
(1-|Z|^2)\,
d\vec x^2 \ .$$
The magnetic charges of 15 vector fields are given by $q^I{}_a$. In a more detailed form and assigning only $\pm 1$ charges we have $$\begin{aligned}
\label{16}
Z^1 &=& {1 \over |\vec x - \vec x_1| } - {1 \over |\vec x - \vec x_{16}| }\ ,
\\
Z^2 &=& {1 \over |\vec x - \vec x_2| } - {1 \over |\vec x - \vec x_{16}| }\ ,
\\
& & \dots \\
Z^{15} &=& {1 \over |\vec x - \vec x_{15}| }- {1 \over |\vec x - \vec x_{16}|
}\ .\end{aligned}$$ The total ADM mass, the total magnetic charge in each of the 15 gauge groups, as well as the total scalar charge in every gauge group vanish. Still we have a rather non-trivial configuration. If we take only one hole, for example the first one, remove all other 15 far away, we will find that it has a non-vanishing positive magnetic as well as the scalar charge. The same with the second and other 13 holes. If we finally consider the 16-th one, we would find that when the other 15 holes are far away, this one happens to be negatively charged (both in magnetic and scalar charges) in all 15 gauge groups. The system of these 16 holes is in equilibrium.
It is natural therefore to move to an alternative picture: associate with each hole a massless hypermultiplet as suggested in [@GMS]. Each hypermultiplet contains two charged complex scalars $h^{a\alpha}$ where $\alpha = 1,2$ is the global $SU(2)_R$ index of N=2 representation. This gives a total of $32$ complex scalar fields The potential describing the interactions between these holes in agreement with N=2 supersymmetry is given by $$\label{pot}
V\sim \sum_{I,J=1}^{15} M_{IJ}D ^{\alpha \beta I}D_{\alpha \beta}{}^{ J} \ ,$$ where $M_{IJ}$ is a positive definite matrix and $$D^{\alpha \beta I} = \sum_{a=1}^{16} q^I{}_a ( h^{* a \alpha} h^{a\beta} +
h^{* a \beta} h^{a \alpha}) \ .$$ A remarkable feature of this potential is the existence of flat directions along which the bilinear combinations of scalar fields $D^{\alpha \beta I}$ vanish, $$D^{\alpha \beta I} = 0 \ .
\label{flat}$$ This gives $45$ real constraints on $32$ complex fields. In addition there are $15$ gauge transformations which rotate the fields, leaving 4 real vacuum parameters. Up to a gauge transformation the general solution of (\[flat\]) is defined by a complex two-vector $v$. $$h^{a\alpha} = v^{\alpha}\qquad \hbox{for all $a$} \ .$$ Moving along the flat direction, the holes condense and their moduli space is parametrized by a single hypermultiplet. The point $v = 0$ was associated in [@GMS] with the conifold point in the space of quintics (at which all $16$ cycles vanish). Moving away from this point along the flat direction corresponds to giving a vev’s to the charged hypermultiplets which break all $15$ U(1)’s. Thus a second branch of the moduli space corresponding to a charged black hole condensate was discovered in [@GMS]. This branch has $101-15=86$ massless vector multiplets, and $2+1=3$ massless hypermultiplets.
We would like to stress here that the total picture depends crucially on the fact that $D^{\alpha \beta I} = 0
$ when the condensate ansatz is introduced into the expression for the flat direction condition. The reason it works is that $$D^{\alpha \beta I}|_{ h^{a \alpha} = v^{\alpha}} = ( v^{* \alpha} v^{\beta} +
v^{* \beta} v^{ \alpha}) \sum_{a=1}^{16} q^I{}_a = ( v^{* \alpha}
v^{\beta} +
v^{* \beta} v^{ \alpha})\; Q^I =0 \ .$$ Thus it is clear that if the total charge $Q_I$ of all holes in every gauge group would not vanish, we would be unable to have a condensate with $v^{\alpha} \neq 0$. But now that we know that such 16 holes may exist in equilibrium the total picture of condensation of massless holes with vanishing total charge looks much more plausible.
Massless Mode as a Goldstone Boson
==================================
Being supported by the existence of exact massless neutral multi-center solutions, we may study the general case of spontaneous symmetry breaking which brings one theory with $h_{21}$ massless vector multiplets and $ h_{11}$ massless hypermultiplets into the phase with $h'_{21}$ massless vector multiplets and $ h'_{11}$ massless hypermultiplets.
Let us first try to understand if there is any magic here about the numbers 15 and 16 in the example above. If we would start with $n$ gauge groups instead of 15 and $n+1$ holes (hypermultiplets) we would easily construct an $n+1$-hole solution using the example in [@FKS] with ${ SU(1, n ) \over SU(n)} $. This would be the generalization of eqs. (\[15\]), (\[16\]). Thus we can consider the case of $n$ gauge groups and $n+1$-hole solution with the total mass and charges in every gauge group vanishing. However, what will happen with the counting above which worked well for 15,16 case? It actually works in the general case as well. We start with $4 \times
(n+1)$ real scalars (for $n+1$ hypermultiplets). We have to impose $3\times n$ conditions for a direction to be flat. In addition we can use $n$ gauge transformations. Thus we are left with 4 real vacuum parameters $v^\alpha$ as before, since previously we had $$[4 \times (15+1)] - [3\times 15] -[15] = 4 \ ,$$ and now we have $$[4 \times (n+1)] - [3\times n] -[n] = 4 \ .$$ This would describe a condensate of a system of the original $n+1$ holes. The moduli space of the condensed holes is parametrized as before by a single hypermultiplet. The condensate breaks $n$ gauge groups this time.
Thus, suppose we start with $n$ gauge groups and choose the $n+1$-hole solution with the total charge in all groups vanishing. The distribution of the charges has the same structure as before: the first hole has charge 1 in the first group, etc, the $n$-th one has the charge $1$ in the $n$-th group. The last one has the negative charge in all groups. One can represent it as a charge matrix with $n$ columns (for $n$ groups) and $n+1$ rows (for $n+1$ holes) $$\label{config}
\left (\matrix{
{}~~1 & ~0 ~\cdots & 0 \cr
{}~~0 & ~1 ~ \cdots & 0 \cr
{}~~\vdots &~~\vdots ~~\ddots & \vdots \cr
{}~~0 & ~0 ~\cdots & 1 \cr
-1 &\hskip -.3 cm -1 ~\cdots &\hskip -.2 cm -1~ \cr
}\right )$$ The counting above shows that again we get one massless mode and $n$ massive ones. The relevant question to ask is: does this situation allow the interpretation of the massless mode as a Goldstone boson? For this purpose we may check the symmetries of the potential in the form in which we first perform the shift of the fields. The form of the hyper-hole condensate $\langle h^a\rangle = v$ suggests the natural combinations of fields,[^4] $$\phi^{\alpha I} \equiv \sum_{a=1}^{n} q^I{}_a h^{ a \alpha}\ , \qquad
G^\alpha \equiv
h^{(n+1)\alpha}\ ,$$ such that the ground state is defined by $$\langle \phi^{\alpha I}\rangle =0\ , \qquad \langle G^\alpha \rangle = v^\alpha
\ .$$ The potential in these variables is given by eq. (\[pot\]) with $$D^{\alpha \beta I} = \phi ^{* \alpha I} ( \phi ^{ \beta I} + 2 G^\beta) +
( \phi ^{* \beta I} + 2 G^{* \beta}) \phi ^{ \alpha I}\ .$$ In terms of fields with vanishing vacuum expectation values $$G^\alpha = \tilde {G}^\alpha + v^\alpha\ , \qquad
\langle\tilde {G}^\alpha\rangle =0\ ,$$ we have $$D^{\alpha \beta I} = \phi ^{* \alpha I} ( \phi^{ \beta I} + \tilde {G}^\beta
+ v^\beta) +
(\phi ^{* \beta I} + \tilde {G}^{* \beta} + v^{*\beta}) \phi ^{ \alpha I}\ .$$ The relevant Goldstone-type continuous global symmetry of this theory is a $U(2)$ symmetry given by $$\label{sym}
\Delta \phi ^{ \alpha I} = \lambda^{\alpha \beta} \phi^{ \beta I}\ , \qquad
\Delta G^\alpha = \lambda^{\alpha \beta}
G^\beta\ , \qquad \lambda^\dagger = \lambda \ .$$ This symmetry has exactly four parameters in agreement with the fact that one massless hypermultiplet has four massless scalars. This symmetry is broken spontaneously. To prove that there is one massless Goldstone hypermultiplet (four scalars) in such theory one can use the standard procedure. The values of the variations of the fields at the ground state are $$\langle\Delta \phi^{ \alpha I}\rangle = \lambda^{\alpha \beta}
\langle\phi^{ \beta I}\rangle =0\ , \qquad \Delta \langle G^\alpha\rangle =
\lambda^{\alpha \beta}
\langle G^\beta\rangle = \lambda^{\alpha \beta} v^\alpha \ .$$ From the symmetry of the potential we get $${\partial V \over \partial \phi^I} \Delta \phi^I + {\partial V \over
\partial G} \Delta G + {\partial V \over \partial \phi^{I*}} \Delta \phi^{I*}
+ {\partial V \over \partial G^*} \Delta G^* =0 \ .$$ The second derivatives of this equation over $G^*$ and $\phi^I$ taken at the ground state give the consistency conditions for the mass matrix: $$\begin{aligned}
\Big\langle{\partial^2 V \over \partial G^\alpha \partial G^{*\gamma
}}\Big\rangle \lambda^{\alpha \beta} v^\beta + \Big\langle{\partial^2 V
\over \partial G^{*\alpha} \partial G^{*\gamma}}\Big\rangle ( \lambda^{\alpha
\beta} v)^* &=&0 \ , \nonumber\\
\nonumber\\
\Big\langle{\partial^2 V \over \partial G^\alpha \partial \phi^{*\gamma I}
}\Big\rangle \lambda^{\alpha \beta} v + \Big\langle{\partial V \over
\partial G^{* \alpha} \partial \phi^{* I \gamma}}\Big\rangle ( \lambda^{\alpha
\beta} v)^* &=&0 \ .\end{aligned}$$ This tells us that there is one Goldstone massless hypermultiplet $G^\alpha$ with four massless scalars whose mass matrix is decoupled from the other $n$ massive hypermultiplets $\phi_I^\alpha$. This is in agreement with the fact that there is one four-dimensional spontaneously broken continuous symmetry $U(2)$ (\[sym\]). The spontaneously generated mass term for $n$ hypermultiplets is $$\sim [\phi ^{* \alpha I} v^\beta +
v^{*\beta} \phi ^{ \alpha I}] M_{IJ} [\phi_{ \alpha} ^{* J} v_\beta +
v^{*}_{\beta} \phi_{ \alpha}^J] \ .$$ If we would start with the neutral set of holes but with the different distribution of charges over the holes we would have to follow a slightly more complicated procedure of diagonalizing the massive and massless modes. However we will get again $n$ massive and one massless mode.
Thus we have found how to decrease any number $ h_{21} - h_{21}'=n $ of massless vector multiplets and increase the number of massless hypermultiplets by one ($ h_{11}'-
h_{11}=1$). The rule was to create one neutral cluster of $n+1$ massless charged holes with the configuration of charges described in (\[config\]). If one would like to have more than one, $k= h_{11}'-
h_{11}$, massless hypermultiplet as a result of the hole condensation, this is also possible. One has to start with $k$ clusters of neutral system of holes of the type described above. Each cluster has some number of columns $n_k$ and rows $(n_k+1)$. $$\label{matrix}
\left (\matrix{
\left (\matrix{
{}~1 ~~ \cdots ~~0 \cr
{}~\vdots ~~\ddots ~~ \vdots \cr
{}~0 ~~\cdots ~~1 \cr
-1 ~\cdots -1 \cr
}\right )
& \cdots & 0 \cr
\vdots& \ddots & \vdots \cr
\cr
0 & \cdots &
\left (\matrix{
{}~~1 & ~0 ~\cdots & 0 \cr
{}~~0 & ~1 ~ \cdots & 0 \cr
{}~~\vdots &~~\vdots ~~\ddots & \vdots \cr
{}~~0 & ~0 ~\cdots & 1 \cr
-1 &\hskip -.3 cm -1 ~\cdots &\hskip -.2 cm -1~ \cr
}\right )
\cr
}\right )$$ The matrix of charges is a direct product of the matrices of the type given for one neutral set. The total number of columns in this matrix equals the number of vector multiplets $n= n_1+n_2+\dots
+n_k$ which become massive in the process of spontaneous breaking of symmetry. The number of submatrices in (\[matrix\]) equals $k$, the number of massless hypermultiplets. If the hypermultiplets in each cluster interact only with the vector multiplets and hypermultiplets of its own group one can write down the potential with $[U(2) ]^k$ global symmetry, which is spontaneously broken and as the result, $k$ massless hypermultiplets describe the condensed state of the original neutral $k$ clusters. All of such multi-hole configurations are available.
After Condensation
==================
After the condensate of one cluster of massless holes has been formed we end up by a set of holes which are all massive except one. It is interesting to go back from the effective action which was treating the holes as hypermultiplets to the original action of supergravity interacting with $n$ vector multiplets and identify the configuration which describes the state of a system after the condensate has been formed. For simplicity consider the case of two sets of 4 gauge groups and again, the simplest for our purpose Kahler manifold describing ${ SU(1,4+4 ) \over SU(4+4)} $ N=2 black holes [@FKS]. The first four massless vector multiplets will be excited in the initial configuration and the second four in the final[^5]. This will allow us to have an example of 5 holes with the required properties. The original configuration of 5 massless holes is given by $$m_1 = m_2=m_3=m_4=m_5=0\ , \qquad Z^1_\infty =
Z^2_\infty=Z^3_\infty=Z^4_\infty=0 \ .$$ with the charge matrix $$\label{1}
\left (\matrix{
{}~1 & ~0 & ~0 & ~0 \cr
{}~0 & ~1 & ~0 & ~0 \cr
{}~0 & ~0 & ~1 & ~0 \cr
{}~0 & ~0 & ~0 & ~1 \cr
-1 & -1 & -1 & -1 \cr
}\right )$$ The charge of the graviphoton in all 5 holes equals the mass and vanishes. The mass formula is $$m_a = {\sum_{I=1}^{I=4} Z^I_{\infty} q^I{}_a \over (1-|Z_\infty|^2)} \ .$$ The massless solution is $$\label{5crtical}
Z^I = \sum _{a= 1}^{a=5} {q^I{}_a \over |\vec x - \vec x_a| }\, \ , \qquad
\sum _{a= 1}^{a=5} q^I{}_a=0\ , \qquad I=1,2,3,4.$$ $$ds^2 = (1-|Z|^2)^{-1}\, dt^2 -
(1-|Z|^2)\,
d\vec x^2 \ .$$ The magnetic charges of 4 vector fields are given by $q^I{}_a$. The scalars are $$\begin{aligned}
\label{5}
Z^1 &=& {1 \over |\vec x - \vec x_1| } - {1 \over |\vec x - \vec x_{5}| } \ ,
\\
Z^2 &=& {1 \over |\vec x - \vec x_2| } - {1 \over |\vec x - \vec x_{5}| } \ ,\\
Z^3 &=& {1 \over |\vec x - \vec x_3| } - {1 \over |\vec x - \vec x_{5}| } \ ,\\
Z^{4} &=& {1 \over |\vec x - \vec x_{4}| }- {1 \over |\vec x - \vec x_{5}| } \
{}.\end{aligned}$$
After the condensation which is described in the dual picture of interacting hypermultiplets the final system corresponds to the 5-hole solution with 4 massive and one massless hole. One possible example of such configuration is: $$m_1 = m_2=m_3=m_4= {C\over \sqrt {1-4C^2}} \, \qquad m_5=0 \qquad Z^5_\infty
= - Z^6_\infty=Z^7_\infty=-Z^8_\infty=C \ ,$$ with the charge matrix $$\label{2}
\left (\matrix{
{}~1 & ~0 & ~0 & ~0 \cr
{}~0 & -1 & ~0 & ~0 \cr
{}~0 & ~0 & ~1 & ~0 \cr
{}~0 & ~0 & ~0 & -1 \cr
-1 & -1 & -1 & -1 \cr
}\right )$$ The new 5 holes are sitting in different places[^6] $$\label{15}
Z^I = Z^I_\infty + \sum _{a= 1}^{a=5} {q^I{}_a \over |\vec x - \vec
{\underline {x}}_{a}| }\, \ , \qquad Q^5= \sum _{a= 1}^{a=5} q^5{}_a=Q^7= \sum
_{a= 1}^{a=5} q^7{}_a=0 \ ,$$
$$Q^6 = \sum _{a= 1}^{a=5} q^6{}_a= -2\ , \qquad Q^8 =\sum _{a= 1}^{a=5}
q^8{}_a=-2 \ .$$
The metric is $$\label{39}
ds^2 = \left({1-|Z|^2 \over 1-|Z_\infty |^2}\right)^{-1}\, dt^2 -
\left({1-|Z|^2 \over 1-|Z_\infty |^2}\right) \,
d\vec x^2 \ , \qquad I=5,6,7,8 \ .$$ $$\begin{aligned}
\label{5}
Z^5 &=& +C + {1 \over |\vec x -\vec {\underline {x}}_{1}| } - {1 \over |\vec x
- \vec {\underline {x}}_{5}| }\ ,\\
Z^6 &= & -C - {1 \over |\vec x - \vec {\underline {x}}_{2}| } - {1 \over |\vec
x - \vec {\underline {x}}_{5}| }\ ,\\
Z^7 &=& +C+ {1 \over |\vec x - \vec {\underline {x}}_{3}| } - {1 \over |\vec x
- \vec {\underline {x}}_{5}| }\ ,\\
Z^8 &=& -C - {1 \over |\vec x - \vec {\underline {x}}_{4}| }- {1 \over |\vec x
- \vec {\underline {x}}_{5}| }\ .\end{aligned}$$ There are many other solutions with 4 massive and 1 massless holes but with the different total charge in various gauge groups. All of these configurations have the limit when the scalars in the vector multiplets at infinity vanish $C\rightarrow 0$ and all five holes become massless. However, these solutions still carry a non-vanishing total charge in some of the gauge groups. In the example above, the charge in the 6-th and 8-th direction is not vanishing. If we would try to study this system as we did with the previous one we would find that $$<D^{\alpha \beta \,6}> = \sum_{a=1}^{5} q^6{}_a <( h^{* a \alpha} h^{a\beta} +
h^{* a \beta} h^{a \alpha})> = -2 \, ( v^{* \alpha} v^{\beta} +
v^{* \beta} v^{ \alpha})) \neq 0 .$$ Thus $v\neq 0$ is not a ground state of these theory. Thus the condensation consistent with N=2 supersymmetric potential of hypermultiplets only occurs when we start with an $n$-hole solution [*with the total mass and all charges vanishing*]{}. For this configuration one can write down the potential for the hypermultiplets with the global $U(2)$ symmetry with flat directions along which the holes condense. This symmetry is broken explicitly either when the mass terms for the hypermultipets is added to the action or spontaneously, as considered in this paper. After the spontaneous generation of the condensate (vacuum expectation value of the scalar part of the hypermultiplets in the effective theory) the new state may be described as an $n$-hole solution of the same theory but with the different values of masses and charges. The values of the masses are defined by the condensate.
Discussion
==========
The main result of this work is the realization of the fact that supersymmetric multi-hole solutions with the vanishing mass and total charges in all gauge groups do exist. They are massless and neutral, still it is not a flat space: the solution has an arbitrary number of centers with a particular distribution of charges. Such configurations plays the central role in the picture of the black hole condensation. This picture reflects nicely various properties of unbroken supersymmetry, in particular, the subtleties of the supersymmetric balance of forces in different multiplets.
We may qualify the condensation of holes as a process which drives the transition from one multi-hole configuration to another. After the formation of the condensate in the neutral system of $(n+1)$ types of massless holes, we obtain a new vacuum state with $n$ types of massive holes and one massless hole. The original gravitational system has various solutions of this kind which are also supersymmetric but whose total mass and charges are not vanishing anymore. The initial as well as the final configurations are both exact multi–hole solutions of the same effective Lagrangian. However these two sets have a dramatic difference in the masses and charges. In particular one can show that the distribution of charges in the final state differs from the original one by specific sign-flip of charges in some holes and by different asymptotic behavior of the scalars in the vector multiplets. Therefore both the original as well as the final set of holes are in equilibrium according to the balance of forces analysis in supersymmetric systems. The picture of transition between two different sets of supersymmetric holes presented above relies on the effective theory and may suggest the possible link between various supersymmetric solitons which are known to exist in supergravities and in string theories. The ultimate importance of the spontaneous generation of the central charges in supersymmetric theories reveals itself in this picture: the massive supersymmetric configurations are created in the process of spontaneous breaking of symmetries of the theory. The mass parameters of the theory are not fixed at the fundamental level, but appear in the theory with the various choices of the ground states. We have found that the picture of condensation of N=2 supersymmetric BPS states is related to the spontaneous breaking of a global $U(2)$ symmetry. The presence of central charges in $N$-extended supersymmetry is known to break the global $U(N)$ symmetry. One can expect that the dynamics of the massive BPS states which form short massive multiplets in general may be understood via spontaneous breaking of the $U(N)$ symmetry which is present in massless theories with global extended $N$ supersymmetries.
We would like to add, from a somewhat different perspective, that the existence of flat directions often has important cosmological implications such as inflation, Polonyi field problem, etc. It would be very interesting to study this question in the context of the theory of BPS condensation. As a first step one may try to obtain solutions describing massless multi center BPS states in de Sitter space, like it was done in [@KT] for extreme Reissner-Nordström black holes. Since massless black holes do not modify metric of space-time far away from them, it may happen that they will not affect de Sitter expansion, and, [*vice versa*]{}, flat directions which we discussed above, will remain flat in de Sitter space. If this is the case, then one may expect that during inflation quantum fluctuations will move the BPS field $h^{a\alpha}$ along all possible flat directions in different causally disconnected parts of the universe. This would divide the universe after inflation into exponentially large domains corresponding to different stringy vacua. One should note that it is not so easy to obtain inflation in string theory. But here again the existence of flat directions of a new type may occur to be very useful. The best way of unifying string theory and inflation would be to consider inflation along flat directions. If some of the BPS flat directions become not exactly flat either due to expansion of the universe or because of the supersymmetry breaking, one may try to investigate the possibility that inflation occurs during the rolling of the black hole condensate towards the minimum of its effective potential. This is admittedly a very speculative possibility, but it sounds so interesting that it would be hard not to mention it here.
Acknowledgements {#acknowledgements .unnumbered}
================
We are grateful to L. Susskind for a stimulating discussion. This work was supported by NSF grant PHY-8612280.
[1000]{} K. Behrndt, “About a class of exact string backgrounds,” Humboldt–Universität preprint HUB-EP-9516 (1995), hep–th/9506106. R. Kallosh, “Duality symmetric quantization of superstring,” Stanford University preprint SU-ITP-95-12 (1995), hep-th/9506113, to be published in Phys. Rev. D. R. Kallosh and A. Linde, “Exact Supersymmetric Massive and Massless White Holes,” preprint SU-ITP-95-14 (1995), hep-th/9507022, to be published in Phys. Rev. D. M. Cvetič and D. Youm, Phys. Lett. [**B359**]{}, 87 (1995). S. Ferrara, R. Kallosh, and A. Strominger, “N=2 extremal black holes," preprint CERN-TH-95-211 (1995), hep-th/9508072, to be published in Phys. Rev. D. P. Townsend, “Supergravity solitons and non-perturbative superstrings," preprint DAMTP-R-95-52 (1995), hep-th/9510190. C. Hull and P. Townsend, Nucl. Phys. [**B438**]{}, 109 (1995); Nucl. Phys. [**B451**]{}, 525 (1995). E. Witten, Nucl. Phys. [**B443**]{}, 85 (1995). A. Strominger, Nucl. Phys. [**B451**]{}, 96 (1995). R. Kallosh, A. Linde, T. Ortín, A. Peet, and A. van Proeyen, Phys. Rev. D [**46**]{}, 5278 (1992). K. Behrndt and R. Kallosh, “O(6,22) BPS configurations of the heterotic string,” SU-ITP-95-19 (1995), hep-th/9509102. A. Sen, Int. J. Mod. Phys. [**A8**]{}, 5079 (1993); Nucl. Phys. [**B440**]{}, 421 (1995). B.R. Greene, D.R. Morrison, and A. Strominger, Nucl. Phys. [**B451**]{}, 109 (1995). D. Kastor and J. Traschen, Phys. Rev. D[**47**]{}, 5370 (1993).
[^1]: E-mail: [email protected]
[^2]: E-mail: [email protected]
[^3]: This solution with same sign charges was found in [@BK], however it remained unnoticed that alternating signs are also possible.
[^4]: One can choose different forms of the hyper-hole condensate, e.g. $\langle h^1\rangle = v,~~ \langle
h^2\rangle = - v $, etc. For all of this choices one can find the natural combinations of fields in terms of which the theory near each of these ground states is described as the one with $\langle h^a\rangle = v$.
[^5]: We need two sets of vector multiplets since the fields in the first group will become massive when the condensate is formed.
[^6]: The positions of holes in the original multi-center solutions $x_a$ is arbitrary. We choose a different set of centers $\vec {\underline {x}}_{a}$ for the “after phase transition solution” to stress the fact that the positions do not have to coincide with the original ones.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We have conducted a three band ($BVI$) variability survey of the globular cluster M3. This is the first three band survey of the cluster using modern image subtraction techniques. Observations were made over 9 nights in 1998 on the 1.2 m. telescope at the F. L. Whipple Observatory in Arizona. We present photometry for 180 variable stars in the M3 field, of which 12 are newly discovered. New discoveries include six SX Phe type variables which all lie in the blue straggler region of the color magnitude diagram, two new first overtone RR Lyrae, a candidate multi-mode RR Lyrae, a detached eclipsing binary, and two unclassified variables. We also provide revised periods for 52 of the 168 previously known variables that we observe. The catalog and photometry for the variable stars are available via anonymous ftp at ftp://cfa-ftp.harvard.edu/pub/kstanek/M3/.'
author:
- 'J. D. Hartman, J. Kaluzny, A. Szentgyorgyi & K. Z. Stanek'
title: 'BVI Photometric Variability Survey of M3.'
---
Introduction
============
The globular cluster M3 (NGC 5272) is one of the most studied clusters in the Galaxy. Since the work of Bailey (1913) and Shapley (1914), there have been numerous surveys for photometric variability in the cluster. In 1973, Sawyer-Hogg compiled the most complete catalog of variables in the cluster prior to the modern era. Recently Bakos, Benko, & Jurcsik (2000) compiled a catalog of 274 variable stars in the cluster, and published improved identification and astrometry for them. M3 is particularly known for containing the largest number of RR Lyrae variables, with more than 182 such stars (Clement et al. 2001). Many groups have focused on studying the population of RR Lyrae variables. Corwin et al. (2001) and Clementini et al. (2004) have published a catalog of 222 confirmed or suspected RR Lyrae variables, and have provided a detailed analysis of the 8 known double-mode RR Lyraes (RRd). Other surveys have focused specifically on searches for variability in blue straggler stars (BSS), notably Kaluzny et al. (1998) discovered only one SX Phe star out of 25 monitored BSS. They also discovered a contact binary star near the base of the red giant branch.
In this contribution we present the data from a $BVI$ variability survey of the cluster performed in the spring of 1998. This is the first three band survey of the cluster using modern image subtraction techniques. New results include the discovery of 6 new SX Phe variables, 2 new first-overtone RR Lyraes, a candidate multi-mode RR Lyrae, a detached eclipsing binary, and 2 new variables that we do not classify. We also provide revised periods for 52 of the 168 previously known variables that we observe.
We describe our observations and data reduction in the following section. In §3 we present the catalog of variables, in §4 we analyze the light curve of the eclipsing binary NV296, in §5 we briefly describe the population of variables using color magnitude diagrams, and in §6 we discuss the results.
Observations and Data Reduction
===============================
The observations were made at the F. L. Whipple Observatory (FLWO) 1.2 m telescope using the 4-Shooter CCD mosaic containing four thinned, back-side-illuminated, AR-coated Loral 2048x2048 CCDs. The camera has a pixel scale of $0\farcs333$ pixel$^{-1}$ and a field of view (FOV) of $11\farcm4\times 11\farcm4$ for each chip. We obtained observations on 9 nights between the dates of April 16-20, 1998 and May 28-31, 1998. These observations consist of 4 x 300 s, and 67 x 420 s $B$-band exposures, 14 x 240 s, 198 x 300 s, and 13 x 450 s $V$-band exposures and 4 x 180 s, and 72 x 240 s $I$-band exposures. The sampling cadence was as low as 7 minutes, allowing for the detection of very short period SX Phe variables. The center of M3 at $(\alpha,\delta)=(13^{\rm h}42^{\rm
m}11\fs2,+28\arcdeg22\arcmin32\arcsec)$ (J2000.0) was positioned in the lower, right corner of chip 3. A mosaic of all four chips is shown in Fig \[fov\].
The preliminary CCD reductions were performed using the standard routines in the IRAF CCDPROC package[^1].
To obtain photometry we used the image subtraction techniques due to Alard & Lupton (1998; also Alard 2000) as implemented in the ISIS 2.1[^2] package. To detect variables we first took the absolute value of all the subtracted images and stacked them to form a single image. We then searched this image for strong point sources, identifying these as variables. We obtained light curves only for sources identified as variable in this way by performing weighted-aperture photometry on the subtracted images. This procedure provides differential photometry in Analog-to-Digital Units (ADUs), to convert to magnitudes we obtained profile photometry for the stars in our field using the DAOPHOT/ALLSTAR package (Stetson 1987, 1991). As a sanity check, we also generated light curves using profile photometry which resulted in generally worse precision and yielded no additional variables. The method that we have just described is the same technique used by Kaluzny et al. (2001); the procedure is discussed in more detail in that paper.
We also generated light curves for blue straggler stars, examining them individually for variability. We identified a handful of low-amplitude ($0.04$ mag) SX Phe variables in this fashion.
To transform our light curves to standard BVI magnitudes we observed a total of 128 stars from 22 Landolt (1992) fields so that the transformation for each chip utilized typically 30 stars from 5 fields. The airmass of these observations ranged from 1.08 to 1.93. The resulting uncertainty in the zero-point of our observations is 0.02 magnitudes. As a consistency check we calculated the median colors for the turnoff stars independently for each chip. The results shown in Table \[tab:tab1\] are consistent with an uncertainty of 0.02 magnitudes.
We obtained astrometry for the variables by first matching to the catalog of Bakos et al. (2000). We used at least 10 preliminary matches on each chip to obtain the transformation between rectangular and equatorial coordinates. The residuals of the matched stars were less than $0\farcs1$. Using the RA/DEC from this transformation we proceded to match our sources to the Two Micron All Sky Survey ([*2MASS*]{}; Skrutskie et al. 1997) point-source catalog using a matching radius of $0\farcs7$ for a total of 98 matches. The median residual was $0\farcs43$. All but three of these matches are RR Lyrae. Both of the unclassified newly discovered variables and one of the newly discovered SX Phe variables matched to sources in [*2MASS*]{}. Figure \[2mass\] shows the location of 93 of the matched sources on the [*2MASS*]{} $J$ vs. $J-K$ color-magnitude-diagram (CMD) for M3. The five matched sources not shown in this figure were missing $K$ photometry. The fact that all but a few sources lie clumped near a region of constant $J$ (the horizontal branch) suggests that these matches to RR Lyrae are correct, with the scatter likely due to variability of the RR Lyrae and blending with nearby sources. In Figure \[RRLyr\_location\] we plot the position on the sky of all RR Lyrae in our catalog, together with the positions of those that match to [*2MASS*]{}. From this figure it is clear that the non-matches are due to incompleteness in the [*2MASS*]{} catalog near the crowded center of the globular cluster.
Catalog of Variables
====================
Following the above procedures we identify 180 variable sources of which 12 are newly identified as variables. The newly identified variables include 6 new SX Phe stars, 2 first overtone RR Lyrae (RR1), a candidate multi-mode RR Lyrae (RR01), one detached eclipsing binary (EB), and 2 unclassified variables. We also provide revised periods for 52 of the known variables. The revisions were determined based on visual comparisons between the light curves phased with the published period and the light curves phased with the period calculated using the Schwarzenberg-Czerny (1996) algorithm. We consider the period to be necessary of revision if the published period is incompatible with the observed light curve given the formal errors from the photometry. We have recovered one of the previously identified SX Phe stars (V237), we note however that the coordinates given for this variable appear to have been switched with the coordinates for V238 in the discovery paper by Kaluzny et al. (1998), this error has been propagated through the catalogs of Bakos et al. (2000) and Clement et al. (2001), we also revise the period for this variable. Finding charts for the newly discovered variables are shown in Figure \[finder\_chart\]. A machine-readable version of the catalog, as well as $V$, $B$, $I$, $B-V$ and $V-I$ light curves for the variable stars are available via anonymous ftp[^3]. We present the catalog in Tables \[tab:tab2\] and \[tab:tab3\], the columns are as follows:
- [*ID*]{}: The ID number of the variable. V1-V285 is taken from Clement et al. (2001). New identifications are denoted by “NV.” The ID number roughly corresponds to discovery order.
- [*2MASS-flag*]{}: Integer denoting whether or not the variable matched to a [*2MASS*]{} source. The flag is 0 for no match, and 1 for a match.
- [*RA*]{}: Right ascension for epoch J2000.0. This value is taken from the [*2MASS*]{} catalog for sources with a [*2MASS*]{} match, otherwise it is obtained from the rectangular to equatorial transformation derived using the Bakos et al. (2000) catalog.
- [*DEC*]{}: Declination for epoch J2000.0. This value is taken from the [*2MASS*]{} catalog for sources with a [*2MASS*]{} match, otherwise it is obtained from the rectangular to equatorial transformation derived using the Bakos et al. (2000) catalog.
- [*V$_{flag}$*]{}: Integer denoting the type of V-band observations. Values are 1 for observations that could be converted to magnitudes or 2 for observations that were left in differential count units.
- [*B$_{flag}$*]{}: Integer denoting the type of B-band observations. Values are 0 for no B-band detection, 1 for observations that could be converted to magnitudes, or 2 for observations that were left in differential count units.
- [*I$_{flag}$*]{}: Integer denoting the type of I-band observations. Values are 0 for no I-band detection, 1 for observations that could be converted to magnitudes, or 2 for observations that were left in differential count units.
- [*A$_{V}$*]{}: Observed full-amplitude in V-band, defined to be the faintest observed magnitude minus the brightest observed magnitude on the cleaned light curve.
- $\langle V\rangle$: Flux averaged mean V magnitude of the star. For the eclipsing binary this is the out of eclipse magnitude determined with EBOP(see §4).
- [*A$_{B}$*]{}: Observed full-amplitude in B-band, defined to be the faintest observed magnitude minus the brightest observed magnitude on the cleaned light curve.
- $\langle B\rangle$: Flux averaged mean B magnitude of the star. For the eclipsing binary this is the out of eclipse magnitude determined with EBOP(see §4).
- [*A$_{I}$*]{}: Observed full-amplitude in I-band, defined to be the faintest observed magnitude minus the brightest observed magnitude on the cleaned light curve.
- $\langle I\rangle$: Flux averaged mean I magnitude of the star. For the eclipsing binary this is the out of eclipse magnitude determined with EBOP(see §4).
- [*Period*]{}: The observed best period of variability in days. This was derived using the [ANOVA]{} statistic of Schwarzenberg-Czerny (1996). In cases where aliasing allowed for a number of acceptable periods, we chose the period corresponding to the peak in the periodogram nearest to the published period, where available.
- [*Published Period*]{}: The published period for the star taken from Clementini et al. (2004), Corwin & Carney (2002) or Clement et al. (2001) in that order of priority.
- [*Period Revision Flag*]{}: Integer denoting whether or not the observed best period should be taken as a revision of the published period. This determination is based on a visual comparison between the light curve phased with the published period and the light curve phased with the observed best period. Values are 1 for a revision, and 0 for no revision.
- [*JD of minimum V*]{}: Julian Date (in 2450000) of the first minimum in V-band to occur after the first observation. This field is null for unclassified variables.
- [*Phase of minimum B*]{}: Phase of the minimum in B-band, where 0 phase is set to the minimum in V-band.
- [*Phase of minimum I*]{}: Phase of the minimum in I-band, where 0 phase is set to the minimum in V-band.
- [*Classification*]{}: Classification of variability, the symbols are as follows:
- RR0: fundamental mode RR Lyr (RRab).
- RR1: first overtone RR Lyr (RRc).
- RR01: candidate multi-mode RR Lyr (RRd). Note that the possible presence of multiple periods was determined by eye and was not the result of a systematic search.
- SXP: SX Phe type variable.
- N/A: Unclassified variable.
- EB: Eclipsing binary.
- [*Remarks*]{}: Here we denote the possible presence of multiple modes (mp) for SXP variables as well as the presence of the Blazhko effect (Bl) within the observations.
Figure \[lc\] shows the $V$-band, $B-V$ and $V-I$ light curves for 12 of the 130 previously identified variables in the catalog for which these 3 measurements are available. We display $V$, $B-V$ and $V-I$ light curves, where available, for the 2 unclassified new variables and the 3 new RRs in Figure \[LPVRR1\] and light curves for the 6 new SX Phe variables in Figure \[SXP\]. The $BVI$ light curves for the eclipsing binary (NV297) are shown in Figure \[EB\] together with a model fit.
Analysis of the Eclipsing Binary NV296
======================================
Parameters for the eclipsing binary NV296 were determined using the Eclipsing Binary Orbit Program (EBOP) model for detached eclipsing binaries (Nelson & Davis 1972; Popper & Etzel 1981) (see Figure \[EB\]). The first step was to obtain a fit to the V-band light curve. In doing so we varied the luminosity ratio, the radius of the primary, the inclination of the orbit, and the out-of-eclipse luminosity. We then obtained a second fit varying the ratio of the radii in place of the radius of the primary. We found empirically that the solution would not converge when the radius of the primary and the ratio of the radii were allowed to vary simultaneously. We held the mass ratio fixed at 1.0, and assumed that gravity darkening and the reflection effect were negligible. Because the period is relatively short ($0.445955$ days) we assumed that the orbits would have circularized, so we held the eccentricity fixed to 0. We note that when we allowed this parameter to vary the results were consistent with $e=0$. Having obtained a model for the V-band light curve, we proceeded to fit a model to the B-band light curve assuming the same orbital parameters and radii from the V-band model. We did not analyze the I-band light curve since noise and poor phase coverage during the eclipses conspired to prevent us from obtaining an acceptable fit.
Without spectra it is difficult to constrain the spectral types and luminosity classes of the components, and hence the limb darkening coefficients. As a preliminary analysis we fixed the limb darkening coefficients for both components to 0.57 in $V$ and 0.72 in $B$, which is consistent with an F2V star in the limb darkening tables from Claret & Gimenez (1990). We caution that the results from this fit should be treated as preliminary until spectra can be obtained for this variable..
The parameters from the best fit model are shown in Table \[Tab:EB\]. To test the effect of the unconstrained mass ratio on the fit, we obtained an independent model assuming a mass ratio of 0.7. The parameters for this model are also shown in Table \[Tab:EB\], we note that the results appear to be consistent, in particular the position of the components on the CMD is unaffected by the mass ratio. We find that the primary has $B-V=0.349 \pm 0.042$ mag, and $V=19.331 \pm 0.028$ mag, while the secondary has $B-V=0.395 \pm 0.064$ mag, and $V=19.747 \pm 0.042$ mag. The errors listed here and in Table \[Tab:EB\] are propagated from the standard errors produced by the EBOP fits, we caution that for all parameters except the color these errors are likely to be optimistic. In the case of the $B-V$ colors the errors listed are calculated assuming $B$ and $V$ are uncorrelated when in reality the two magnitudes will be correlated as a result of the fitting procedure. We note that for three different choices of the limb darkening coefficients $B-V$ changed by less than a percent for both components whereas the other parameters varied by amounts consistent with the standard errors.
Color Magnitude Diagrams
========================
In Figures \[cmbv\] and \[cmvi\] we plot the $B-V$ vs. $V$ and $V-I$ vs. $V$ CMDs for the cluster. In these figures we use filled circles to show RR Lyr variables, open squares for SXP variables, stars for unclassified variables and open triangles for the two components of the EB. Blue straggler stars that were examined for variability are shown with filled triangles. Note that these stars were selected based on their position in the $B-V$ CMD. Where available, the locations of these stars in the $V-I$ CMD are also shown.
The SXP variables for which colors could be obtained all lie within the blue straggler region of the CMD as a result of our selection procedure. Although their location on the CMD may indicate that these variables are indeed likely members of the cluster, one must be cautious in such claims following the identification with quasars of three variables in the M3 field lying in the blue straggler region of the CMD (Meusinger, Sholz & Irwin, 2001).
The components of the eclipsing binary NV296 appear to lie near the ZAMS at a distance of $(m-M)_V=15.04$ mag (Harris 1996). The photometry for the primary is consistent with an F2V dwarf, and the secondary is consistent with an F3-4V dwarf. If the eclipsing binary is indeed a member of the cluster then both components (particularly the primary) appear to lie slightly below the cluster main sequence, closer to the ZAMS for the cluster.
The location of the unclassified variables on the CMD is completely uncertain since we have not observed a full period for either of these stars.
To further study the RR Lyr population we plot the horizontal branch regions of the CMDs in detail in Figures \[cmbv\_hb\] and \[cmvi\_hb\]. Here we distinguish between the types of RR Lyr variables using filled circles for RRab, open circles for RRc and open stars for RRd candidates. The evolution from RRab to RRc along the horizontal branch is clearly demonstrated in the CMD for M3. We note that some RR Lyrae may appear to lie outside of the instability strip as a result of uncertainties in their colors. We also note that blending near the center of the cluster may cause a few variables to appear artificially bright.
Discussion
==========
There has been a great deal of interest in studying the variability of BSS in globular clusters. A number of globular clusters show a substantial population of variable blue stragglers. Gilliland et al. (1998) found that 9 out of 47 monitored BSS in 47 Tucanae showed variability above 0.02 mag. Recently Kaluzny et al (2004) identified 35 new SX Phe variables in $\omega$ Centauri. However, despite having a sizeable population of BSS, Kaluzny et al. (1998) found only a single SX Phe star in M3. We confirm the relative under-abundance of SX Phe stars in M3, finding only 7 variables out of 122 monitored BSS. We note that with image subtraction we would have been able to detect single-mode, short-period variability in the monitored BSS with amplitudes greater than 0.02 magnitudes (see for example NV289 in Figure \[lc\]).
Eclipsing binaries are another interesting target for globular cluster variability surveys. Besides the possibility that binaries play an important role in the dynamical evolution of globular clusters (Hut et al. 1992) and the formation of the BSS (Leonard 1989; Leonard & Fahlman 1991), eclipsing binaries can also be used as a tool to determine accurate ages and distances to the clusters (Paczyński 1997; see Kaluzny et al. 2004 for a discussion of a systematic program to search for detached eclipsing binaries in nearby globular clusters). The location of the components of NV296 on the CMD suggests that the binary may indeed be a member of the cluster. If it is a member, then there must be some mechanism that has allowed the components to avoid evolution into a giant. One hypothesis may be mass transfer onto the primary, however the fact that the light curve seems well fit by a detached eclipsing binary model suggests that mass transfer is not currently occurring at a significant rate. Moreover, the secondary star itself appears to be younger than other F3-4V stars in the cluster. It is necessary to obtain spectra for this system before anything more can be said about its membership. If it is a member, then spectra will be useful in determining the masses of the components.
As mentioned in the introduction, many of the recent studies of M3 have focused on the RR Lyrae population. Most recently Cacciari, Corwin & Carney (2004) used data from previous $BVI$ surveys to conduct a detailed analysis of the RR Lyrae population in M3. Other authors have pointed to inconsistencies between existing models for horizontal branch evolution and the observed population (Catelan 2004; Clementini et al. 2004 discuss the observation of rapid evolution among the double mode RR Lyrae). Although we do not perform a systematic analysis of the RR Lyrae population, the data that we present should be useful for further investigations of these stars.
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[lrr]{} 1 & 0.450 & 0.592\
2 & 0.456 & 0.580\
3 & 0.450 & 0.561\
4 & 0.473 & 0.590\
\[tab:tab1\]
[lrrrr]{} Mass Ratio & 1.0 & 0.0 & 0.7 & 0.0\
V Primary & 19.331 & 0.028 & 19.351 & 0.024\
V Secondary & 19.747 & 0.042 & 19.719 & 0.036\
(B-V) Primary & 0.349 & 0.042 & 0.348 & 0.036\
(B-V) Secondary & 0.395 & 0.064 & 0.396 & 0.055\
V $J_{s}/J_{p}$ & 0.65922 & 0.02325 & 0.64884 & 0.02280\
B $J_{s}/J_{p}$ & 0.63197 & 0.03088 & 0.62028 & 0.02917\
$R_{p}/a$ & 0.28594 & 0.00357 & 0.28499 & 0.00360\
$R_{s}/R_{p}$ & 1.01456 & 0.02725 & 1.00764 & 0.02126\
V Limb Darkening & 0.57 & 0.00 & 0.57 & 0.00\
B Limb Darkening & 0.72 & 0.00 & 0.72 & 0.00\
Inclination & $75.66777\arcdeg$ & $0.50846\arcdeg$ & $75.86569\arcdeg$ & $0.47571\arcdeg$\
Eccentricity & 0.00 & 0.00 & 0.00 & 0.00\
V Scale Factor & 18.76666 & 0.00214 & 18.76719 & 0.00228\
B Scale Factor & 19.13395 & 0.00230 & 19.13484 & 0.00237\
\[Tab:EB\]
[lcrrcccrrrrrr]{} V001 & 1 & 13:42:11.12 & +28:20:33.8 & 1 & 1 & 1 & 01.090 & 15.722 & 01.381 & 16.114 & 00.645 & 15.226\
V004 & 1 & 13:42:08.21 & +28:22:33.3 & 1 & 0 & 1 & 00.784 & 14.304 & & & 00.441 & 14.259\
V005 & 1 & 13:42:31.29 & +28:22:20.7 & 1 & 1 & 1 & 00.848 & 15.780 & 01.045 & 16.143 & 00.504 & 15.295\
V006 & 1 & 13:42:02.08 & +28:23:41.6 & 1 & 1 & 1 & 01.108 & 15.782 & 01.362 & 16.155 & 00.687 & 15.301\
V007 & 0 & 13:42:11.09 & +28:24:10.6 & 1 & 1 & 1 & 01.241 & 15.515 & 01.470 & 15.818 & 00.755 & 15.229\
V009 & 1 & 13:41:49.51 & +28:19:13.3 & 1 & 1 & 1 & 01.058 & 15.686 & 01.300 & 16.017 & 00.647 & 15.216\
V010 & 1 & 13:42:23.10 & +28:25:00.6 & 1 & 1 & 1 & 00.948 & 15.613 & 01.097 & 15.919 & 00.594 & 15.167\
V011 & 1 & 13:41:59.98 & +28:19:11.8 & 1 & 1 & 1 & 01.214 & 15.753 & 01.503 & 16.174 & 00.767 & 15.271\
V012 & 1 & 13:42:11.26 & +28:20:17.0 & 1 & 1 & 1 & 00.540 & 15.591 & 00.632 & 15.876 & 00.294 & 15.272\
V017 & 1 & 13:42:22.42 & +28:15:22.7 & 1 & 1 & 1 & 00.824 & 15.684 & 00.972 & 15.972 & 00.474 & 15.188\
V018 & 1 & 13:42:18.95 & +28:17:47.3 & 1 & 1 & 1 & 01.151 & 15.703 & 01.306 & 15.979 & 00.673 & 15.263\
V019 & 1 & 13:42:38.12 & +28:18:37.8 & 1 & 1 & 1 & 00.507 & 15.745 & 00.589 & 16.119 & 00.290 & 15.162\
V020 & 1 & 13:42:36.83 & +28:18:11.8 & 1 & 1 & 1 & 01.025 & 15.588 & 01.209 & 15.846 & 00.583 & 15.186\
V021 & 1 & 13:42:37.78 & +28:23:01.4 & 1 & 1 & 1 & 01.084 & 15.778 & 01.343 & 16.070 & 00.669 & 15.271\
V022 & 1 & 13:42:25.92 & +28:22:32.1 & 1 & 1 & 1 & 01.220 & 15.913 & 01.427 & 16.236 & 00.801 & 15.443\
V023 & 1 & 13:42:02.84 & +28:27:20.9 & 1 & 1 & 1 & 00.885 & 15.642 & 01.047 & 15.932 & 00.583 & 15.112\
V024 & 1 & 13:42:00.31 & +28:22:51.9 & 1 & 1 & 1 & 00.713 & 15.523 & 00.885 & 15.939 & 00.431 & 15.001\
V027 & 1 & 13:42:03.17 & +28:20:58.9 & 1 & 1 & 1 & 00.863 & 15.588 & 01.078 & 15.974 & 00.528 & 15.107\
V030 & 0 & 13:42:08.73 & +28:23:40.3 & 1 & 0 & 0 & 01.051 & 15.312 & & & &\
V031 & 1 & 13:42:13.97 & +28:23:47.4 & 1 & 1 & 1 & 01.150 & 15.537 & 01.418 & 15.877 & 00.686 & 15.135\
V032 & 1 & 13:42:12.39 & +28:23:42.3 & 1 & 1 & 1 & 01.098 & 15.430 & 01.279 & 15.732 & 00.701 & 15.163\
V034 & 1 & 13:42:21.71 & +28:25:32.5 & 1 & 1 & 1 & 00.488 & 15.682 & 00.622 & 16.060 & 00.274 & 15.221\
V036 & 1 & 13:42:24.55 & +28:22:07.4 & 1 & 1 & 1 & 01.146 & 15.581 & 01.308 & 15.926 & 00.738 & 15.211\
V037 & 1 & 13:41:53.56 & +28:25:25.6 & 1 & 1 & 1 & 00.493 & 15.679 & 00.614 & 15.958 & 00.300 & 15.304\
V038 & 1 & 13:41:56.04 & +28:24:49.1 & 1 & 1 & 1 & 01.002 & 15.601 & 01.292 & 15.975 & 00.618 & 15.168\
V039 & 1 & 13:41:52.97 & +28:24:42.4 & 1 & 1 & 1 & 00.918 & 15.677 & 01.160 & 16.070 & 00.570 & 15.177\
V040 & 1 & 13:41:50.93 & +28:24:33.1 & 1 & 1 & 1 & 00.960 & 15.727 & 01.158 & 16.060 & 00.602 & 15.216\
V043 & 1 & 13:42:19.07 & +28:23:07.0 & 1 & 1 & 1 & 01.112 & 15.625 & 01.265 & 15.847 & 00.672 & 15.188\
V044 & 1 & 13:42:24.37 & +28:24:22.0 & 1 & 1 & 1 & 01.201 & 15.623 & 01.240 & 15.986 & 00.684 & 15.207\
V045 & 1 & 13:41:53.23 & +28:20:31.1 & 1 & 1 & 1 & 00.899 & 15.786 & 01.164 & 16.093 & 02.607 & 13.893\
V050 & 1 & 13:42:12.24 & +28:18:47.8 & 1 & 1 & 1 & 00.688 & 15.581 & 00.914 & 15.821 & 00.430 & 15.180\
V051 & 1 & 13:42:13.87 & +28:18:55.9 & 1 & 1 & 1 & 00.881 & 15.713 & 01.025 & 16.109 & 00.498 & 15.191\
V053 & 0 & 13:42:10.94 & +28:24:45.1 & 1 & 1 & 1 & 01.184 & 15.695 & 01.389 & 15.945 & 00.729 & 15.319\
V054 & 1 & 13:42:08.98 & +28:24:28.5 & 1 & 1 & 1 & 01.029 & 15.751 & 01.216 & 16.094 & 00.579 & 15.295\
V055 & 1 & 13:41:55.91 & +28:28:05.7 & 1 & 1 & 1 & 01.059 & 15.809 & 01.350 & 16.120 & 00.663 & 15.274\
V056 & 1 & 13:42:00.67 & +28:28:39.9 & 1 & 1 & 1 & 00.488 & 15.629 & 00.613 & 15.904 & 00.299 & 15.269\
V057 & 1 & 13:42:23.25 & +28:22:42.3 & 1 & 1 & 1 & 00.682 & 15.875 & 00.932 & 16.292 & 00.440 & 15.347\
V059 & 1 & 13:42:03.25 & +28:18:53.2 & 1 & 1 & 1 & 00.889 & 15.686 & 00.979 & 16.107 & 00.473 & 15.182\
V060 & 1 & 13:41:49.05 & +28:17:25.5 & 1 & 1 & 1 & 00.699 & 15.561 & 00.882 & 15.974 & 00.446 & 15.023\
V061 & 1 & 13:42:25.78 & +28:28:45.7 & 1 & 1 & 1 & 00.705 & 15.732 & 00.885 & 16.103 & 00.403 & 15.262\
V062 & 1 & 13:42:18.22 & +28:29:39.1 & 1 & 1 & 1 & 00.534 & 15.645 & 00.607 & 16.025 & 00.346 & 15.093\
V063 & 1 & 13:42:14.21 & +28:28:24.1 & 1 & 1 & 1 & 00.721 & 15.711 & 00.868 & 16.082 & 00.428 & 15.221\
V064 & 1 & 13:42:20.11 & +28:28:12.5 & 1 & 1 & 1 & 00.719 & 15.689 & 00.845 & 16.058 & 00.441 & 15.169\
V065 & 1 & 13:42:20.92 & +28:28:10.0 & 1 & 1 & 1 & 00.933 & 15.566 & 01.109 & 15.852 & 00.591 & 15.056\
V066 & 1 & 13:42:03.77 & +28:24:42.8 & 1 & 0 & 0 & 00.542 & 15.680 & & & &\
V067 & 1 & 13:42:01.51 & +28:24:44.3 & 1 & 1 & 1 & 01.019 & 15.680 & 01.249 & 15.981 & 00.638 & 15.156\
V068 & 1 & 13:42:13.09 & +28:25:37.1 & 1 & 1 & 1 & 00.803 & 15.619 & 00.961 & 15.880 & 00.465 & 15.238\
V069 & 1 & 13:42:17.57 & +28:25:03.3 & 1 & 1 & 1 & 00.929 & 15.679 & 01.062 & 16.083 & 00.568 & 15.229\
V070 & 1 & 13:42:14.32 & +28:25:14.4 & 1 & 1 & 1 & 00.394 & 15.404 & 00.473 & 15.723 & 00.246 & 14.961\
V071 & 1 & 13:42:23.65 & +28:22:40.5 & 1 & 1 & 1 & 00.970 & 15.692 & 01.188 & 15.936 & 00.591 & 15.210\
V072 & 1 & 13:42:45.25 & +28:22:41.3 & 1 & 1 & 1 & 01.270 & 15.733 & 01.521 & 15.912 & 00.793 & 15.351\
V073 & 1 & 13:42:44.72 & +28:23:45.7 & 1 & 1 & 1 & 00.266 & 15.707 & 00.352 & 16.097 & 00.181 & 15.130\
V074 & 1 & 13:42:18.15 & +28:25:13.1 & 1 & 1 & 1 & 01.125 & 15.950 & 01.489 & 16.329 & 00.583 & 15.435\
V075 & 1 & 13:42:15.13 & +28:25:21.4 & 1 & 1 & 1 & 00.541 & 15.652 & 00.610 & 15.867 & 00.314 & 15.299\
V078 & 1 & 13:42:15.06 & +28:23:48.5 & 1 & 1 & 1 & 00.872 & 15.539 & 01.064 & 15.884 & 00.510 & 15.043\
V079 & 1 & 13:42:14.69 & +28:28:31.4 & 1 & 1 & 1 & 00.559 & 15.703 & 00.565 & 16.000 & 00.310 & 15.287\
V080 & 1 & 13:42:43.02 & +28:27:27.8 & 1 & 1 & 1 & 01.020 & 15.790 & 01.240 & 16.141 & 00.639 & 15.273\
V081 & 1 & 13:42:37.36 & +28:28:34.3 & 1 & 1 & 1 & 01.069 & 15.740 & 01.234 & 16.007 & 00.657 & 15.248\
V083 & 1 & 13:41:38.00 & +28:24:33.4 & 1 & 1 & 1 & 01.166 & 15.822 & 01.396 & 16.170 & 00.764 & 15.318\
V084 & 1 & 13:42:16.31 & +28:25:27.1 & 1 & 1 & 1 & 00.734 & 15.681 & 00.844 & 16.043 & 00.452 & 15.168\
V085 & 1 & 13:42:34.65 & +28:26:29.0 & 1 & 1 & 1 & 00.506 & 15.516 & 00.613 & 15.757 & 00.316 & 15.209\
V087 & 1 & 13:42:19.84 & +28:23:42.6 & 1 & 1 & 1 & 00.571 & 15.535 & 00.631 & 15.826 & 00.345 & 15.157\
V090 & 1 & 13:42:18.92 & +28:19:34.2 & 1 & 1 & 1 & 01.096 & 15.658 & 01.335 & 15.971 & 00.648 & 15.219\
V091 & 0 & 13:42:10.59 & +28:13:32.2 & 2 & 2 & 2 & & & & & &\
V093 & 1 & 13:41:47.40 & +28:16:04.1 & 1 & 1 & 1 & 02.461 & 15.768 & 00.947 & 16.013 & 02.472 & 15.252\
V094 & 1 & 13:41:34.54 & +28:18:55.1 & 1 & 1 & 1 & 01.141 & 15.775 & 01.342 & 16.107 & 00.681 & 15.256\
V096 & 1 & 13:41:59.12 & +28:18:47.3 & 1 & 1 & 1 & 00.786 & 15.941 & 00.451 & 16.370 & 00.574 & 15.254\
V097 & 1 & 13:42:01.70 & +28:19:24.6 & 1 & 1 & 1 & 00.445 & 15.685 & 00.550 & 15.995 & 00.262 & 15.305\
V099 & 1 & 13:42:26.76 & +28:21:48.0 & 1 & 1 & 1 & 00.549 & 15.588 & 00.700 & 15.873 & 00.334 & 15.210\
V100 & 1 & 13:42:16.75 & +28:24:19.7 & 1 & 1 & 1 & 00.616 & 15.713 & 00.721 & 16.086 & 00.377 & 15.178\
V101 & 1 & 13:42:14.98 & +28:24:05.6 & 1 & 1 & 1 & 00.627 & 15.671 & 00.714 & 16.036 & 00.401 & 15.167\
V104 & 1 & 13:42:09.49 & +28:25:07.3 & 1 & 1 & 1 & 01.170 & 15.495 & 01.365 & 15.789 & 00.771 & 15.039\
V105 & 1 & 13:42:09.85 & +28:25:53.2 & 1 & 1 & 1 & 00.336 & 15.591 & 00.417 & 15.776 & 00.193 & 15.320\
V108 & 1 & 13:41:54.77 & +28:27:51.9 & 1 & 1 & 1 & 01.122 & 15.661 & 01.398 & 15.892 & 00.716 & 15.238\
V117 & 1 & 13:42:18.39 & +28:14:54.0 & 1 & 1 & 1 & 00.674 & 15.647 & 00.852 & 15.988 & 00.383 & 15.156\
V118 & 1 & 13:42:22.50 & +28:17:50.6 & 1 & 1 & 1 & 01.205 & 15.886 & 01.444 & 16.204 & 00.730 & 15.391\
V119 & 1 & 13:42:30.67 & +28:24:28.8 & 1 & 1 & 1 & 01.121 & 15.452 & 01.387 & 15.736 & 00.712 & 15.144\
V120 & 1 & 13:41:48.99 & +28:26:32.2 & 1 & 1 & 1 & 00.437 & 15.683 & 00.545 & 16.076 & 00.277 & 15.103\
V121 & 0 & 13:42:08.17 & +28:23:38.1 & 1 & 0 & 1 & 01.165 & 15.327 & & & 00.906 & 15.484\
V122 & 0 & 13:42:09.03 & +28:21:55.7 & 2 & 2 & 2 & & & & & &\
V125 & 1 & 13:42:25.64 & +28:20:30.1 & 1 & 1 & 1 & 00.458 & 15.654 & 00.534 & 15.917 & 00.260 & 15.253\
V128 & 0 & 13:42:20.12 & +28:24:53.9 & 1 & 1 & 1 & 00.546 & 15.605 & 00.645 & 15.856 & 00.319 & 15.334\
V129 & 0 & 13:42:08.21 & +28:23:59.7 & 1 & 0 & 0 & 00.439 & 15.034 & & & &\
V130 & 1 & 13:42:11.77 & +28:24:06.2 & 1 & 1 & 1 & 00.439 & 15.548 & 00.515 & 15.876 & 00.266 & 15.060\
V134 & 0 & 13:42:09.78 & +28:23:35.0 & 1 & 1 & 1 & 00.642 & 15.644 & 00.725 & 15.975 & 00.402 & 15.166\
V135 & 1 & 13:42:09.43 & +28:23:20.4 & 1 & 1 & 1 & 00.820 & 15.539 & 01.096 & 15.885 & 00.490 & 15.052\
V136 & 0 & 13:42:09.57 & +28:23:16.5 & 1 & 1 & 1 & 00.558 & 15.311 & 00.656 & 15.797 & 00.303 & 14.746\
V137 & 1 & 13:42:15.47 & +28:22:23.2 & 1 & 1 & 1 & 00.898 & 15.572 & 00.968 & 15.910 & 00.578 & 15.139\
V139 & 0 & 13:42:14.11 & +28:23:10.8 & 1 & 1 & 1 & 01.240 & 15.531 & 01.376 & 15.865 & 00.803 & 15.211\
V140 & 1 & 13:42:10.28 & +28:24:30.7 & 1 & 1 & 1 & 00.507 & 15.515 & 00.618 & 15.815 & 00.259 & 15.176\
V142 & 1 & 13:42:09.26 & +28:21:43.6 & 1 & 1 & 1 & 01.075 & 15.775 & 01.212 & 16.115 & 00.772 & 15.483\
V143 & 1 & 13:42:08.88 & +28:22:59.0 & 1 & 1 & 1 & 01.073 & 15.416 & 01.321 & 15.769 & 00.598 & 14.989\
V145 & 0 & 13:42:13.61 & +28:22:51.3 & 1 & 1 & 1 & 00.943 & 15.429 & 01.207 & 15.820 & 00.566 & 14.968\
V146 & 0 & 13:42:18.48 & +28:21:44.3 & 1 & 1 & 1 & 01.090 & 15.649 & 01.410 & 16.067 & 00.737 & 15.273\
V147 & 0 & 13:42:09.83 & +28:23:29.7 & 1 & 1 & 1 & 00.474 & 15.693 & 00.475 & 15.963 & 00.265 & 15.354\
V148 & 1 & 13:42:10.95 & +28:23:20.0 & 1 & 1 & 1 & 00.832 & 15.194 & 01.108 & 15.558 & 00.404 & 14.479\
V149 & 0 & 13:42:14.10 & +28:23:35.5 & 1 & 1 & 1 & 01.148 & 15.642 & 01.289 & 15.966 & 00.594 & 15.070\
V150 & 0 & 13:42:16.68 & +28:23:20.8 & 1 & 1 & 1 & 00.827 & 15.798 & 00.956 & 16.166 & 00.499 & 15.268\
V151 & 0 & 13:42:11.99 & +28:22:01.3 & 1 & 1 & 1 & 00.909 & 15.459 & 01.038 & 15.757 & 00.566 & 15.116\
V152 & 1 & 13:42:17.47 & +28:23:33.6 & 1 & 1 & 1 & 00.458 & 15.490 & 00.509 & 15.733 & 00.272 & 15.235\
V156 & 0 & 13:42:09.99 & +28:22:01.8 & 1 & 1 & 1 & 01.005 & 15.415 & 01.163 & 15.726 & 00.511 & 15.039\
V157 & 1 & 13:42:10.20 & +28:23:18.8 & 1 & 1 & 1 & 01.078 & 15.249 & 01.261 & 15.707 & 00.412 & 14.505\
V159 & 0 & 13:42:10.34 & +28:22:59.1 & 1 & 1 & 1 & 00.432 & 15.312 & 00.586 & 15.841 & 00.388 & 14.747\
V160 & 0 & 13:42:10.80 & +28:21:59.4 & 1 & 1 & 1 & 01.046 & 15.567 & 01.205 & 15.932 & 00.668 & 15.115\
V161 & 0 & 13:42:12.80 & +28:21:45.2 & 1 & 1 & 1 & 00.681 & 15.730 & 00.895 & 16.132 & 00.368 & 15.112\
V165 & 0 & 13:42:17.04 & +28:23:03.2 & 1 & 1 & 2 & 01.076 & 15.468 & 01.401 & 16.006 & &\
V168 & 1 & 13:42:08.09 & +28:22:49.9 & 1 & 0 & 1 & 00.473 & 15.192 & & & 00.462 & 15.595\
V170 & 1 & 13:42:09.33 & +28:23:15.2 & 1 & 1 & 1 & 00.466 & 15.246 & 00.595 & 15.453 & 00.271 & 14.901\
V171 & 0 & 13:42:09.51 & +28:22:59.6 & 1 & 1 & 1 & 00.567 & 15.580 & 00.618 & 15.703 & 00.380 & 15.381\
V172 & 0 & 13:42:09.89 & +28:23:08.8 & 1 & 1 & 0 & 01.021 & 15.590 & 01.108 & 15.927 & &\
V173 & 0 & 13:42:10.50 & +28:23:21.8 & 1 & 1 & 1 & 00.526 & 14.881 & 00.563 & 15.170 & 00.294 & 14.520\
V174 & 0 & 13:42:10.84 & +28:22:08.9 & 1 & 1 & 1 & 01.174 & 15.735 & 01.343 & 16.080 & 00.761 & 15.404\
V175 & 0 & 13:42:14.62 & +28:23:09.4 & 1 & 1 & 0 & 00.932 & 15.713 & 00.844 & 15.714 & &\
V176 & 0 & 13:42:14.98 & +28:23:16.2 & 1 & 1 & 0 & 01.004 & 15.717 & 01.141 & 15.989 & &\
V177 & 1 & 13:42:16.28 & +28:22:13.9 & 1 & 1 & 1 & 00.557 & 15.507 & 00.674 & 15.765 & 00.330 & 15.250\
V178 & 0 & 13:42:17.47 & +28:23:29.9 & 1 & 1 & 1 & 00.406 & 15.712 & 00.448 & 15.908 & 00.245 & 15.475\
V180 & 0 & 13:42:10.09 & +28:22:13.5 & 1 & 1 & 0 & 00.770 & 15.735 & 00.832 & 16.071 & &\
V181 & 0 & 13:42:09.21 & +28:22:29.3 & 1 & 1 & 0 & 00.410 & 15.188 & 00.475 & 15.636 & &\
V184 & 0 & 13:42:09.57 & +28:22:27.8 & 1 & 1 & 1 & 01.337 & 15.770 & 01.467 & 16.039 & 01.045 & 15.277\
V187 & 0 & 13:42:09.64 & +28:22:52.0 & 2 & 2 & 2 & & & & & &\
V188 & 0 & 13:42:09.49 & +28:23:06.9 & 1 & 1 & 1 & 00.449 & 15.742 & 00.477 & 15.982 & 00.337 & 15.506\
V189 & 0 & 13:42:09.59 & +28:22:22.0 & 1 & 1 & 1 & 00.478 & 15.146 & 00.629 & 15.609 & 00.304 & 14.603\
V190 & 0 & 13:42:10.88 & +28:23:11.4 & 1 & 1 & 1 & 01.095 & 15.557 & 01.276 & 15.905 & 00.713 & 15.177\
V191 & 0 & 13:42:11.60 & +28:23:06.0 & 2 & 2 & 2 & & & & & &\
V192 & 0 & 13:42:11.31 & +28:22:46.9 & 1 & 1 & 1 & 00.308 & 14.328 & 00.454 & 14.868 & 00.153 & 13.523\
V193 & 0 & 13:42:12.61 & +28:22:35.8 & 1 & 1 & 1 & 00.811 & 15.255 & 01.000 & 15.674 & 00.615 & 15.010\
V194 & 0 & 13:42:12.74 & +28:22:30.2 & 1 & 1 & 2 & 00.621 & 15.250 & 00.476 & 15.294 & &\
V195 & 0 & 13:42:10.48 & +28:22:14.8 & 1 & 1 & 0 & 00.389 & 15.444 & 00.376 & 15.914 & &\
V197 & 0 & 13:42:15.91 & +28:22:52.2 & 1 & 1 & 1 & 01.347 & 15.505 & 01.455 & 15.586 & 00.764 & 15.089\
V200 & 0 & 13:42:11.19 & +28:23:04.1 & 2 & 2 & 0 & & & & & &\
V201 & 0 & 13:42:11.78 & +28:22:34.0 & 1 & 1 & 1 & 00.847 & 15.345 & 01.058 & 15.778 & 00.554 & 15.079\
V202 & 1 & 13:41:42.83 & +28:24:17.5 & 1 & 1 & 1 & 00.142 & 15.574 & 00.214 & 16.007 & 00.099 & 14.971\
V207 & 1 & 13:42:14.19 & +28:22:11.9 & 1 & 1 & 1 & 00.366 & 15.381 & 00.397 & 15.675 & 00.196 & 14.966\
V208 & 0 & 13:42:11.72 & +28:21:44.5 & 1 & 1 & 1 & 00.397 & 15.502 & 00.522 & 15.860 & 00.256 & 15.135\
V212 & 0 & 13:42:09.87 & +28:22:04.4 & 1 & 1 & 0 & 00.858 & 15.475 & 00.972 & 15.938 & &\
V213 & 0 & 13:42:09.57 & +28:22:12.8 & 1 & 1 & 1 & 00.468 & 15.461 & 00.529 & 15.753 & 00.396 & 15.046\
V214 & 0 & 13:42:13.94 & +28:22:48.9 & 1 & 2 & 1 & 00.920 & 15.292 & & & 00.549 & 15.237\
V215 & 0 & 13:42:10.43 & +28:22:41.7 & 1 & 1 & 1 & 00.793 & 15.423 & 00.797 & 15.664 & 00.501 & 15.188\
V216 & 0 & 13:42:13.60 & +28:22:31.6 & 1 & 1 & 1 & 00.525 & 15.703 & 00.517 & 15.863 & 00.393 & 15.542\
V218 & 0 & 13:42:13.62 & +28:22:13.2 & 1 & 1 & 1 & 00.806 & 15.738 & 00.936 & 16.037 & 00.471 & 15.343\
V220 & 1 & 13:42:13.98 & +28:22:27.1 & 2 & 2 & 2 & & & & & &\
V221 & 0 & 13:42:10.21 & +28:22:28.9 & 1 & 1 & 1 & 00.358 & 15.020 & 00.445 & 15.418 & 00.601 & 14.672\
V223 & 0 & 13:42:13.28 & +28:22:36.6 & 1 & 1 & 0 & 00.605 & 15.647 & 00.673 & 15.950 & &\
V229 & 0 & 13:42:09.00 & +28:21:56.8 & 2 & 0 & 1 & & & & & 00.869 & 15.272\
V234 & 0 & 13:42:13.09 & +28:22:02.9 & 1 & 1 & 1 & 01.137 & 15.800 & 01.260 & 16.184 & 00.897 & 15.476\
V235 & 0 & 13:42:13.77 & +28:23:19.8 & 1 & 1 & 1 & 00.507 & 15.516 & 00.588 & 15.886 & 00.309 & 14.948\
V237 & 0 & 13:42:15.73 & +28:18:17.3 & 1 & 1 & 1 & 00.232 & 18.019 & 00.228 & 18.288 & 00.243 & 17.743\
V239 & 0 & 13:42:09.86 & +28:22:16.0 & 2 & 2 & 0 & & & & & &\
V240 & 0 & 13:42:09.46 & +28:22:35.3 & 1 & 1 & 1 & 00.511 & 15.624 & 00.479 & 15.785 & 00.258 & 15.200\
V241 & 0 & 13:42:10.76 & +28:22:38.0 & 2 & 1 & 2 & & & 00.962 & 14.912 & &\
V243 & 0 & 13:42:12.25 & +28:22:15.6 & 1 & 1 & 1 & 00.514 & 15.335 & 00.612 & 15.758 & 00.327 & 14.805\
V245 & 0 & 13:42:09.89 & +28:23:00.0 & 1 & 1 & 0 & 00.481 & 15.414 & 00.576 & 15.713 & &\
V246 & 0 & 13:42:12.76 & +28:22:40.5 & 1 & 1 & 0 & 00.600 & 15.698 & 00.472 & 15.630 & &\
V247 & 0 & 13:42:14.81 & +28:22:15.6 & 2 & 2 & 2 & & & & & &\
V249 & 0 & 13:42:10.31 & +28:22:48.2 & 2 & 2 & 0 & & & & & &\
V250 & 0 & 13:42:10.51 & +28:22:52.5 & 1 & 1 & 0 & 00.823 & 14.958 & 00.986 & 15.468 & &\
V252 & 0 & 13:42:10.99 & +28:22:44.0 & 1 & 1 & 1 & 00.878 & 15.728 & 00.785 & 15.905 & 00.512 & 15.128\
V253 & 0 & 13:42:12.14 & +28:22:32.8 & 1 & 1 & 1 & 00.531 & 15.562 & 00.548 & 15.805 & 00.397 & 15.225\
V254 & 0 & 13:42:12.42 & +28:22:53.6 & 2 & 2 & 0 & & & & & &\
V255 & 0 & 13:42:12.62 & +28:22:43.5 & 2 & 2 & 0 & & & & & &\
V256 & 1 & 13:42:13.08 & +28:22:59.0 & 2 & 2 & 2 & & & & & &\
V258 & 0 & 13:42:14.31 & +28:23:31.5 & 1 & 1 & 1 & 00.552 & 15.600 & 00.733 & 16.004 & 00.390 & 15.081\
V259 & 0 & 13:42:14.57 & +28:22:54.9 & 1 & 1 & 1 & 00.293 & 15.114 & 00.397 & 15.571 & 00.139 & 14.541\
V261 & 0 & 13:42:10.08 & +28:22:40.4 & 1 & 1 & 1 & 00.361 & 15.077 & 00.409 & 15.355 & 00.220 & 14.659\
V264 & 0 & 13:42:10.87 & +28:22:29.8 & 2 & 1 & 0 & & & 00.288 & 14.493 & &\
V269 & 0 & 13:42:12.79 & +28:22:33.0 & 2 & 2 & 2 & & & & & &\
V270 & 1 & 13:42:11.93 & +28:23:32.2 & 1 & 1 & 1 & 00.507 & 14.503 & 00.553 & 14.751 & 00.284 & 14.096\
V271 & 1 & 13:42:12.16 & +28:23:18.6 & 2 & 2 & 0 & & & & & &\
NV286 & 1 & 13:41:51.55 & +28:17:00.6 & 1 & 1 & 1 & 00.171 & 17.406 & 00.104 & 17.816 & 00.128 & 16.304\
NV287 & 1 & 13:41:41.00 & +28:20:55.5 & 1 & 1 & 1 & 00.060 & 16.964 & 00.107 & 17.747 & 00.064 & 15.935\
NV288 & 0 & 13:42:17.39 & +28:13:35.1 & 1 & 1 & 1 & 00.087 & 17.489 & 00.079 & 17.731 & 00.051 & 17.204\
NV289 & 1 & 13:42:12.17 & +28:19:20.9 & 1 & 1 & 1 & 00.109 & 17.399 & 00.095 & 17.692 & 00.084 & 17.079\
NV290 & 1 & 13:42:21.30 & +28:23:45.0 & 1 & 0 & 0 & 00.060 & 15.670 & & & &\
NV291 & 0 & 13:42:18.07 & +28:22:39.6 & 1 & 1 & 1 & 00.838 & 17.462 & 00.978 & 17.697 & 00.519 & 17.141\
NV292 & 0 & 13:42:11.18 & +28:21:54.0 & 1 & 1 & 0 & 00.267 & 15.704 & 00.251 & 15.809 & &\
NV293 & 0 & 13:42:20.59 & +28:28:32.8 & 1 & 0 & 0 & 00.087 & 17.813 & & & &\
NV294 & 0 & 13:42:13.69 & +28:25:30.3 & 1 & 0 & 0 & 00.093 & 18.024 & & & &\
NV295 & 0 & 13:41:58.56 & +28:27:59.4 & 1 & 1 & 1 & 00.196 & 18.272 & 00.164 & 18.529 & 00.162 & 17.985\
NV296 & 0 & 13:41:42.98 & +28:24:04.0 & 1 & 1 & 1 & 00.466 & 18.767 & 00.310 & 19.134 & 00.398 & 18.307\
NV297 & 1 & 13:41:31.72 & +28:24:10.9 & 1 & 1 & 1 & 00.046 & 15.844 & 00.067 & 16.811 & 00.024 & 14.465\
\[tab:tab2\]
[lrrcrrrrl]{} V001 & 0.520300 & 0.520596 & 0 & 920.705400 & 0.268883 & 0.305977 & RR0 & Bl?\
V004 & 0.585033 & 0.584900 & 0 & 920.753434 & & 0.109739 & RR0 & Bl?\
V005 & 0.501451 & 0.504178 & 1 & 920.760300 & 0.107879 & 0.990228 & RR0 & Bl\
V006 & 0.514333 & 0.514333 & 0 & 920.835834 & 0.672006 & 0.009527 & RR0 &\
V007 & 0.497396 & 0.497429 & 0 & 920.866716 & 0.805403 & 0.974837 & RR0 &\
V009 & 0.541545 & 0.541553 & 0 & 921.072365 & 0.006666 & 0.002899 & RR0 &\
V010 & 0.569430 & 0.569544 & 0 & 920.748500 & 0.720984 & 0.048996 & RR0 &\
V011 & 0.507910 & 0.507894 & 0 & 920.871500 & 0.999291 & 0.015948 & RR0 &\
V012 & 0.317968 & 0.317540 & 1 & 920.737396 & 0.980589 & 0.070523 & RR1 &\
V017 & 0.575998 & 0.576159 & 1 & 921.191002 & 0.066674 & 0.016840 & RR0 & Bl?\
V018 & 0.516411 & 0.516451 & 0 & 920.928300 & 0.078831 & 0.072229 & RR0 &\
V019 & 0.631976 & 0.631972 & 0 & 920.803000 & 0.001000 & 0.009766 & RR0 & Bl?\
V020 & 0.491302 & 0.490476 & 1 & 920.785288 & 0.992062 & 0.130673 & RR0 &\
V021 & 0.515786 & 0.515756 & 0 & 920.939100 & 0.286425 & 0.116502 & RR0 &\
V022 & 0.481424 & 0.481424 & 0 & 920.986068 & 0.069577 & 0.987745 & RR0 & Bl\
V023 & 0.595376 & 0.595376 & 0 & 920.942172 & 0.849769 & 0.016292 & RR0 & Bl\
V024 & 0.663372 & 0.663372 & 0 & 920.808884 & 0.998667 & 0.022937 & RR0 & Bl\
V027 & 0.579087 & 0.579073 & 0 & 921.214913 & 0.974076 & 0.988993 & RR0 &\
V030 & 0.512092 & 0.512092 & 0 & 920.833332 & & & RR0 &\
V031 & 0.580720 & 0.580720 & 0 & 921.008860 & 0.061716 & 0.013776 & RR0 & Bl?\
V032 & 0.495350 & 0.495350 & 0 & 921.107500 & 0.791662 & 0.006258 & RR0 &\
V034 & 0.559050 & 0.560963 & 1 & 920.860200 & 0.016725 & 0.130579 & RR0 & Bl\
V036 & 0.545599 & 0.545599 & 0 & 921.000907 & 0.945748 & 0.981488 & RR0 &\
V037 & 0.326639 & 0.326639 & 0 & 920.967091 & 0.056919 & 0.057786 & RR1 &\
V038 & 0.558185 & 0.558011 & 0 & 921.025545 & 0.909089 & 0.891219 & RR0 & Bl?\
V039 & 0.587067 & 0.587067 & 0 & 921.348533 & 0.002783 & 0.025438 & RR0 &\
V040 & 0.551535 & 0.551535 & 0 & 920.761430 & 0.989919 & 0.030533 & RR0 &\
V043 & 0.540532 & 0.540510 & 0 & 920.975472 & 0.053333 & 0.014800 & RR0 &\
V044 & 0.506254 & 0.337850 & 0 & 920.924892 & 0.953762 & 0.037728 & RR0 & Bl\
V045 & 0.536073 & 0.536073 & 0 & 920.886560 & 0.073397 & 0.672720 & RR0 & Bl\
V050 & 0.512891 & 0.513170 & 1 & 921.143445 & 0.934918 & 0.994740 & RR0 &\
V051 & 0.583970 & 0.583970 & 0 & 920.869800 & 0.993150 & 0.002654 & RR0 & Bl\
V053 & 0.504881 & 0.504881 & 0 & 920.851000 & 0.969696 & 0.001218 & RR0 &\
V054 & 0.506247 & 0.506247 & 0 & 920.804800 & 0.992703 & 0.943901 & RR0 & Bl?\
V055 & 0.529822 & 0.529822 & 0 & 920.891956 & 0.994043 & 0.010203 & RR0 &\
V056 & 0.329605 & 0.329600 & 0 & 920.839785 & 0.002670 & 0.065487 & RR1 &\
V057 & 0.512191 & 0.512191 & 0 & 920.852656 & 0.979695 & 0.070478 & RR0 &\
V059 & 0.588727 & 0.588826 & 0 & 921.113473 & 0.002257 & 0.989809 & RR0 & Bl\
V060 & 0.707722 & 0.707727 & 0 & 921.270814 & 0.985164 & 0.000814 & RR0 &\
V061 & 0.520941 & 0.520926 & 0 & 921.142959 & 0.958763 & 0.201435 & RR0 & Bl?\
V062 & 0.652418 & 0.652418 & 0 & 921.309430 & 0.915695 & 0.075022 & RR0 &\
V063 & 0.570282 & 0.570382 & 0 & 920.951790 & 0.009820 & 0.042435 & RR0 &\
V064 & 0.605465 & 0.605465 & 0 & 921.317740 & 0.878928 & 0.005285 & RR0 &\
V065 & 0.668349 & 0.668347 & 0 & 921.393302 & 0.987002 & 0.007501 & RR0 &\
V066 & 0.620055 & 0.619100 & 1 & 921.305315 & & & RR0 & Bl\
V067 & 0.568333 & 0.568333 & 0 & 921.186767 & 0.008622 & 0.996189 & RR0 & Bl\
V068 & 0.358690 & 0.358690 & 0 & 920.998700 & 0.015612 & 0.077755 & RR01 &\
V069 & 0.566615 & 0.566615 & 0 & 921.012215 & 0.938203 & 0.930535 & RR0 &\
V070 & 0.486276 & 0.486093 & 1 & 921.023112 & 0.961314 & 0.101580 & RR1 &\
V071 & 0.549053 & 0.549053 & 0 & 920.855294 & 0.865707 & 0.053314 & RR0 & Bl?\
V072 & 0.456078 & 0.456078 & 0 & 920.761354 & 0.747315 & 0.986384 & RR0 &\
V073 & 0.673659 & 0.670799 & 1 & 920.748500 & 0.982702 & 0.941698 & RR0 & Bl\
V074 & 0.492152 & 0.492152 & 0 & 920.884772 & 0.988418 & 0.310481 & RR0 &\
V075 & 0.314080 & 0.314080 & 0 & 920.860200 & 0.024134 & 0.885061 & RR1 &\
V078 & 0.611965 & 0.611965 & 0 & 920.930100 & 0.936434 & 0.956615 & RR0 & Bl?\
V079 & 0.361269 & 0.357269 & 1 & 921.049955 & 0.984222 & 0.990569 & RR01 &\
V080 & 0.538331 & 0.537556 & 1 & 921.207445 & 0.995843 & 0.171571 & RR0 & Bl?\
V081 & 0.529122 & 0.529122 & 0 & 921.025146 & 0.794478 & 0.053137 & RR0 &\
V083 & 0.501264 & 0.501264 & 0 & 920.849544 & 0.728933 & 0.009289 & RR0 &\
V084 & 0.595732 & 0.595732 & 0 & 920.923804 & 0.009400 & 0.990096 & RR0 &\
V085 & 0.355808 & 0.355817 & 0 & 920.895076 & 0.989106 & 0.003541 & RR1 &\
V087 & 0.354368 & 0.357478 & 0 & 921.064864 & 0.953427 & 0.078156 & RR01 &\
V090 & 0.517031 & 0.517031 & 0 & 920.900000 & 0.036340 & 0.092138 & RR0 &\
V091 & 0.529369 & 0.529369 & 0 & 920.829773 & 0.217140 & 0.168763 & RR0 & Bl?\
V093 & 0.602294 & 0.602294 & 0 & 921.282644 & 0.746954 & 0.013283 & RR0 & Bl?\
V094 & 0.523716 & 0.523696 & 0 & 920.813968 & 0.984251 & 0.962644 & RR0 &\
V096 & 0.499416 & 0.499416 & 0 & 920.938568 & 0.061472 & 0.088704 & RR0 &\
V097 & 0.334944 & 0.334933 & 0 & 920.805572 & 0.056224 & 0.055317 & RR1 &\
V099 & 0.482199 & 0.361696 & 1 & 921.013804 & 0.914143 & 0.113858 & RR01 & Bl?\
V100 & 0.618813 & 0.618813 & 0 & 920.813061 & 0.815114 & 0.005171 & RR0 &\
V101 & 0.643886 & 0.643886 & 0 & 921.284314 & 0.967451 & 0.073370 & RR0 & Bl\
V104 & 0.569946 & 0.569931 & 0 & 921.171862 & 0.989999 & 0.061220 & RR0 &\
V105 & 0.287744 & 0.287744 & 0 & 921.001260 & 0.007187 & 0.058274 & RR1 &\
V108 & 0.519615 & 0.519615 & 0 & 921.163810 & 0.987683 & 0.024586 & RR0 &\
V117 & 0.600529 & 0.597263 & 1 & 921.078313 & 0.061159 & 0.028905 & RR0 & Bl?\
V118 & 0.499391 & 0.499391 & 0 & 920.871036 & 0.014818 & 0.042051 & RR0 &\
V119 & 0.517692 & 0.517692 & 0 & 921.188872 & 0.043431 & 0.994754 & RR0 &\
V120 & 0.640139 & 0.640139 & 0 & 920.758900 & 0.917830 & 0.003601 & RR0 &\
V121 & 0.535211 & 0.535211 & 0 & 921.211745 & & 0.091366 & RR0 & Bl\
V122 & 0.509621 & 0.516090 & 0 & 920.918108 & 0.274808 & 0.078937 & RR0 &\
V125 & 0.349780 & 0.349823 & 0 & 920.924420 & 0.035165 & 0.998227 & RR1 &\
V128 & 0.292044 & 0.292040 & 0 & 920.748500 & 0.869759 & 0.978962 & RR1 &\
V129 & 0.406102 & 0.406102 & 0 & 920.957600 & & & RR0 &\
V130 & 0.565181 & 0.567840 & 1 & 921.182519 & 0.897622 & 0.930886 & RR0 & Bl\
V134 & 0.618057 & 0.618057 & 0 & 921.283896 & 0.015023 & 0.996254 & RR0 &\
V135 & 0.568591 & 0.568397 & 0 & 921.171075 & 0.779662 & 0.978107 & RR0 &\
V136 & 0.617153 & 0.617186 & 0 & 921.176847 & 0.031759 & 0.971968 & RR0 &\
V137 & 0.575161 & 0.575161 & 0 & 921.070864 & 0.893209 & 0.887390 & RR0 &\
V139 & 0.559970 & 0.559970 & 0 & 920.860200 & 0.007697 & 0.957783 & RR0 &\
V140 & 0.333499 & 0.333499 & 0 & 920.813726 & 0.999409 & 0.121362 & RR1 & Bl?\
V142 & 0.568628 & 0.568628 & 0 & 920.790800 & 0.023045 & 0.014069 & RR0 &\
V143 & 0.596739 & 0.596535 & 0 & 921.340661 & 0.034987 & 0.019235 & RR0 &\
V145 & 0.514456 & 0.514456 & 0 & 920.757864 & 0.733178 & 0.072504 & RR0 &\
V146 & 0.502149 & 0.596745 & 1 & 920.939100 & 0.011152 & 0.934490 & RR0 &\
V147 & 0.346495 & 0.346479 & 0 & 921.092130 & 0.000029 & 0.966839 & RR1 &\
V148 & 0.467252 & 0.467290 & 0 & 920.988996 & 0.984154 & 0.016907 & RR0 &\
V149 & 0.548252 & 0.548150 & 0 & 920.971600 & 0.966512 & 0.221876 & RR0 &\
V150 & 0.523919 & 0.523919 & 0 & 920.811480 & 0.070358 & 0.156440 & RR0 & Bl\
V151 & 0.516497 & 0.517767 & 1 & 921.141115 & 0.975617 & 0.018757 & RR0 &\
V152 & 0.326130 & 0.326135 & 0 & 920.865910 & 0.017171 & 0.964094 & RR1 &\
V156 & 0.532003 & 0.531987 & 0 & 921.113154 & 0.630164 & 0.956626 & RR0 &\
V157 & 0.543086 & 0.542820 & 1 & 921.236948 & 0.064435 & 0.043076 & RR0 &\
V159 & 0.534461 & 0.533260 & 1 & 921.009442 & 0.211989 & 0.118682 & RR0 & Bl?\
V160 & 0.656928 & 0.657324 & 0 & 921.116672 & 0.022377 & 0.080551 & RR0 &\
V161 & 0.526380 & 0.521640 & 1 & 921.221320 & 0.955925 & 0.073217 & RR0 &\
V165 & 0.483630 & 0.483630 & 0 & 920.925300 & 0.980936 & 0.278767 & RR0 &\
V168 & 0.274279 & 0.275970 & 1 & 920.865384 & & 0.849482 & RR1 &\
V170 & 0.432339 & 0.432260 & 0 & 921.088722 & 0.871899 & 0.978309 & RR0 &\
V171 & 0.303307 & 0.303828 & 1 & 920.786330 & 0.965052 & 0.026376 & RR1 &\
V172 & 0.542261 & 0.542288 & 0 & 920.859959 & 0.997274 & & RR0 &\
V173 & 0.607016 & 0.607016 & 0 & 921.186984 & 0.020296 & 0.103780 & RR0 &\
V174 & 0.592278 & 0.587030 & 1 & 920.802710 & 0.829377 & 0.863071 & RR0 & Bl\
V175 & 0.569703 & 0.569700 & 0 & 921.105385 & 0.958908 & & RR0 &\
V176 & 0.539702 & 0.540593 & 1 & 921.105838 & 0.066155 & & RR0 & Bl?\
V177 & 0.348338 & 0.348749 & 1 & 921.021524 & 0.931721 & 0.018993 & RR1 &\
V178 & 0.266981 & 0.267387 & 1 & 920.748500 & 0.061053 & 0.025845 & RR1 &\
V180 & 0.609131 & 0.615930 & 1 & 921.312176 & 0.010697 & & RR0 &\
V181 & 0.664073 & 0.663840 & 0 & 920.813281 & 0.075114 & & RR0 &\
V184 & 0.531176 & 0.532130 & 1 & 920.896900 & 0.852682 & 0.994819 & RR0 &\
V187 & 0.585874 & 0.584126 & 1 & 921.311214 & 0.015553 & 0.952741 & RR0 &\
V188 & 0.266510 & 0.266288 & 1 & 920.857460 & 0.983828 & 0.029642 & RR1 &\
V189 & 0.612931 & 0.612940 & 0 & 920.984800 & 0.888405 & 0.869643 & RR0 &\
V190 & 0.522735 & 0.522803 & 0 & 921.165465 & 0.639846 & 0.960018 & RR0 &\
V191 & 0.525798 & 0.520030 & 1 & 920.770800 & 0.962122 & 0.308864 & RR0 &\
V192 & 0.497111 & 0.497900 & 1 & 920.929278 & 0.017658 & 0.692467 & RR0 &\
V193 & 0.747840 & 0.747840 & 0 & 920.970360 & 0.059104 & 0.059959 & RR0 & Bl?\
V194 & 0.489200 & 0.489200 & 0 & 920.889000 & 0.848937 & 0.229967 & RR0 &\
V195 & 0.643940 & 0.643940 & 0 & 920.898380 & 0.133118 & & RR0 & Bl?\
V197 & 0.499904 & 0.499904 & 0 & 920.764676 & 0.016379 & 0.016003 & RR0 &\
V200 & 0.358013 & 0.359820 & 1 & 921.002184 & 0.970811 & & RR01 &\
V201 & 0.540571 & 0.541410 & 1 & 920.749258 & 0.917894 & 0.842581 & RR0 & Bl?\
V202 & 0.773562 & 0.773562 & 0 & 921.023538 & 0.999889 & 0.074996 & RR0 &\
V207 & 0.345329 & 0.344580 & 1 & 920.861604 & 0.042568 & 0.977804 & RR1 &\
V208 & 0.338234 & 0.338020 & 0 & 920.898494 & 0.995867 & 0.961275 & RR1 &\
V212 & 0.542136 & 0.542196 & 0 & 920.804712 & 0.977128 & & RR0 & Bl?\
V213 & 0.299944 & 0.299706 & 1 & 920.865036 & 0.002547 & 0.904996 & RR1 &\
V214 & 0.539302 & 0.540450 & 1 & 921.266446 & 0.990758 & 0.232990 & RR0 &\
V214 & 0.539489 & 0.540450 & 1 & 921.227229 & & & RR0 &\
V215 & 0.528949 & 0.527100 & 1 & 921.184451 & 0.835527 & 0.892058 & RR0 &\
V216 & 0.346484 & 0.346484 & 0 & 920.992228 & 0.956627 & 0.030443 & RR1 &\
V218 & 0.544870 & 0.544870 & 0 & 920.770010 & 0.959623 & 0.864848 & RR0 & Bl\
V220 & 0.600110 & 0.600110 & 0 & 920.975090 & 0.073636 & 0.096449 & RR0 &\
V221 & 0.378752 & 0.378752 & 0 & 920.843884 & 0.118019 & 0.087656 & RR1 &\
V223 & 0.329210 & 0.329586 & 1 & 920.855640 & 0.894201 & & RR1 &\
V229 & 0.503635 & 0.687700 & 1 & 920.978435 & & 0.974515 & RR0 &\
V234 & 0.507925 & 0.549500 & 1 & 920.785750 & 0.922823 & 0.900182 & RR0 &\
V235 & 0.759901 & 0.765190 & 1 & 920.924695 & 0.978616 & 0.039742 & RR0 & Bl?\
V237 & 0.041984 & & 1 & 920.693148 & 0.209223 & 0.111185 & SXP & mp\
V239 & 0.497584 & 0.669490 & 1 & 921.034288 & 0.080003 & & RR0 &\
V240 & 0.276010 & 0.276010 & 0 & 921.004000 & 0.071012 & 0.135394 & RR1 &\
V241 & 0.596172 & 0.594080 & 1 & 920.756216 & 0.766980 & 0.946418 & RR0 &\
V243 & 0.634288 & 0.629910 & 1 & 921.108612 & 0.925611 & 0.947210 & RR0 & Bl?\
V245 & 0.284037 & 0.284030 & 0 & 920.872941 & 0.858371 & & RR1 &\
V246 & 0.339171 & 0.339140 & 0 & 920.993319 & 0.032376 & & RR1 &\
V247 & 0.605350 & 0.605350 & 0 & 921.152200 & 0.010209 & 0.031635 & RR0 &\
V249 & 0.533010 & 0.533010 & 0 & 921.201700 & 0.985666 & & RR0 & Bl?\
V250 & 0.590310 & 0.590310 & 0 & 920.847190 & 0.823415 & & RR0 & Bl\
V252 & 0.501550 & 0.501550 & 0 & 920.950300 & 0.855647 & 0.832519 & RR01 &\
V253 & 0.332661 & 0.331610 & 1 & 920.850651 & 0.965971 & 0.931402 & RR01 &\
V254 & 0.605519 & 0.604550 & 0 & 921.140032 & 0.873781 & & RR0 &\
V255 & 0.572640 & 0.572640 & 0 & 921.203710 & 0.651841 & & RR0 &\
V256 & 0.320459 & 0.318360 & 1 & 920.887774 & 0.945157 & 0.909177 & RR1 &\
V258 & 0.713374 & 0.713409 & 0 & 921.050404 & 0.897383 & 0.981948 & RR0 &\
V259 & 0.333463 & 0.332870 & 1 & 920.823344 & 0.037054 & 0.092532 & RR1 &\
V261 & 0.444942 & 0.444140 & 1 & 920.945216 & 0.988843 & 0.972994 & RR1 &\
V264 & 0.356490 & 0.356490 & 0 & 920.978930 & 0.903980 & & RR1 &\
V269 & 0.355657 & 0.290000 & 1 & 921.053598 & 0.902282 & 0.952330 & RR1 &\
V270 & 0.690170 & 0.690170 & 0 & 921.052730 & 0.249736 & 0.178637 & RR01 &\
V271 & 0.632472 & 0.630430 & 1 & 921.344022 & 0.004165 & & RR0 &\
NV286 & & & 0 & & & & N/A &\
NV287 & & & 0 & & & & N/A &\
NV288 & 0.036600 & & 1 & 920.682300 & 0.147541 & 0.874317 & SXP & mp\
NV289 & 0.035789 & & 1 & 920.703144 & 0.045321 & 0.296544 & SXP & mp\
NV290 & 0.240425 & & 1 & 920.904200 & & & RR01 &\
NV291 & 0.071629 & & 1 & 920.770188 & 0.924779 & 0.940555 & SXP &\
NV292 & 0.296579 & & 1 & 920.798310 & 0.887012 & & RR1 &\
NV293 & 0.029462 & & 1 & 920.771768 & & & SXP &\
NV294 & 0.038827 & & 1 & 920.781886 & & & SXP &\
NV295 & 0.036081 & & 1 & 920.763793 & 0.842549 & 0.853940 & SXP &\
NV296 & 0.445955 & & 1 & 921.127710 & 0.006637 & 0.950825 & EB &\
NV297 & 0.402037 & & 1 & 920.948141 & 0.146715 & 0.748595 & RR1 &\
\[tab:tab3\]
[^1]: IRAF is distributed by the National Optical Astronomy Observatories, which is operated by the Association of Universities for Research in Astronomy, Inc., under agreement with the National Science Foundation.
[^2]: ISIS package is available from C. Alard’s web-site at http://www2.iap.fr/users/alard/package.html
[^3]: ftp://cfa-ftp.harvard.edu/pub/kstanek/M3/
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The Panofsky-Wenzel theorem connects the transverse deflecting force in an rf structure with the existence of a longitudinal electric field component. In this paper it is shown that a transverse deflecting force is always accompanied by an additional longitudinal magnetic field component which leads to an emittance growth in the direction perpendicular to the transverse force. Transverse deflecting waves can thus not be described by pure TM or TE modes, but require a linear combination of basis modes for their representation. The mode description is preferably performed in the HM–HE basis to avoid converge problems, which are fundamental for the TM–TE basis.'
author:
- 'Valentin V. Paramonov'
- Klaus Floettmann
title: |
Report 18-103, DESY, Hamburg\
Fundamental characteristics of transverse deflecting fields
---
Introduction
============
Transverse deflecting rf fields find numerous applications in modern particle accelerators for example as particle separators [@Edwards; @2001] or fast rf deflectors [@Alesini; @2009], as streaking device for diagnostics purposes [@Ratner; @2015], in emittance exchange beam lines [@Emma; @2011] or as crab cavities in circular colliders as the LHC [@Burt; @2014]. It is well known, that a beam passing through a transverse deflecting field will not only receive the desired phase dependent transverse momentum, but it will also change its energy spread due to a longitudinal electric field component which varies over the transverse size of the beam. The fundamental relation of the transverse gradient of the longitudinal electric field component and the transverse momentum was formulated by Panofsky and Wenzel in their seminal paper in 1956 [@Pif]. Originally derived in the context of transverse deflecting rf structures, which is also the focus of this paper, the Panofsky-Wenzel theorem became a fundamental relation also for the discussion of wake potentials [@Wilson] and devices as pickups and kickers [@Lambertson].\
Complementary to the longitudinal electric field component a longitudinal magnetic field component exists in transverse deflecting rf fields which has been widely ignored so far. The existence of the longitudinal magnetic component requires to revisit the general mode description of transverse deflecting rf structures. Due to the coupling of the transverse motion to the longitudinal magnetic field it leads to a small, but fundamental contribution to the transverse beam emittance in the direction perpendicular to the transverse force.
Supporting structure
====================
To provide an effective interaction between a particle, moving with the velocity $v_z=\beta_z c$, and an electromagnetic rf field it is necessary to match the phase velocity $v_{ph}$ of a harmonic field component to the velocity of the particle. Since the phase velocity in a simple waveguide is higher than the speed of light $c$, while $\beta_z \leq 1.0$ it is necessary to slow down the wave in an appropriate structure.\
![Supporting structures for slow deflecting waves: periodical iris loaded circular waveguide, left, and dielectric lined waveguide, right.[]{data-label="fig_1"}](geometry_new_small.eps){width="85mm"}
Commonly periodically iris loaded structures as the example shown in Fig. \[fig\_1\], left, are employed to achieve this. The field distribution in a periodical structure represents a sum of spatial harmonics and by proper selection of the period length $d= \frac{\beta_z \lambda \theta}{2 \pi}$, $0 \leq \theta \leq \pi$ a synchronous harmonic can be generated. $\theta$ is the phase advance per period and the wavelength $\lambda$ for this kind of structures is typically in the range of 20-3 cm (L– to X–band). A fundamental property of periodical structures is the appearance of spatial harmonics which lead to nonlinearities in the field distribution. The nonlinearities can not be completely eliminated but they can be minimized in the region occupied by the beam by a proper design of the structure [@FlPa2014; @Paramonov2013].\
Spatial harmonics do not appear in structures which are uniform in the longitudinal $z$ direction. To slow down the wave the structure can be partially filled with a dielectric medium (see Fig. \[fig\_1\], right) but also thin metallic layers [@Tsakanov] or even a rough surface [@Novokhatsky] can lead to slow waves.\
Dielectric lined waveguides have been discussed already in the 1960s [@Vagin; @Dawson] as candidates for beam separators operating at typical rf frequencies (3 GHz). They gain now interest again as streaking device for diagnostics purposes operating in the sub-THz to THz range [@dlw_tds], where the dimensions are so small that the production of periodical structures reaches technical limits. The demand to resolve ever shorter bunch length and also the progress in the generation of THz pulses of sufficient power and pulse length [@Ahr2017] are driving forces of these developments. Moreover, the absence of spatial harmonics makes dielectric lined waveguides also attractive from the beam dynamics point of view, because undesired side effects, like undesired emittance contributions, are minimized to their fundamental limits.
General relations and mode description
======================================
The transverse deflecting force $\vec F_{\perp}$ acting on a particle moving along the longitudinal axis $z$ with the velocity $\vec V= \vec i_z v_z$ is defined by the transverse components of the Lorentz force $$\begin{gathered}
\vec F_{\perp} =e(E_x-v_z B_y) \vec i_x+e(E_y+v_z B_x) \vec i_y, \hfill \\
\vec F_{\perp} =e(E_r-v_z B_{\vartheta}) \vec i_r+e(E_{\vartheta}+v_z B_r) \vec i_{\vartheta}, \hfill
\end{gathered}
\label{Eq. 0}$$ where $\vec i_x, \vec i_y, \vec i_z$ and $\vec i_{\vartheta}, \vec i_r, \vec i_z$ are the basis vectors of the Cartesian and cylindrical coordinate system, respectively.\
The fundamental relations of deflecting field components and the corresponding longitudinal electric and magnetic field components follow directly from Maxwell’s equations $\text{curl} \vec E = - \frac{\partial }{{\partial t}} \vec B$ and $\text{curl} \vec B = \frac{\partial }{{\partial t}} \frac{\vec E}{c^2}$ as: $$\begin{gathered}
\frac{\partial}{\partial x} E_z =\frac{\partial}{\partial z} E_x+ \frac{\partial}{\partial t} B_y, \hfill \\
\frac{\partial}{\partial y} cB_z =\frac{\partial}{\partial t} \frac{E_x}{c}+ \frac{\partial}{\partial z} c B_y. \hfill
\end{gathered}
\label{Eq. 1}$$ For a wave oscillating with frequency $\omega$ and wavenumber $k_z=\omega/v_{ph}$ as $\propto \text{e}^{i(\omega t - k_z z)}$ the relation $\frac{1}{\partial t}=-\frac{v_{ph}}{\partial z}$ holds, thus $$\begin{gathered}
\frac{\partial }{{\partial x}}{ E_z} = \frac{\partial }{\partial z}\left( E_x-v_{ph}B_y \right), \hfill \\
-\frac{\partial }{{\partial y}}{cB_z} = \frac{\partial }{{\partial z}} \left(\frac{v_{ph}}{c}E_x-cB_y \right).\hfill
\end{gathered}
\label{Eq. 2}$$ Equivalent transformations can be applied to the $E_y$ and $B_x$ components, resulting in $$\begin{gathered}
\frac{\partial }{{\partial y}}{ E_z} = \frac{\partial }{\partial z}\left( E_y+v_{ph}B_x \right), \hfill \\
\frac{\partial }{{\partial x}}{cB_z} = \frac{\partial }{{\partial z}} \left(\frac{v_{ph}}{c}E_y+cB_x \right).\hfill
\end{gathered}
\label{Eq. 3}$$ For a particle, traveling with longitudinal velocity component $v_z=v_{ph}$ synchronously with the wave, the first equations in Eqs \[Eq. 2\] and \[Eq. 3\] mean: $$\frac{1}{e} \frac{\partial}{\partial z} \vec F_{\perp}= -\frac{1}{e v_{ph}} \frac{\partial}{\partial t} \vec F_{\perp}=
-i \frac{k_z}{e} \vec F_{\perp}= \vec{\nabla}_{\perp} E_z,
\label{Eq. 4}$$ where $\vec{\nabla}_{\perp} = \vec i_x \frac{\partial }{\partial x} +\vec i_y \frac{\partial }{\partial y}$, see [@Garault] and references therein.\
A synchronous transverse force $\vec F_t \neq 0$ can thus only exist together with a longitudinal component of the electric field $E_z$ in a deflecting rf field. The second equations in Eqs \[Eq. 2\] and \[Eq. 3\] necessarily require the simultaneous existence of a longitudinal magnetic field component and for the particular case $v_z=v_{ph}=c$ the relation $$c \left [ \vec i_z \times \vec{\nabla} B_z \right ]_{\perp} = -i \frac{k_z}{e} \vec F_{\perp}= \vec{\nabla}_{\perp} E_z,
\label{Eq. 5}$$ holds.\
A transverse force $\vec F_{\perp}$ is thus accompanied by both, a longitudinal electric, as well as a longitudinal magnetic field component.\
A general cylindrical symmetric representation of the electro-magnetic fields in vacuum in a finite domain including the symmetry axis can be based on Hertzian basis vectors, see reference [@Hahn] and references therein for a general discussion. Table \[Tab\_1\] summarizes the transverse TM–TE and the hybrid HM–HE basis vectors, as introduced in [@Hahn], with a modified notation (e.g. $\vec H$ has been replaced by $\vec B$). Here $k_0=\omega/c$, $k_z=\omega/v_{ph}$ and $k_r^2=k_0^2-k_z^2$ is used, while $n \in \mathbb{N}$ defines the azimuthal dependence.\
[ l c c c c c c]{}\
& TM & TE & HM & HE & azimuthal & spatio-temporal\
& & & & & dependence & dependence\
\
$\displaystyle E_r $ & $\displaystyle -k_z\frac{J'_n}{k_r^{n - 1}}$ & $\displaystyle -\frac{n k_0}{r}\frac{J_n}{k_r^n}$ & $\displaystyle k_0 k_z\frac{J_{n + 1}}{k_r^{n + 1}}$ & $\displaystyle k_z^2\frac{J_{n + 1}}{k_r^{n + 1}}+\frac{n}{r}\frac{J_n}{k_r^n}$ & $\displaystyle \cos(-n \vartheta)$ & $\displaystyle i\text{e}^{i(\omega t-k_z z)} $\
$\displaystyle E_\vartheta $ & $\displaystyle \frac{n k_z}{r}\frac{J_n}{k_r^n}$ & $\displaystyle k_0\frac{J'_n}{k_r^{n - 1}}$ & $\displaystyle k_0 k_z\frac{J_{n + 1}}{k_r^{n + 1}}$ & $\displaystyle k_0^2\frac{J_{n + 1}}{k_r^{n + 1}}- \frac{n}{r}\frac{J_n}{k_r^n} $ & $\displaystyle \sin(n \vartheta)$ & $\displaystyle i\text{e}^{i(\omega t-k_z z)}$\
$\displaystyle E_z $ & $\displaystyle k_r^2\frac{J_n}{k_r^n}$ & 0 & $\displaystyle k_0\frac{J_n}{k_r^n}$ & $\displaystyle k_z\frac{J_n}{k_r^n}$ & $\displaystyle \cos(-n \vartheta)$ & $ \displaystyle \text{e}^{i(\omega t-k_z z)}$\
$\displaystyle cB_r $ & $\displaystyle -\frac{n k_0}{r}\frac{J_n}{k_r^n}$ & $\displaystyle -k_z\frac{J'_n}{k_r^{n - 1}}$ & $\displaystyle -k_z^2\frac{J_{n + 1}}{k_r^{n + 1}} -\frac{n}{r}\frac{J_n}{k_r^n} $ & $\displaystyle -k_0 k_z\frac{J_{n + 1}}{k_r^{n + 1}}$ & $\displaystyle \sin(n \vartheta)$ & $\displaystyle i\text{e}^{i(\omega t-k_z z)}$\
$\displaystyle cB_\vartheta $ & $\displaystyle -k_0\frac{J'_n}{k_r^{n - 1}}$ & $\displaystyle -\frac{n k_z}{r}\frac{J_n}{k_r^n}$ & $\displaystyle k_0^2\frac{J_{n + 1}}{k_r^{n + 1}}-\frac{n}{r}\frac{J_n}{k_r^n} $ & $\displaystyle k_0 k_z\frac{J_{n + 1}}{k_r^{n + 1}}$ & $\displaystyle \cos(-n \vartheta) $ & $\displaystyle i\text{e}^{i(\omega t-k_z z)}$\
$\displaystyle cB_z $ & 0 & $\displaystyle k_r^2\frac{J_n}{k_r^n}$ & $\displaystyle -k_z\frac{J_n}{k_r^n}$ & $\displaystyle -k_0\frac{J_n}{k_r^n}$ & $\displaystyle \sin(n \vartheta)$ & $\displaystyle \text{e}^{i(\omega t-k_z z)}$\
\[Tab\_1\]
The expressions of the basis vectors in Table \[Tab\_1\] are presented for $v_{ph} > c$ and $k_r^2 >0$. For waves with $v_{ph} < c$, $k_r$ becomes imaginary and the modified Bessel functions should be used to describe the radial dependencies of the field components.\
In the domain under consideration any field component $G$ can be described by a linear combination of the basis vectors as: $$G = A \times TM +B \times TE\quad \text{or} \quad G= P\times HM+ Q\times HE
\label{Eq. 6}$$ The coefficients $A$, $B$, $P$ and $Q$ are determined by the boundary conditions and the energy balance of the problem under consideration.\
The traditional nomenclature of modes builds up on the TM–TE basis of transverse waves, which is well suited and usually applied for the field description in rf engineering and for modes with $n=0$ symmetry, as e.g. accelerating modes. TM (Transverse Magnetic) and TE (Transverse Electric) waves have each only one longitudinal field component; for TM waves $E_z \neq 0, B_z=0$ and for TE waves $E_z = 0, B_z \neq 0$. It is common practice to assume that the longitudinal field components and thus the coefficients $A$ and $B$ are independent; the discussion above shows however that this is not the general case for $n \ge 1$.\
The general solution of the boundary conditions has thus six field components and requires the linear combination of two basis vectors, either TM and TE or HM and HE. Only for $n=0$ two separate solutions with three field components each can be formulated.\
The TM-TE basis exhibits a methodical convergence problem when the phase velocity approaches $c$. For $v_{ph} \rightarrow c$, $k_z \rightarrow k_0$, $k_r \rightarrow 0$ the longitudinal components $E_z$ and $cB_z$ vanish as $k_r^2$. In addition all transverse field components vanish for $n=0$ but remain finite for $n \geq 1$, cf. Table \[Tab\_1\].\
Comparing the longitudinal field components by means of Eq. \[Eq. 6\] yields the following relations of the coefficients of the two basis: $$A = - \frac{P k_0 + Q k_z}{k_r^2}, \quad
B = \frac{P k_z + Q k_0}{k_r^2}.
\label{Eq. 7}$$ $A$ and $B$ are thus divergent $\propto k_r^{-2}$ when $k_r$ approaches zero. Thus, while the basis vectors converge to zero, the product of basis vectors with the vector coefficients does not converge to zero and the field description is possible also in the limit $v_{ph}=c$.\
By means of the relations Eq. \[Eq. 7\] and the identity $ J'_n=\frac{n J_n}{r k_r} - J_{n + 1}$ the complete equivalence of the two basis can be shown. The deflecting field representation for the transverse modes in a dielectric lined waveguide by Chang and Dawson [@Dawson] in the TM–TE basis is hence essentially the same as the earlier result of Vagin and Kotov [@Vagin] in the HM–HE basis.\
The HM–HE basis has no convergence problem, i.e. no component of the basis converges to zero in the limit $k_r \rightarrow 0$ irrespective of $n$. Since each basis vector contains simultaneously electric and magnetic longitudinal vector components both, linear combinations and also pure HM or HE modes can satisfy the boundary conditions as well as Eq. \[Eq. 4\] and \[Eq. 5\] for $n \ge 1$.\
The description of pure TM or TE modes in the HM–HE basis leads to fixed relations of the vector coefficients as $B=0$, $k_z P=-k_0 Q$, $A= -\frac{P}{k_0}$ for TM modes and $A=0$, $k_0 P=k_z Q$, $B= \frac{Q}{k_0}$ for TE modes. Still $A$ or $B$ need to be divergent in the limit $k_r \rightarrow 0 $ thus also $P$ and $Q$ get divergent while the vector components of the hybrid basis don’t converge to zero. Thus only the sum of HM and HE remains finite in this case.\
The TM–TE basis is therefore preferable for the field description in the case $n=0$, while the HM–HE basis is advantageous for $n \ge 1$.
Deflecting field for the relativistic case
==========================================
For the relativistic case $\beta_z =1, v_{ph}=c, k_0=k_z, k_r=0$ the Helmholtz wave equation reduces to the Laplace equation. In the HM–HE representation all basis vectors remain $\ne 0$ and continuous with respect to $k_r$. Expressions for the field components are found as limits for $k_r \rightarrow 0$ from the general form in Table \[Tab\_1\] by expanding the Bessel function as: $$\lim_{k_r \rightarrow 0} \frac{J_n(k_r r)}{k_r^n} = \frac{r^n}{2^n n!}.
\label{Eq. 7a}$$ The results of these transformations are summarized in Table \[Tab\_2\]. Similar expressions can be found in [@Garault], [@Hahn] and [@Vagin].\
Following Table \[Tab\_2\] the field distribution of the synchronous deflecting wave, $n=1, v_{ph}= c$ in the region of the interaction with the beam is: $$\begin{gathered}
E_z=\left [ P+Q \right ] \frac{k_0 r}{2}\cos(\vartheta) \text{e}^{i(\omega t - k_0z)}, \hfill \\
E_r=i\left [ P \frac{k_0^2r^2}{8} + Q \left( \frac{k_0^2 r^2}{8}+\frac{1}{2} \right) \right ] \cos(\vartheta)\text{e}^{i(\omega t - k_0z)},\hfill \\
E_{\vartheta}= i\left [ P \frac{k_0^2r^2}{8} +Q \left( \frac{k_0^2r^2}{8}-\frac{1}{2} \right) \right ] \sin(\vartheta)\text{e}^{i(\omega t - k_0z)}, \hfill \\
cB_z=-\left [ P+Q \right ] \frac{k_0 r}{2} \sin(\vartheta) \text{e}^{i(\omega t - k_0z)},\hfill \\
cB_r=-i\left [ P \left(\frac{k_0^2r^2}{8}+\frac{1}{2} \right) + Q \frac{k_0^2 r^2}{8} \right ] \sin(\vartheta) \text{e}^{i(\omega t - k_0z)},\hfill \\
cB_{\vartheta}=i\left [ P \left(\frac{k_0^2r^2}{8}-\frac{1}{2} \right)+Q \frac{k_0^2r^2}{8} \right ] \cos(\vartheta) \text{e}^{i(\omega t - k_0z)}, \hfill
\end{gathered}
\label{Eq. 8}$$ Eq. \[Eq. 8\] describes for example the field in a dielectric lined waveguide [@Vagin].\
For periodically iris loaded structures approximated expressions for the field components of the fundamental spatial harmonics inside the aperture $0 \leq r \leq a$ were obtained by means of the so-called small pitch approximation as (cf. [@Garault]): $$\begin{gathered}
E_z = {\hat E} \frac{k_0 r}{2} \cos(\vartheta) \text{e}^{i(\omega t-k_0 z)}, \hfill \\
E_r=i {\hat E} \left[ \frac{k_0^2r^2+k_0^2 a^2}{8} \right]\cos(\vartheta) \text{e}^{i(\omega t-k_0 z)}, \hfill \\
E_{\vartheta}= i {\hat E} \left[ \frac{k_0^2r^2 - k_0^2 a^2}{8} \right]\sin(\vartheta) \text{e}^{i(\omega t-k_0 z)}, \hfill\\
cB_z=-{\hat E} \frac{k_0 r}{2} \sin(\vartheta) \text{e}^{i(\omega t-k_0 z)}, \hfill \\
cB_r=-i {\hat E} \left[\frac{4+k_0^2 r^2- k_0^2a^2}{8} \right]\sin(\vartheta) \text{e}^{i(\omega t-k_0 z)}, \hfill\\
cB_{\vartheta}= i {\hat E} \left[\frac{4-k_0^2r^2-k_0^2a^2}{8} \right]\cos(\vartheta) \text{e}^{i(\omega t-k_0 z)}, \hfill \\
\end{gathered}
\label{Eq. 9}$$ The small pitch approximation requires that $v_{ph}=c$, that the cell length $d$ is shorter than the wavelength, $d \ll \lambda$, and that the iris thickness $t$ is smaller than the cell length, $t \ll d$, see Fig. \[fig\_1\], left. Furthermore it demands that the boundary condition $E_{\vartheta}=0$ is met at the aperture radius of the iris $r=a$.\
A comparison of Eq. \[Eq. 8\] and \[Eq. 9\] yields $$\begin{gathered}
P+Q={\hat E}, \quad P/Q=\frac{4}{k_0^2a^2}-1, \hfill \\
Q={\hat E} \frac{k_0^2 a^2}{4},\quad P={\hat E} \left ( 1- \frac{k_0^2 a^2}{4} \right ).
\end{gathered}
\label{Eq. 10}$$ The expression of the field components in Eq. \[Eq. 9\] are thus a particular case of the more general relations in Eq. \[Eq. 8\], i.e. the expression for the deflecting field components in Eq. \[Eq. 8\] is valid for both, the wave in the longitudinally homogeneous dielectric lined waveguide and for the synchronous spatial harmonics in the periodically iris loaded structure.\
The transverse force, Eq. \[Eq. 0\], follows from Eq. \[Eq. 8\] for $v_z=c$ as: $$\begin{gathered}
F_r= i e \frac{P+Q}{2} \cos(\vartheta) \text{e}^{i(\omega t - k_0 z)}= i e \frac{\hat E}{2} \cos(\vartheta) \text{e}^{i(\omega t - k_0 z)},\hfill \\
F_{\vartheta}=-i e \frac{P+Q}{2} \sin(\vartheta) \text{e}^{i(\omega t - k_0 z)}= -i e \frac{\hat E}{2} \sin (\vartheta) \text{e}^{i(\omega t - k_0 z)}, \hfill
\end{gathered}
\label{Eq. 11}$$ or, transferring to Cartesian coordinates, as: $$\begin{gathered}
F_x= i e \frac{\hat E}{2} \text{e}^{ i(\omega t -k_0 z)}, \hfill \\
F_y=0, \hfill \\
E_z = \frac{\hat E}{2} k_0 x \text{e}^{ i(\omega t -k_0 z)}, \hfill \\
cB_z= - \frac{\hat E}{2} k_0 y \text{e}^{ i(\omega t -k_0 z)}. \hfill
\end{gathered}
\label{Eq. 12}$$ For the synchronous relativistic case $v_{ph}=v_z=c$ the deflecting force, or equivalently the deflecting field, is in the region of of the interaction $0 \leq r \leq a$ constant. The longitudinal field components are shifted by $\pi/2$ in the spatio-temporal phase and rise linearly with the distance from the axis.\
According to Eq. \[Eq. 10\] is the deflecting field distributions in axially symmetric iris loaded structures for typical values of wave number $k_0$ and aperture radius $a$ dominated by the HM mode, because $P/Q > 1$. The ratio $P/Q $ depends on the design of the supporting structure and defines also other structure parameters, such as frequency, group velocity, effective shunt impedance and, in case of periodically loaded structures, the level of higher spatial harmonics. Periodical structures with complexer geometry than the simple iris loaded structure give additional freedom for optimizations and allows to reach an overall more attractive set of parameters [@FlPa2014].
[ l c c c c]{}\
& HM & HE & azimuthal & spatio-temporal\
& & & dependence & dependence\
\
$\displaystyle E_r $ & $\displaystyle \frac{k_0^2 r^{n+1}}{2^{n+1}(n+1)!}$ & $\displaystyle \frac{k_0^2 r^{n + 1}}{2^{n + 1}(n+1)!}+\frac{r^{n-1}}{2^n (n-1)!} $ & $\displaystyle \cos(-n\vartheta)$ & $\displaystyle i\text{e}^{i(\omega t-k_z z)} $\
$\displaystyle E_\vartheta $ & $\displaystyle \frac{k_0^2 r^{n+1}}{2^{n+1}(n + 1)!}$ & $\displaystyle \frac{k_0^2 r^{n + 1}}{2^{n+1} (n + 1)!}- \frac{r^{n-1}}{2^n (n-1)!} $ & $\displaystyle \sin(n\vartheta)$ & $\displaystyle i\text{e}^{i(\omega t-k_z z)}$\
$\displaystyle E_z $ & $\displaystyle \frac{k_0 r^n}{2^n n!}$ & $\displaystyle \frac{k_0 r^n}{2^n n!}$ & $\displaystyle \cos(-n\vartheta)$ & $ \displaystyle \text{e}^{i(\omega t-k_z z)}$\
$\displaystyle cB_r $ & $\displaystyle -\frac{k_0^2 r^{n + 1}}{2^{n+1}(n + 1)!} -\frac{r^{n-1}}{2^n (n-1)!} $ & $\displaystyle -\frac{k_0^2 r^{n + 1}}{2^{n + 1}(n+1)!}$ & $\displaystyle \sin( n\vartheta)$ & $\displaystyle i\text{e}^{i(\omega t-k_z z)}$\
$\displaystyle cB_\vartheta $ & $\displaystyle \frac{k_0^2 r^{n + 1}}{2^{n + 1}(n+1)!}-\frac{r^{n-1}}{2^n (n-1)!} $ & $\displaystyle \frac{k_0^2 r^{n + 1}}{2^{n + 1}(n+1)!}$ & $\displaystyle \cos(-n\vartheta) $ & $\displaystyle i\text{e}^{i(\omega t-k_z z)}$\
$\displaystyle cB_z $ & $\displaystyle -\frac{k_0 r^n}{2^n n!}$ & $\displaystyle -\frac{k_0 r^n}{2^n n!}$ & $\displaystyle \sin(n\vartheta)$ & $\displaystyle \text{e}^{i(\omega t-k_z z)}$\
\[Tab\_2\]
Influence of the longitudinal field components on the particle dynamics
=======================================================================
From Eq. \[Eq. 4\] follows that the integrated transverse momentum transfered to a particle moving on a straight line (rigid beam approximation) through a region with an arbitrary electromagnetic field is related to the integrated longitudinal field by $$p_{\perp}=-i \frac{e}{k_0 c}\int \limits_0^L \nabla_{\perp} E_z {dz}.
\label{Eq. 12a}$$ Eq. \[Eq. 12a\] is referred to as Panofsky-Wenzel theorem. It is valid in this strict form only for synchronous motion with $v_z=v_{ph}=c$. (Asynchronous field components average to zero when the integration length is long enough.) The rigid beam approximation excludes ponderomotive forces, however, for a force like derived in Eq. \[Eq. 12\] (fundamental spatial harmonics, $n=1$, $v_{ph}=c$) ponderomotive forces are anyhow zero, because $F_y$ is everywhere zero – not only one average – and $F_x$ does not depend on $x$.\
In accordance to Eq. \[Eq. 12a\] the transverse momentum $p_{\perp}=p_x$ and energy change of a bunch of particles passing through a deflecting structure of length $L$ following Eq. \[Eq. 12\] read as: $$\begin{gathered}
p_x=\frac{eV}{c}\left(\sin(\varphi)+\cos(\varphi)\Delta \varphi \right) \hfill \\
E=e k_0 V x \left( \cos(\varphi)-\sin(\varphi)\Delta \varphi \right), \hfill \\
\end{gathered}
\label{Eq. 13}$$ where $V=\frac{{\hat E}L}{2}$ is the integrated deflecting voltage and a first order Taylor expansion of the phase $\varphi=\omega t -k_0 z$ has been made. $\Delta \varphi$ denotes the position of a particle relative to the bunch center; $\Delta \varphi =-k_0 \Delta z$.\
While the first term in the momentum equation describes the average momentum gained by the bunch, the second term describes the spread due to the differences experienced by particles in the head and the tail of the bunch. In the fully deflecting mode, $\varphi=\pi/2$, the momentum spread is to first order zero, while it is maximal at $\varphi=0$, the standard operation phase for cavity applications as diagnostics, crabbing and emittance exchange.\
Due to the dependence of the longitudinal field on the transverse coordinate the induced energy spread is on all phases uncorrelated: $$\sigma_E= \;\left\{ \begin{gathered}
e k_0 V \sigma_x \quad \text{for} \quad \varphi=0, \hfill \\
e k_0^2 V \sigma_x \sigma_z \quad \text{for}\quad \varphi=\frac{\pi}{2}, \hfill \\
\end{gathered} \right.
\label{Eq. 14}$$ with the transverse rms beam size in the streaking direction $\sigma_x$ and the longitudinal rms bunch length $\sigma_z$. The Panofsky-Wenzel theorem, Eq. \[Eq. 12a\], as well as Eq. \[Eq. 13\] and Eq. \[Eq. 14\] describe the beam dynamics to first order. In second order the induced transverse momentum couples to the longitudinal magnetic field and the transverse particle position changes, which leads to an additional correlated energy spread of [@Emma; @2011] $$\frac{{\Delta {E^{cor}}}}{{\Delta z}} = \frac{{\left( {ek_0V} \right)}^2}{c p_z}\frac{L}{6},
\label{Eq. 15}$$ where $p_z$ denotes the longitudinal momentum of the particle.\
The second order effects combine the cosine-like transverse momentum with the sine-like longitudinal field components, the Taylor expansion of the product is thus of the form $\frac{1}{2} \sin(2\varphi)+\cos(2\varphi)\Delta \varphi$ and Eq. \[Eq. 15\] is therefore valid for $\varphi=0$ and for $\varphi=\pi/2$.\
The momentum in the streaking direction leads in combination with the longitudinal magnetic field also to a force in direction perpendicular to the streaking plane: $${\hat F_y}=e\frac{p_x}{\gamma m_0}B_z,
\label{Eq. 16}$$ with the rest mass of the particle $m_0$ and the relativistic Lorentz factor $\gamma$. Thus $$p_y = \frac{1}{c}\int {{\hat F_y}} \;dz =\frac{{\left( {ek_0V} \right)}^2}{2c^2 p_z} y \Delta z.
\label{Eq. 17}$$ The transverse momentum $p_y$ is linear in the transverse position $y$, i.e. it is a focusing force, which depends however on the longitudinal position $\Delta z$ in the bunch. It leads thus to a projected emittance contribution of $$\varepsilon_y=\frac{{\left( {ek_0V} \right)}^2}{2m_0c^2 p_z} \sigma_y^2 \sigma_z,
\label{Eq. 18}$$ with the transverse rms beam size perpendicular to the streaking direction $\sigma_y$. Eq. \[Eq. 17\] and \[Eq. 18\] are again valid for $\varphi=0$ and for $\varphi=\pi/2$.\
The emittance growth Eq. \[Eq. 18\] in the direction perpendicular to the streaking force is a fundamental property of the deflecting field which doesn’t depend on the design of the supporting structure or the operating mode of the cavity.\
For present day beam and cavity parameters the emittance growth is very small, but it can not be eliminated. Additional transverse forces can still appear due the spatial harmonics especially in the end cells of rf structures and due to the backward traveling component in standing wave cavities. These forces average out in a first order approximation of the particle motion, but are relevant for second order effects [@Rei; @97; @Opanasenko] and often dominate the beam dynamics in the plane perpendicular to the streaking direction.
Summary
=======
Transverse deflecting rf structures find nowaday numerous applications in particle accelerators. Regardless of the design and operating mode of the supporting structure, a synchronous transverse force, generated by the common interaction of the transverse electric and magnetic field components, is always accompanied by both, electric as well as magnetic, longitudinal field components. The complete deflecting rf field has necessarily six field components and requires the representation by a linear combination of two basis vectors. The description in the HM–HE basis avoids convergence problems which are characteristic for the usual TM–TE basis for waves matched to the velocity of light.\
Both, the longitudinal electric and the longitudinal magnetic field component lead to undesired and fundamental beam dynamics effects.\
The longitudinal magnetic field, which has been ignored so far, in combination with the induced transverse momentum in the direction of the deflecting force, results in a small, but fundamental, emittance contribution in the direction perpendicular to the deflecting force.
Acknowledgments
===============
The author VP is supported by RFMEFI62117X0014 program.
[99]{}
L. Bellantoni, H. T. Edwards, M. McAshan et al., ’Design and Measurement of a Deflecting Mode Cavity for an RF Separator’, in *Proceedings of the 2001 Particle Accelerator Conference, Chicago, IL, (IEEE, New York, 2001).*
D. Alesini, F. Marcellini, A. Ghigo et al., ’The New RF Deflectors for the CTF3 Combiner Ring’, in *Proceedings of the 2009 Particle Accelerator Conference, Vancouver, BC,(TRIUMPF, Vancouver, 2011).*
D. Ratner, C. Behrens, Y. Ding et al., ’Time-resolved imaging of the microbunching instability and energy spread at the Linac Coherent Light Source’, Phys. Rev. ST Accel. Beams 3, 030704, 2015. P. Emma, M. Cornacchia, ’Transverse to longitudinal emittance exchange’, Phys. Rev. ST Accel. Beams 5, 084001, 2002. D. R. Brett, R. B. Appleby, R. De Maria et al., ’Accurate crab cavity modeling for the high luminosity Large Hadron Collider’, Phys. Rev. ST Accel. Beams 17, 104001, 2014. W. K. H. Panofsky, W. A. Wenzel, ’Some considerations concerning the transverse deflection of charged particles in Radio-Frequency fields’ Rev. Sci. Inst. **27**, 967, 1956. P. B. Wilson, ’Introduction to wake fields and wake potentials’, AIP Conf. Proc. 184, Vol. 1, edited by M. Month and M. Dienes (AIP, New York, 1989), pp. 525.
O. Lambertson, ’Dynamic Devices-Pickups and Kickers’, in Physics of Accelerators, eds. M. Month and M. Dienes, AIP Conf. Proc. 153, 1414 (1987).
K. Floettmann, V. Paramonov, ’Beam dynamics in transverse deflecting rf structures’, Phys. Rev. ST Accel. Beams 17, 024001, 2014. V. V. Paramonov ’Field distribution analysis in deflecting structures’ DESY 18-13, arXiv:1302.5306v1 (2013). M. Ivanyan, A. Grigoryan, A. Tsakanian et al., ’Narrow-band impedance of a round metallic pipe with a low conductive thin layer’, Phys. Rev. ST Accel. Beams 17, 021302, 2014. A. Novokhatsky, M. Timm, T. Weiland, ’The surface roughness wake field effect’, Proceedings of the 1998 International Computational Accelerator Physics Conference, Monterey, 1998. V. A. Vagin, V. I. Kotov, ’Investigation of hybrid waves in a circular waveguide partially filled with dielectric’, Journal of Technical Physics, Vol. 35, no. 7, p. 1273, 1965.\
Transl. in Soviet Physics - Technical Physics, Vol. 10, no. 7, p. 987, 1966.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The so-called unidentified infrared emission (UIE) features at 3.3, 6.2, 7.7, 8.6, and 11.3${\,{\rm \mu m}}$ ubiquitously seen in a wide variety of astrophysical regions are generally attributed to polycyclic aromatic hydrocarbon (PAH) molecules. Astronomical PAHs may have an aliphatic component as revealed by the detection in many UIE sources of the aliphatic C–H stretching feature at 3.4${\,{\rm \mu m}}$. The ratio of the observed intensity of the 3.4${\,{\rm \mu m}}$ feature to that of the 3.3${\,{\rm \mu m}}$ aromatic C–H feature allows one to estimate the aliphatic fraction of the UIE carriers. This requires the knowledge of the intrinsic oscillator strengths of the 3.3${\,{\rm \mu m}}$ aromatic C–H stretch (${A_{3.3}}$) and the 3.4${\,{\rm \mu m}}$ aliphatic C–H stretch (${A_{3.4}}$). Lacking experimental data on ${A_{3.3}}$ and ${A_{3.4}}$ for the UIE candidate materials, one often has to rely on quantum-chemical computations. Although the second-order M$\o$ller-Plesset (MP2) perturbation theory with a large basis set is more accurate than the B3LYP density functional theory, MP2 is computationally very demanding and impractical for large molecules. Based on methylated PAHs, we show here that, by scaling the band strengths computed at an inexpensive level (e.g., [B3LYP/6-31G$^{\ast}$]{}) we are able to obtain band strengths as accurate as that computed at far more expensive levels (e.g., MP2/[6-311+G(3df,3pd)]{}). We calculate the model spectra of methylated PAHs and their cations excited by starlight of different spectral shapes and intensities. We find $\left({I_{3.4}}/{I_{3.3}}\right)_{\rm mod}$, the ratio of the model intensity of the 3.4${\,{\rm \mu m}}$ feature to that of the 3.3${\,{\rm \mu m}}$ feature, is insensitive to the spectral shape and intensity of the exciting starlight. We derive a straightforward relation for determining the aliphatic fraction of the UIE carriers (i.e., the ratio of the number of C atoms in aliphatic units $N_{\rm C,ali}$ to that in aromatic rings $N_{\rm C,aro}$) from the observed band ratios ${\left(I_{3.4}/I_{3.3}\right)_{\rm obs}}$: ${N_{\rm C,ali}}/{N_{\rm C,aro}}\approx0.57\times{\left(I_{3.4}/I_{3.3}\right)_{\rm obs}}$ for neutrals and ${N_{\rm C,ali}}/{N_{\rm C,aro}}\approx0.26\times{\left(I_{3.4}/I_{3.3}\right)_{\rm obs}}$ for cations.'
author:
- 'X.J. Yang, Aigen Li, R. Glaser, and J.X. Zhong'
title: |
\
Polycyclic Aromatic Hydrocarbons with Aliphatic Sidegroups: Intensity Scaling for the C–H Stretching Modes and Astrophysical Implications
---
Introduction\[sec:intro\]
=========================
The infrared (IR) spectra of a wide range of galactic and extragalactic objects with associated dust and gas are dominated by a series of emission features at 3.3, 6.2, 7.7, 8.6, 11.3, and 12.7${\,{\rm \mu m}}$ (see Peeters 2014). Collectively known as the “unidentified” IR emission (IUE) features due to the fact that the exact nature of their carriers remains unknown (see Peeters et al. 2003, Yang et al. 2017), the hypothesis of polycyclic aromatic hydrocarbon (PAH) molecules as the carriers of the UIE features has gained widespread acceptance and extreme popularity. The PAH hypothesis attributes the UIE features to the stretching and bending vibrational modes of PAH molecules (Léger & Puget 1984, Allamandola et al. 1985). While PAH is a precisely defined chemical term (i.e., PAHs are fused benzene rings made up of carbon and hydrogen atoms), the PAH hypothesis does not really postulate that astronomical PAHs are pure aromatic compounds as strictly defined by chemists. Instead, PAH molecules in astronomical environments may include ring defects (e.g., see Yu & Nyman 2012), substituents (e.g., N in place of C, see Hudgins et al. 2005, Mattioda et al. 2008, Alvaro Galué et al. 2010, Gruet et al. 2016, Gao et al. 2016; O in place of C, see Bauschlicher 1998; Fe in place of C, see Szczepanski et al. 2006, Bauschlicher 2009, Simon & Joblin 2010), partial deuteration (e.g., see Allamandola et al. 1989, Hudgins et al. 2004, Peeters et al. 2004, Draine 2006, Onaka et al. 2014), partial dehydrogenation (e.g., see Tielens et al. 1987, Malloci et al. 2008) and sometimes superhydrogenation (e.g., see Bernstein et al. 1996, Thrower et al. 2012, Sandford et al. 2013).
Astronomical PAHs may likely also include an aliphatic component, as revealed by the detection in many UIE sources of a weak satellite emission feature at 3.4${\,{\rm \mu m}}$ always which accompanies the 3.3${\,{\rm \mu m}}$ emission feature (e.g., see Geballe et al. 1985, 1989, Jourdain de Muizon et al. 1986, 1990, Nagata et al. 1988, Allamandola et al. 1989, Sandford et al. 1991, Joblin et al. 1996, Sloan et al. 1997). For illustration, we show in Figure \[fig:Aro\_Ali\_Obs\] the 3.3 and 3.4${\,{\rm \mu m}}$ emission features of several representative astrophysical regions. The 3.4${\,{\rm \mu m}}$ feature is generally thought to arise from the C–H stretching vibration of aliphatic hydrocarbon materials, while the 3.3${\,{\rm \mu m}}$ feature is due to the C–H stretching mode of aromatic hydrocarbons. Also detected in some UIE sources are the aliphatic C–H deformation bands at 6.85 and 7.25${\,{\rm \mu m}}$ (see Sloan et al. 2014, and see Table 3 in Yang et al. 2016a for a summary). In recent years, the aliphatic fraction of the UIE carriers — the ratio of the number of C atoms in aliphatic units ($N_{\rm C,ali}$) to that in aromatic rings ($N_{\rm C,aro}$) — has received much attention (e.g., see Kwok & Zhang 2011, Li & Draine 2012, Rouillé et al. 2012, Steglich et al. 2013, Yang et al. 2013, 2016a,b). Kwok & Zhang (2011) argued that the material responsible for the UIE features may have a substantial aliphatic component and therefore, by definition, PAHs can not be the UIE carrier. This argument can be tested by examining the ratio of the observed intensity of the 3.3${\,{\rm \mu m}}$ feature (${I_{3.3}}$) to that of the 3.4${\,{\rm \mu m}}$ feature (${I_{3.4}}$) of UIE sources. If the intrinsic oscillator strengths (per chemical bond) of the 3.3${\,{\rm \mu m}}$ aromatic C–H stretch (${A_{3.3}}$) and the 3.4${\,{\rm \mu m}}$ aliphatic C–H stretch (${A_{3.4}}$) are known, one could drive the aliphatic fraction of the UIE carriers from $N_{\rm C,ali}/N_{\rm C,aro}\approx
0.3\times\,\left(I_{3.4}/I_{3.3}\right)
\times\,\left(A_{3.3}/A_{3.4}\right)$ (see Li & Draine 2012). Here the factor 0.3 arises from the assumption of one aliphatic C atom corresponding to 2.5 aliphatic C–H bonds (intermediate between methylene –CH$_2$ and methyl –CH$_3$) and one aromatic C atom corresponding to 0.75 aromatic C–H bond (intermediate between benzene C$_6$H$_6$ and coronene C$_{24}$H$_{12}$).
Unfortunately, there is little experimental information on ${A_{3.3}}$ and ${A_{3.4}}$ for the UIE candidate materials. Therefore, one often has to rely on quantum-chemical computations based on density functional theory or second-order perturbation theory. To this end, one often uses the Gaussian09 software (Frisch et al. 2009) and employs the hybrid density functional theoretical method (B3LYP) in conjunction with a variety of basis sets. In the order of increasing accuracy and computational demand, the commonly adopted basis sets are (see Pople et al. 1987): [6-31G$^{\ast}$]{}, [6-31+G$^{\ast}$]{}, [6-311+G$^{\ast}$]{}, [6-311G$^{\ast\ast}$]{}, [6-31+G$^{\ast\ast}$]{}, [6-31++G$^{\ast\ast}$]{}, [6-311+G$^{\ast\ast}$]{}, [6-311++G$^{\ast\ast}$]{}, [6-311+G(3df,3pd)]{}, and [6-311++G(3df,3pd)]{}. One also often employs second-order M$\o$ller-Plesset perturbation theory (hereafter abbreviated as MP2) in conjunction with these basis sets. The MP2 method is thought to be more accurate in computing band intensities than B3LYP (see Cramer et al. 2004). Indeed, as demonstrated in §\[sec:B3LYP\], the IR intensities calculated at the [B3LYP/6-31G$^{\ast}$]{} level for the 3.3${\,{\rm \mu m}}$ aromatic C–H stretches of benzene, naphthalene, anthracene, pyrene, and coronene are much higher compared to their gas-phase experimental results. This is also true for methylated species (e.g., methylated benzene or toluene, see §\[sec:B3LYP\]). Using better basis sets in conjunction with the B3LYP method, we find that the IR intensities still differ by a factor of ${\sim\,}$30% compared to the experimental results. In contrast, Pavlyuchko et al. (2012) reported that the IR intensities calculated for benzene and toluene at the level [MP2/6-311G(3df,3pd)]{} would match the experimental results very well. Ideally, in order to compute ${A_{3.3}}$ and ${A_{3.4}}$ as accurately as possible, one should study the candidate UIE carriers at the most pertinent levels \[e.g., MP2 in conjunction with [6-311++G$^{\ast\ast}$]{}, [6-311+G(3df,3pd)]{}, or [6-311++G(3df,3pd)]{}\]. However, the huge computational demand required by these techniques often makes it impractical to compute ${A_{3.3}}$ and ${A_{3.4}}$, particularly for large molecules. In this work, based on methylated aromatic hydrocarbon molecules (with the methyl group taken to represent the aliphatic component of the UIE carriers), we present in §\[sec:scaling\] an intensity scaling approach which, by scaling the intensities computed at an inexpensive level (e.g., [B3LYP/6-31G$^{\ast}$]{}) we are able to obtain intensities as accurate as that computed at far more expensive levels (e.g., MP2/[6-311+G(3df,3pd)]{}). We calculate in §\[sec:astro\] the model emission spectra of PAHs containing various numbers of methyl sidegroups, excited by starlight of different spectral shapes and intensities. We derive $\left({I_{3.4}}/{I_{3.3}}\right)_{\rm mod}$, the ratio of the model intensity of the 3.4${\,{\rm \mu m}}$ feature to that of the 3.3${\,{\rm \mu m}}$ feature. We explore the variation of $\left({I_{3.4}}/{I_{3.3}}\right)_{\rm mod}$ with the spectral shape and intensity of the exciting starlight. We summarize the principal results in §\[sec:summary\].
B3LYP IR Intensities for C–H Stretching Modes \[sec:B3LYP\]
===========================================================
To derive the intrinsic oscillator strengths of the 3.3${\,{\rm \mu m}}$ aromatic C–C stretch (${A_{3.3}}$) and the 3.4${\,{\rm \mu m}}$ aliphatic C–H stretch (${A_{3.4}}$), we have employed density functional theory and second-order perturbation theory to compute the IR vibrational spectra of seven PAH species (benzene C$_6$H$_6$, naphthalene C$_{10}$H$_8$, anthracene C$_{14}$H$_{10}$, phenanthrene C$_{14}$H$_{10}$, pyrene C$_{16}$H$_{10}$, perylene C$_{20}$H$_{12}$, and coronene C$_{24}$H$_{12}$), as well as all of their methyl derivatives (see Yang et al. 2013). All of the molecules have been studied in all conformations at the [B3LYP/6-31G$^{\ast}$]{} level. The calculations always show three methyl C–H stretches for all the methyl derivatives of all the molecules, and we always describe these three bands as ${\nu_{\rm Me,1}}$, ${\nu_{\rm Me,2}}$, and ${\nu_{\rm Me,3}}$.
For benzene, the gas-phase experimental spectrum of the [*National Institute of Standards and Technology*]{} (NIST)[^1] gives an absorption intensity of ${\sim\,}$54.4${\,{\rm km}}{\,{\rm mol}}^{-1}$ for the aromatic C–H stretches, in close agreement with the intensity of ${\sim\,}$55${\,{\rm km}}{\,{\rm mol}}^{-1}$ computed by Pavlyuchko et al. (2012) at the [MP2/6-311G(3df,3pd)]{} level,[^2] but much lower than the computed intensity of ${\sim\,}$104${\,{\rm km}}{\,{\rm mol}}^{-1}$ derived at the [B3LYP/6-31G$^{\ast}$]{} level. The gas-phase intensity measurements of the aromatic C–H stretches have been reported for naphthalene (${\sim\,}$96${\,{\rm km}}{\,{\rm mol}}^{-1}$; Cané et al. 1996), anthracene (${\sim\,}$161${\,{\rm km}}{\,{\rm mol}}^{-1}$; Cané et al. 1997), pyrene (${\sim\,}$122${\,{\rm km}}{\,{\rm mol}}^{-1}$; Joblin et al. 1994), and coronene (${\sim\,}$161${\,{\rm km}}{\,{\rm mol}}^{-1}$; Joblin et al. 1994). To our knowledge, no gas phase IR intensities have been published for phenanthrene and perylene, although the IR absorption spectra of various matrix-isolated PAH species including phenanthrene and perylene have been obtained (e.g., see Hudgins & Allamandola 1995a,b, 1997; Hudgins & Sandford 1998a,b; Szczepanski & Vala 1993a,b). Similar to benzene, the experimental intensities are much lower than our calculated results for the aromatic C–H stretches at the [B3LYP/6-31G$^{\ast}$]{} level which are respectively ${\sim\,}$139, 178, 188 and 257${\,{\rm km}}{\,{\rm mol}}^{-1}$ for naphthalene, anthracene, pyrene and coronene, exceeding their experimental values by ${\sim\,}$45%, 11%, 54% and 60%, respectively. For toluene, we digitize the NIST experimental spectra and integrate over the range of 3000–3200${\,{\rm cm}}^{-1}$ to obtain the intensity of the aromatic C–H stretch ($A_{\rm aro}$). Similarly, we integrate over the range of 2800–3000${\,{\rm cm}}^{-1}$ to obtain the intensity of the aliphatic C–H stretch ($A_{\rm ali}$). The relative intensity of the methyl (aliphatic) signal to that of the aromatic band is ${A_{\rm ali}/A_{\rm aro}}\approx 0.79$. A similar analysis of the experimental spectrum of Wilmshurst & Bernstein (1957) results in ${A_{\rm ali}/A_{\rm aro}}\approx0.71$.[^3] Our integration of the NIST spectrum of toluene gives a total intensity of ${\sim\,}$97.2${\,{\rm km}}{\,{\rm mol}}^{-1}$ for all the C–H stretches (both methyl and aromatic) and is in excellent agreement with the value of ${\sim\,}$95${\,{\rm km}}{\,{\rm mol}}^{-1}$ calculated by Pavlyuchko et al. (2012) and by Galabov et al. (1992) at the [MP2/6-311G(3df,3pd)]{} level. According to our ratio of the measured intensities for the methyl to aromatic regions (${A_{\rm ali}/A_{\rm aro}}\approx 0.79$), this overall intensity corresponds to intensities of ${\sim\,}$42.9${\,{\rm km}}{\,{\rm mol}}^{-1}$ for the methyl bands and of ${\sim\,}$54.3${\,{\rm km}}{\,{\rm mol}}^{-1}$ for the aromatic bands. The intensities computed at the [B3LYP/6-31G$^{\ast}$]{} level for toluene are ${\sim\,}$165.3${\,{\rm km}}{\,{\rm mol}}^{-1}$ for the entire region and ${\sim\,}$70.4 and ${\sim\,}$94.9${\,{\rm km}}{\,{\rm mol}}^{-1}$ for the methyl and aromatic sections, respectively. Again, we see that the computed intensities are much higher than the experimental values from the gas phase measurements.
In the absence of absolute intensity experimental data for naphthalene, anthracene, phenanthrene, perylene, pyrene and coronene, we are unfortunately not able to compare the experimental intensities of the C–H stretches of these molecules with that computed at the [B3LYP/6-31G$^{\ast}$]{} level.
Scaling Approaches for the Computed Total Intensities of C–H Stretching Modes \[sec:scaling\]
==============================================================================================
As we have seen in §\[sec:B3LYP\], the IR intensities calculated at the [B3LYP/6-31G$^{\ast}$]{} level are much higher compared to the experimental results. Using better basis sets in conjunction with the B3LYP method, we found that the IR intensities still differ by a factor of ${\sim\,}$30% compared to the experiment results. Pavlyuchko et al. (2012) reported that the IR intensities calculated for benzene and toluene at the level [MP2/6-311G(3df,3pd)]{} would match the experimental results very well. We have tried to reproduce their data for benzene and toluene by performing both MP2(fc) and MP2(full) computations with the 6-311G(3df,3pd) basis set.[^4]
While the MP2(full)/6-311+G(3df,3pd) level data reproduce the measured IR intensities reasonably well, such calculations are far too expensive especially for large molecules. The MP2(full) computations of the naphthalene systems with the large basis sets including the (3df,3pd) polarization functions each requires several days of computer time on eight processors. Considering that the absolute values computed at all of the MP2 levels are better than the respective values computed at the B3LYP levels, one would be inclined to explore scaling approaches of the MP2 data computed with modest basis sets. However, we will show below that scaling approaches that are based on the B3LYP data can be just as successful in spite of the fact that the absolute numbers computed at the [B3LYP/6-31G$^{\ast}$]{} level differ much more from experiment than do the [MP2/6-31G$^{\ast}$]{} data.
Before we proceed, it is useful to clarify the meaning of scaling approaches. In the most typical approach to scaling, it is attempted to reproduce a set of experimental data with a set of data obtained at a level ${L_i}$ such that $p({\rm exp}) \approx f \cdot p({L_i})$, that is, one scaling factor $f$ is applied to all values in the data set and this scale factor depends on the level, $f = f({L_i})$. This kind of scaling is commonly employed for vibrational frequencies. For intensities, however, we will see that approaches of the type $p({\rm exp}) \approx f \cdot p({L_i})+C({L_i})$ are more successful, that is, there will be a non-zero offset.
Let $ML1$, $ML2$ and $ML3$ respectively represent the [MP2(full)]{} computations with the [6-31G$^{\ast}$]{}, 6-311+G(d,p), and 6-311+G(3df,3pd) basis sets. Let $BL1$, $BL2$ and $BL3$ respectively represent the B3LYP computations with the [6-31G$^{\ast}$]{}, 6-311+G(d,p), and 6-311+G(3df,3pd) basis sets. As can be seen from Figure \[fig:Int\_LevelDep\] (top left), the total intensities ($A$) computed at the MP2 level but with different basis sets \[i.e., $A(ML1)$, $A(ML2)$, and $A(ML3)$\] are linearly related:
\[eq:MP2\_LS\_SSMS\] $$\begin{aligned}
A(ML3)&\approx0.7615\,A(ML1)~~,~~(r^{2}\approx0.9575)
\label{eq:MP2_LS_SS_a} \\
A(ML3)&\approx0.9382\,A(ML1)-20.4880~~,~~(r^{2}\approx0.9949)
\label{eq:MP2_LS_SS_b} \\
A(ML3)&\approx0.8089\,A(ML2)~~,~~(r^{2}\approx0.9984)
\label{eq:MP2_LS_MS}\end{aligned}$$
where $r^{2}$ is the linear-correlation coefficient. While eq.\[eq:MP2\_LS\_MS\] describes an excellent linear correlation between the intensities computed with the $ML3$ method and that with the $ML2$ method without any need for an offset, the analogous eq.\[eq:MP2\_LS\_SS\_a\] is less successful and an excellent linear correlation between $A(ML3)$ and $A(ML1)$ only is achieved when a non-zero offset is allowed in eq.\[eq:MP2\_LS\_SS\_b\]. The analogous relations also hold at the B3LYP level (eq.\[eq:B3LYP\_LS\_SS\]) and they are shown in Figure \[fig:Int\_LevelDep\] (top right), where $A(BL1)$, $A(BL2)$, and $A(BL3)$ are respectively the intensities computed at the $BL1$, $BL2$ and $BL3$ levels.
\[eq:B3LYP\_LS\_SS\] $$\begin{aligned}
A(BL3)&\approx0.7306\,A(BL1)~,~~(r^{2}\approx0.9610)
\label{eq:B3LYP_LS_MS_a} \\
A(BL3)&\approx0.8838\,A(BL1)-26.1670~,~~(r^{2}\approx0.9924)
\label{eq:B3LYP_LS_MS_b}\\
A(BL3)&\approx0.8089\,A(BL2)~,~~(r^{2}\approx0.9984)
\label{eq:B3LYP_LS_SS_a} \\
A(BL3)&\approx0.8395\,A(BL2)-3.3861~,~~(r^{2}\approx0.9998)
\label{eq:B3LYP_LS_SS_b}\end{aligned}$$
Also shown in Figure \[fig:Int\_LevelDep\] (bottom left) are the nearly linear relations between the IR intensities computed at the B3LYP and MP2(full) levels with a common basis set. The data are very well described by linear regression and there is no need for a non-zero offset in any of the following equations (see eqs.\[eq:B3LYP\_MP2\_SS\], \[eq:B3LYP\_MP2\_MS\], and \[eq:B3LYP\_MP2\_LS\]). It is remarkable that these slopes are rather similar for the various basis sets.
\[eq:B3LYP\_MP2\_scale\] $$\begin{aligned}
A(ML1)&\approx0.6769\,A(BL1)~,~~(r^{2}\approx0.9971)
\label{eq:B3LYP_MP2_SS} \\
A(ML2)&\approx0.7877\,A(BL2)~,~~(r^{2}\approx0.9966)
\label{eq:B3LYP_MP2_MS} \\
A(ML3)&\approx0.7056\,A(BL3)~,~~(r^{2}\approx0.9949)
\label{eq:B3LYP_MP2_LS}\end{aligned}$$
In light of these linear correlations, it is clear that there must be a strong linear correlation between the lowest DFT level, our standard level [B3LYP/6-31G${^{\ast}}$]{} (i.e., $BL1$), and the best MP2 level, the level MP2(full)/6-311+G(3df,3pd) (i.e., $ML3$). Eqs.\[eq:MP2\_LS\_SS\_a\] and \[eq:B3LYP\_MP2\_SS\] suggest a correlation coefficient of $\approx 0.7615\times0.6769\approx 0.5154$ and the actual correlation coefficient of eq.\[eq:B3LYP\_MP2\_a\] is ${\sim\,}$0.5152 and it is essentially the same (see Figure \[fig:Int\_LevelDep\], bottom right). Considering the need for non-zero offset in eq.\[eq:MP2\_LS\_SS\_b\], we also explore eq.\[eq:B3LYP\_MP2\_b\] and achieve an excellent linear correlation:
\[eq:B3LYP\_MP2\] $$\begin{aligned}
A(ML3)&\approx0.5152\,A(BL1)~,~~(r^{2}\approx0.9428)
\label{eq:B3LYP_MP2_a} \\
A(ML3)&\approx0.6655\,A(BL1)-25.6770~,~~(r^{2}\approx0.9964)
\label{eq:B3LYP_MP2_b}\end{aligned}$$
This tells that, by applying this scaling relation (eq.\[eq:B3LYP\_MP2\_b\]), we just need to perform computations at an inexpensive level (e.g., [B3LYP/6-31G$^{\ast}$]{}) and we are still able to obtain intensities as accurate as that computed at far more advanced levels \[e.g., MP2/[6-311+G(3df,3pd)]{}\].
Astrophysical Implications \[sec:astro\]
========================================
As shown in Yang et al. (2013), the aromatic C–H stretch band strength does not vary significantly for different molecules. It has an average value (per aromatic C–H bond) of $\langle {A_{3.3}}\rangle \approx 14.03{\,{\rm km}}{\,{\rm mol}}^{-1}$, with a standard deviation of $\sigma({A_{3.3}})\approx 0.89{\,{\rm km}}{\,{\rm mol}}^{-1}$. On the other hand, the aliphatic C–H stretch band strength is more dependent on the nature of the molecule and also on the specific isomer. The average band strength (per aliphatic C–H bond) is $\langle {A_{3.4}}\rangle \approx 23.68{\,{\rm km}}{\,{\rm mol}}^{-1}$, and the standard deviation is $\sigma({A_{3.4}})\approx 2.48{\,{\rm km}}{\,{\rm mol}}^{-1}$. All of these values are calculated for neutral PAHs at the [B3LYP/6-311+G$^{\ast\ast}$]{} (i.e., $BL2$) level. As discussed in §\[sec:scaling\], these values need to be scaled. By taking [MP2(full)/6-311+G(3df,3pd)]{} (i.e., $ML3$) to be the level which gives the most reliable band strength, the intensities need to be scaled with two formulae: eqs.\[eq:MP2\_LS\_MS\] and \[eq:B3LYP\_MP2\_MS\]. Thus, we derive for neutral PAHs $\langle{A_{3.3}}\rangle \approx 14.03
\times 0.7877 \times 0.8089
\approx 8.94{\,{\rm km}}{\,{\rm mol}}^{-1}$ (i.e., ${\sim\,}$$1.49\times10^{-18}{\,{\rm cm}}$ per C–H bond), $\langle {A_{3.4}}\rangle \approx 23.68
\times 0.7877 \times 0.8089
\approx 15.09{\,{\rm km}}{\,{\rm mol}}^{-1}$ (i.e., ${\sim\,}$$2.50\times10^{-18}{\,{\rm cm}}$ per C–H bond), and $\langle{A_{3.4}}\rangle/\langle{A_{3.3}}\rangle \approx 1.69$. Similarly, we obtain for PAH cations $\langle{A_{3.3}}\rangle
\approx 0.92{\,{\rm km}}{\,{\rm mol}}^{-1}$, $\langle {A_{3.4}}\rangle
\approx 3.20{\,{\rm km}}{\,{\rm mol}}^{-1}$, and $\langle{A_{3.4}}\rangle/
\langle{A_{3.3}}\rangle \approx 3.48$. We note that, although these results were derived from the mono-methyl derivatives of small PAH molecules, it has been shown in Yang et al. (2016b) that the ${A_{3.4}/A_{3.3}}$ ratios determined from the PAH molecules attached with a wide range of sidegroups (including ethyl, propyl, and butyl) as well as dimethyl-substituted pyrene are close to that of mono-methyl PAHs.
In addition to the 3.4${\,{\rm \mu m}}$ C–H stretch, PAHs with aliphatic sidegroups also have two aliphatic C–H deformation bands at 6.85${\,{\rm \mu m}}$ and 7.25${\,{\rm \mu m}}$. Yang et al. (2016a) have derived ${A_{6.85}}$ and ${A_{7.25}}$, the intrinsic oscillator strengths of the 6.85 and 7.25${\,{\rm \mu m}}$ aliphatic C–H deformation bands for both neutral and ionized methyl-substituted PAHs. They obtained lower limits of ${A_{6.85}}/{A_{6.2}}\approx5.0$ and ${A_{7.25}}/{A_{6.2}}\approx0.5$ for neutrals, ${A_{6.85}}/{A_{6.2}}\approx0.5$ and ${A_{7.25}}/{A_{6.2}}\approx0.25$ for cations, where ${A_{6.2}}$ is the intrinsic oscillator strength of the 6.2${\,{\rm \mu m}}$ aromatic C–C stretch.
With ${A_{3.4}}/{A_{3.3}}$, ${A_{6.85}}/{A_{6.2}}$ and ${A_{7.25}}/{A_{6.2}}$ derived for both neutral and ionized PAHs, we now calculate the emission spectra of methyl PAHs excited by starlight and the corresponding model band ratios ${I_{3.4}}/{I_{3.3}}$. Lets consider a PAH molecule containing $N_{\rm C,aro}$ aromatic C atoms and $N_{\rm C,ali}$ aliphatic C atoms (i.e., $N_{\rm C,ali}$ methyl sidegroups). We approximate their absorption cross sections by adding three Drude functions to that of PAHs of ${N_{\rm C,aro}}$ aromatic C atoms, with these Drude functions respectively representing the 3.4${\,{\rm \mu m}}$ aliphatic C–H stretch, and the 6.85 and 7.25${\,{\rm \mu m}}$ aliphatic C–H deformations: $$\begin{aligned}
C_{\rm abs}({N_{\rm C}},\lambda) & = & {C^{\scriptscriptstyle\rm PAH}_{\rm abs}}({N_{\rm C,aro}},\lambda)\\
& + & {N_{\rm C,ali}}\frac{2}{\pi}
\frac{\gamma_{3.4} \lambda_{3.4} \sigma_{\rm int,3.3}
\left(A_{3.4}/A_{3.3}\right)}
{(\lambda/\lambda_{3.4}-\lambda_{3.4}/\lambda)^2
+\gamma_{3.4}^2}\\
&+& {N_{\rm C,ali}}\frac{2}{\pi}
\frac{\gamma_{6.85} \lambda_{6.85}
\sigma_{\rm int,6.2} \left({A_{6.85}}/{A_{6.2}}\right)}
{(\lambda/\lambda_{6.85}-\lambda_{6.85}/\lambda)^2
+\gamma_{6.85}^2}\\
&+& {N_{\rm C,ali}}\frac{2}{\pi}
\frac{\gamma_{7.25} \lambda_{7.25}
\sigma_{\rm int,6.2} \left({A_{7.25}}/{A_{6.2}}\right)}
{(\lambda/\lambda_{7.25}-\lambda_{7.25}/\lambda)^2
+\gamma_{7.25}^2} ,\end{aligned}$$ where ${N_{\rm C}}={N_{\rm C,aro}}+{N_{\rm C,ali}}$; $\lambda_{3.4}=3.4{\,{\rm \mu m}}$, $\lambda_{6.85}=6.85{\,{\rm \mu m}}$, and $\lambda_{7.25}=7.25{\,{\rm \mu m}}$ are respectively the peak wavelengths of the 3.4, 6.85 and 7.25${\,{\rm \mu m}}$ features; $\gamma_{3.4}\lambda_{3.4}=0.03{\,{\rm \mu m}}$, $\gamma_{6.85}\lambda_{6.85}=0.2{\,{\rm \mu m}}$, and $\gamma_{7.25}\lambda_{7.25}=0.2{\,{\rm \mu m}}$ are respectively the mean FWHMs of the astronomical 3.4, 6.85 and 7.25${\,{\rm \mu m}}$ features (Yang et al. 2003, 2016a),[^5] and $\sigma_{{\rm int},3.3}$ and $\sigma_{{\rm int},6.2}$ are respectively the integrated strengths per (aromatic) C atom of the 3.3${\,{\rm \mu m}}$ aromatic C–H stretch and 6.2${\,{\rm \mu m}}$ aromatic C–C stretch (see Draine & Li 2007). Due to their small size (and therefore small heat capacity), PAHs are heated sporadically by single starlight photons. Unless exposed to an extremely intense radiation field, PAHs will undergo strong temperature fluctuations and will not attain an equilibrium temperature (see Li 2004). We take the “thermal-discrete” technique developed by Draine & Li (2001) to calculate the temperature probability distribution functions and the resulting emission spectra of methyl PAHs. Let $dP$ be the probability that the temperature of the molecule will be in $[T,T+dT]$. The emissivity of this molecule (of $N_{\rm C}$ C atoms) becomes $$j_\lambda({N_{\rm C}}) = \int C_{\rm abs}({N_{\rm C}},\lambda)\,
4\pi B_\lambda(T)\,\frac{dP}{dT}\,dT ~,$$ where $B_\lambda\left(T\right)$ is the Planck function at wavelength $\lambda$ and temperature $T$. As shown in Figures 6, 7 of Draine & Li (2007), the 3.3${\,{\rm \mu m}}$ interstellar UIE emitters are in the size range of ${N_{\rm C}}$${\sim\,}$20–30 C atoms. For illustrative purpose, we therefore consider $N_{\rm C,aro}=24$ (like coronene). For a coronene-like molecule, up to 12 methyl sidegroups can be attached to it. We thus consider methyl PAHs of ${N_{\rm C,ali}}=0, 1, 2, ..., 12$ aliphatic C atoms. For all molecules, we fix $N_{\rm C,aro}=24$. In Figure \[fig:modspec\_U\] we show the IR emission spectra of both neutral and ionized methyl PAHs of ${N_{\rm C,ali}}=0, 2, 6$ illuminated by the solar neighbourhood interstellar radiation field (ISRF) of Mathis, Mezger & Panagia (1983; MMP83). Figure \[fig:modspec\_U\] shows that, the 3.4 and 6.85${\,{\rm \mu m}}$ features are clearly visible in the IR emission spectra for ${N_{\rm C,ali}}=2$, while the 7.25${\,{\rm \mu m}}$ feature remains hardly noticeable even for ${N_{\rm C,ali}}=6$. This is because the intrinsic strength of the 7.25${\,{\rm \mu m}}$ feature is weaker than that of the 6.85${\,{\rm \mu m}}$ feature by a factor of ${\sim\,}$8 for neutral methyl PAHs and by a factor of ${\sim\,}$3 for their cations (Yang et al. 2016a). In the following discussions, we will focus on the 3.3 and 3.4${\,{\rm \mu m}}$ features since the molecules considered here are too small to be the dominant UIE emitters at ${\sim\,}$6–8${\,{\rm \mu m}}$ (see Figures 6, 7 of Draine & Li 2007).
We have also explored the effects of starlight intensities on the IR emission spectra of methyl PAHs by increasing the MMP ISRF by a factor of $U$. As shown in Figure \[fig:modspec\_U\], the resulting IR emission spectra for $U=1, 100, 10^4, 10^6$, after scaled by $U$, are essentially identical. This is not unexpected. The single-photon heating nature of these molecules assures that their IR emission spectra (scaled by the starlight intensity) to remain the same for different starlight intensities. Single-photon heating implies that the shape of the high-$T$ end of the temperature probability distribution function $dP/dT$ for a methyl PAH is the same for different levels of starlight intensity, and what only matters is the mean photon energy (which determines to what peak temperature a molecule will reach, upon absorption of such a photon; see Draine & Li 2001, Li 2004).
For a given ${N_{\rm C,ali}}$, we derive ${\left(I_{3.4}/I_{3.3}\right)_{\rm mod}}$, the model intensity ratio of the 3.4${\,{\rm \mu m}}$ band to the 3.3${\,{\rm \mu m}}$ band, from $$\label{eq:Iratiomod}
\left(\frac{I_{3.4}}{I_{3.3}}\right)_{\rm mod}
= \frac{\int_{3.4}\Delta j_\lambda({N_{\rm C}})\,d\lambda}
{\int_{3.3}\Delta j_\lambda({N_{\rm C}})\,d\lambda} ~~,$$ where $\int_{3.3}\Delta j_\lambda({N_{\rm C}})\,d\lambda$ and $\int_{3.4}\Delta j_\lambda({N_{\rm C}})\,d\lambda$ are respectively the feature-integrated excess emission of the 3.3 and 3.4${\,{\rm \mu m}}$ features of the methyl PAH molecule. In Figure \[fig:Iratio\_U\] we show the model intensity ratios ${\left(I_{3.4}/I_{3.3}\right)_{\rm mod}}$ as a function of ${N_{\rm C,ali}}/{N_{\rm C,aro}}$ for neutral and ionized methyl PAHs. It is encouraging to see in Figure \[fig:Iratio\_U\] that, with ${N_{\rm C,ali}}/{N_{\rm C,aro}}=0.5$, ${\left(I_{3.4}/I_{3.3}\right)_{\rm mod}}$ reaches ${\sim\,}$0.9 for neutrals and ${\sim\,}$2.0 for cations, demonstrating that the unusually high ${\left(I_{3.4}/I_{3.3}\right)_{\rm obs}}$ ratios observed in some protoplanetary nebulae (e.g., IRAS 04296+3429 with ${\left(I_{3.4}/I_{3.3}\right)_{\rm obs}}\approx1.54$) can be accounted for by a mixture of neutral and ionized methyl PAHs, with a reasonable fraction of C atoms in methyl sidegroups. In Figure \[fig:Iratio\_U\] we also compare the model band ratios with the ratios computed from the simple relation ${\left(I_{3.4}/I_{3.3}\right)_{\rm mod}^{\prime}}= 1.76\times\left({N_{\rm C,ali}}/{N_{\rm C,aro}}\right)$ for neutrals or ${\left(I_{3.4}/I_{3.3}\right)_{\rm mod}^{\prime}}= 3.80\times\left({N_{\rm C,ali}}/{N_{\rm C,aro}}\right)$ for cations. Figure \[fig:Iratio\_U\] shows that this simple, straightforward relation does an excellent job in accurately predicting ${\left(I_{3.4}/I_{3.3}\right)_{\rm mod}}$. This is nice because in future studies one can simply use this convenient relation to determine the aliphatic fraction ${N_{\rm C,ali}}/{N_{\rm C,aro}}$ of the UIE carrier from the observed band ratio ${\left(I_{3.4}/I_{3.3}\right)_{\rm obs}}$: ${N_{\rm C,ali}}/{N_{\rm C,aro}}\approx0.57\times{\left(I_{3.4}/I_{3.3}\right)_{\rm obs}}$ for neutrals and ${N_{\rm C,ali}}/{N_{\rm C,aro}}\approx0.26\times{\left(I_{3.4}/I_{3.3}\right)_{\rm obs}}$ for cations. There is no need to compute the temperature probability distribution functions and the IR emission spectra of methyl PAHs as long as one is only interested in the aliphatic fraction of the UIE carrier.
So far, we have only considered methyl PAHs excited by the MMP83-type starlight. To examine whether and how the spectral shape of the exciting starlight affects the model IR emission spectra and the band ratios ${\left(I_{3.4}/I_{3.3}\right)_{\rm mod}}$, we consider methyl PAHs of ${N_{\rm C,ali}}=0, 1, 2, ...12$ aliphatic C atoms and ${N_{\rm C,ali}}=24$ aromatic C atoms excited by stars with an effective temperature of ${T_{\star}}=6,000{\,{\rm K}}$ like our Sun and by stars of ${T_{\star}}=22,000{\,{\rm K}}$ like the B1.5V star HD37903 which illuminates the reflection nebula NGC2023. We fix the starlight intensity in the 912${\,{\rm \AA}}$–1${\,{\rm \mu m}}$ wavelength range to be that of the MMP83 ISRF (i.e., $U=1$): $$\int_{1\mu {\rm m}}^{912{\rm {\,{\rm \AA}}}}
4\pi J_\star(\lambda,{T_{\star}})\,d\lambda
= \int_{1\mu {\rm m}}^{912{\rm {\,{\rm \AA}}}}
4\pi J_{\rm ISRF}(\lambda)\,d\lambda ~~,$$ where $J_\star(\lambda, {T_{\star}})$ is the intensity of starlight approximated by the Kurucz model atmospheric spectrum, and $J_{\rm ISRF}(\lambda)$ is the MMP83 ISRF starlight intensity. As shown in Figure \[fig:modspec\_T\], for a given ${N_{\rm C,ali}}/{N_{\rm C,aro}}$, the ${T_{\star}}=6,000{\,{\rm K}}$ model results in a lower emissivity level than that of the MMP83 ISRF model. In contrast, the ${T_{\star}}=22,000{\,{\rm K}}$ model results in a higher emissivity level than that of the MMP83 ISRF model. This is because, exposed to a [*softer*]{} radiation field, PAHs absorb individual photons with a [*lower*]{} mean energy than that of a [*harder*]{} radiation field and therefore emit less (because they absorb less). Nevertheless, the emission spectral profiles are very similar to each other. This is also illustrated in Figure \[fig:Iratio\_T\] which shows that the model band ratios ${\left(I_{3.4}/I_{3.3}\right)_{\rm mod}}$ differ very little for methyl PAHs excited by starlight of different spectral shapes. So far, we have confined ourselves to coronene-like PAHs with ${N_{\rm C,aro}}=24$. To examine the effects of the PAH size on the model IR emission spectra and the band ratios ${\left(I_{3.4}/I_{3.3}\right)_{\rm mod}}$, we consider methyl PAHs of ${N_{\rm C,aro}}=20$ aromatic C atoms (like perylene) and ${N_{\rm C,ali}}=0, 1, 2, ...12$ aliphatic C atoms, as well as methyl PAHs of ${N_{\rm C,aro}}=32$ aromatic C atoms (like ovalene) and ${N_{\rm C,ali}}=0, 1, 2, ...14$ aliphatic C atoms.[^6] As shown in Figures \[fig:modspec\_NC\],\[fig:Iratio\_NC\], neither the IR emission spectra in the C–H stretch region nor the band ratios ${\left(I_{3.4}/I_{3.3}\right)_{\rm mod}}$ appreciably differ from each other.
Finally, we compare in Figure \[fig:Iratio\_mod\] the band ratios ${\left(I_{3.4}/I_{3.3}\right)_{\rm obs}}$ observed in the eight representative astrophysical environments shown in Figure \[fig:Aro\_Ali\_Obs\] with that calculated from methyl PAHs. It is seen that the observed band ratios ${\left(I_{3.4}/I_{3.3}\right)_{\rm obs}}$ of all sources except the protoplanetary nebula IRAS 04296+3429 all fall below the model ${\left(I_{3.4}/I_{3.3}\right)_{\rm mod}}$ curve of neutral PAHs with ${N_{\rm C,aro}}=24$ and ${N_{\rm C,ali}}/{N_{\rm C,aro}}{\lesssim}0.5$. For IRAS 04296+3429, the unusually high ratio of ${\left(I_{3.4}/I_{3.3}\right)_{\rm obs}}\approx1.54$ falls below the model ${\left(I_{3.4}/I_{3.3}\right)_{\rm mod}}$ curve of PAH cations. This demonstrates that a mixture of neutral and ionized methyl PAHs are capable of accounting for all the observed band ratios, including those of protoplanetary nebulae some of which exhibit an extremely strong 3.4${\,{\rm \mu m}}$ feature.
Summary {#sec:summary}
=======
We have presented an intensity scaling scheme for scaling the band strengths of the C–H stretching features of PAHs with a methyl side chain computed with B3LYP which is less accurate and computationally less demanding. Such an intensity scaling approach allows us to obtain accurate band strengths, as accurate as that computed with MP2 in conjunction with large basis sets which is known to be more accurate than B3LYP but computationally very expensive. It is found that the band intensities calculated with [B3LYP/6-31G$^{\ast}$]{} for a number of molecules are much higher than their gas-phase experimental values. Using better basis sets in conjunction with the B3LYP method, the computed intensities are still considerably higher (by ${\sim\,}$30%) compared to their experimental results. The MP2 method with the basis set of [6-311+G(3df,3pd)]{} reproduces the measured intensities reasonably well. However, such calculations are far too expensive especially for large molecules. It is shown that intensity scaling approaches that are based on the B3LYP data can be just as successful. We have also calculated the model spectra of methylated PAHs and their cations of different sizes and various numbers of methyl sidegroups, excited by starlight of different spectral shapes and intensities. We find that the ratio of the model intensity of the 3.4${\,{\rm \mu m}}$ feature to that of the 3.3${\,{\rm \mu m}}$ feature is insensitive to the PAH size and the spectral shape and intensity of the exciting starlight. We have derived a simple, convenient, and straightforward relation for determining the aliphatic fraction ${N_{\rm C,ali}}/{N_{\rm C,aro}}$ of the 3.3${\,{\rm \mu m}}$-band carriers from the observed band ratios ${\left(I_{3.4}/I_{3.3}\right)_{\rm obs}}$: ${N_{\rm C,ali}}/{N_{\rm C,aro}}\approx0.57\times{\left(I_{3.4}/I_{3.3}\right)_{\rm obs}}$ for neutrals and ${N_{\rm C,ali}}/{N_{\rm C,aro}}\approx0.26\times{\left(I_{3.4}/I_{3.3}\right)_{\rm obs}}$ for cations.
Rationale for A Non-Zero Offset in the Intensity Scaling Relation
=================================================================
We show here that the non-zero offset in the intensity scaling relation (see §\[sec:scaling\]) comes from the fact that the intensities of methyl (aliphatic) and aromatic C–H stretches do not scale alike (i.e., ${f_{\rm ali}}\neq {f_{\rm aro}}$). Eqs.\[eq:I\_L12\_define\_a\] and \[eq:I\_L12\_define\_b\] show the total intensities of the C–H stretching regions as a function of the numbers of methyl ($n_{{3.4}}$) and aromatic ($n_{{3.3}}$) C–H bonds and the average IR intensities of a methyl (${A_{3.4}}$) or of an aromatic (${A_{3.3}}$) C–H stretching bond for two theoretical levels ${L_i}$ and ${L_j}$:
\[eq:I\_L12\_define\] $$\begin{aligned}
A({L_i})&~=~n_{{3.4}}\,A_{{3.4}}({L_i}) + n_{{3.3}}\,A_{{3.3}}({L_i})
\label{eq:I_L12_define_a} \\
A({L_j})&~=~n_{{3.4}}\,A_{{3.4}}({L_j}) + n_{{3.3}}\,A_{{3.3}}({L_j})
\label{eq:I_L12_define_b}\end{aligned}$$
where $A_{{3.4}}({L_i})$ and $A_{{3.3}}({L_i})$ are respectively the strengths of one aliphatic or one aromatic C–H bond computed at the ${L_i}$ level, and $A_{{3.4}}({L_j})$ and $A_{{3.3}}({L_j})$ are the same parameters but computed at the ${L_j}$ level.
Assuming that the intensities of the methyl (aliphatic) and aromatic C–H stretches scale with factors ${f_{\rm ali}}$ and ${f_{\rm aro}}$, respectively, one can express the total intensity at level ${L_j}$ as a function of the average IR intensities of a methyl (aliphatic) or of an aromatic C–H stretching bond at theoretical levels ${L_i}$ \[i.e., $A_{{3.4}}({L_i})$ and $A_{{3.3}}({L_i})$; see eq.\[eq:I\_L12\_scale\_1a\]\]. By addition and subtraction of the term ${f_{\rm aro}}\,n_{{3.4}}\,A_{{3.4}}({L_i})$, it is possible to rewrite eq.\[eq:I\_L12\_scale\_1a\] such that $A({L_j})$ is expressed as a function of $A({L_i})$ and $A_{{3.4}}({L_i})$ (see eq.\[eq:I\_L12\_scale\_1d\]). Using instead the analogous term ${f_{\rm ali}}\,n_{{3.3}}\,A_{{3.3}}({L_i})$ gives $A({L_j})$ as a function of $A({L_i})$ and $A_{{3.3}}({L_i})$ (see eq.\[eq:I\_L12\_scale\_2d\]).
\[eq:I\_L12\_scale1\] $$\begin{aligned}
A({L_j})&~=~{f_{\rm ali}}\,n_{{3.4}}\,A_{{3.4}}({L_i}) + {f_{\rm aro}}\,n_{{3.3}}\,A_{{3.3}}({L_i})
\label{eq:I_L12_scale_1a} \\
&~=~{f_{\rm ali}}\,n_{{3.4}}\,A_{{3.4}}({L_i}) + {f_{\rm aro}}\,n_{{3.3}}\,A_{{3.3}}({L_i})
+ {f_{\rm aro}}\,n_{{3.4}}\,A_{{3.4}}({L_i}) - {f_{\rm aro}}\,n_{{3.4}}\,A_{{3.4}}({L_i})
\label{eq:I_L12_scale_1b} \\
&~=~{f_{\rm aro}}\,[n_{{3.4}}\,A_{{3.4}}({L_i}) + n_{{3.3}}\,A_{{3.3}}({L_i})]
+ {f_{\rm ali}}\,n_{{3.4}}\,A_{{3.4}}({L_i}) - {f_{\rm aro}}\,n_{{3.4}}\,A_{{3.4}}({L_i})
\label{eq:I_L12_scale_1c} \\
&~=~{f_{\rm aro}}\,A({L_i}) + \underline{({f_{\rm ali}}- {f_{\rm aro}})\,n_{{3.4}}\,A_{{3.4}}({L_i})}
\label{eq:I_L12_scale_1d}\end{aligned}$$
or
\[eq:I\_L12\_scale2\] $$\begin{aligned}
A({L_j})&~=~{f_{\rm ali}}\,n_{{3.4}}\,A_{{3.4}}({L_i}) + {f_{\rm aro}}\,n_{{3.3}}\,A_{{3.3}}({L_i})
\label{eq:I_L12_scale_2a} \\
&~=~{f_{\rm ali}}\,n_{{3.4}}\,A_{{3.4}}({L_i}) + {f_{\rm aro}}\,n_{{3.3}}\,A_{{3.3}}({L_i})
+ {f_{\rm ali}}\,n_{{3.3}}\,A_{{3.3}}({L_i}) - {f_{\rm ali}}\,n_{{3.3}}\,A_{{3.3}}({L_i})
\label{eq:I_L12_scale_2b} \\
&~=~{f_{\rm aro}}\,[n_{{3.4}}\,A_{{3.4}}({L_i}) + n_{{3.3}}\,A_{{3.3}}({L_i})]
+ {f_{\rm aro}}\,n_{{3.3}}\,A_{{3.3}}({L_i}) - {f_{\rm ali}}\,n_{{3.3}}\,A_{{3.3}}({L_i})
\label{eq:I_L12_scale_2c} \\
&~=~{f_{\rm ali}}\,A({L_i}) + \underline{({f_{\rm aro}}- {f_{\rm ali}})\,n_{{3.3}}\,A_{{3.3}}({L_i})}
\label{eq:I_L12_scale_2d}\end{aligned}$$
where the underlined terms in eqs.\[eq:I\_L12\_scale\_1d\] and \[eq:I\_L12\_scale\_2d\] are responsible for the offset in the correlations between the total intensities at levels ${L_i}$ and ${L_j}$, and these offsets vanish only when ${f_{\rm aro}}= {f_{\rm ali}}$. This condition never holds and, in addition, it also is not trivial to determine at what level ${f_{\rm aro}}$ and ${f_{\rm ali}}$ converge. We have extensively studied the basis set effects at the B3LYP level for toluene and the three isomers of methylpyrene (see Yang et al. 2013). There is a very large basis set dependency in that $A_{{3.3}}$ is greatly reduced with the improvements of the basis set. The typical $A_{{3.3}}$ value at the [B3LYP/6-31G$^{\ast}$]{} level is ${\sim\,}$18–20${\,{\rm km}}{\,{\rm mol}}^{-1}$ and this value drops to ${\sim\,}$12.5–13.3${\,{\rm km}}{\,{\rm mol}}^{-1}$ at the highest level [6-311++G(3df,3pd)]{}, i.e., a scaling factor of ${f_{\rm aro}}\approx0.7$. In contrast, the basis set dependency of $A_{{3.4}}$ is less than that of $A_{{3.3}}$. A typical $A_{{3.4}}$ value at the [B3LYP/6-31G$^{\ast}$]{} level is ${\sim\,}$23–27${\,{\rm km}}{\,{\rm mol}}^{-1}$ and this value drops to ${\sim\,}$19–24${\,{\rm km}}{\,{\rm mol}}^{-1}$ at the [6-311++G(3df,3pd)]{} level, i.e., a scaling factor of ${f_{\rm ali}}\approx0.85$. This confirms the need for non-zero offset in intensity scaling because ${f_{\rm ali}}\neq {f_{\rm aro}}$.
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[^1]: The intensities for benzene are taken from the 3-term Blackman-Harris entries with a resolution of 0.125${\,{\rm cm}}^{-1}$.
[^2]: Bertie & Keefe (1994) gave a significantly higher value of $A_{\rm aro}(\nu_{12})\approx 73\pm{9}{\,{\rm km}}{\,{\rm mol}}^{-1}$ based on their integration over the range of 3175–2925${\,{\rm cm}}^{-1}$. Note that this region contains some intensity from the (weak) combination bands.
[^3]: Note that $A_{\rm aro}$ ($A_{\rm ali}$) is the strength of all the aromatic (aliphatic) C–H stretches while ${A_{3.3}}$ (${A_{3.4}}$) is the strength of the aromatic (aliphatic) stretch per C–H bond. For toluene, $A_{\rm aro} = 5{A_{3.3}}$ and $A_{\rm ali} = 3{A_{3.4}}$ and therefore we have ${A_{3.4}/A_{3.3}}= \left(5/3\right)\,{A_{\rm ali}/A_{\rm aro}}$.
[^4]: The MP2 computations are performed either with the full active space of all core and valence electrons considered in the correlation energy computation, denoted MP2(full), or with the frozen core approximation and the consideration of just the valence electrons in the correlation treatment, denoted MP2(fc). With MP2/6-311G(3df,3pd), Pavlyuchko et al. (2012) calculated the C–H stretch intensities of benzene and toluene to be ${\sim\,}$53${\,{\rm km}}{\,{\rm mol}}^{-1}$ and ${\sim\,}$98${\,{\rm km}}{\,{\rm mol}}^{-1}$, respectively. We have tried both MP2(fc)/6-311G(3df,3pd) and MP2(full)/6-311G(3df,3pd). With MP2(fc)/6-311G(3df,3pd), we obtained ${\sim\,}$53.8${\,{\rm km}}{\,{\rm mol}}^{-1}$ and ${\sim\,}$97.1${\,{\rm km}}{\,{\rm mol}}^{-1}$ for benzene and toluene, respectively, while with MP2(full)/6-311G(3df,3pd) these intensities become ${\sim\,}$52.4${\,{\rm km}}{\,{\rm mol}}^{-1}$ and ${\sim\,}$94.7${\,{\rm km}}{\,{\rm mol}}^{-1}$. Although the MP2(fc) results closely match that of Pavlyuchko et al. (2012), the MP2(full) results are closer to the experimental results (${\sim\,}$55${\,{\rm km}}{\,{\rm mol}}^{-1}$ for benzene and ${\sim\,}$95${\,{\rm km}}{\,{\rm mol}}^{-1}$ for toluene). Since MP2(full) considers all the core and valence electrons and thus should be more accurate than MP2(fc), we therefore calculate all other vibrational spectra with MP2(full) in conjunction with the standard basis set 6-31G$^{\ast}$ and the extended basis sets 6-311+G$^{\ast\ast}$ and 6-311+G(3df,3pd) for benzene, naphthalene and their mono-methyl derivatives as test cases.
[^5]: As defined by Draine & Li (2007), $\gamma_{3.4}$, $\gamma_{6.85}$, and $\gamma_{7.25}$ are dimentionless parameters.
[^6]: We note that it is not necessary to consider larger PAHs since the 3.3${\,{\rm \mu m}}$ C–H feature is predominantly emitted by small neutral PAHs of ${\sim\,}$20–30 C atoms (see Figures 6,7 of Draine & Li 2007).
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The fundamental relations in the dynamics of single diffraction dissociation and elastic scattering at high energies are discussed.'
author:
- |
[A.A. Arkhipov]{}\
*[Institute for High Energy Physics]{}\
*[Protvino, 142284 Moscow Region, Russia]{}**
title: 'WHAT CAN WE LEARN FROM THE STUDY OF SINGLE DIFFRACTIVE DISSOCIATION AT HIGH ENERGIES?[^1]'
---
PACS numbers: 11.80.-m, 13.85.-t, 21.30.+y
Keywords: inclusive reactions, diffraction dissociation, three-body forces, elastic scattering, total cross-sections, slope of diffraction cone, numeric calculations, fit to the data, interpretation of experiments.
Introduction
============
I shall start my talk from the end with the short answer to the question in the title. The study of the inclusive reactions in the region of diffraction dissociation at high energies provides a unique possibility to learn on a new type of interactions between elementary particles or a new type of fundamental forces, which the three-body forces are. What was the beginning on?
In 1994 the CDF group at Fermilab published new results on the measurements of $p\bar p$ single diffraction dissociation at $\sqrt{s} = 546$ and $1800\ GeV$. They observed that a popular supercritical Pomeron model did not describe new measured values. The statement, made in [@1], is as follows: The value of $\sigma_{sd}^{p\bar p} = (7.89\pm 0.33)\ mb$, measured at $\sqrt{s} =
546\ GeV$, is extrapolated by the supercritical Pomeron model to $\sigma_{sd}^{p\bar p} = (13.9\pm 0.9)\ mb$ at $\sqrt{s} = 1800\
GeV$, while the measured value at this energy is equal to $\sigma_{sd}^{p\bar p} =
(9.45\pm 0.44)\ mb$. The ratio of the measured $\sigma_{sd}^{p\bar
p}$ to that obtained by extrapolation is ( = 1800 GeV) = 0.680.05 . \[1\] Moreover, at $\sqrt{s} = 20 \ GeV$ the experimental $\sigma_{sd}^{p\bar p} = (4.9\pm 0.55)\ mb$ is 4.5 times larger than the value $\sigma_{sd}^{p\bar p} = (1.1\pm 0.17)\ mb$, obtained by the extrapolation of the measured value of $\sigma_{sd}^{p\bar p}$ at $\sqrt{s} = 546\ GeV$ down to $\sqrt{s} = 20\ GeV$ with the help of the supercritical Pomeron model. So, the latest experimental measurement of $p\bar p$ single diffraction dissociation at c.m.s. energies $\sqrt{s} = 546$ and $1800\ GeV$, carried out by the CDF group at the Fermilab Tevatron collider, has shown that the popular model of supercritical Pomeron does not describe the existing experimental data.
We called the emerged situation as a supercrisis for the supercritical Pomeron model (SCPM).[^2] The supercrisis is illustrated on Fig. 1 extracted from paper [@3].
The attempts undertaken in Refs. [@3; @4] to save the SCPM are also shown on this figure. Unfortunately GLM paper [@4] contains a crude mathematical mistake. The mistake was observed by B.V. Struminsky and E.S. Martynov from Kiev [@5]. Besides, in our opinion, an eikonalization procedure cannot be considered as a saving ring for SCPM because this procedure is outside the original Regge ideology. The idea of renormalized Pomeron flux proposed by Goulianos is a good physical idea for an experimentalist, but this idea cannot be a satisfactory one for a theorist because the idea is not grounded by the underlying Regge theory.
Obviously, the foundations of the Pomeron model require a further theoretical study and the construction of newer, more general phenomenological framework, which would enable one to remove the discrepancy between the model predictions and the experiment.
Although nowadays we have in the framework of local quantum field theory a gauge model of strong interactions formulated in terms of the known QCD Lagrangian, its relations to the so called “soft" (interactions at large distances) hadronic physics are far from desired. The understanding of this physics is high interest because it has an intrinsically fundamental nature.
In 1970 the experiments at the Serpukhov accelerator revealed that the $K^+p$ total cross section increased with energy. The increase of the $pp$ total cross section was discovered at the CERN ISR and then the effect of rising total cross sections was confirmed at the Fermilab accelerator.
In spite of more than 25 years after the formulation of QCD we still cannot obtain from the QCD Lagrangian the answer to the question why all the hadronic total cross-sections grow with energy. We cannot predict total cross-sections in an absolute way starting from the fundamental QCD Lagrangian as well mainly because it is not a perturbative problem.
It is well known, e.g., that nonperturbative contributions to the gluon propagator influence the behaviour of “soft" hadronic processes and the knowledge of the infrared behaviour of QCD is certainly needed to describe the “soft" hadronic physics in the framework of QCD. Unfortunately, today we don’t know the whole picture of the infrared behaviour of QCD, we have some fragments of this picture though (see e.g. Ref. [@6]).
At the same time it is more or less clear now that the rise of the total cross-sections is just the shadow (not antishadow!) of particle production.
Through the optical theorem the total cross-section is related to the imaginary part of the elastic scattering amplitude in the forward direction. That is why the theoretical understanding of elastic scattering has the fundamental importance.
From the unitarity relation it follows that the imaginary part of the elastic scattering amplitude contains the contribution of all possible inelastic channels in two-particle interaction. It is clear therefore that we cannot understand the elastic scattering without understanding the inelastic interaction.
Among all the possible inelastic interactions there is a special class of processes which are called a single diffraction dissociation. The single diffraction dissociation is the scattering process where one of two particles in the initial state breaks up during the interaction producing a system of particles in a limited region of (pseudo)rapidity.[^3]
Good and Walker have shown [@7] that the single diffraction dissociation is predicted by the basic principles of quantum mechanics. However both the elastic scattering and single diffraction dissociation cannot correctly be calculated in QCD due to the non-perturbative nature of the interactions.
The popular Regge phenomenology represents elastic and diffractive scattering by the exchange of the Pomeron, a color singlet Reggeon with quantun numbers of the vacuum. It should be noted that the definition of the Pomeron as Reggeon with the highest Regge trajectory $\alpha_P(t)$, carrying the quantum numbers of the vacuum, is not the only one.[^4] There are many other definitions of the Pomeron: Pomeron is a gluon “ladder" [@8]; Pomeron is a bound state of two reggezied gluons – BFKL-Pomeron [@9]; soft and hard Pomerons [@10; @11]; etc.[^5] This leapfrog is because of the exact nature of the Pomeron and its detailed substructure remains such as that no one knows what it is. The difficulty of establishing the true nature of the Pomeron in QCD is almost obviously related to the calculations of non-perturbative gluon exchange.
Nevertheless in the near past simple formulae of the Regge phenomenology provided good parameterization of experimental data on “soft" hadronic physics and pragmatic application of Pomeron phenomenology had been remarkably successful (see e.g. the latest issue of the Review of Particles Properties).
That was the case before the appearance of the above-mentioned CDF data on single diffractive dissociation and recent results from HERA. Of course, it is good that we have a simple and compact form for representing a great variety of data for different hadronic processes, but it is certainly bad that power behaved total cross-sections violate unitarity. Often and often encountered claim, that the model with power behaved total cross-sections is valid in the non-asymptotic domain which has been explored up today, is not correct because the supercritical Pomeron model is an asymptotic one by definition.
We suggested another approach to the dynamical description of one-particle inclusive reactions [@12]. The main point of our approach is that new fundamental three-body forces are responsible for the dynamics of particle production processes of inclusive type. Our consideration revealed several fundamental properties of one-particle inclusive cross-sections in the region of diffraction dissociation. In particular, it was shown that the slope of the diffraction cone in $p\bar p$ single diffraction dissociation is related to the effective radius of three-nucleon forces in the same way as the slope of the diffraction cone in elastic $p\bar p$ scattering is related to the effective radius of two-nucleon forces. It was also demonstrated that the effective radii of two- and three-nucleon forces, which are the characteristics of elastic and inelastic interactions of two nucleons, define the structure of the $p\bar p$ total cross-sections in a simple and physically clear form. I’ll touch upon these properties later on.
First of all let me tell you a few words what I mean by three-body forces about.
Three-body forces in relativistic quantum theory
================================================
Using the LSZ or the Bogoljubov reduction formulae in quantum field theory [@13] we can easily obtain the following cluster structure for $3\rightarrow 3$ scattering amplitude (see diagram below) \_[123]{} = [F]{}\_[12]{} + [F]{}\_[23]{} + [F]{}\_[13]{} + [F]{}\_[123]{}\^C \[2\] where ${\cal F}_{ij} , (i,j = 1,2,3)$ are $2 \rightarrow 2$ scattering amplitudes, ${\cal F}_{123}^C$ is called the connected part of the $3 \rightarrow 3$ scattering amplitude.
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In the framework of single-time formalism in quantum field theory [@14] we construct the $3 \rightarrow 3$ off energy shell scattering amplitude $T_{123}(E)$ with the same (cluster) structure as (\[2\]) T\_[123]{}(E) = T\_[12]{}(E) + T\_[23]{}(E) + T\_[13]{}(E) + T\_[123]{}\^C(E). \[3\] Following the tradition we’ll call the kernel describing the interaction of three particles as the three particle interaction quasipotential. The three particle interaction quasipotential $V_{123}(E)$ is related to the off-shell $3 \rightarrow 3$ scattering amplitude $T_{123}(E)$by the Lippmann-Schwinger type equation T\_[123]{}(E) = V\_[123]{}(E) + V\_[123]{}(E)G\_0(E)T\_[123]{}(E). \[4\] There exists the same transformation between two particle interaction quasipotentials $V_{ij}$ and off energy shell $2
\rightarrow 2$ scattering amplitudes $T_{ij}$ T\_[ij]{}(E) = V\_[ij]{}(E) + V\_[ij]{}(E)G\_0(E)T\_[ij]{}(E). \[5\] It can be shown that in the quantum field theory the three particle interaction quasipotential has the following structure [@15] V\_[123]{}(E) = V\_[12]{}(E) + V\_[23]{}(E) + V\_[13]{}(E) + V\_0(E). \[6\] The quantity $V_0(E)$ is called the three-body forces quasipotential. The $V_0(E)$ represents the defect of three particle interaction quasipotential over the sum of two particle interaction quasipotentials and describes the true three-body interactions. The three-body forces quasipotential is an inherent connected part of total three particle interaction quasipotential which cannot be represented by the sum of pair interaction quasipotentials.
The three-body forces scattering amplitude is related to the three-body forces quasipotential by the equation T\_0(E) = V\_0(E) + V\_0(E)G\_0(E)T\_0(E). \[7\]
It should be stressed that the three-body forces appear as a result of consistent consideration of three-body problem in the framework of local quantum field theory.
Global analyticity of the three-body forces
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Let us introduce the following useful notations = 2i\^4(\_[i=1]{}\^[3]{}p’\_i-\_[j=1]{}\^[3]{}p\_j) [F]{}\_[123]{}(s;[e]{}’,e), \[8\] $$s = (\sum_{i=1}^{3}p'_i)^2 = (\sum_{j=1}^{3}p_j)^2.$$ The ${\hat e}', \hat e \in S_5$ are two unit vectors on five-dimensional sphere describing the configuration of three-body system in the initial and final states (before and after scattering).
We will denote the quantity $T_0$ restricted on the energy shell as $$T_0\mid_{on\, energy\, shell}\, = {\cal F}_0.$$ The unitarity condition for the quantity ${\cal F}_0$ with account for the introduced notations can be written in form [@16; @17]\
&=& A\_3(s)d\_[5]{}([e]{}”)[F]{}\_0(s;[e]{}’,[e]{}”) \_0(s;e,[e]{}”) + H\_0(s;[e]{}’,e), \[9\]
$$Im{\cal F}_0(s;{\hat e}',\hat e)=\frac{1}{2i}\left[{\cal F}_0(s;{\hat
e}',\hat e)
-\stackrel{*}{\cal F}_0(s;\hat e,{\hat e}')\right],$$ where $$A_3(s) = {\Gamma}_3(s)/S_5 ,$$ ${\Gamma}_3(s)$ is the three-body phase-space volume, $S_5$ is the volume of unit five-dimensional sphere. $H_0$ defines the contribution of all the inelastic channels emerging due to three-body forces.
Let us introduce a special notation for the scalar product of two unit vectors ${\hat e}'$ and $\hat e$ = [e]{}\^[’]{}e. \[10\] We will use the other notation for the three-body forces scattering amplitude as well $${\cal F}_0(s;{\hat e}',\hat e) = {\cal F}_0(s;\eta,\cos\omega),$$ where all other variables are denoted through $\eta$.
Now we are able to go to the formulation of our basic assumption on the analytical properties for the three-body forces scattering amplitude [@16; @17].
We will assume that for physical values of the variable $s$ and fixed values of $\eta$ the amplitude ${\cal F}_0(s;\eta,\cos\omega)$ is an analytical function of the variable $\cos\omega$ in the ellipse $E_0(s)$ with the semi-major axis z\_0(s) = 1 + \[11\] and for any $\cos\omega \in E_0(s)$ and physical values of $\eta$ it is polynomially bounded in the variable $s$. $M_0$ is some constant having mass dimensionality.
Such analyticity of the three-body forces amplitude was called a global one. The global analyticity may be considered as a direct geometric generalization of the known analytical properties of two-body scattering amplitude strictly proved in the local quantum field theory [@18; @19; @20; @21; @22].
At the same time the global analyticity results in the generalized asymptotic bounds.
$\framebox{\bf GLOBAL ANALYTICITY}\enspace \& \enspace \framebox{\bf
UNITARITY}$\
2ex $\Downarrow$\
2ex $\framebox{\bf GENERALIZED ASYMPTOTIC BOUNDS}$
For example the generalized asymptotic bound for $O(6)$-invariant three-body forces scattering amplitude looks like [@16; @17] $$Im\,{\cal F}_0(s;...) \leq \mbox{Const}\, s^{3/2}
\bigl(\frac{\ln s/s'_0}{M_0}\bigr)^5 = \mbox{Const}\,
s^{3/2}R_0^5(s),
\label{12}$$ where $R_0(s)$ is the effective radius of the three-body forces introduced according to [@22] where the effective radius of two-body forces has been defined, R\_0(s) = = , (s) = ,s, \[13\] $r_0$ is defined by the power of the amplitude ${\cal F}_0$ growth at high energies [@17], $M_0$ defines the semi-major axis of the global analyticity ellipse (\[11\]), ${\Lambda}_0$ is the effective global orbital momentum, $\Pi(s)$ is the global momentum of three-body system, $s'_0$ is a scale defining unitarity saturation of three-body forces.
It is well known that the Froissart asymptotic bound [@23] can be experimentally verified, because with the help of the optical theorem we can connect the imaginary part of $2 \rightarrow 2$ scattering amplitude with the experimentally measurable quantity which is the total cross-section. So, if we want to have a possibility for the experimental verification of the generalized asymptotic bounds $(n\geq3)$, we have to establish a connection between the many-body forces scattering amplitudes and the experimentally measurable quantities. For this aim we have considered the problem of high energy particle scattering from deuteron and on this way we found the connection of the three-body forces scattering amplitude with the experimentally measurable quantity which is the total cross-section for scattering from deuteron [@24]. Moreover the relation of the three-body forces scattering amplitude to one-particle inclusive cross-sections has been established [@25].
I shall briefly sketch now the basic results of our analysis of high-energy particle scattering from deuteron.
Scattering from deuteron
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The problem of scattering from two-body bound states was treated in [@24; @25] with the help of dynamic equations obtained on the basis of single-time formalism in QFT [@15]. As has been shown in [@24; @25], the total cross-section in the scattering from deuteron can be expressed by the formula \_[hd]{}\^[tot]{}(s) = \_[hp]{}\^[tot]{}(s) +\_[hn]{}\^[tot]{}(s) - (s), \[14\] where $\sigma_{hd}, \sigma_{hp}, \sigma_{hn}$ are the total cross-sections in scattering from deuteron, proton and neutron, $$\delta\sigma(s) = \delta\sigma_G(s) +\delta\sigma_0(s),\label{15}$$ $$\delta\sigma_G(s) =
\frac{\sigma_{hp}^{tot}(\hat s) \sigma_{hn}^{tot}(\hat s)}{4\pi(
R^2_d+B_{hp}(\hat s)+B_{hn}(\hat s))}
\equiv \frac{\sigma_{hp}^{tot}(\hat s) \sigma_{hn}^{tot}(\hat
s)}{4\pi
R^2_{eff}(s)},\ \ \hat s = \frac{s}{2},
\label{16}$$ $B_{hN}(s)$ is the slope of the forward diffraction peak in the elastic scattering from nucleon, $1/R_d^2$ is defined by the deuteron relativistic formfactor , q, \[17\] $\delta\sigma_G$ is the Glauber correction or shadow effect. The Glauber shadow correction originates from elastic rescatterings of an incident particle on the nucleons inside the deuteron.
The quantity $\delta\sigma_0$ represents the contribution of the three-body forces to the total cross-section in the scattering from deuteron. The physical reason for the appearance of this quantity is directly connected with the inelastic interactions of an incident particle with the nucleons of deuteron. Paper [@25] provides for this quantity the following expression: $$\delta\sigma_0(s) = -\frac{(2\pi)^3}{q}
\int \frac{d\vec\Delta\Phi(\vec\Delta)}
{2E_p(\vec\Delta /2)2E_n(\vec\Delta /2)}
Im\,R\bigl(s;-\frac{\vec\Delta}{2},\frac{\vec\Delta}{2},\vec q;
\frac{\vec\Delta}{2},-\frac{\vec\Delta}{2},\vec q \bigr),\label{18}$$ where $q$ is the incident particle momentum in the lab system (rest frame of deuteron), $\Phi(\vec\Delta)$ is the deuteron relativistic formfactor, normalized to unity at zero, $$E_N(\vec\Delta)=\sqrt{\vec\Delta^2 + M^2_N}\quad N = p, n,$$ $M_N$ is the nucleon mass. The function $R$ is expressed via the amplitude of the three-body forces $T_0$ and the amplitudes of elastic scattering from the nucleons $T_{hN}$ by the relation $$R = T_0 + \sum_{N=p,n}(T_0G_0T_{hN} + T_{hN}G_0T_0).\label{19}$$ In [@24] the contribution of three-body forces to the scattering amplitude from deuteron was related to the processes of multiparticle production in the inelastic interactions of the incident particle with the nucleons of deuteron. This was done with the help of the unitarity equation. The character of the energy dependence of $\delta\sigma_0$ was shown to be governed by the energy behaviour of the corresponding inclusive cross-sections.
Here, for simplicity, let us consider the model where the imaginary part of the three-body forces scattering amplitude has the form $$Im\,{\cal F}_0(s; \vec p_1, \vec p_2, \vec p_3; \vec q_1, \vec q_2,
\vec q_3)
= f_0(s)
\exp \Biggl\{-\frac{R^2_0(s)}{4} \sum^{3}_{i=1} (\vec p_i-\vec
q_i)^2\Biggr\},\label{20}$$ where $f_0(s)$, $R_0(s)$ are free parameters which, in general, may depend on the total energy of three-body interaction. Note that the quantity $f_0(s)$ has the dimensionality $[R^2]$.
In case of unitarity saturation of the three-body forces, we have from the generalized asymptotic theorems f\_0(s) \~ s\^[3/2]{}[()]{}\^5 = s\^[3/2]{}R\_0\^5(s),\[21\] R\_0(s) = s/s’\_0 s.\[22\]
In the model all the integrals can be calculated in the analytical form. As a result, we obtain for the quantity $\delta\sigma_0$ [@25] $$\delta\sigma_0(s) = \frac{(2\pi)^{6}f_0(s)}{sM_N }
\Biggl\{\frac{\sigma_{hN}(s/2)}{2\pi[B_{hN}(s/2)+R^2_0(s)-R^4_0(s)/4(
R^2_
0(s)+R^2_d)]}-1\Biggr\}$$ .\[23\] If the condition R\_0\^2(s) B\_[hN]{}(s/2) R\_d\^2 \[24\] is realized, then we obtain from expression (\[23\]) \_0(s) = (2)\^[9/2]{},\[25\] where (s) = - 1,\[26\] and we suppose that asymptotically $$B_{hp}=B_{hn}\equiv B_{hN},\quad \sigma_{hp}^{tot}=\sigma_{hn}^{tot}
\equiv \sigma_{hN}^{tot}.$$
It follows from the Froissart theorem and generalized asymptotic bounds (\[12\]) that the following asymptotic behaviour is admitted for the $\chi(s)$: (s) \~,s . \[27\]
Three-body forces in single diffraction dissociation
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From the analysis of the problem of high-energy particle scattering from deuteron we have derived the formula connecting one-particle inclusive cross-section with the imaginary part of the three-body forces scattering amplitude. This formula looks like , \[28\] $$Im{\cal F}_0^{scr}(\bar s;-\vec{\Delta}, \vec{\Delta}, \vec q;
\vec{\Delta}, -\vec{\Delta}, \vec q\,) = Im{\cal F}_0(\bar
s;-\vec{\Delta}, \vec{\Delta}, \vec q;
\vec{\Delta}, -\vec{\Delta}, \vec q\,)-$$ $$-
4\pi\int d\vec{\Delta}'\frac{\delta\left[E_N(\vec{\Delta} -
\vec{\Delta}') + \omega_h(\vec q+\vec{\Delta}') - E_N(\vec{\Delta}) -
\omega_h(\vec q)\right]}{2\omega_{h}(\vec q + \vec{\Delta}')2E_N
(\vec{\Delta} - \vec{\Delta}')}\times$$ Im[F]{}\_[hN]{}(s; , q; -’, q + ’)Im[F]{}\_0(|s;-, -’, q + ’; , -, q), \[29\] $$E_N(\vec{\Delta})=\sqrt{{\vec{\Delta}}^2+M_N^2},\quad
\omega_h(\vec q)=\sqrt{{\vec q}\,^2+m_h^2},$$ $$I(s) = 2{\lambda}^{1/2}(s,m_h^2,M_N^2),\quad \hat s = \frac{\bar s +
m_h^2 - 2M_N^2}{2},$$ $$\bar s = 2(s + M_N^2) - M_X^2,\quad t = - 4{\vec\Delta}^2.$$ I’d like to draw the attention to the minus sign in the R.H.S. of Eq. (\[28\]). The simple model for the three-body forces considered above (see Eq. (\[20\])) gives the following result for the one-particle inclusive cross-section in the region of diffraction dissociation $$\frac{s}{\pi}\frac{d\sigma_{hN\rightarrow NX}}{dtdM_X^2} =
\frac{(2\pi)^3}{I(s)}\chi(\bar s)Im{\cal F}_0(\bar
s;-\vec{\Delta}, \vec{\Delta}, \vec q;
\vec{\Delta}, -\vec{\Delta}, \vec q\,)$$ = (|s)f\_0(|s)\[30\] where $$\chi(\bar s) = \frac{\sigma^{tot}_{hN}({\bar s}/2)}{2\pi[B_{hN}({\bar
s}/2) + R_0^2(\bar s)]} -1.$$ The configuration of particles momenta and kinematical variables are shown in Fig. 2. The variable $\bar s$ in the R.H.S. of Eq. (\[30\]) is related to the kinematical variables of one-particle inclusive reaction by the equation |s = 2(s + M\_N\^2) - M\_X\^2,\[31\] $$t = - 4{\Delta}^2.$$
There is a temptation to call the quantity $I(s)\chi^{-1}(\bar s)$ a renormalized flux. However, it should be pointed out that in our case we have a flux of real particles and function $\chi (s)$ has quite a clear physical meaning. The function $\chi (s)$ originates from initial and final states interactions and describes the effect of screening the three-body forces by two-body ones [@25].
If we take the usual parameterization for one-particle inclusive cross-section in the region of diffraction dissociation = A(s.M\_X\^2), \[32\] then we obtain for the quantities $A$ and $b$ A(s,M\_X\^2) = (|s)f\_0(|s), b(s,M\_X\^2) = \[33\].
Eq. (\[33\]) shows that the effective radius of three-body forces is related to the slope of diffraction cone for inclusive diffraction dissociation processes in the same way as the effective radius of two-body forces is related to the slope of diffraction cone in elastic scattering processes. Moreover, it follows from the expressions $$R_0(\bar s) = \frac{r_0}{M_0} \ln \bar s/s'_0,\quad \bar s =
2(s+M_N^2) - M_X^2$$ that the slope of diffraction cone for inclusive diffraction dissociation processes at fixed energy decreases with the growth of missing mass. This property agrees well qualitatively with the experimentally observable picture.
Hence physically tangible notion of the effective radius of three-body forces introduced previously provides a clear physical interpretation that helps one to create a visual picture and representation for inclusive diffraction dissociation processes at the same level as one can understand and represent elastic scattering processes at high energies. Besides, relation (\[28\]) together with linear equation (\[7\]) for the three-body forces scattering amplitude may be the basis of powerful dynamic apparatus for constructing the dynamical models for the theoretical description of the inclusive reactions.
In the case of unitarity saturation of the three-body forces, we have from generalized asymptotic theorems $$f_0(s) \sim s^{3/2}\left(\frac{\ln s/s'_0}{M_0}\right)^5,\quad
\chi(s) \sim \frac{1}{\sqrt{s}{\ln}^3 s},\quad s\rightarrow
\infty.$$ This means that
.\[34\]
On the other hand, comparing formulae (\[25\]) and (\[33\]) we see that one and the same combination $\chi f_0$ enters in the equations. Therefore, we can extract this combination and express it through experimentally measurable quantities. We have in this way A(s,M\_X\^2) = \_0(|s). \[35\] In that case it would be very desirable to think about the creation of accelerating deuterons beams instead of protons ones at the now working accelerators and colliders.
On the structure of hadronic total cross-sections
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Let’s rewrite the equation for $\chi(s)$ $$\chi(s) = \frac{\sigma_{hN}^{tot}(s/2)}{2\pi[B_{hN}(s/2)+R_0^2(s)]} -
1$$ in the form
\_[hN]{}\^[tot]{}(s) = 2(1+).\[36\] From the Froissart and generalized asymptotic bounds we have $$\chi(s) = O\left(\frac{1}{\sqrt{s}\ln^3s}\right),\quad s\rightarrow
\infty.$$ We also know that [@20] \_[hN]{}\^[tot]{}(s,s\_0) \~\^2(s/s\_0)B\_[hN]{}(s,s\_0)\~\^2(s/s\_0),\[37\] and Eq. (\[36\]) gives $$R_0^2(2s,s'_0)\sim \ln^2(2s/s'_0)\sim \ln^2(s/s_0),\quad
s\rightarrow\infty.$$ Therefore, we come to the following asymptotic consistency condition: .\[38\] The asymptotic consistency condition tells us that we have not any new scale. The scale defining unitarity saturation of three-body forces is unambiguously expressed by the scale which defines unitarity saturation of two-body forces. In that case we have $$R_0^2(2s,s'_0) = R_0^2(s,s_0)$$ and \[39\] with a common scale $s_0$.
Reminding the relation between the effective radius of two-body forces and the slope of diffraction cone in elastic scattering B\_[hN]{}(s) = R\_[hN]{}\^2(s), \[40\] we obtain \_[hN]{}\^[tot]{}(s) = R\_[hN]{}\^2(s) + 2R\_0\^2(s),s.\[41\] Equations (\[39\]) and (\[41\]) define a new nontrivial structure of hadronic total cross-section. It should be emphasized that the coefficients staying in the R.H.S. of Eq. (\[41\]) in front of effective radii of two- and three-body forces are strongly fixed.
It is useful to compare the new structure of total hadronic cross-section with the known structure. We have from unitarity
\_[hN]{}\^[tot]{}(s) = \_[hN]{}\^[el]{}(s) + \_[hN]{}\^[inel]{}(s).\[42\] If we put \_[hN]{}\^[el]{}(s) = R\_[hN]{}\^[el\^2]{}(s),\_[hN]{}\^[inel]{}(s) = 2R\_[hN]{}\^[inel\^2]{}(s),\[43\] then we come to the similar formula
\_[hN]{}\^[tot]{}(s) = R\_[hN]{}\^[el\^2]{}(s) + 2R\_[hN]{}\^[inel\^2]{}(s).\[44\] But it should be borne in mind R\_[hN]{}\^2(s) = R\_[hN]{}\^[el\^2]{}(s),R\_0\^2(s) = R\_[hN]{}\^[inel\^2]{}(s).\[45\] In fact, we have \_[hN]{}\^[el]{}(s) = = ,\[46\] \_[hN]{}\^[inel]{}(s) = \_[hN]{}\^[tot]{}(s).\[47\] Of course, Eqs. (\[43\]) are the definitions of $R_{hN}^{el}$ and $R_{hN}^{inel}$. The definition of $R_{hN}^{el}$ corresponds to our classical imagination, the definition of $R_{hN}^{inel}$ corresponds to our knowledge of quantum mechanical problem for scattering from the black disk. Let us suppose that \_[hN]{}\^[tot]{}(s\_m) R\_[hN]{}\^2(s\_m), s\_m,(R\_0\^2(s\_m)R\_[hN]{}\^2(s\_m)),\[48\] then we obtain \_[hN]{}\^[el]{}(s\_m) = R\_[hN]{}\^2(s\_m),\_[hN]{}\^[inel]{}(s\_m) = R\_[hN]{}\^2(s\_m).\[49\] This simple example shows that the new structure of total hadronic cross-sections is quite different from that given by Eq. (\[42\]). The reason is that the structure (\[39\]) is of the dynamical origin. We have mentioned above that the coefficients, staying in the R.H.S. of Eq. (\[41\]) in front of effective radii of two- and three-body forces, are strongly fixed. In fact, we found here the answer to the old question: Why the constant ($\pi/m_{\pi}^2
\approx 60\,mb $) staying in the Froissart bound is too large in the light of the existing experimental data. The constant in the R.H.S. of Eq. (\[41\]), staying in front of effective radius of hadron-hadron interaction, is 4 times smaller than the constant in the Froissart bound. But this is too small to correspond to the experimental data. The second term in the R.H.S. of Eq. (\[41\]) fills an emerged gap.
It is a remarkable fact that the quantity $R_0^2$, which has the clear physical interpretation, at the same time, is related to the experimentally measurable quantity which the total cross-section is. This important circumstance gives rise to the new nontrivial consequences which are discussed in the next section.
We made an attempt to check up the structure (\[39\]) on its correspondence to the existing experimental data and I’d like to present the preliminary results here.
At the first step, we made a weighted fit to the experimental data on the proton-antiproton total cross-sections in the range $\sqrt{s}>10\, GeV$. The data were fitted with the function of the form predicted by Froissart bound in the spirit of our approach[^6] \^[tot]{}\_[asmpt]{} = a\_0 + a\_2 \^2(/) \[50\] where $a_0, a_2, \sqrt{s_0}$ are free parameters. We accounted for experimental errors $\delta x_i$ (statistical and systematic errors added in quadrature) by fitting to the experimental points with the weight $w_i=1/(\delta x_i)^2$. Our fit yielded a\_0 = (42.04790.1086)mb,a\_2 = (1.75480.0828)mb,\[51\] = (20.741.21)GeV.\[52\] The fit result is shown in Fig. 3.
After that we made a weighted fit to the experimental data on the slope of diffraction cone in elastic $p\bar p$ scattering. The experimental points and the references, where they have been extracted from, are listed in [@26]. The fitted function of the form B = b\_0 + b\_2 \^2(/20.74), \[53\] which is also suggested by the asymptotic theorems of local quantum field theory, has been used. The value $\sqrt{s_0}$ has been fixed by (\[52\]) from the fit to the $p\bar p$ total cross-sections data. Our fit yielded b\_0 = (11.920.15)GeV\^[-2]{} ,b\_2 = (0.30360.0185)GeV\^[-2]{}.\[54\] The fitting curve is shown in Fig. 4.
At the final stage we build a global (weighted) fit to all the data on proton-antiproton total cross-sections in a whole range of energies available up today. The global fit was made with the function of the form \^[tot]{}\_[p|p]{}(s) = \^[tot]{}\_[asmpt]{}(s) \[55\] where $m_N$ is proton (nucleon) mass, R\^2\_0(s) = (GeV\^[-2]{}),\[56\] \^[tot]{}\_[asmpt]{}(s) = 42.0479 + 1.7548 \^2(/20.74),\[57\] B(s) = 11.92 + 0.3036 \^2(/20.74),\[58\] $c, d_1, d_2, d_3$ are free parameters. Function (\[55\]) corresponds to the structure given by Eq. (\[39\]).
In fact, we have for the function $\chi (s)$ in the R.H.S. of Eq. (\[39\]) theoretical expression in the form (s) = \[59\] where \^4 (s) = \_a\^b dx ,\[60\] $$a = 2m_N,\quad b = \sqrt{2s+m_N^2}-m_N.$$ It can be proved that $\kappa (s)$ has the following asymptotics[^7] $$\kappa (s)\sim \sqrt{s},\quad s\rightarrow \infty,$$ $$\kappa (s)\sim \sqrt{s-4m_N^2},\quad s\rightarrow 4m_N^2.$$ We used at the moment the simplest function staying in the R.H.S. of Eq. (\[55\]) which described these two asymptotics.
Our fit yielded $$d_1 = (-12.12\pm 1.023)GeV,\quad d_2 = (89.98\pm 15.67)GeV^2,$$ d\_3 = (-110.5121.60)GeV\^3,c = (6.6551.834)GeV\^[-2]{}.\[61\] The fitting curve is shown in Figs. 5, 6.
The experimental data on proton-proton total cross-sections display a more complex structure at low energies than the proton-antiproton ones. To describe this complex structure we, of course, have to modify formula (\[55\]) without destroying the general structure given by Eq. (\[39\]). The modified formula looks like $$\sigma_{pp}^{tot}(s) = \sigma^{tot}_{asmpt}(s) \times$$ ,\[62\] where $\sigma^{tot}_{asmpt}(s)$ is the same as in proton-antiproton case (Eq. (\[57\])) and d(s) = \_[k=1]{}\^[8]{},Resn(s) = \_[i=1]{}\^[8]{}. \[63\] Compared to Eq. (\[55\]) we introduced here an additional term $Resn(s)$ describing diproton resonances which have been extracted from [@27; @28]. The positions of resonances and their widths are listed in Table I. The $c_1, c_2, s_{thr}, d_i, C_R^i
(i=1,...,8)$ were considered as free fit parameters. The fitted parameters obtained by fit are listed below (see $C_R^i$ in Table I.) $$c_1=(192.85\pm 1.68) GeV^-2,\quad c_2=(186.02\pm 1.67) GeV^-2,$$ $$s_{thr}=(3.5283\pm 0.0052) GeV^2,$$ $$d_1=(-2.197\pm 1.134)10^2 GeV,\quad d_2=(4.697\pm 2.537)10^3 GeV^2,$$ $$d_3=(-4.825\pm 2.674)10^4 GeV^3,\quad d_4=(28.23\pm 15.99)10^4 GeV^4,$$ $$d_5=(-98.81\pm 57.06)10^4 GeV^5,\quad d_6=(204.5\pm 120.2)10^4 GeV^6,$$ d\_7=(-230.2137.3)10\^4 GeV\^7,d\_8=(108.2665.44)10\^4 GeV\^8.\[64\] The fitting curve is shown in Figs. 7-10. It should be pointed out that our fit revealed that the resonance with the mass $m_R=2106\,
MeV$ should be odd parity. Our fit indicates that this resonance is strongly confirmed by the set of experimental data on proton-proton total cross-sections. That is why a further study of diproton resonances is very desirable.
The figures 4-11 display a very good correspondence of theoretical formula (\[39\]) to the existing experimental data on proton-proton and proton-antiproton total cross-sections.
I’d like to emphasize the following attractive features of formula (\[39\]). This formula represents hadronic total cross-section in a factorized form. One factor describes high-energy asymptotics of total cross-section and it has the universal energy dependence predicted by the Froissart theorem. Other factor is responsible for the behaviour of total cross-section at low energies and this factor has also a universal asymptotics at the elastic threshold. It is a remarkable fact that the low-energy asymptotics of total cross-section at the elastic threshold is dictated by the high-energy asymptotics of three-body (three-nucleon in that case) forces. This means that we undoubtedly faced very deep physical phenomena here. The appearance of new threshold $s_{thr}=3.5283\,
GeV^2$ in proton-proton channel, which is near to the elastic threshold, is a nontrivial fact too. It’s clear that the difference of two identical terms with different thresholds in the R.H.S. of Eq. (\[62\]) is a tail of crossing symmetry which was not actually taken into account in our consideration. What physical entity does this new threshold correspond to? This interesting question is still open.
Anyway we have established that simple theoretical formula (\[39\]) described the global structure of $pp$ and $p\bar p$ total cross-sections in the whole range of energies available up today. Of course, our results concerning a global description of hadronic total cross-sections are to be considered as preliminary ones. We know the ways how they can be refined later on.
On the slope of diffraction cone in single diffraction dissociation
===================================================================
We have shown above that the slope of diffraction cone in the single diffraction dissociation is related to the effective interaction radius for three-body forces b\_[SD]{}(s,M\_X\^2) = R\_0\^2(|s,s’\_0),\[65\] $$\bar s = 2(s + M_N^2) - M_X^2, \quad s'_0 = 2s_0.$$
Let us define the slope of diffraction cone in the single diffraction dissociation at a fixed point over the missing mass B\^[sd]{}(s) = b\_[SD]{}(s,M\_X\^2)\_[M\_X\^2 = 2M\_N\^2]{}.\[66\] Now taking into account Eqs. (\[40\],\[41\]) where the effective interaction radius for three-body forces can be extracted from R\_0\^2(2s,2s\_0) = R\_0\^2(s,s\_0) = \^[tot]{}(s) - B\^[el]{}(s),\[67\] and the equation \^[el]{}(s) = ,(= 0)\[68\] we come to the fundamental relation between the slopes in the single diffraction dissociation and elastic scattering ,\[69\] where X . The quantity $X$ has a clear physical meaning, it has been introduced in the papers of C.N. Yang and his collaborators [@29; @30].
We found $X = 0.25$ at $\sqrt{s} = 1800\, GeV$ (see the CDF paper mentioned in Introduction). Hence, in that case we have $B^{sd} =
B^{el}/2$ which is confirmed not so badly in the experimental measurements.
In the limit of the black disk $(X = 1/2)$ we obtain ,\[70\] and .
So, we observe that there is quite a nontrivial dynamics in the slopes of diffraction cone in the single diffraction dissociation and elastic scattering processes. In particular, we can study an intriguing question on the black disk limit not only in the measurements of total hadronic cross-sections compared with elastic ones but in the measurements of the slopes in single diffraction dissociation processes together with elastic scattering ones.
There is a more general formula which can be derived with account of the real parts for the amplitudes. This formula looks like .\[71\] If $\rho_{el} = 0$ or $\rho_0 = -\rho_{el}$, then we come to Eq. (\[69\]). In the case when $\rho_{el} \not= 0$, we can rewrite Eq. (\[71\]) in the form \_0 = .\[72\] Eq. (\[72\]) can be used for the calculation of the new quantity $\rho_0$. Anyway, the measurements of real parts for the amplitudes seem to play an important role in the future high energy hadronic physics.
On total single diffractive dissociation\
cross-section
=========================================
For the total single diffractive dissociation cross-section defined as \_[sd]{}(s) = 2\_[M\_[min]{}\^2]{}\^[0.1s]{}\_[t\_[-]{}(M\_X\^2)]{}\^[t\_[+]{} (M\_X\^2)]{} dt A(s,M\_X\^2)\[73\] we obtained the following asymptotic formula \_[sd]{}(s) = ,\[74\] where $c_0, c_2$ are related to effective interaction radius for three-body forces $$R_0^2(s) = c_0 + c_2\ln^2(\sqrt{s}/\sqrt{s_0}),$$ and $A_0, A_2$ to be found from the fit to the experimental data on $\sigma_{sd}$. The experimental values for $p\bar p$ single diffraction dissociation cross-sections, which were used, are listed in Table II. Our fit yielded [@26; @31] $$A_0 = 23.395\pm 2.664\, mbGeV^{-2},\quad A_2 = 4.91\pm 0.26\,
mbGeV^{-2}.$$ The fit result is shown in Fig. 11. As you can see, the fitting curve goes excellently over the experimental points of the CDF group at Fermilab.
Thus, we have shown that from the generalized asymptotic theorems a là Froissart there follows a simple formula which allows one to match the experimental data on $p\bar p$ single diffraction dissociation cross-sections at high energies including lower energies as well. At present only the suggested approach allows one to quantitatively describe the observed behaviour of $p\bar p$ single diffraction dissociation cross-sections.
Some time ago many high energy physicists thought that the increase of total cross-sections was due to the same increase of single diffraction dissociation cross-sections. Now we know that this thought is wrong and, moreover, we understand why this is the case.
As it has been shown above the phenomenon of exceedingly moderate energy dependence of single diffraction dissociation cross-sections on $s$ observed by CDF at Fermilab is a manifestation of unitarity saturation of three-nucleon forces at Fermilab Tevatron energies. This phenomenon is confirmed in the dynamics consistent with unitarity becoming apparent in the effect of screening of three-body forces by two-body ones. It is to be compared with the discovery of the increase of $pp$ total cross-sections at CERN ISR and of the growth of $K^{+}p$ total cross-sections revealed at Serpukhov accelerator. In this context, the CDF data are the ones of the most significant experimental results obtained in the last years.
In fact, we have found the bound (like Froissart bound!) .\[75\]
I’d like especially to point out that analyticity and unitarity together with the dynamic apparatus of single-time formalism in QFT provide the clear answers to the asymptotic behaviour both the elastic scattering and single diffraction dissociation at high energies, which correspond to the experimentally observable picture.
It is very nice that the understanding of “soft" physics based on general principles of QFT, such as analyticity and unitarity, is so fine confirmed by the experimentally observable picture compared to the models where the general principles have been broken down.
I hope that it will be possible to test the obtained results at higher energies, such as those of the LHC collider and even higher ones.
On the forms of strong interaction\
dynamics
===================================
Conditionally there are two forms of strong interaction dynamics: t-channel form and s-channel one. 0.1 true in
**t-channel form**
0.1 true in The fundamental quantity here is some set of Regge trajectories: t-channel form \_R(t).\[76\] Here subscript $R$ enumerates different Regge trajectories which are the poles in the t-channel partial wave amplitudes for the given process. There are a lot of people who work in the field of t-channel dynamics of strong interactions.
Some part of scientific community works in the field of s-channel form of strong interaction dynamics. 0.1 true in
**s-channel form**
0.1 true in The fundamental quantity here is an effective interaction radius of fundamental forces: s-channel form R\_(s).\[77\] Here subscript $\alpha$ enumerates different types of hadrons and fundamental forces acting between them. The s-channel form of dynamics allows one to create a physically transparent and visual geometric picture of strong interactions for hadrons. I’d like to emphasize the attractive features of this form of strong interaction dynamics. 0.1 true in
- Universality (existence of pion with $m_{\pi}\not=0$): $$\fbox{$\displaystyle R_{\alpha}(s) \sim \frac{r_{\alpha}}{m_{\pi}}\ln
\frac{s}{s_0},\quad s \rightarrow \infty$}\,.$$
- Compatibility with the general principles of relativistic quantum theory.
- Fine mathematical structures are given by the global analyticity together with single-time formalism in QFT.
That is why, in our opinion, the s-channel form of strong interaction dynamics is more preferable than the t-channel one.
Conclusion
==========
In Commemoration of the 200th Anniversary of Alexander S. Pushkin I’d like to conclude my talk with the Ode: 0.3 true in wncyr10 at 12.0pt [ **** ]{} 0.3 true in
Acknowledgements {#acknowledgements .unnumbered}
================
It is my pleasure to express thanks to the Organizing Committee for the given opportunity to attend the VIIIth Blois Workshop and present the report there. I am indebted to V.V. Ezhela for the access to the computer readable files on total proton-proton and proton-antiproton cross-sections in the IHEP COMPAS database. The friendly encouragement and many pieces of good advice on computer usage from A.V. Razumov are gratefully acknowledged.
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(288,388) (-50,-110)[{width="14cm"}]{}
Figure 1: The total single diffraction cross-sections for $p(\bar
p)+p\rightarrow p(\bar p)+X$ vs $\sqrt{s}$ compared with the predictions of the renormalized Pomeron flux model of Goulianos [@3] (solid line) and of the model Gostman, Levin and Maor [@4] (dashed line, labelled GLM).
(275,315) (-40,-50)[{width="12cm"}]{}
Figure 2: Kinematical notations and configuration of momenta in the relation of one-particle inclusive cross-section to the three-body forces scattering amplitude.
(288,198) (15,10)[]{} (144,0)[$\sqrt{s}\, (GeV)$]{} (0,77)
Figure 3: The total proton-antiproton cross-section versus $\sqrt{s}$ compared with formula (\[50\]). Solid line represents our fit to the data. Statistical and systematic errors added in quadrature.
(288,188) (15,10)[]{} (144,0)[$\sqrt{s}\, (GeV)$]{} (0,77)
Figure 4: Slope $B$ of diffraction cone in $p\bar p$ elastic scattering. Solid line represents our fit to the data.
(288,204) (15,10)[]{} (144,0)[$\sqrt{s}\, (GeV)$]{} (0,87)
Figure 5: The total proton-antiproton cross-section versus $\sqrt{s}$ compared with formula (\[55\]). Solid line represents our fit to the data.
(288,194) (15,10)[]{} (144,0)[$\sqrt{s}\, (GeV)$]{} (0,87)
Figure 6: The total proton-antiproton cross-section versus $\sqrt{s}$ compared with formula (\[55\]) in the range $\sqrt{s}<10\, GeV$ (fragment of Fig. 5). Solid line represents our fit to the data.
(288,204) (15,10)[]{} (144,0)[$\sqrt{s}\, (GeV)$]{} (0,87)
Figure 7: The total proton-proton cross-section versus $\sqrt{s}$ compared with formula (\[62\]). Solid line represents our fit to the data. Statistical and systematic errors added in quadrature.
(288,194) (15,10)[]{} (144,0)[$\sqrt{s}\, (GeV)$]{} (0,87)
Figure 8: The total proton-proton cross-section (vs $\sqrt{s}$) including a point from cosmic rays experiment [@36] compared with formula (\[62\]). Solid line represents our fit to the data.
(288,194) (15,10)[]{} (144,0)[$\sqrt{s}\, (GeV)$]{} (0,87)
Figure 9: The total proton-proton cross-section in the range $\sqrt{s}<30\, GeV$ compared with formula (\[62\]). Solid line represents our fit to the data.
(288,182) (15,10)[]{} (144,0)[$\sqrt{s}\, (GeV)$]{} (0,87)
Figure 10: The total proton-proton cross-section at low energies compared with formula (\[62\]). Solid line represents our fit to the data.
(388,314) (18,18)[{width="370bp"}]{} (185,0)[$\sqrt{s}\, (GeV)$]{} (0,150)
Figure 11: Total single diffraction dissociation cross-section compared with formula (\[74\]). Solid line represents our fit to the data.
[^1]: The talk presented at the VIIIth Blois Workshop on Elastic and Diffractive Scattering. Protvino, Russia, June 28–July 2, 1999.
[^2]: Recent experimental results from HERA [@2] lead us to the same conclusion. The soft Pomeron phenomenology as currently developed cannot incorporate the HERA data on structure function $F_2$ at small $x$ and total $\gamma^{*}p$ cross section from $F_2$ measurements as a function of $W^2$ for different $Q^2$.
[^3]: Pseudorapidity is defined as $\eta = -\ln \tan (\theta /2)$ where $\theta$ is the polar angle of the produced particle with respect to the beam direction. Pseudorapidity is frequently used as an approximation to rapidity.
[^4]: For supercritical Pomeron $\alpha_P(0) - 1 = \Delta \ll 1,\, \Delta > 0$ is responsible for the growth of hadronic cross-sections with energy.
[^5]: At the Workshop I heard new definition of Pomeron from N.N. Nikolaev: Pomeron is (neither more nor less!) a label of diffraction.
[^6]: Recently, from a careful analysis of the experimental data and a comparative study of the known characteristic parameterizations, Bueno and Velasco have shown (Phys. Lett. B[**380**]{}, 184 (1996)) that statistically a “Froissart-like" type parameterization for proton-proton and proton-antiproton total cross-sections is strongly favoured.
[^7]: Integral in R.H.S. of Eq.(\[60\]) can be expressed in terms of the Appell function.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Recent HBT results from the CERES experiment at the SPS are reviewed. Emphasis is put on the centrality and beam energy dependence, and the results are put into perspective with results at lower and higher beam energies. The rather weak beam energy dependence of the HBT radii may be understood in terms of a transition from baryon to pion dominated freeze-out. The observed short lifetimes and emission durations are presently in contradiction to results from model calculations.'
---
[**Pion Interferometry:**]{} [**Recent results from SPS**]{}\
Harald Appelshäuser\
[*Physikalisches Institut der Universität Heidelberg\
Philosophenweg 12, D-69120 Heidelberg, Germany\
*]{}
Introduction
============
While single particle momentum distributions give only indirect insight to the space-time evolution of the system created in high energy heavy ion collisions, the lifetime and the spatial extent of the pion source as well as the existence of collective velocity fields at the time of thermal freezeout can be probed by the study of Bose-Einstein momentum correlations of identical pions via HBT interferometry [@pratt1]. The width of the correlation peak at vanishing relative momenta reflects the so-called length of homogeneity of the pion emitting source. Only in static sources can the length of homogeneity, in the following also called ’source radius’, be interpreted as the true geometrical size of the system. In a dynamic system, the occurence of space-momentum correlations of the emitted particles due to collective expansion generally leads to a reduction of the observed source radii, depending on the strength of the expansion and the thermal velocity of the pions $\sqrt{T_{f}/m_{t}}$ at thermal freeze-out [@maksin]. A differential analysis of the HBT correlations in bins of the pair transverse momentum $k_{t}$=$\frac{1}{2}|\vec{p_{t,1}}+\vec{p_{t,2}}|$ thus provides valuable information about the properties of the collective expansion of the system [@prattcso].
To obtain most detailed information about the space-time evolution the three-momentum difference vector $\vec{q}$ of two like-sign pions is decomposed into components, $\vec{q}$=$(q_{\rm long},q_{\rm side},q_{\rm out})$. Following Bertsch and Pratt [@berpra], $q_{\rm long}$ is the momentum difference along the beam direction, calculated in the longitudinal rest frame (LCMS) of the pair, $q_{\rm out}$ is parallel to the pair transverse momentum $\vec{k_{t}}$ and $q_{\rm side}$ is perpendicular to $q_{\rm long}$ and $q_{\rm out}$.
The correlation function is defined as the ratio $C_{2}(\vec{q})$=$A_{2}(\vec{q})/B_{2}(\vec{q})$ where the ’signal’ $A_{2}(\vec{q})$ is the probability to find a pair with momentum difference $\vec{q}$ in a given event and the ’background’ $B_{2}(\vec{q})$ is the corresponding mixed-event distribution.
The normalized correlation functions are fit by a Gaussian: $$\begin{aligned}
C_{2}(\vec{q})&=& 1+\lambda \exp(-R_{\rm long}^{2}q_{\rm long}^{2}
-R_{\rm side}^{2}q_{\rm side}^{2} \nonumber \\
&& -R_{\rm out}^{2}q_{\rm out}^{2}
-2R_{\rm outlong}^{2}q_{\rm out}q_{\rm long}),\end{aligned}$$
with $R_{\rm long}$, $R_{\rm side}$, $R_{\rm out}$ being the Gaussian source radii and $\lambda$ the correlation strength. The cross-term $R_{\rm outlong}^{2}$ appears as a consequence of space-time correlations in non-boost-invariant systems.
Results from SPS
================
Pion HBT interferometry analyses have been performed by a number of experiments at SPS [@na49hbt; @na44hbt; @wa98hbt; @wa97hbt]. A systematic study of the centrality dependence of HBT radii around midrapidity from Pb+Au collisions at all three presently available SPS energies, 40, 80, and 158 AGeV, was recently presented by the CERES collaboration [@na45hbt]. Details of their analysis can be found in [@heinz].
{width="15cm"}\
The $k_{t}$-dependence of the longitudinal source parameter $R_{\rm long}$ is shown in Fig. 1 for the three different beam energies and as function of the centrality of the collision. Similar to previous studies, a strong decrease of $R_{\rm long}$ with $k_{t}$ is observed, characteristic for a strong collective longitudinal expansion, where the length of homogeneity is entirely saturated by the thermal length scale $\sim\sqrt{T_{f}/m_{t}}$. For the case of longitudinal boost-invariance, $R_{\rm long}$ can be connected to the lifetime $\tau_{s}$ of the system, the time elapsed between the onset of expansion and kinetic freeze-out by $R_{\rm long}$=$\tau_{s}(T_{f}/m_{t})^{\frac{1}{2}}$ [@maksin].
The slight but systematic overall increase of $R_{\rm long}$ with centrality and beam energie is reflected in a correspondingly longer lifetime when applying the above expression to the data and assuming $T_{f}$=120 MeV (see fits in Fig. 1). The observed lifetimes at the SPS are 6-8 fm/c.
{width="15cm"}\
{width="15cm"}\
\[fig:rout\]
The transverse source parameter $R_{\rm side}$ is closely related to the [*true*]{} geometrical transverse extension of the system at freeze-out. Consequently, a weaker $k_{t}$-dependence is observed for $R_{\rm side}$ (Fig. 2), indicating the presence of radial flow. A slight but systematic increase of $R_{\rm side}$ with centrality is measured at all beam energies, as expected from simple collision geometry. The beam energy dependence is surprising: largest $R_{\rm side}$ and the strongest $k_{t}$-dependence are observed at the lowest energy. This led previously to the conclusion that radial flow may reach a maximum at the lower SPS energies [@ceresqm]. We will discuss below that the extraction of the radial flow velocity from the $k_{t}$-dependence of $R_{\rm side}$ at the lower SPS energies might be questionable. At 158 AGeV, a radial flow velocity of about 0.5$c$ was obtained from the analysis of single particle $m_{t}$-spectra and $R_{\rm side}(k_{t})$ [@na49hbt; @tomwiedheinz].
Many authors have discussed that a strong first order phase transition with a large latent heat can lead to a retardation of hadronization during the mixed phase and consequently to a long duration of pion emission [@berpra; @bergong; @berbrown; @rischke; @rigyu]. This should be observable via a strong increase of the outward source parameter $R_{\rm out}$ with respect to $R_{\rm side}$. However, no long-lived source has been observed so far. At SPS, $R_{\rm out}$ (Fig. 3) is similar to $R_{\rm side}$, indicating a short pion emission duration and a sudden freeze-out.
Beam energy dependence
======================
A large amount of pion interferometry data have been published by experiments at AGS, SPS, and RHIC. This allows for a systematic study of the source parameters over a wide range of beam energies. It has been argued that the [*HBT null effect*]{} [@gyulint], the absence of any beam energy dependence, in particular when going to RHIC, is the actual surprising result from HBT interferometry.
In Fig. 4 are shown the $k_{t}$-dependences of $R_{\rm long}$,$R_{\rm side}$, and $R_{\rm out}$ in central Pb(Aa)+Pb(Au) collisions at different beam energies [@e895hbt; @heinz; @starhbt]. Indeed, no dramatic variation of any of the source parameters can be observed. However, a closer inspection reveals some interesting features. $R_{\rm long}$ is approximately constant from AGS to the lower SPS energies, but develops a significant increase within the SPS regime and towards RHIC, indicating a smooth increase of the lifetime.
Most interesting is the behaviour of $R_{\rm side}$: It is gradually [*decreasing*]{} at small $k_{t}$ up to top SPS energy, connected with a continuous [*flattening*]{} of the $k_{t}$-dependence. At RHIC, $R_{\rm side}$ is again larger than at SPS while the shape is not yet well measured, at least the $\pi^{-}\pi^{-}$ sample looks rather flat. Naivly, the flattening indicates a [*decrease*]{} of the radial flow velocity as function of beam energy, in contradiction to the present understanding [@xuqm].
$R_{\rm out}$ shows a rather weak energy dependence, possibly indicating a shallow minimum at the lower SPS energy.
{width="15cm"}\
\[fig:edep\]
A straight-forward investigation of the freeze-out properties can be performed by relating the measured source parameters to an effective freeze-out volume, $V_{f}$=$2\pi R_{\rm long}R_{\rm side}^{2}$. Assuming freeze-out at constant density [@pom], we expect $V_{f}$ to scale linearly with the charged particle multiplicity. Fig. 5 shows $V_{f}$ as function of the number of participants at 40, 80, and 158 AGeV. A linear scaling with $N_{\rm part}$ is indeed observed at all three energies, consistent with the assumption of a constant freeze-out density, since the number of charged particles was found to scale close to linear with $N_{\rm part}$ at SPS [@wa98mult].
{width="15cm"}\
{width="15cm"}\
\[fig:vol\_edep\]
But the beam energy dependence is surprising: There is no clear hierarchy visible in Fig. 5 as expected from the increase of multiplicity by about 50% between 40 AGeV and 158 AGeV; the smallest $V_{f}$ are observed at 80 AGeV. Obviously, the freeze-out volume scales with multiplicity as long as multiplicity is controlled via centrality, but it does not scale accordingly as multiplicity changes with beam energy.
The comparison of $V_{f}$ at different beam energies from AGS to RHIC sheds some light on this: The freeze-out volume $V_{f}$ is gradually decreasing within the AGS energy range, reaches a minimum at SPS and then increases towards RHIC, as demonstrated in Fig. 6.
Clearly, a simple relation between multiplicity and freeze-out volume cannot hold. On the other hand, it is plausible to assume that at low energies pions interact mainly with nucleons, while at high energies pion-pion scattering dominates. Is the transition from nucleon to pion dominated freeze-out characterized by a minimum in $V_{f}$ at SPS?
{width="15cm"}\
\[fig:dndy\]
In Fig. 7 (left) the midrapidity density of pions [@cebra; @seyboth; @starmult; @starqm] and protons [@e895prot; @e917prot; @ceresqm; @na49prot; @na49aprot; @staraprot] in central Pb(Au)+Pb(Au) collisions is shown as function of $\sqrt{s}$. The pion yield increases monotonically with beam energy. The [*total*]{} proton (p+$\bar{\rm p}$) yield at midrapidity drops from AGS to SPS and stays approximately constant between SPS and RHIC because the decreasing number of net protons is compensated by p-$\bar{\rm p}$ production. The sum of pions and protons, however, is still a monotonic function of $\sqrt{s}$.
At this point, the different cross sections $\sigma_{\pi\pi}$ and $\sigma_{\pi N}$ have to be considered. In Fig. 7 (upper right) the cross section weighted sum of pions and protons $N_{\rm eff}$=$2\cdot N_{{\rm p}+\bar{\rm p}} \cdot \sigma_{\pi N}$+ $3\cdot N_{\pi^{-}} \cdot \sigma_{\pi \pi}$ is shown [^1]. For the cross sections $\sigma_{\pi\pi}$=10 mb and $\sigma_{\pi N}$=65 mb are assumed. Indeed, the cross section weighted sum of pions and nucleons at midrapidity is [*non*]{}-monotonic and exhibits a minimum at SPS. As a consequence, the ratio $V_{f}/N_{\rm eff}$, which has the dimension of a length, is approximately constant (Fig. 7, lower right). This result suggests $V_{f}/N_{\rm eff}$$\approx$1 fm as a universal freeze-out condition. In this picture, the relatively weak beam energy dependence of HBT source parameters can be understood as an interplay between the decreasing (and eventually levelling off) (p+$\bar{\rm p}$) yield and the increasing pion multiplicity at midrapidity, if their different cross sections with pions are taken into account. It is interesting to obtain a detailed understanding of the role of protons for pion freeze-out at low energies. The strong momentum dependence of the $\sigma_{\pi N}$ cross section may affect the $k_{t}$-dependence of $R_{\rm side}$ at low beam energies and cause the flattening of the $k_{t}$-dependence with increasing beam energy, as the importance of protons ceases. At the lower SPS energies, where protons are still important, the interpretation of the $k_{t}$-dependence of $R_{\rm side}$ in terms of radial flow may be questionable, and possibly breaks down at AGS.
Discussion
==========
Single particle spectra and azimuthal anisotropies at SPS and RHIC have been well reproduced by state-of-the-art hydro+cascade calculations [@soff; @soffhirsch; @teaneyqm]. However, they fail to reproduce the measured HBT parameters, in particular the lifetime of the system is grossly overestimated.
In these models, the early plasma phase is described by hydrodynamics assuming an ideal plasma EOS with a first order phase transition and a mixed phase with an adjustable latent heat. At the end of the mixed phase, hadrons are created assuming a thermal phase space population superimposed by the collective velocity field created by the plasma pressure. After hadronization, hadrons are rescattered using a hadronic cascade code.
In [@teaney] the response of the hydro+cascade calculation to different choices of the latent heat is well documented. From this investigation, it becomes obvious that the choice of a relatively large latent heat of 0.8 GeV/fm$^3$, needed to describe the SPS single particle data, is mainly driven by the observed mass dependence of the inverse slope parameters. Smaller latent heat produces larger flow velocities [*before*]{} hadronization and therefore too hard spectra, in particular for the multi-strange baryons which do not follow the linear scaling behaviour observed for $\pi$, K, p [@xu]. On the other hand, a small latent heat would lead to an early acceleration of the matter and therefore to a faster dilution and shorter lifetime, in particular of the hadronic phase, which is suggested by the HBT measurements.
This leads to a closer inspection of the data as represented in Fig. 8 (left). For some of the particle species, the observed inverse slope parameter scales linearly with the particle mass, consistent with the assumption of a common flow velocity. There are, however, a number of exceptions which do not follow this rule ($\phi$, $\Xi$, $\Omega$). This was explained by their small cross sections with the surrounding matter, hence they are expected to participate less in the collective motion.
{width="15cm"}\
\[fig:mdep\]
On the other hand, it is evident that the data separate into two groups: All mesons have similar slopes, and the same is true for baryons. One may therefore as well correlate the inverse slope with the number of constituent quarks inside the hadron, rather than with the hadron mass (Fig. 8, right). Also here a scaling behaviour can be observed, and almost all particles are consistent with this trend. Note that the large inverse slope of the $\phi$ observed by NA49 [@na49phi] is experimentally still under debate, since there is also a measurement by NA50 [@na50phi], giving a much smaller number.
The physical picture which arises from this representation is different from the common interpretation: collective motion has completely developed [*before*]{} hadronization, with constituent quarks being the flowing objects. Hadrons are formed by coalescence of constituent quarks, preferentially if they are close in phase space, thereby adding their momenta. When hadrons acquire mass, momentum is conserved which leaves the momentum spectra unchanged. In this picture, the data are consistent with a kinetic freeze-out temperature of 0.12 GeV, a collective quark flow velocity of $\sqrt{1/3}$ and an effective quark mass of 0.33 GeV: $T$=$0.12+\frac{1}{2} \frac{1}{3}\cdot 0.33 \cdot n_{\rm quarks}$ (dashed line in Fig. 8, right). Most of the explosive power of the system would be assigned to the early, pre-hadronic phase of the collision, thereby qualitatively explaining the short lifetimes observed by HBT. It also suggests that the hadronic mass scale is no more relevant in the early phase of the collision, as naively expected for a deconfined partonic system.
[99]{} S. Pratt, Phys. Rev. Lett. [**53**]{} (1984) 1219. A.N. Makhlin and Yu.M. Sinyukov, Z. Phys. [**C39**]{} (1988) 69. S. Pratt, T. Csörgö, and J. Zimanyi, Phys. Rev. [**C42**]{} (1990) 2646. G. Bertsch, Nucl. Phys. [**A498**]{}, (1989) 173c, S. Pratt, Phys. Rev. [**D33**]{} (1986) 1314. NA49 Collaboration, H. Appelshäuser [*et al.*]{}, Eur. Phys. J. [**C2**]{} (1998) 661. NA44 Collaboration, I.G. Bearden [*et al.*]{}, Eur. Phys. J. [**C18**]{} (2000) 317. WA98 Collaboration, M.M. Aggerwal [*et al.*]{}, Eur. Phys. J. [**C16**]{} (2000) 445. WA97 Collaboration, F. Antinori [*et al.*]{}, J. Phys. [**G27**]{} (2001) 2325. H. Appelshäuser, INT-RHIC Winter Workshop 2002, Seattle. H. Tilsner, PhD Thesis, Universität Heidelberg (2002), publication in preparation. CERES Collaboration, D. Adamova [*et al.*]{}, Nucl. Phys. [**A698**]{} (2002) 253c. B. Tomasik, U.A. Wiedemann, and U. Heinz, [*nucl-th/9907096*]{}. G. Bertsch, M. Gong, and M. Tohyama, Phys. Rev. [**C37**]{} (1988) 1896. G. Bertsch and G.E. Brown, Phys. Rev. [**C40**]{} (1989) 1830. D. Rischke, Nucl. Phys. [**A610**]{} (1996) 88. D. Rischke and M. Gyulassy, Nucl. Phys. [**A608**]{} (1996) 479. M. Gyulassy, INT-RHIC Winter Workshop 2002, Seattle. E895 Collaboration, M.A. Lisa [*et al.*]{}, Phys. Rev. Lett. [**84**]{} (2000) 2798. STAR Collaboration, C. Adler [*et al.*]{}, Phys. Rev. Lett. [**87**]{} (2001). N. Xu and M. Kaneta, Nucl. Phys. [**A698**]{} (2002) 306c. I. Pomeranchuk, Dokl. Akad. Nauk SSSR [**78**]{} (1951) 884. WA98 Collaboration, M.M. Aggerwal [*et al.*]{}, Eur. Phys. J. [**C18**]{} (2001) 651. E895 Collaboration, J.L. Klay [*et al.*]{}, publication in preparation. NA49 Collaboration, M. Gazdzicki, Proceedings of this Conference. STAR Collaboration, C. Adler [*et al.*]{}, Phys. Rev. Lett. [**87**]{} (2001). STAR Collaboration, C. Adler [*et al.*]{}, Nucl. Phys. [**A698**]{} (2002) 64c. E895 Collaboration, J.L. Klay [*et al.*]{}, Phys. Rev. Lett. [**88**]{} (2002). E917 Collaboration, B.B. Back [*et al.*]{}, Phys. Rev. Lett. [**86**]{} (2001) 1970. NA49 Collaboration, H. Appelshäuser [*et al.*]{}, Phys. Rev. Lett. [**82**]{} (1999) 2471. NA49 Collaboration, G. Veres [*et al.*]{}, Nucl. Phys. [**A661**]{} (1999) 383c. STAR Collaboration, C. Adler [*et al.*]{}, Phys. Rev. Lett. [**86**]{} (2001) 4778. S. Soff, S. Bass, and A. Dumitru, Phys. Rev. Lett. [**86**]{} (2001) 3981. S. Soff, Proceedings of this Conference. D. Teaney, J. Lauret, and E.V. Shuryak, Nucl. Phys. [**A698**]{} (2002) 479. D. Teaney, J. Lauret, and E.V. Shuryak, [*nucl-th/0110037*]{}. NA44 Collaboration, N. Xu [*et al.*]{}, Nucl. Phys. [**A610**]{}, (1996) 175c. NA49 Collaboration, S.V. Afanasev [*et al.*]{}, Phys. Lett. [**B491**]{} (2000) 59. NA50 Collaboration, M.C. Abreu [*et al.*]{}, J. Phys. [**G27**]{} (2001) 405.
[^1]: Note that there are a few simplifications: the protons are multiplied by 2 to account for the neutrons, which is not completely correct for the lowest beam energies. Also the role of light nuclei and other produced particles is neglected.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'According to the classification using projective representations of the SO(3) group, there exist two topologically distinct gapped phases in spin-1 chains. The symmetry-protected topological (SPT) phase possesses half-integer projective representations of the SO(3) group, while the trivial phase possesses integer linear representations. In the present work, we implement non-Abelian symmetries in the density matrix renormalization group (DMRG) method, allowing us to keep track of (and also control) the virtual bond representations, and to readily distinguish the SPT phase from the trivial one by evaluating the multiplet entanglement spectrum. In particular, using the entropies $S^I$ ($S^H$) of integer (half-integer) representations, we can define an entanglement gap $G = S^I - S^H$, which equals 1 in the SPT phase, and $-1$ in the trivial phase. As application of our proposal, we study the spin-1 models on various 1D and quasi-1D lattices, including the bilinear-biquadratic model on the single chain, and the Heisenberg model on a two-leg ladder and a three-leg tube. Among others, we confirm the existence of an SPT phase in the spin-1 tube model, and reveal that the phase transition between the SPT and the trivial phase is a continuous one. The transition point is found to be critical, with conformal central charge $c=3$ determined by fits to the block entanglement entropy.'
author:
- 'Wei Li, Andreas Weichselbaum, and Jan von Delft'
title: 'Identifying Symmetry-Protected Topological Order by Entanglement Entropy'
---
introduction
============
Symmetry-protected topological (SPT) phases have attracted enormous research interest recently. [@Wen_1; @Wen_2; @Wen_3; @Wen_4; @Meng; @Burnell; @Cirac; @Pollmann_1; @Pollmann_2; @Pollmann_3; @Haegeman; @Thomas] Among the interesting models exhibiting SPT order, a remarkable example is the spin-1 chain. The generic spin-1 bilinear-biquadratic (BLBQ) model can be written down as $$H_{blbq} = J \sum_{<i,j>} [\cos(\theta) S_i S_j + \sin(\theta) (S_i S_j)^2],
\label{eq-bbq}$$ where the coupling $J=1$ sets the energy scale, and $\theta$ is a tunable parameter. The phase diagram of the spin-1 BLBQ model with respect to various $\theta$’s is well known (except for a subtlety in the thin region near $\theta=-5/4 \pi$).[@Lauchli; @Tu] When $-\pi/4 < \theta < \pi/4$, the system is in the Haldane phase. [@Haldane] Although this phase has been intensively studied, it has been realized to be an SPT phase only very recently.[@Wen_1; @Cirac; @Pollmann_1] At $\theta=\arctan(1/3)$, an exactly solvable point within the Haldane phase, the ground state is termed AKLT state,[@Affleck] which can be exactly expressed as a matrix product state (MPS) with bond dimension $D=2$. The Haldane phase has a nonzero spin gap, called Haldane gap, which can be interpreted in terms of spinon confinement. [@Rachel] No local order parameter can be found to distinguish the Haldane phase from a trivial gapped phase, nevertheless, there exists a nonlocal string order parameter (SOP), [@Rommelse] $$O_{\alpha} = - \lim_{j-i \to \infty} [S_i^{\alpha} \exp(i \pi \sum_{i<l<j} S_l^{\alpha}) S_j^{\alpha}],
\label{eq-SOP}$$ where $\alpha=x, z$. This string order parameter characterizes the topological order in the Haldane phase. Further studies show that the string order parameter $O_{x,z}$ can be transformed to two ordinary ferromagnetic order parameters through a nonlocal unitary transformation $U_{KT} = \prod_{k < l} \exp(i \pi S_k^z S_l^x)$. Therefore, a nonzero string order parameter actually reveals a hidden $Z_2 \times Z_2$ symmetry breaking. [@KT; @Oshikawa]
![(Color online) (a) Spin chain with coupling constant $J$. (b) Two-leg spin ladder model with $J_L$ and $J_R$ for couplings along chain and rung directions, respectively. (c) Three-leg spin tube model, $J_L$ is the coupling along the leg. Each isosceles triangle contains two kinds of couplings, $J_R$ for the two equal sides and $\alpha J_R$ for the third. (d) The SU(2)-invariant matrix product state describing ground state of spin-1 chain (a), ladder (b) or tube (c). $|S=n\rangle$ represents a multiplet with quantum number $S=n$. For the spin-1 model, each local space is a $|S=1\rangle$ triplet. The input bond multiplets on both open ends can be controlled, and three common choices are shown in (d). (e) shows the phase diagram of the spin-1 BLBQ chain,[@Lauchli; @Tu] H, C, FM, and D stand for Haldane, critical, ferro-magnetic, and dimerized phases, respectively. There is a narrow region near $\theta \approx -3/4 \pi$ with possible spin nematic order, whose existence still remains debatable.[]{data-label="fig-sketch"}](fig1.eps){width="1\linewidth"}
The Haldane phase is protected by several global symmetries. According to the valence bond solid (VBS) picture, the gapped Haldane phase only possesses short-range entanglement, hence it is not an intrinsic topological phase.[@Wen_2] Its nontrivial topological properties are protected by parity symmetry, time reversal symmetry, and $Z_2 \times Z_2$ rotational symmetry around the $x$ and $z$ axes.[@Wen_1; @Pollmann_1] The Haldane phase can not be adiabatically connected to the trivial one as long as one of the above symmetries is preserved along the path; instead, a quantum phase transition (QPT) must occur along the way. As is well known, the Landau paradigm classifies the various symmetry-breaking phases according to symmetry groups.[@Landau; @Landau_2] Nevertheless, the existence of a QPT between SPT phases and trivial ones shows that gapped phases without symmetry breaking in one dimension (and also in higher dimensions) can still be distinct and classified by the group cohomology of symmetries.[@Wen_2; @Wen_3; @Cirac]
To be specific, we consider the gapped phases in SO(3) Heisenberg chains, which can be generally classified by different projective representations of the rotational SO(3) group, i.e., the corresponding group cohomology $H^2(\rm{SO(3), U(1))=Z_2}$. (even) representations of SU(2) are linear representations of SO(3); half-integer (odd) representation, which involve an additional minus sign after $2 \pi$ rotations (owing to the SU(2) double covering, SO(3) = SU(2)/$\rm{Z}_2$) are projective representations of SO(3). Based on this observation, the classification theory states: there are two distinct gapped phases in spin-1 chains corresponding to two different kinds of representations of SO(3), linear and projective. They correspond to the trivial phase and the Haldane phase, respectively. [@Wen_2; @Cirac]
It has recently been discovered that these two phases also differ strikingly in the structure of their entanglement spectra. The entanglement spectrum consists of the eigenvalues of the entanglement Hamiltonian $H_E = -\log(\rho)$, where $\rho$ is the reduced density matrix of a subsystem.[@Li] The entanglement spectrum of the bulk has an intimate relationship with the real excitation spectrum on the boundary.[@Cirac2] Closely related with the group cohomology classification, an interesting feature has been found: For the spin-1 chain, the entanglement spectrum is found to show at least two-fold degeneracy for the Haldane phase, while it is generally non-degenerate for the trivial phase.[@Pollmann_1] The occurrence of the two-fold degeneracy can be used to numerically identify the Haldane phase.
Actually, this degeneracy in the entanglement spectrum is a signature of the appearance of half-integer-spin multiplets in the MPS geometric bond, which support projective representations of the SO(3) group. Take the AKLT point as an intuitive example: according to the construction of the AKLT state, each local spin-1 is decomposed into two spin-1/2 ancillas. The AKLT state can be exactly expressed as MPS with bond dimension 2, hence only one $|S=1/2\rangle$ doublet appears on each of its geometric bonds, and the entanglement spectrum is two-fold degenerate. For other generic states in the Haldane phase, the multiplets on the geometric bonds are generally $S$=half-integer, which leads to at least two-fold degeneracy and supports projective representations. This key feature can be used to differentiate the SPT phase from the trivial one, the latter instead has integer bond multiplets that support linear representations. Therefore, if we could *directly identify the virtual spins* on the geometric bonds of the MPS, it would be straightforward to see whether the representation is projective or linear, and thus to identify the SPT or trivial phase.
One powerful numerical method for solving 1D quantum spin models is the density matrix renormalization group (DMRG).[@White; @Schollwoeck] In order to further improve its efficiency and stability, Abelian and non-Abelian symmetries have been implemented in the DMRG algorithm.[@McCulloch] In particular, the SU(2) DMRG technique enables us to identify the spin of the multiplets on the virtual bonds. Note, though, that if open boundary conditions are adopted for SU(2) DMRG, because only integer-spin sectors are allowed by adding spin-1’s together, the renormalized bases on the virtual bonds are automatically integer multiplets, i.e., linear representations of SO(3). This would imply the absence of two-fold degeneracy within each multiplet (every multiplet contains odd number of individual states) even in Haldane phase, which seems paradoxical.
To solve this problem, we propose a protocol algorithm in this paper which automatically determines the proper bond representations. In addition, by defining and calculating the integer and half-integer entanglement entropies, we elucidate why this protocol algorithm works, and obtain a simple criterion for identifying the SPT phase. We test these ideas in three spin-1 lattice models, and show that they succeed in telling the SPT phase from the trivial one.
The paper is organized as follows. In Sec. II, the MPS and related DMRG algorithms are briefly introduced. In Secs. III-V, we show that the entanglement entropies can be used to identify the SPT phase, by studying three examples including the single spin-1 chain, 2-leg ladder and 3-leg tube models. In particular, in the spin-1 tube model, the transition between the SPT and trivial phases is verified to be a continuous QPT. Sec VI offers a summary.
![(Color online) (a) The multiplet entanglement spectrum of a spin-1 BLBQ chain, calculated using the protocol algorithm. Multiplets $|S=0\rangle \oplus |S=1/2 \rangle$ are put on the end bonds, and the converged spectra are obtained after several DMRG sweeps. $\theta=0$ (asterisks) is in the Haldane phase, with half-integer-spin bond multiplets; $\theta=-\pi/2$ (circles) is in the trivial phase, with integer-spin bond multiplets. Every data point represents a multiplet (not as usual a single state within a multiplet). Therefore, a multiplet with symmetry label $S$ corresponds to $2S+1$ degenerate states. (b) Multiplet spectrum of integer and half-integer representations for the Haldane phase calculated for $\theta=0$, and using $|S=1/2\rangle$ (asterisks) or $|S=0\rangle$ (circles) respectively, on the end bonds. The half-integer spectrum has been shifted by $\log(2)$, in order to reveal the one-to-one correspondence between each multiplet in the half-integer spectrum and a pair of degenerate multiplets in the integer spectrum.[]{data-label="fig-ent-spec"}](fig2.eps){width="1.0\linewidth"}
SU(2) invariant Matrix Product States and Multiplet Entanglement Spectrum
=========================================================================
The variational MPS ground state of 1D Heisenberg systems with Hamiltonians like Eq. (\[eq-bbq\]) can be written in an SU(2)-invariant form. Corresponding bond spaces are factorized into two parts,[@Weich] $$| \tilde{Q} \tilde{n}; \tilde{Q}_z \rangle = \sum_{Q n, Q_z} \sum_{q l,q_z} (A_{Q,\tilde{Q}}^q)_{n, \tilde{n}}^{l}(C_{Q, \tilde{Q}}^q)_{Q_z, \tilde{Q}_z}^{q_z} |Q n; Q_z\rangle |q l; q_z\rangle,$$ where $Qn$ (and $\tilde{Q}\tilde{n}$, $ql$) are composite multiplet indices. $Q$ specifies the symmetry sector, $n$ distinguishes different multiplets with the same $Q$, and $Q_z$ ($\tilde{Q_z}$, $q_z$) labels the individual states within a given multiplet in symmetry sector $Q$ ($\tilde{Q}$, $q$).
The $A$-tensors can be regarded as physical tensors which combine the input multiplets $(Q n)$ with the local space $(q l)$, and transform (and possibly truncate) them into the output multiplets $(\tilde{Q} \tilde{n})$; the $C$-tensors are the Clebsch-Gordan coefficients (CGC) which take care of the underlying mathematical symmetry structure. The tensor product of physical tensor $A$ (reduced multiplet space) and its related mathematical tensor $C$ (CGC space) has been called the QSpace,[@Weich] which is a generic representation used in practice to describe all symmetry-related tensors. [@Weich] The QSpace is a very useful concept not only for MPS wavefunctions, but also for calculating the matrix elements of irreducible tensor operators, which can be treated in the same framework according to the Wigner-Eckart theorem.
By implementing the QSpace in our DMRG code, we need to determine only the physical $A$-tensors variationally as in plain DMRG, while the underlying CGC space ($C$-tensors) are fully determined by symmetry. The $A$-tensors manipulate multiplets $(Q n)$ only on the reduced multiplet level, which leads to a large gain in numerical efficiency. In this work, by adopting the SU(2)-invariant MPS, we are able to keep track of the quantum numbers $S$ of the bond multiplets, and hence to distinguish the SPT phase and the trivial phase straightforwardly.
Given an SU(2)-invariant MPS, it is natural to consider its *multiplet entanglement spectrum*, defined of multiplets, rather than individual states. To be explicit, we note that any SU(2)-invariant MPS can be written in the following form: $$\begin{aligned}
&&|\psi \rangle = \sum_{\{q_i^z\}} {\rm{Tr}} [(A^{q_1}_{Q_1, Q_2})^{l_1}_{n_1,n_2} (\Lambda_{Q_2})_{n_2} ... (\Lambda_{Q_{L-1}})_{n_{L-1}} (A^{q_{L}}_{Q_{L-1}, Q_{L}})^{l_L}_{n_{L-1}, n_{L}} \notag \\
&&(C^{q_1}_{Q_1, Q_2})^{q_1^z}_{Q_1^z, Q_2^z} (\lambda_{Q_2})_{Q_2^z} ... (\lambda_{Q_{L-1}})_{Q_{L-1}^z} (C^{q_L}_{Q_{L-1}, Q_L})^{q_L^z}_{Q_{L-1}^z, Q_L^z} ] |q_1^z ... q_L^z \rangle. \notag \\
\label{eq-su2-mps}\end{aligned}$$ The trace includes all the quantum labels $(Q_i n_i,Q_i^z)$, while $q_i, l_i$ all equal 1 in the present spin-1 case. Eq. (\[eq-su2-mps\]) is an SU(2)-invariant version of Eq. (4) in Ref. . The difference is that the conventional MPS matrix $A$ is represented in the factorized form of a direct product, i.e., $A_{Q_{i-1}, Q_i}^{q_i} \otimes C_{Q_{i-1}, Q_i}^{q_i}$. In Eq. \[eq-su2-mps\] above, we have assumed the canonical MPS forms in both the reduced multiplet space and the CGC space. Notice that since the $C$-matrices store CGC’s, they automatically fulfill the left- and right-canonical conditions. Therefore, the diagonal $\lambda$-matrices are identity matrices, and nontrivial diagonal matrices $\Lambda$ exist only on the multiplet level. Their eigenvalues $\Lambda_i$ determine the multiplet entanglement spectrum defined as $$E_i=- \log(\rho_i) = -2\log(\Lambda_i),$$ where $\rho_i = \Lambda_i^2$ is the reduced-density-matrix eigenvalue corresponding to each multiplet.
In order to illustrate the above concepts, let us now consider the spin-1 BLBQ model on a single chain \[see Fig. \[fig-sketch\] (a) for the lattice geometry and (d) for corresponding MPS\]. We use generalized boundary conditions on both ends of the MPS, in that the left (right) input bases of $A_1$ ($A_L$) can be specified as desired. The most natural choice in DMRG is to take the input basis to be a singlet $|S=0\rangle$, as usually done for open boundary conditions. In that case, however, the spin quantum number $S$ of the virtual bond multiplets would automatically be integer, as only integer $S$ results when adding two integer spins together. For this reason, SU(2) DMRG calculations with conventional open boundary conditions will never yield the half-integer bond (projective) representation of the SO(3) symmetry, but always a “trivial" state without the expected (at least) two-fold degeneracy in each bond multiplet expected for the Haldane phase.
On the other hand, the boundary can also be set up by taking both end bonds to be $|S=1/2\rangle$ doublets, instead of singlets $|S=0\rangle$.[@expl-sbc] Since then only half-integer multiplets appear in the virtual bonds, this always yields an “SPT" state possessing doubly degenerate entanglement spectrum. In particular, for the spin-1 chain of Hamiltonian Eq. (\[eq-bbq\]), this choice of boundary condition produces an “SPT" state for any $\theta \in [-\pi/2, \pi/4]$. However, this seemingly contradicts the well-established fact that Haldane phase is confined to $\theta \in (-\pi/4, \pi/4)$. In order to resolve this apparent paradox, we here also study a more general situation, where we input the direct sum $|S=0\rangle \oplus |S=1/2\rangle$ on the two boundary bonds. This gives rise to the possibility of both integer and half-integer multiplets on the bonds, and allows us to do actually parallel DMRG calculations in two independent symmetry sections, i.e., integer and half-integer bond spaces.
We thus adopt the following *protocol algorithm for determining the bond representations*: we input both integer and half-integer multiplets on the boundary virtual bonds, and perform several DMRG sweeps back and forth. In the presence of state space truncation along the bonds, depending on the Hamiltonian parameters, the system will eventually converge to the half-integer projective representation or the integer linear representation of SO(3), thus telling the SPT phase from a trivial one.
Two typical “multiplet entanglement spectra" selected through DMRG sweeps, and calculated using $|S=0\rangle \oplus |S=1/2\rangle$ boundary states, are shown in Fig. \[fig-ent-spec\] (a). Here each data point represents a multiplet, in contrast to the traditional state entanglement spectrum, where each data point corresponds to an individual state. $\theta=0$ corresponds to the conventional Heisenberg model, whose ground state belongs to the Haldane phase. The converged multiplet spectrum obtained is shown using asterisks: all points in the spectrum correspond to half-integer quantum numbers $S$, and each asterisk with quantum number $S$ represents $2S+1$ (an even number) degenerate U(1) states, as expected for an SPT phase. On the other hand, the system with $\theta=-\pi/2$ is in the dimerized phase, a trivial gapped phase. Its SU(2) multiplet spectrum is plotted using open circles in Fig. \[fig-ent-spec\] (a). In contrast to the $\theta=0$ case, the circles are all located at integer $S$, as expected for a trivial (non-SPT) phase.
In the protocol algorithm, where $|S=0 \rangle \oplus |S=1/2\rangle$ is used as auxiliary boundary state, DMRG allows the “correct" bond representation to be found, as long as the system is not very close to the phase transition point. In the following, in order to compare the multiplet spectra between the integer and half-integer representations, we now change strategy and enforce the representation by specifying one of the two boundary state types on both ends of the chain, i.e., $|S=0\rangle$ ($|S=1/2 \rangle$) for integer (half-integer) representation.
In Fig. \[fig-ent-spec\] (b), we choose $\theta=0$ (corresponding for the Haldane phase), and compare the multiplet entanglement spectra $E^I_{i}$ (circle) and $E^H_i + \log(2)$ (asterisks), which are obtained by enforcing either integer or half-integer representations, respectively. The integer-spin multiplet spectrum evidently displays a two-fold degeneracy, whereas the half-integer-spin multiplet spectrum does not. Instead, we observe a one-to-one correspondence between each multiplet in $E^H_i + \log{(2)}$ and a pair of degenerate multiplets in $E^I_i$. The shift value $\log(2)$ is chosen because the two representations have different numbers of states with nonzero weights in their reduced density matrices. The nonzero individual states in the integer representation are twice as many as those in the half-integer one.
This different behavior of the degeneracies in the integer and half-integer multiplet entanglement spectra can be understood as follows: in the presence of space inversion symmetry, time reversal symmetry, or some $Z_2 \times Z_2$ rotational symmetry, etc., which protects the Haldane phase, it has been proven that $\Lambda_{Q_i} \otimes \lambda_{Q_i}$ has an even degeneracy of at least 2. [@Pollmann_1] Therefore in the Haldane phase, either $\Lambda_{Q_i}$ or $\lambda_{Q_i}$ should have even degeneracy. For the half-integer bond representation, the $Q_i$’s are half-integer and therefore the $\lambda_{Q_i}$’s are identity matrices with an even number of diagonal elements, implying that an even degeneracy appears in the CGC space; thus the $\Lambda_{Q_i}$ in the reduced multiplet space is not necessarily two-fold degenerate, which explains the absence of degeneracies in the multiplet spectrum $E_i^H$ (asterisks). On the other hand, for integer bond representations, the $\lambda_{Q_i}$’s are identity matrices of odd-number rank, therefore an even degeneracy must instead appear on the multiplet level, which explains the two-fold degeneracy obtained in $E_i^I$ (circles). This difference between integer and half-integer representations has an important consequence in the entanglement entropy, which will be discussed in the next section.
To summarize, the lesson learnt from Fig. \[fig-ent-spec\] is as follows. In Fig. \[fig-ent-spec\] (a) we showed that, if mixed boundary $|S=0 \rangle \oplus |S=1/2\rangle$ is adopted, DMRG sweep can select the half-integer-spin representation in the Haldane phase and integer-spin representation in the trivial phase. Fig. \[fig-ent-spec\] (b) illustrates that if one studies the Haldane phase using auxiliary spin $|S=0\rangle$ or $|S=1/2\rangle$ on the external bond, respectively, then the general requirement of having an entanglement spectrum of even degeneracy is satisfied by having the multiplet spectrum being degenerate or non-degenerate for the case of integer-spin or half-integer-spin representation, respectively.
![(Color online) Integer and half-integer entanglement entropies, $S^I$ and $S^H$, of the spin-1 BLBQ model, for $L=200$ (dash-dotted lines) and $L=500$ (solid lines). Results for different system sizes coincide for $\theta$-values far from the critical point at $\theta_c=-\pi/4$ (vertical dash-dotted line), but differ in the intermediate region between the two vertical dashed lines. $S^I$ and $S^H$ cross at a “pseudo-transition" point $\theta_c^{L}$, which moves towards the critical point as the system size is increased (shown in panel (d), the extrapolated point is very close to the true critical one). In the above calculations, 400 multiplets (about 1600 individual states) have been retained, the truncation errors are of the order $10^{-10}$ around critical point, and are negligible ($10^{-14}$ or less) for the rest parameters. The entanglement entropies are evaluated at the center of the chain. Panel (a) also shows the entanglement entropy obtained by the iTEBD algorithm[@iTEBD] (asterisks), which favors the minimally entangled states, and always follows the lower entanglement entropies. (b) The entanglement gap $G=S^I-S^H$, which equals $\pm1$ in the SPT phase and the trivial phase respectively. The dashed vertical lines mark the intermediate region, where $G$ is not a constant owing to finite size effects. Panel (c) shows the string order parameter (SOP $O_z$) of Eq. (\[eq-SOP\]), obtained by iTEBD calculations, which retain up to 200 states.[]{data-label="fig-chain"}](fig3.eps){width="0.95\linewidth"}
Entanglement Gap and Symmetry-Protected Topological Phase
=========================================================
During the DMRG sweeps in the protocol using $|S=0 \rangle \oplus |S=1/2\rangle$ as boundary, as long as the doublet $|S=1/2\rangle$ is not physically coupled to the bulk (the coupling strength between the auxiliary boundary spin-1/2 and the spin-1 chain can be set to be very weak or even turned off), the integer and half-integer symmetry sectors have exactly the same ground-state energy. Therefore, the energy is irrelevant in selecting the symmetry sector in the protocol algorithm. Instead, since the two-site update scheme of DMRG is adopted during the sweeps, the truncation and hence the entanglement entropy is important in selecting the symmetry sector.
In order to uncover this mechanism underlying the protocol algorithm, we now study the bipartite entanglement entropies in the integer and half-integer symmetry sectors, respectively, by enforcing different boundary states. The entanglement entropies are defined as $$S^{X} = - \sum_{Q} {\rm{Tr}_{Q}} [(\rho^X_{Q} \otimes D^X_{Q}) \log_2(\rho^X_{Q} \otimes D^X_{Q})],
\label{eq-ent-ent}$$ where $X= H$ or $I$ for half-integer or integer representations, and the difference $G=S^I - S^H$ will be called the “entanglement gap". In Eq. (\[eq-ent-ent\]), $\rho^X$ is the reduced density matrix on the multiplet level. It is block-diagonal, with blocks $\rho^X_{Q}$ labeled by $Q$ and matrix elements $(\rho^X_{Q})_{n,n'}$. $D^X$ is an identity matrix, with matrix elements $(D^X_Q)_{Q_z,Q'_z} = \delta_{Q_z,Q'_z}$ whose trace thus equals the inner dimension of each multiplet.. Consequently, ${\rm{Tr}}_Q[\cdot]$ refers to the trace over both, the multiplet index $n$ as well as as the internal multiplet space $Q_z$ of a given symmetry sector $Q$. Note that the logarithm to base 2 ($\log_2$) is adopted in evaluating the entanglement entropy. $\rho^X$ and $D^X$ are readily obtained from DMRG simulations. We note the SU(2) multiplet language used to formulate Eq. (6) for the von Neumann entropy can easily be applied to also calculate the Renyi entropy. Very recently, the latter has been employed to study the local differential convertibility and thereby probe the SPT phase.[@Cui] Though we here focus only on the von Neumann entropy, our analysis can be be generalized straightforwardly to the Renyi entropy.
In Fig. \[fig-chain\], $S^I$ and $S^H$ of the spin-1 BLBQ chain \[Eq. (\[eq-bbq\])\] are plotted in (a), and the entanglement gap $G$ is shown in (b). For the Haldane phase ($-\pi/4<\theta<\pi/4$ in Fig. \[fig-chain\]) we find $S^I > S^H$, thus the half-integer bond representation has lower entanglement than the integer one, although the ground-state energies in both representations are the same. On the other hand, for the dimerized phase ($-0.6 \pi<\theta<-\pi/4$ in Fig. \[fig-chain\]) we find $S^I < S^H$. This observation explains why the protocol using $|S=0\rangle \oplus |S=1/2\rangle $ on end bonds employed for Fig. \[fig-ent-spec\] (a) succeeded in selecting the “correct" bond representations: DMRG always favors lower entanglement, and the representation (integer or half-integer) with higher entanglement would be discarded by truncations during the sweeps.
Another interesting observation is that the entanglement gap $G = S^I-S^H$ is found to be a constant +1 ($-1$) in the SPT (trivial) phase. It is rather robust and almost independent of different Hamiltonian parameters and system sizes, except for the intermediate region near the critical point, where the finite-size effects become significant. This region is marked by vertical dashed lines in Fig. \[fig-chain\]. The entanglement curves cross within this region, and the crossing point moves to the true critical point $\theta_c = -\pi/4$, the exactly soluble Takhtajan-Bubujian point, [@TB] with increasing system sizes.
The value $G = \pm1$ actually originates from the different topology of the SU(2) and SO(3) groups, and hence can be regarded as a topological invariant in each phase. In order to understand this, let us again consider the exactly solvable AKLT model with $\theta=\arctan(1/3)$. The reduced tensor at the multiplet level is $A_{S=1/2, S=1/2}^{S=1}=1$, a simple tensor with bond dimension 1, i.e., a scalar number. The corresponding CGC tensor is $C(S=1/2, S=1 | S=1/2)$, which combines a spin doublet with a triplet into an output spin doublet.
The corresponding reduced density matrix of half-infinite AKLT chain is a $2\times 2$ diagonal matrix, $ \begin{pmatrix} 1/2 & & 0 \\ 0 & & 1/2 \end{pmatrix}$, fully encoded in the CGC space only, and resulting in an entanglement entropy $S^H=2[-1/2\log_2(1/2)]=1$. However, for the integer bond representation, we instead have a 2 $\times$ 2 diagonal matrix $$\begin{pmatrix}
A_{S=0, S=0}^{S=1} = 1/4 & & 0 \\ 0 & & A_{S=1, S=1}^{S=1} = 1/4
\end{pmatrix}$$ in the reduced multiplet space. The two degenerate multiplets contain 4 degenerate states in total, and the full reduced density matrix is a $4 \times 4$ diagonal matrix with all elements $1/4$. The entanglement entropy is $S^I=4[-1/4\log_2(1/4)]=2$, which is larger than corresponding $S^H$ and the gap $G = S^I- S^H = 1$.
Next, we consider a generic state in the SPT phase away from the special AKLT point. As shown in Fig. \[fig-ent-spec\], there exists a one-to-one correspondence between one $\tilde{S}=(2n+1)/2$ multiplet in the half-integer sector and one pair of degenerate multiplets with $S=n$ and $n+1$ in the integer sector ($n = 0, 1, 2, ...$). For the latter, the degeneracy on the multiplet level cannot be trivially lifted owing to the protection of the symmetry. Consequently, this multiplet degeneracy enhances the entanglement entropies and opens an entanglement gap of $G = S^I- S^H = 1$, as shown in Fig. \[fig-chain\]. On the other hand, adopting integer virtual bonds would preferably lower the entropy by 1 for the trivial dimer phase. As shown in Fig. \[fig-chain\], in this case entanglement gap is $G=-1$.
Fig. \[fig-chain\](c) presents the nonlocal string order parameter obtained by iTEBD calculations; it is nonzero in the Haldane phase and vanishes in the trivial phase. The comparison of our entanglement entropy results with the SOP data validates that $G$ can be used to distinguish SPT phase from the trivial one.
Lastly, we remark that the results in Fig. \[fig-chain\] were obtained by evaluating finite-size systems. When the system is close to the critical point, the entanglement entropies $S^I$ and $S^H$ are shown to cross each other. In Figs. \[fig-chain\](a), the lower values of these two entropy curves can be regarded as giving the “true" entanglement entropies. The combined curve shows a sharp peak, which is missed when considering either $S^I$ or $S^H$ alone. In Fig. \[fig-chain\](a), the results obtained by iTEBD are also shown, which always favor the low entanglement curves. The iTEBD data coincide with the SU(2) DMRG results (except for the region near the critical point), which validates our arguments above. The crossing point of the $S^I$ and $S^H$ curves (as the peak of the low entanglement curve) can be viewed as a “pseudo-transition" point. As the system size increases, the pseudo-transition point approaches the true critical point $\theta_c = -\pi/4$ \[see Fig. \[fig-chain\](d)\]. In the thermodynamic limit, the gap $G$ are supposed to show a jump between 1 and $-1$ just at the critical point, and the peaks of the entanglement entropies are expected to diverge.
Spin-1 Heisenberg Tube Model
============================
![(Color online) (a) Integer and half-integer entanglement entropies $S^I$ and $S^H$ for the spin-1 tube model. The critical point estimated from their crossing point is $\alpha_c = 0.571(1)$. (b) Entanglement gap $G = S^I-S^H$. $G=1$ when $\alpha<\alpha_c$, identifying the existence of an SPT phase, and $G=-1$ when $\alpha<\alpha_c$, corresponding to a trivial phase. The system size is $3 \times 100$, 400 multiplets are reserved, which lead to maximum truncation error of $10^{-8}$ (at the critical point).[]{data-label="fig-tube"}](fig4.eps){width="0.95\linewidth"}
![(Color online) Analysis of the block entanglement entropy, $S(x)$, of the ground state of the spin-1 three-leg tube Heisenberg model, in the vicinity of $\alpha_c$. (a) and (b) show the entanglement entropy between boundary block of length x and the rest of the system, for several different system sizes and $\alpha$-values, on (a) a linear scale and (b) a log(sin) scale on the horizontal axis. Curves are vertically offset by 1 unit for clarity. Here we show the data on one of the three sublattices in tube model, which contains entanglement entropies cut at the $i$-th bond \[$\rm{mod}(i,3)=1$\]; the other two curves give the same fitting results and are not present here. The conformal central charge is determined as $c \simeq 3$, (c) shows how the fitted $c$’s vary with $\alpha$, for three fixed system sizes. (d) and (e) show, respectively, the maximal $c$-values and corresponding $\alpha$-values obtained for 5 different $N$-values. The system size ranges from $N=50\times3$ to $100\times3$ ($N$ is the total site number), and up to 450 bond multiplets are reserved in the calculations. A half-integer bond representation was adopted in the calculations; the fittings of integer-representation entanglement entropies lead to the same conclusion.[]{data-label="fig-fitc"}](fig5.eps){width="0.9\linewidth"}
![(Color online) (a) The ground state energy per site $e_0$ of a spin-1 tube versus the coupling ratio $\alpha$. The system size varies from 60$\times$3 to 120$\times$3. For the largest size $120 \times 3$, 500 SU(2) multiplets ($\approx 2000$ equivalent U(1) states) are retained in the calculations, truncation errors are less than $10^{-9}$. The inset shows the first-order derivatives of energies $de_o/d\alpha$, which are substantially converged with different system sizes, and are shown clearly to be continuous through the critical point. (b) The second-order derivative $d^2 e_0/ d \alpha^2$, which shows a diverging peak at $\alpha_c = 0.5715(5)$. The inset in (b) shows $d^2e_0/d\alpha^2$ in the vicinity of critical point on a log-log scale. The data points fall into two linear lines (except for the points very close to the critical point $\alpha_c$, owing to the finite-size effects near the critical point), which implies algebraic divergence. The dashed lines in the inset are fits to the form $d^2 e_0/d \alpha^2 \propto (\alpha - \alpha_c)^{-\nu}$, with $\nu \simeq 0.87$ and $0.6$, approaching critical point from left and right sides, respectively.[]{data-label="fig-eng-curv"}](fig6.eps){width="1\linewidth"}
In this section, we study the SPT phase in a spin-1 tube model. This model has been studied by Charrier et. al. in Ref. . Following their conventions, schematically depicted in Fig. \[fig-sketch\] (c), the Hamiltonian is given by: $$\begin{aligned}
& H_{tube} & = H_L + H_R, \notag \\
& H_L & = J_L \sum_{i, a = \{1,2,3\}} \bold{S}_{i, a} \bold{S}_{i+1, a}, \notag \\
& H_R & = J_R \sum_{i} (\bold{S}_{i,1} \bold{S}_{i,2} + \bold{S}_{i, 2} \bold{S}_{i,3} + \alpha \bold{S}_{i, 1} \bold{S}_{i,3}).\end{aligned}$$ $H_L$ and $H_R$ are the intra- and inter-chain coupling terms, respectively. In Ref. , the authors found a Haldane phase existing for $0< \alpha <0.57$, with $J_L=0.1$ and $J_R=1$, where each triangle contains an effective spin-1. For $0.57<\alpha<1.5$, they found a trivial disordered phase, with each isosceles triangle carries an effective spin-0, leading to a spin-0 chain (note that the combined product space 1 $\otimes$ 1 $\otimes$ 1 allows for exactly one spin-0 singlet). At the critical point $\alpha_c$, the system undergoes a quantum phase transition between the Haldane and the trivial phase. For $0< J_L/J_R < 0.65$, there are still phase transitions separating two phases, but at different $\alpha_c$; if $J_L/J_R > 0.65$, no phase transition occurs because the trivial phase no longer exists. [@Charrier]
Next, we revisit this model using SU(2) DMRG calculations, and study it by evaluating the entanglement entropies $S^I$ and $S^H$. In Fig. \[fig-tube\], the entropies $S^I$ and $S^H$ intersect at $\alpha_c\approx 0.571(1)$. For $\alpha < \alpha_c$, $G = S^I - S^H = 1$, the half-integer representation has lower entanglement. The dominating bond multiplets are doublets ($S$=1/2), and the system is in an SPT (Haldane) phase. In contrast, for $\alpha > \alpha_c$, $G = S^I - S^H = -1$, the ground state favor integer bond representations. The energy results show that the energy per triangle is uniform along the leg direction, without any translational symmetry breaking. The leading bond multiplet in the entanglement spectrum is found to be a singlet ($S$=0), and the system is in a trivial disordered phase. In addition, we remark that the proper definition of a SOP in this spin-1 tube has been discussed by the authors in Ref. . The SPT phase that we have here identified by entanglement entropy, indeed also possesses a nonzero SOP.
Compared with the spin-1 BLBQ model, finite-size effects are much less significant in the spin tube model. For a system size of $100 \times 3$, the values at which the peaks of integer and half-integer entropies occur lie quite close together. By combining $S^I$ of $\alpha>\alpha_c$ and $S^H$ of $\alpha<\alpha_c$, we can see a very sharp peak in the joint low entanglement curve, which suggests a second-order quantum phase transition.
Next, we address the order of the phase transition in more details by checking the criticality at $\alpha_c$. The block entanglement entropy of size $x$ can be fitted with the following form: $$S(x) = \frac{c}{6} \log_2[ \frac{N}{\pi} \sin(\pi \frac{x}{N})] + \rm{const.},
\label{eq-CC}$$ where $N$ is the total number of sites. $N=3L$ for the tube of length $L$. This is the Cardy-Calabrese formula [@Holzhey; @Vidal_2; @Calabrese] with open boundary condition, showing that the block entanglement entropy has a logarithmic correction to the entanglement area law at the critical point.[@Eisert] $c$ is the conformal central charge, which characterizes the criticality. The fitting results are shown in Fig. \[fig-fitc\], which strongly suggests that the transition point is critical or very close to some gapless point (quasi-critical). The central charge obtained from the fits is $c\simeq3$.
By the DMRG ordering of sites into one linear sequence, the 3-leg tube has three different sublattices (and hence three kinds of bonds), two of which are equivalent. Therefore, when cutting the systems in different ways, we can get three block entanglement entropy curves, one of which is shown in Fig. \[fig-fitc\]. The fittings of the other two curves lead to the same results. Fig. \[fig-fitc\](a) and (b) show fits for 5 different system sizes and $\alpha$-values. Fig. \[fig-fitc\](c) shows that the $c$-values obtained from each fit exhibit, for given tube with total site number $N$, a clear maximum as function of $\alpha$. This maximal value (located at $\alpha_c^N$) can be regarded as the best estimation of $c$. Note, the system is most close to critical at $\alpha_c^N$, and away from critical when $\alpha < \alpha_c^N$ and $\alpha > \alpha_c^N$; Cardy-Calabrese formula (Eq. \[eq-CC\]) gradually loses its legitimacy in the latter case, and fitted value of $c$ is reduced away from $\alpha_c^N$. Collecting these maximal points, in Figs. \[fig-fitc\](d) and (e) we plot, respectively, how the fitted $c$’s and estimated transition points $\alpha_c^N$’s vary with different system sizes (from $50\times3$ to $100\times3$). The fitted $c$’s (estimated transition points from entanglement) tend towards 3 (critical point estimated from energy derivatives) when $N$ is increased. Moreover, in the fits, we follow the same strategy as in Refs. and fit the central charge in the central region of the chain. Typically we omit 10 to 20 sites (depending on the total system sizes) from both ends, and take $c$ to be the limiting value obtained when increasing the omitted site number.
The ground-state energy curves and their derivatives with respect to $\alpha$ are presented in Fig. \[fig-eng-curv\]. The energy per site is defined as $e_o = E_{\rm{tot}}/N$, where $E_{\rm{tot}}$ is the total energy and $N$ is the number of sites. The first-order derivatives of energies do not show any discontinuities at the transition point, but the second-order derivatives have very sharp peaks at $\alpha_c$. In the inset of Fig. \[fig-eng-curv\] (b), we also plot $d^2 e_0/d \alpha^2$ on a log-log scale. The observed power law behavior implies the algebraic divergence of $d^2 e_0/d \alpha^2$ approaching $\alpha_c = 0.5715(5)$, i.e., $d^2 e_0/d \alpha^2 \propto (\alpha-\alpha_c)^{-\nu}$. The exponent $\nu$ has two different values, depending from which side $\alpha_c$ is approached. Both, though, are less than 1, which implies that $de_0/d\alpha$ maintains a smooth behavior at $\alpha_c$. Therefore, the results of entanglement entropies, block entropy fittings, along with the energy derivatives, all support the conclusion that there is a continuous phase transition at $\alpha_c$. This contradicts the conclusion in Ref. , where the transition is argued to be of weakly first-order.
In order to thoroughly clarify the transition order, more detailed studies of the correlation functions and excitation gaps are needed, which we leave as future studies. The parameters could also be tuned (say, take $J_R/J_L$ different from $0.1$ studied above) and investigate the nature of the phase transition; or introduce some other parameters in the Hamiltonian (say bilinear-biquadratic parameter $\theta$) and inspect the transition along some other paths in the parameter space. We have done some preliminary calculations along these lines (not shown in this paper), which reinforce the conclusion of a second-order phase transition.
Absence of Symmetry-Protected Topological Phase in Spin-1 Heisenberg Ladder
===========================================================================
![(Color online) The entanglement entropies of spin-1 two-leg ladder system with system size $80 \times 2$. 200 multiplets are retained in the calculations, and the maximum truncation errors $ \approx 10^{-8}$. (a) $S^I$ and $S^H$ represent integer and half-integer entropies, respectively. $S^I(0)$ or $S^I(1)$ means that $|S=0 \rangle$ or $|S=1 \rangle$ dominates in the multiplet spectrum, respectively. The dashed line is a guide for the eye.[]{data-label="fig-ladder"}](fig7.eps){width="0.95\linewidth"}
Lastly, let us consider the spin-1 two-leg ladder model, $$H = J_L \sum_{i, a=\{1,2\}} S_{i, a} S_{i+1, a} + J_R \sum_{i} S_{i, 1} S_{i,2}.
\label{eq-ladder}$$ There are two kinds of couplings in this model \[see Fig. \[fig-sketch\] (b)\], $J_L$ along the chain direction and $J_R$ on the rungs. In Fig. \[fig-ladder\], the entropies $S^I$ and $S^H$ are plotted. Two versions of $S^I$ are shown, $S^I(0)$ and $S^I(1)$, both obtained with integer bond representations, but with different leading (lowest) multiplets in the entanglement spectrum: $|S=0\rangle$ for $S^I(0)$ and $|S=1 \rangle$ for $S^I(1)$. The latter can be obtained by attaching auxilliary spin-1’s on both ends in our SU(2) DMRG.
For $J_R>0$, $G = S^I(0)-S^H>0$ in Fig. \[fig-ladder\], thus the ground state favors integer-spin representation, verifying the triviality of the ground state. Indeed, for the limiting case $J_R/J_L \to \infty$, the ground state is a simple direct product of rung singlets. On the other side, for $J_R<0$ the system is in the same phase as the spin-2 antiferromagnetic Heisenberg chain (reached in the limiting case $J_R/J_L \to -\infty$). Fig. \[fig-ladder\] shows that the ground states in this region also favor integer representations. However, the lowest multiplet in the entanglement spectrum is the spin triplet $|S=1\rangle$, rather than the singlet $|S=0 \rangle$, consistent with the results of Ref. . The two low-entanglement curves from the $S=0$ and $S=1$ symmetry sectors together form a smooth line in Fig. \[fig-ladder\] (a) (indicated by a dashed line), which represents the “true" entanglement entropy of the system.
No sign of criticality can be seen from the entanglement entropies, and it is hence believed that only one disordered phase exists in the spin-1 Heisenberg ladder model. Our observation is in agreement with the conclusion in Ref., that the model does not undergo any phase transition from $J_R<0$ to $J_R>0$. The fact that there does not exist an SPT phase in the spin-1 Heisenberg ladder model studied above can be ascribed to the triviality of the standard $S=2$ AKLT states, which can be adiabatically connected to the topologically trivial state without any phase transition. [@Pollmann_2; @Oshikawa; @Tonegawa] The triviality of the standard $S=2$ AKLT state can be also be intuitively understood as follows: it has *two* valence bonds (corresponding to two virtual spin-1/2) living on each geometric bond, since these two virtual spin-1/2 couple to either spin 0 or 1, the total spin forms integer-spin representations of SO(3) on the geometric bond, leading to a conclusion of a topologically trivial phase. This argument also applies to the spin-1 Heisenberg ladder studied above (especially when $J_R<0$). The bond states are more complicated for the general two-leg Heisenberg ladder model, nevertheless, they form integer representations of SO(3) and the corresponding groundstate belongs to a trivial phase.
Conclusion
==========
We have proposed a novel way to identify SPT phases in one dimension by evaluating entanglement entropies. With SU(2) DMRG method, we can keep track of the bond multiplets, and readily tell half-integer-spin projective representation from integer-spin ones by checking the multiplet entanglement spectrum introduced in this paper. In addition, we have shown that auxiliary boundary spins attached on both ends of the chain can be used to control the bond representations; this significantly changes the entanglement entropies in the bulk, depending on the topological properties of the phase.
In the SPT phase, we showed that a two-fold degeneracy for the overall entanglement spectrum appears either in the reduced multiplet space or in the CGC space, depending on whether the integer or half-integer bond representations are adopted, respectively. In the latter case, the two-fold degeneracy occurs in CGC space, which reduces the entanglement entropy $S^H$ relative to $S^I$ (entanglement gap $G=1$), providing a practical criterion for identifying SPT phases. The existence of an entanglement entropy gap also allows us to automatically select the “correct" representation (integer or half-integer) through DMRG sweeps, which always favor low entanglement representation. The entanglement gap closes at the critical point, which can be used to detect the quantum phase transitions.
Several 1D and quasi-1D systems have been studied in this work; the SPT phase in the spin-1 chain and the spin-1 tube model are successfully identified by evaluating the entanglement entropies. For the spin-1 tube model, the numerical results indicate that the phase transition between the SPT phase and the trivial phase is a continuous one. The fact that the two-leg spin-1 Heisenberg ladder has no SPT phase for any $J_R$ is also validated by our entropy results.
Acknowledgement
===============
WL would like to thank H.-H. Tu and T. Quella for helpful discussions on symmetry-protected topological order and the projective representations of symmetry groups. WL was also indebted to Shou-Shu Gong for useful discussions on the numerical results and DMRG techniques. This work was supported by the DFG through SFB-TR12, SFB631, the NIM Cluster of Excellence, and also WE4819/1-1 (AW).
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{
"pile_set_name": "ArXiv"
}
|
---
author:
- |
Xunpeng Huang\
Bytedance AI Lab\
`[email protected]` Runxin Xu\
Peking University\
`[email protected]` Hao Zhou\
Bytedance AI Lab\
`[email protected]` Zhe Wang\
Ohio State University\
`[email protected]` Zhengyang Liu\
Beijing Institute of Technology\
`[email protected]` Lei Li[^1]\
Bytedance AI Lab\
Beijing, China\
`[email protected]`
bibliography:
- 'references.bib'
title: '[ACMo]{}: Angle-Calibrated Moment Methods for Stochastic Optimization'
---
Introduction {#sec:intro}
============
Related Work {#sec:related}
============
Our Proposed [ACMo]{} {#sec:method}
=====================
Theoretical Results {#sec:analysis}
===================
Experiments {#sec:experiments}
===========
Conclusion {#sec:conclu}
==========
[^1]: Corresponding author
|
{
"pile_set_name": "ArXiv"
}
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---
author:
- George Wallerstein
- Wenjin Huang
title: ' The Composition of RR Lyrae Stars: Start-line for the AGB '
---
Introduction
============
This paper reports on research by Sergei M. Andrievsky, Valentin V. Kovtyukh, Marcio Catelan, Dana Casetti-Dinescu, and Gisella Clementini as well as ourselves.
The RR Lyrae stars are of great value in studies of the structure of our Galaxy and the history of its composition as modified by stellar nucleosynthesis over the ages. The fact that they show a narrow range of luminosity from about Mv=0.4 to 0.8, as derived from their membership in globular clusters, allows their distances to be derived and their orbits calculated from their proper motions and radial velocities. Their effective temperatures vary from approximately 6000 to 7000 K during their pulsation. Their spectra often show lines of many elements without excessive blending.
The chemical compostion of RR Lyrae stars is a combination of their original composition when they were main sequence stars near Mv=4.5 with important changes induced by nuclear reactions whose products may be convected to the stellar surface. The first mixing event to affect their atmospheres is the deepening of convection as the star leaves the subgiant branch and begins its evolution up the almost vertical giant branch [@hoy55]. CNO-processing converts some carbon into nitrogen and reduces the 12C/13C ratio from its initial value to about 20. As the star’s evolution slows down at the red giant clump additional mixing further reduces the carbon and lowers the 12C/13C ratio to about 4-8 [@gil91; @cha94; @cha98]. These changes have been seen in many globular clusters.
At the tip of the red giant branch important events take place. The triple-alpha reaction starts in the degenerate core producung additional carbon. At first the tempature rises exponentially because the pressure and density do not respond until the degeneracy is removed by the temperature rise and the reaction rate depends approximately on the temperature raised to the 30th power. This sets up a grossly superadiabatic temperature gradient and initiates violent convection [@moc09]. Calculations indicate that the products of helium burning do not reach the surface but observations of the carbon abundance in RR Lyrae stars are useful to confirm the accuracy of the calculations and to describe the initial conditions for stars that will eventually evolve up the AGB, perhaps to become carbon stars. In addition mass-loss on the red giant branch plus additional mass-loss at the time of the helium-flash reduces the stellar mass from its original value of about 0.8 Msun to about 0.55 Msun as derived from multi-periodic RR Lyrae stars. Such mass-loss may give us a glimpse into the interior of what had once been a red giant.
Our first project to determine the composition of RR Lyrae stars was suggested by M. Catelan who noted that a few RR Lyrae and related stars showed kinematics similar to that of Omega Cen. The first is VY Ser, a genuine RR Lyrae star of period 0.714 days, while 2 more, V716 Oph and XX Vir have periods slightly longer than 1.0 days and are usually called short-period type II cepheids, but might be referred to as long-period RR Lyrae stars. Dana Casetti-Dinescu confirmed that their galactic orbits do indeed relate them to Omega Cen. The period of VY Ser is typical of RR Lyrae stars in Omega Cen, but only 7 short-period cepheids are known in Omega Cen [@cle01]. In fact it is remarkable that astronomers have defined the break between RR Lyraes and short period cepheid to equal to the period of rotation of the earth. Our observations consisted of 4 echelle spectra of VY Ser, 5 of V716 Oph, and 2 of XX Vir that have been analysed by Andrievsky and Kovtyukh. All 3 stars show \[Fe/H\] close to $-1.6$ which is typical of Omega Cen but cannot be used to reveal the strong gradient of \[s/Fe\] when plotted against \[Fe/H\] in Omega Cen [@van94; @nor95].
Our second group of targets was RR Lyrae stars with period greater than 0.75 days. Such stars are very rare in globular clusters with the conspicuous exception of the two unusual clusters, NGC 6388 and 6441 [@pri01; @pri02]. These relatively metal rich clusters with \[Fe/H\] near $-0.8$ have numerous long-period RR Lyrae stars. In the general field such stars are rare but some are known and their light curves have been obtained by @sch02. Only a few are sufficiently bright for highres spectroscopy with the 3.5-m telescope of the Apache Point Observatory. They are listed in @wal09. \[Fe/H\] values were found for 4 stars but their metallicities ranged from $-1.8$ to $+0.2$. Hence our analyses of these stars failed to relate them to NGC 6388 and 6441. KP Cyg remains an almost unique RR Lyrae star with a period of 0.856 days and \[Fe/H\] $= +0.2$ on the basis of 5 spectra well distributed in phase. It had already been recognized as having delta-S $= 0$ by @pre59. Our spectra show a typical velocty curve for an RR Lyrae star with H-alpha emission at phases 0.43, 0.72, and 0.80.
The third aspect of our RR Lyrae analyses was the recognition that the relatively metal-rich stars show an apparant excess of carbon [@wal09]). The carbon abundance was derived from the lines around 7115Å of multiplets 108 and 109 of @wie98. These lines are sufficiently weak to be found only in stars with \[Fe/H\] $> -1.0$. For metal-poor stars stronger lines must be employed which suffer from NLTE [@fab06]. Fabbian has told us (private communication) that the lines around 7115Å should not suffer from significant NLTE effects. The best strong lines are one at 8335Å and from multiplet 62 beyond 9000Å. That spectral region contains many moderately strong atmospheric H$_{\rm 2}$O lines.
Observation and Data Reduction
==============================
New Observations were obtained with the echelle spectrograph on the Apache Point 3.5-m telescope in March, November and December, 2009. The resolving power is about 30,000 and the signal-to-noise ratio of the reduced spectra is usually 100-150. The cold temperatures and altitude of 9200 ft permit very effective division by the spectrum of a hot rapidly rotating star so as to cancel out the atmospheric H$_{\rm 2}$O absorption lines, especially beyond 8900Å. We have used the H$\gamma$ line to establish Teff and the known mass and luminosity of the RR Lyrae stars for the derivation of the surface gravity (without introducing the perturbation of $\log g$ by the acceleration induced by pulsation). In addition to deriving the iron abundance from Fe II lines we have compared the C I lines with lines of S I whose excitation and ionization potentials are similar to those of C I. We added Si II to the data base as a check on the alpha element excess usually seen in metal-poor stars. The observed lines and their atomic properties are shown in Table 1. The abundances were derived using the updated version of the line synthesis code, MOOG, with the Kurucz atmosphere models [@cas04]. In Table 2 we show the derived abundances of C, S, Si, and Fe. We also show the Delta-S value from @pre59 or other sources such as @lay94. The correlation of the Fe abundance with delta-S is very good.
[crrr]{}\
Wavelength & & $\chi$ &\
(Å) & Ions & (eV) & $\log gf$\
\
4932.05 & 6.00 & 7.68 & $-$1.658\
5052.17 & 6.00 & 7.68 & $-$1.303\
5380.34 & 6.00 & 7.68 & $-$1.616\
7111.47 & 6.00 & 8.63 & $-$1.085\
7113.18 & 6.00 & 8.64 & $-$0.773\
7115.17 & 6.00 & 8.63 & $-$0.824\
7116.99 & 6.00 & 8.64 & $-$0.907\
7119.66 & 6.00 & 8.63 & $-$1.148\
8335.15 & 6.00 & 7.68 & $-$0.437\
9061.44 & 6.00 & 7.47 & $-$0.347\
9062.49 & 6.00 & 7.47 & $-$0.455\
9078.29 & 6.00 & 7.47 & $-$0.581\
9088.51 & 6.00 & 7.47 & $-$0.430\
9094.83 & 6.00 & 7.48 & 0.151\
9111.81 & 6.00 & 7.48 & $-$0.297\
9405.73 & 6.00 & 7.68 & 0.286\
3853.66 & 14.01 & 6.86 & $-$1.341\
5957.56 & 14.01 & 10.07 & $-$0.225\
5978.93 & 14.01 & 10.07 & 0.084\
6347.09 & 14.01 & 8.12 & 0.149\
6371.36 & 14.01 & 8.12 & $-$0.082\
8694.70 & 16.00 & 7.87 & 0.154\
9212.91 & 16.00 & 6.52 & 0.430\
9237.54 & 16.00 & 6.52 & 0.030\
[lccrrrrrrrr]{}\
Star & $P$(days) & $T_{\rm eff}$/$\log g$/$V_{\rm t}$ & $\Delta S$& \[Fe/H\] & \[C/Fe\] & \[S/Fe\] & \[Si/Fe\] & \[C/S\] & \[C/Si\] & Note\
\
V445OPH & 0.397 & 6500/2.5/2.2 & 1 & 0.24 & $-$0.39 & $-0.22$ & $-0.07$ & $-0.17$ & $-0.32$ & 1,3\
RRGEM & 0.397 & 6750/2.5/3.1 & 3 & 0.01 & $-$0.39 & $-0.44$ & $-0.20$ & $ 0.05$ & $-0.19$ & 1,3\
SWAND & 0.442 & 6500/2.5/4.0 & 0 & $-$0.16 & $-$0.39 & $-0.56$ & $-0.06$ & $ 0.17$ & $-0.33$ & 3\
DXDEL & 0.473 & 6500/2.5/3.1 & 2 & $-$0.21 & $-$0.28 & $-0.20$ & $-0.18$ & $-0.08$ & $-0.10$ & 1,3\
ARPER & 0.426 & 6500/2.5/4.0 & 0 & $-$0.32 & $-$0.32 & $-0.51$ & $ 0.01$ & $ 0.19$ & $-0.33$ & 3\
KXLYR & 0.441 & 7000/3.0/3.1 & 0 & $-$0.57 & $-$0.22 & $-0.27$ & $ 0.35$ & $ 0.05$ & $-0.57$ & 2,3\
XZDRA & 0.476 & 6500/2.5/3.0 & 3 & $-$0.75 & 0.06 & $ 0.09$ & $ 0.33$ & $-0.03$ & $-0.30$ & 2,3\
UUVIR & 0.476 & 6250/2.5/3.2 & 2 & $-$0.90 & $-$0.42 & $ 0.11$ & $ 0.52$ & $-0.53$ & $-0.94$ &\
BHPEG & 0.641 & 6500/2.5/2.2 & 6 & $-$1.17 & $-$0.53 & $ 0.10$ & $ 0.40$ & $-0.63$ & $-0.93$ &\
WCVN & 0.552 & 6250/2.5/3.0 & 7 & $-$1.22 & $-$0.46 & $-0.01$ & $ 0.42$ & $-0.45$ & $-0.88$ &\
VZHER & 0.440 & 6250/2.5/2.6 & 4 & $-$1.30 & $-$0.23 & $ 0.21$ & $ 0.60$ & $-0.44$ & $-0.83$ &\
RVUMA & 0.468 & 6500/2.5/2.3 & 3.5 & $-$1.31 & $-$0.05 & $ 0.22$ & $ 0.58$ & $-0.27$ & $-0.63$ &\
RRLEO & 0.452 & 6500/2.5/2.8 & 5 & $-$1.39 & $-$0.04 & $ 0.22$ & $ 0.61$ & $-0.26$ & $-0.65$ &\
TTLYN & 0.597 & 6500/2.5/3.4 & 7 & $-$1.41 & $-$0.98 & $-0.24$ & $ 0.12$ & $-0.74$ & $-1.10$ &\
RRLyr & 0.567 & 6500/2.5/4.0 & 6 & $-$1.44 & $-$0.91 & $-0.03$ & $ 0.14$ & $-0.88$ & $-1.05$ &\
TUUMA & 0.558 & 6500/2.5/3.4 & 6 & $-$1.46 & $-$0.32 & $-0.04$ & $ 0.69$ & $-0.28$ & $-1.01$ &\
VXHER & 0.455 & 6000/2.5/2.4 & 5 & $-$1.48 & $-$0.34 & $ 0.16$ & $ 0.59$ & $-0.50$ & $-0.93$ &\
DHPEG & 0.256 & 6500/2.5/3.0 & 0 & $-$1.53 & $-$0.12 & $ 0.23$ & $ 0.75$ & $-0.35$ & $-0.87$ &\
RRCET & 0.553 & 6500/2.5/3.7 & 5 & $-$1.61 & 0.15 & $ 0.09$ & $ 0.85$ & $ 0.06$ & $-0.70$ &\
STBOO & 0.622 & 6250/2.5/4.0 & 9 & $-$1.77 & $-$0.50 & $-0.15$ & $ 0.46$ & $-0.35$ & $-0.96$ &\
SVERI & 0.714 & 6500/2.5/3.0 & 9 & $-$1.94 & $-$0.44 & $-0.15$ & $ 0.12$ & $-0.29$ & $-0.56$ &\
RUPSC & 0.390 & 6500/2.5/3.5 & 7 & $-$2.04 & $-$0.62 & $-0.12$ & $ 0.51$ & $-0.50$ & $-1.13$ &\
RZCEP & 0.511 & 6500/2.5/3.0 & 5 & $-$2.10 & $-$0.41 & $ 0.01$ & $ 0.57$ & $-0.42$ & $-0.98$ &\
XARI & 0.651 & 6250/2.5/3.8 & 10 & $-$2.68 & $<-$0.46 & $ 0.01$ & $ 0.55$ & $<-0.47$ & $<-1.01$ &\
1. Si II 5957, 5978 only
2. Si II 3853, 5957, 5978 only
3. C I 7100 lines only
Results
=======
In Fig. 1(a) we show \[Si/Fe\] plotted against \[Fe/H\]. The pattern is similar to that which is usually found for metal-poor stars. In Fig. 1(b) the behavior of \[S/Fe\] is qualitatively similar to that of \[Si/Fe\] but is displaced downward by about 0.4 dex. We do not fully understand the low \[S/Fe\] values because we have already applied the NLTE corrections of @tak05 to the derived sulfur abundance. Fig. 1(c) shows \[C/Fe\] vs. \[Fe/H\] for these stars. The scatter is unfortunately large possibly because we have ignored the pulsation effect on $\log g$. However, in Fig. 1(c), it is evident that relatively high carbon abundances appear in the high \[Fe/H\] region. Note that we also used the NLTE corrections of @fab06 in our carbon abundance calculations. Since both the ionization potential and the excitation potentials of the lines are similar, the effects of $\log g$ and possible NLTE uncertainties should be diminished in the abundance ratios of \[C/S\] and \[C/Si\]. In Fig. 1(d), a clear trend of increasing \[C/S\] is present. The same trend is seen even more clearly in the plot of \[C/Si\] vs. \[Fe/H\] (Fig. 1(e)). The Si abundances come from the ionized lines of similar excitation to the C I lines (see Table 1).
We are grateful to Damian Fabbian and Elisabetta Caffau for helpful advice regarding NLTE in C I and S I, and to Sergei Andrievsky for helpful suggestions. This research was supported by the Kenilworth Fund of the New York Community Trust.
Castelli, F., & Kurucz, R. L. 2004, astro-ph/0405087 Charbonnel, C. 1994, , 282, 811 Charbonnel, C., Brown, J. A., & Wallerstein, G. 1998, , 332, 204 Clement, C. M. et al. 2001, , 122, 2587 Fabbian, D., Asplund, M., Carlsson, M., & Kiselman, D. 2006, , 458, 899 Gilroy, K. K., & Brown, J. A. 1991, , 371, 578 Hoyle, F., & Schwarzschild, M. 1955, , 2, 1 Layden, A. C. 1994, , 108, 1016 Mocák, M., Müller, E., Weiss, A., & Kifonidis, K. 2009, , 501, 659 Norris, J. E., & Da Costa, G. S. 1995, , 447, 680 Preston, G. W. 1959, , 130, 507 Pritzl, B. J., Smith, H. A., Catelan, M., & Sweigart, A. V. 2001, , 122, 2600 Pritzl, B. J., — 2002, , 124, 949 Schmidt, E. G. 2002, , 123, 965 Takeda, Y. et al. 2005, PASJ, 57, 751 Vanture, A. D., Wallerstein, G., & Brown, J. A. 1994, , 106, 835 Wallerstein, G., Kovtyukh, V. V., & Andrievsky, S. M. 2009, , 692, 127 Wiese, W. L., Fuhr, J. R., & Deters, T. M. 1998, NIST Monograph No. 7
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We show in this paper that every domain in a separable Hilbert space, say $\cH$, which has a $C^2$ smooth strongly pseudoconvex boundary point at which an automorphism orbit accumulates is biholomorphic to the unit ball of $\cH$. This is the complete generalization of the Wong-Rosay theorem to a separable Hilbert space of infinite dimension. Our work here is an improvement from the preceding work of Kim/Krantz \[KIK\] and subsequent improvement of Byun/Gaussier/Kim \[BGK\] in the infinite dimensions.'
address:
- 'Department of Mathematics, Pohang University of Science and Technology, Pohang 790-784 Korea'
- 'Department of Mathematics, Wichita State University, Wichita, KS 67260-0033 U.S.A.'
author:
- 'Kang-Tae Kim and Daowei Ma'
title: |
Characterization of the Hilbert ball\
by its Automorphisms
---
\[section\] \[theorem\][Definition]{} \[theorem\][Proposition]{} \[theorem\][Condition]{} \[theorem\][Lemma]{} \[theorem\][Corollary]{} \[theorem\][Example]{} \[theorem\][Problem]{}
[^1]
Introduction
============
The primary goal of this article is to establish the following theorem, which gives a full generalization, to a separable Hilbert space of infinite dimension, of the Wong-Rosay Theorem of finite dimension.
\[T:main\] If a domain $\Omega$ in a separable Hilbert space $\cH$ admits a $C^2$ strongly pseudoconvex boundary point at which a holomorphic automorphism orbit accumulates, then $\Omega$ is biholomorphic to the open unit ball in $\cH$.
Since there are many subtleties in setting up the necessary terminology in the infinite dimensions, we shall present the precise definitions in the next section.
There have been several important contributions by several authors concerning this line of research. Chronologically speaking, Wong \[WON\] proved in 1977 the above theorem in $\CC^n$ with the assumption that the domain $\Omega$ is bounded and strongly pseudoconvex at every boundary point. Then, Rosay \[ROS\] improved it in 1979, using holomorphic peak functions, that the theorem holds if the domain is bounded and an automorphism orbit accumulation point is strongly pseudoconvex. Much later in 1995, Efimov \[EFI\] removed the boundedness assumption from $\Omega$. That argument is now well set up using Sibony’s analysis of plurisubharmonic peak functions. See \[BER\], \[GAU\] and \[BGK\], for details. For the infinite dimension, Kim and Krantz \[KIK\] in 2000 proved the above theorem with an extra assumption that $\Omega$ is bounded and convex. They needed convexity since they were relying upon a weak-normal family argument which they developed. Then, developing an infinite dimensional version of Sibony’s analysis on plurisubharmonic peak functions, Byun, Gaussier and Kim (\[BGK\] in 2002) removed the boundedness assumption from the theorem of Kim and Krantz. In this article, we remove the convexity assumption from the theorem of Byun-Gaussier-Kim, thus arriving at the optimal version of the theorem of this type. The crux of the proof uses a new method, which concerns a principle of strong convergence for certain holomorphic mappings of the infinite dimensional Hilbert space. This new convergence argument seems worth exploring further, with a separate interest. Finally, it is worth noting that a manifold version of Wong-Rosay theorem have been studied also. See \[MAK\] for instance. Now the most general version is known, and is due to Gaussier, Kim and Krantz (\[GKK\]). We also present the Hilbert manifold version in this article.
The rest of the paper is organized as follows. Since the proof uses the ideas developed by Kim and Krantz \[KIK\] and then the localization methods introduced in \[BGK\], we shall introduce the outline of their methods shortly after the notation and basic terminology are introduced. Then we shall present our methods leading to the strong convergence of the scaling sequence in the separable Hilbert space and to the proof of the main theorem.
Terminology
===========
We introduce in this section the concepts of smoothness of mappings of infinite dimensional spaces and the strong pseudoconvexity. Further details in great generality can be found in the books of Mujica \[MUJ\] and Dineen \[DIN\], for instance.
Let $\EE$ and $\FF$ be Banach spaces and let $\Omega$ be an open subset of $\EE$. Let $u:\Omega \to \FF$ be a $C^\infty$ smooth mapping. Then for each point $p \in \Omega$ and vectors $v_1,
\ldots, v_k \in \EE$, we may define inductively the derivative $d^k u$ of order $k$ as follows: $$\begin{aligned}
du (p; v_1) & = & \lim_{\RR \ni r \to 0} \frac1r (u(p+rv_1) - u(p)) \\
d^2 u (p; v_1, v_2) & = & \lim_{\RR \ni r \to 0} \frac1r
(du(p+rv_2; v_1) - du(p; v_1)) \\
& \vdots &\end{aligned}$$ $$\begin{gathered}
d^k u (p; v_1, \ldots, v_k) = \\
\lim_{\RR \ni r \to 0} \frac1r (d^{k-1} u (p+rv_k; v_1, \ldots,
v_{k-1}) - d^{k-1} u (p; v_1, \ldots, v_{k-1}))\end{gathered}$$ Notice that these derivatives are symmetric multi-linear over $\RR$. If one so prefer, these formulae can be used to define $C^k$ smoothness, requiring in that case that the corresponding derivatives are continuous multi-linear tensors.
Then the complex differentials can be defined accordingly: $$\begin{aligned}
\partial u (p; v) & = & \frac12 (du(q;v) - i\, du(q; iv)) \\
\bar\partial u (p; v) & = & \frac12 (du(q;v) + i\, du(q; iv)).\end{aligned}$$
We are now able to introduce the concept of holomorphic maps and the remaining terminology thereof. First, by a [*holomorphic mapping*]{} we mean a $C^1$ smooth map that is annihilated by the $\bar\partial$ operator. See \[MUJ\] for equivalent definitions.
A [*domain*]{} in this article is an open connected subset of a Banach space. An [*automorphism*]{} of a domain $\Omega$ is a bijective holomorphic mapping of $\Omega$ with its inverse holomorphic. The [*automorphism group*]{} $\Aut (\Omega)$ of a domain $\Omega$ is the group of all automorphisms of $\Omega$. An [*automorphism orbit*]{} is a set of the form $Aut(\Omega)q =
\{\varphi(q) \mid \varphi \in \Aut (\Omega)\}$, where $q\in
\Omega$. Thus, a boundary point $p$ of $\Omega$ is said to be an [*orbit accumulation point*]{}, if $p$ is an accumulation point of an automorphism orbit, i.e., if there is a sequence $\{\varphi_j
\}\subset \Aut (\Omega)$ and a point $q\in \Omega$ such that $\displaystyle{\lim_{j\to \infty} \|\varphi_j (q)-p\| = 0}$.
We now introduce the concept of strong pseudoconvexity. Let $\Omega$ be a domain in a Banach space $\EE$. We say that $\Omega$ is strongly pseudoconvex at a boundary point $p \in
\partial\Omega$ if there is an open neighborhood $U$ of $p$ and a $C^2$ smooth local defining function $\rho:U \to \RR$ satisfying the following properties:
- $\Omega \cap U = \{z \in U \mid \rho (z) <
0\}$.
- $\partial\Omega \cap U = \{z \in U \mid
\rho (z) = 0\}$.
- $d\rho (q; \cdot)$ is a non-zero functional for every $q \in \partial\Omega \cap U$.
- There exists a constant $C>0$ such that $\partial\bar\partial\rho (p; v, v) \ge C\|v\|^2$ for every $v \in
\EE$ satisfying $\partial\rho (p; v)=0$.
As in the finite dimensional case, we call $\partial\bar\partial
\rho$ the [*Levi form*]{} of $\rho$.
Localization and Pluri-subharmonic Peak Functions {#Localization}
=================================================
Let $\Omega$ be a domain in a Banach space $\EE$, and let $p \in
\partial\Omega$. A continuous function $\psi:\Omega \to \RR$ is said to be [*pluri-subharmonic*]{} if it is subharmonic along every complex affine line in $\Omega$. A [*pluri-subharmonic peak function*]{} at $p$ of $\Omega$ is a pluri-subharmonic function $\psi_p : U \to \RR$ defined on an open neighborhood $U$ of the closure $\overline{\Omega}$ of $\Omega$ satisfying the following two conditions:
- $h(p)=0$, and $h(z)<0$ for every $z \in
\overline{\Omega} \setminus \{p\}$.
- The sets $V_m := \{z \in \overline{\Omega} \mid
h(z)>-1/m\}$, where $m=1,2,\ldots$, form a neighborhood basis at $p$ in $\overline{\Omega}$.
Unlike the finite dimensional cases, the second condition is essential for the definition in the infinite dimensions. On the other hand, notice that every strongly pseudoconvex boundary point admits a pluri-subharmonic peak function for $\Omega$.
Following the work of Sibony \[SIB\], several investigations from the articles of Efimov \[EFI\], Berteloot \[BER\], Gaussier \[GAU\] and Byun- Gaussier-Kim \[BGK\] have been made. We exploit some of them which pertain to the localization and hyperbolicity.
[(cf. p. 588, \[BGK\])]{} Let $\Omega$ be a domain in a Banach space $\EE$ with a $C^2$ smooth strongly pseudoconvex boundary point at which an automorphism orbit accumulates. Then, $\Omega$ is Kobayashi hyperbolic.
[(cf. p. 588, \[BGK\])]{} Let $\Omega$ be a domin in a Banach space $\EE$ with a $C^2$ smooth strongly pseudoconvex boundary point $p$ which admits a sequence $\varphi_j \in \Aut
(\Omega)$ of automorphisms and a point $q \in \Omega$ such that $\lim_{j\to\infty} \varphi_j (q) = p$. Then, for every Kobayashi distance ball $B_\Omega^K (x;r)$ of radius $r$ centered at $x \in
\Omega$ and for every open neighborhood $U$ of $p$ there exists $N>0$ such that $\varphi_j (B_\Omega^K (x;r)) \subset U$ for every $j > N$.
We choose not to include any details of the proofs, in order to avoid an excessive repetition with the references cited above. However, we consider it appropriate to point out that the localization method using pluri-subharmonic peak functions initiated by Sibony seems indeed more effective than the traditional localization arguments relying upon holomorphic peak functions and normal family arguments.
Scaling Maps and Weak Normal Family
===================================
The contents of this section are mostly from the article of Kim and Krantz \[KIK\]. The scaling method introduced here has its finite dimensional origin in the work of Pinchuk \[PIN\], Frankel \[FRA\], Kim \[KIM\] and others. Some other details are in \[BGK\].
Pinchuk’s scaling sequence
--------------------------
Here, we introduce Pinchuk’s scaling sequence. We begin with some notation. We choose an orthonormal basis $e_1, e_2, \ldots$ for a separable Hilbert space $\cH$. Then for each $z \in \cH$, we write $$z = \sum_{m=1}^\infty z_m e_m,$$ and $$z' = \sum_{m=2}^\infty z_m e_m.$$
Let $\Omega$ be a domain in a separable Hilbert space $\cH$ with a $C^2$ strongly pseudoconvex boundary point $p$. Then, there exist an open neighborhood $U$ of $p$ in $\cH$ and an injective holomorphic mapping $G:U \to G(U) \subset \cH$ such that the following hold:
- $G(p)=0$.
- The domain $\Omega_U := G(\Omega\cap U)=\left\{z \in G(U)\mid \re z_1 > \psi (\im z_1, z') \right\} $ is strictly convex. Moreover, the function $\psi$ is strongly convex and vanishes precisely to the second order at the origin.
Now, as in the hypothesis of Theorem \[T:main\], we work with the assumption that there exist $q \in \Omega$ and a sequence $\varphi_j \in \Aut (\Omega)$ such that $\displaystyle{\lim_{j\to\infty} \|\varphi_j (q) - p\| = 0}$. We may choose a subsequence if necessary so that we have $\varphi_j
(q) \in U$ for every $j=1,2,\ldots$. Let $q_j =
G(\varphi_j (q))$. Choose now for each $j$ the point $p_j \in
\partial \Omega_U$ such that $p_j - q_j = r_j e_1$ for some $r_j >
0$. Note that $p_j' = q_j'$. Let us denote by $p_{j1} = \langle
p_j, e_1\rangle$. Then we consider a complex affine linear isomorphism $H_j : \cH \to \cH : z \mapsto w$ defined by $$\begin{aligned}
w_1 &=& e^{\theta_j} (z_1 - p_{j1}) + T_j (z' - p') \\
w' & = & z' - p'\end{aligned}$$ where the bounded linear functional $T_j : (e_1)^\perp \to \CC$ is chosen for each $j$ to satisfy the following properties:
- $H_j (\Omega_U)$ is supported by the real hyperplane defined by $\re w_1 = 0$ at the origin of $\cH$.
- $0 \in \partial T_j(\Omega_U)$.
- $T_j (q_j) = e^{\theta_j} r_j$.
Note also that $e^{\theta_j}$ can be chosen so that it converges to 1 as $j \to \infty$. See \[KIK\] for an explicit choice for these maps and the values for $\theta_j$.
Now we define the Pinchuk scaling sequence. Define the linear map $L_j : \cH \to \cH$ by $$L_j (w) = \frac{w_1}{r_j} e_1 + \frac{1}{\sqrt{r_j}} w'.$$ Then the Pinchuk scaling sequence is defined by the composition $$\omega_j = L_j \circ H_j \circ G \circ \varphi_j.$$ This map is not well-defined on $\Omega$. However, the localization theorems in Section \[Localization\] implies that there exists an increasing and exhausting sequence of Kobayashi distance open balls $B_\Omega^K (q, R_k)$ ($k=1,2,\ldots; R_1 <
R_2 < \cdots$) so that $\omega_j$ is a well-defined map of $B_\Omega^K (q, R_k)$ whenever $j \ge k$.
Weak Normal Family Theorems
---------------------------
In the finite dimensions, the Pinchuk scaling sequence $\omega_j$ defines a normal family whose subsequential limits are holomorphic embeddings of $\Omega$ into the ambient Euclidean space. However, it is not the case in the infinite dimensions. In this section, we introduce the concept of weak convergence of holomorphic mappings that produces holomorphic limits. This is again from \[KIK\] and \[BGK\].
[(Theorem 4.4 of \[BGK\])]{} Let $\EE$ be a separable Banach space, and let $\FF$ a reflexive Banach space. Let $\Omega_1$ and $\Omega_2$ be domains in $\EE$ and $\FF$, respectively. Assume further that $\Omega_2$ is bounded. Then, for every sequence $h_j:\Omega_1 \to \Omega_2$ of holomorphic mappings, there exist a subsequence $h_{j_k}$ and a holomorphic mapping $\widehat h : \Omega_1 \to \FF$ such that, for each $x \in \Omega_1$, the sequence $h_{j_k} (x)$ converges weakly to $\widehat h (x)$.
Unlike the finite dimensions, this theorem is not so effective. The weak limit $\widehat h$ is holomorphic, but not in general injective. Also, it is not even guaranteed at this point that $\widehat h (\Omega_1)$ is contained in $\Omega_2$. (Although, we do have that $\widehat h (\Omega_1)$ is contained in the closed convex hull of $\Omega_2$ due to the reflexivity of $\FF$.) Nonetheless, this is about the best one can obtain from the general theory.
Before proceeding further, we point out that one can modify the Pinchuk scaling sequence $\omega_j$ so that the range becomes bounded. In \[KIK\], a method is described in detail to modify $\omega_j$ by composing with an explicit linear fractional transformation $\Psi$ so that the map $\Psi \circ \omega_j$ has its image in $(1+\epsilon_j) \BB$ for each $j$. Moreover, the sequence of positive numbers $\epsilon_j$ converges monotonically to zero.
From here on, we shall denote the map $\Psi \circ \omega_j$ by $\tau_j$ for $j=1,2,\ldots$.
Calibration of Derivatives and Kim-Krantz Scaling Sequence
----------------------------------------------------------
In \[KIK\], a new method of modifying the scaling sequence has been introduced. The goal is to have a strong convergence of the derivatives $d\tau_j (q; \cdot)$, as $j \to \infty$. In order to do this, Kim and Krantz have used Hilbert isometries to calibrate each differential $d\tau_j (q; \cdot)$ as follows: first they consider the new basis $d\tau_j (q; e_m)$ ($m=1,2,\ldots$) for the separable Hilbert space $\cH$. Then they apply the Gram-Schmidt process to these vectors such as $$f_{jm} := d\tau_j (q; e_m) - \sum_{k=1}^{m-1} \frac{\langle
d\tau_j(q; e_k), f_{jk} \rangle}{\langle f_{jk}, f_{jk} \rangle}
f_{jk}.$$ It turns out that the vectors $f_{jm}$ have their norms uniformly bounded away from zero by a constant independent of $j$ and $m$. Then they define the Hilbert space isometries $S_j : \cH \to \cH$ arising from the condition $S_j (f_{jm}/\|f_{jm}\|) = e_m$ for every $j,m=1,2, \ldots$. Finally, the Kim/Krantz scaling sequence $$\sigma_j := S_j \circ \tau_j : B_\Omega^K (q; R_j) \to
(1+\epsilon_j) \BB$$ is introduced.
If we use the notation $\Sigma_n = \CC e_1 \oplus \ldots \oplus
\CC e_n$, we can observe at this point immediately that $d\sigma_j
(q; \Sigma_n) = \Sigma_n$ for every positive integer $n$. Moreover the sequences $\|d\sigma_j (q; \cdot)\|$ and $\|d\sigma_j (q;
\cdot)^{-1}\|$ are both uniformly bounded. (See Section 7 of \[KIK\] for details.)
In summary, one obtains the following:
Let $\Omega$ be a domain in a separable Hilbert space $\cH$ with a $C^2$ smooth, strongly pseudoconvex boundary point $p$ at which an automorphism orbit accumulates. Then, there exist a point $q \in
\Omega$, a decreasing sequence $\epsilon_j$ of positive numbers tending to zero, an increasing sequence $R_j$ tending to infinity, and a sequence of holomorphic mappings $\sigma_j : B_\Omega^K (q;
R_j) \to (1+\epsilon_j) \BB$ such that
- $\sigma_j$ converges weakly to a holomorphic mapping $\sigma$ at every point of $\Omega$;
- $\sigma_j(q)=0$ and $\sigma(q)=0$;
- $d\sigma_j(q)$ and $d\sigma(q)$ are [*calibrated*]{} in the sense that they map the flag subspace $\Sigma_n=\CC e_1\oplus \cdots\oplus \CC e_n$ into $\Sigma_n$ for each positive integer $n$;
and
- $d\sigma_j(q)$ converges to $d\sigma(q)$ on every $\Sigma_n$.
Notice that the arguments up to this point are sufficient to prove the main theorem of \[KIK\]. The main theorem of \[BGK\] is also in the same line but uses more modifications for the convergence of the sequence of $\sigma_j^{-1}$ since $\Omega$ may be convex but still unbounded. We would like to remark that our proof, as one can see in the subsequent section, goes around such difficulties establishing directly the two facts: (1) $\sigma$ is injective, and (2) $\sigma (\Omega_1) = \Omega_2$.
Strong Convergence of the Scaling Sequence
==========================================
Techniques for Strong Convergence Arguments
-------------------------------------------
We now demonstrate a new method of strong normal families in the infinite dimensional Hilbert space. We begin with an estimate on the Kobayashi metric and distance. From here on, $d_M$ and $k_M$ will denote the Kobayashi distance and metric of the complex manifold $M$, respectively. Let $u:
[0,1)\to [0,\infty)$ be defined by $u(t)=(1/2)\ln[(1+t)/(1-t)]$, so that $u(t)=d_\Delta(0,t)$ and $u^{-1}(s)=\tanh s$.
\[L:esti\] Let $\Omega\subset H$ be a Kobayashi hyperbolic domain. For $q,
x\in \Omega$, denote by $a=d_\Omega(x,q)$. If $\Omega'$ is a subdomain of $\Omega$ such that $\Omega'\supset \{y\in \Omega:
d_\Omega(y,q)<b\}$, where $b>a$, then $d_{\Omega'}(x,q)\le
a/\tanh(b-a)$, $k_{\Omega'}(x, v)\le k_{\Omega}(x, v)/ \tanh(b-a)$ for $v\in H$.
We first prove the second inequality. Let $s=\tanh(b-a)$ and $\epsilon>0$. Then, there exists a holomorphic map $f: \Delta\to
\Omega$ such that $f(0)=x$ and $f'(0)=v/(k_\Omega(x,v)+\epsilon)$. If $\zeta\in \Delta(0,s)$, then $$\begin{aligned}
d_\Omega(q,f(\zeta)) & \le & d_\Omega(q,x)+d_\Omega(x,f(\zeta))\\
& = & a+d_\Omega(f(0),f(\zeta)) \\
& \le & a+d_\Delta(0,\zeta) \\
& < & a+(b-a)=b.\end{aligned}$$ So $f(\Delta(0,s))\subset \Omega'$. Define $g:\Delta\to\Omega'$ by $g(\zeta)=f(s\zeta)$. Then we have $g(0)=x$ and $g'(0)=sf'(0)=sv/(k_\Omega(x,v)+\epsilon)$. Thus, it holds that $k_{\Omega'}(x,v)\le (k_\Omega(x,v)+\epsilon)/s$. Since $\epsilon$ can be arbitrarily small, we obtain that $k_{\Omega'}(x,v)\le k_\Omega(x,v)/s$.
We now prove the first inequality. Let $\epsilon\in (0, b-a)$. There exists a $C^1$ curve $z: [0,1]\to\Omega$ such that $z(0)=q$, $z(1)=x$, and $\int_0^1k_\Omega(z(t),z'(t))\,dt<a+\epsilon$. It follows that $d_\Omega(q,z(t))<a+\epsilon<b$ and $z(t)\in \Omega'$ for each $t\in [0,1]$. By the inequality that we proved in the preceding paragraph, $k_{\Omega'}(z(t),z'(t))\le
k_{\Omega}(z(t),z'(t))/\tanh(b-a-\epsilon)$. Therefore, $$\begin{aligned}
d_{\Omega'}(x,q) & \le & \int_0^1k_{\Omega'}(z(t),z'(t))\,dt \\
& \le & \frac1{\tanh(b-a-\epsilon)}
\int_0^1k_\Omega(z(t),z'(t))\,dt.\end{aligned}$$ So we see that $d_{\Omega'} (x,q)<
(a+\epsilon)/\tanh(b-a-\epsilon)$. Allowing $\epsilon$ tend to $0$, we obtain the desired conclusion.
The next lemma follows immediately by a standard normal family argument.
\[L:disc\] For each positive number $\epsilon<1$ there exists a constant $\delta>0$ such that for each holomorphic function $f:\Delta\to\Delta$ with $f(0)=0$ and $f'(0)>1-\delta$ it holds that $$|f(z)-z|< \epsilon, \;\;\;\;\;\; \mbox{whenever}\; |z|\le 1-\epsilon.$$
We now present a crucial lemma, which establishes the strong uniform convergence on exhausting open subsets for the Kim/Krantz scaling sequence introduced in Section 4.3. We begin with some notation. For bounded linear operators $S$ and $T$ of the Hilbert space $\cH$, we use the standard notation $S \le T$ (or $T\ge S$, equivalently) which means that $\langle (T-S)x, x\rangle \ge 0$ for each $x\in \cH$. We also use two more standard notation: $I$ for the identity map of $\cH$, and the notation $\BB$ for the open unit ball in $\cH$.
\[L:ball\] Let $\{a_j\}$ be a sequence of positive numbers with $a_j\to 0$ and let $g_j: \BB \to \BB$ be a sequence of holomorphic mappings such that $g_j(0)=0$ and $dg_j(0)\ge (1-a_j)I$. Then the sequence $g_j$ converges to $I$ uniformly on each $r\BB$ with $0<r<1$.
Fix $0<r<1$ and $0<\epsilon<1/8$. Let $\zeta$ be a unit vector in the Hilbert space $\cH$. Define a function $f_j:\Delta\to \Delta$ by $f_j(z)=\langle g_j(z\zeta), \zeta\rangle$. Then $f'_j(0)=\langle dg_j(0)\zeta, \zeta\rangle\ge 1-a_j$. By Lemma \[L:disc\], there is a positive integer $k=k(r,\epsilon)$ such that $|f_j(z)-z|<\epsilon$ whenever $j\ge k$ and $|z|\le r$. Let $h_j(z) := g_j(z\zeta)- f_j(z)\zeta$. Then $\langle
h_j(z),\zeta\rangle =0$. By Schwarz’s Lemma, we have $|z|^2\ge
\|g_j(z\zeta)\|^2$. This implies that $$|z|^2 \ge |f_j(z)|^2+\|h_j(z)\|^2\ge
(|z|-\epsilon)^2+\|h_j(z)\|^2.$$ Consequently, we obtain $\|h_j(z)\|^2<2\epsilon-\epsilon^2$ and $$|g_j(z\zeta)-z\zeta|^2=\|h_j(z)\|^2+\|f_j(z)\zeta-z\zeta\|^2\le
(2\epsilon-\epsilon^2)+\epsilon^2=2\epsilon$$ for $j\ge k$ and $|z|\le r$. Therefore, we see immediately that the sequence $g_j$ converges to $I$ uniformly on each $r\BB$.
We introduce one more technical lemma before we present the proof of our main theorem.
\[L:final\] Let $\psi_j: \BB \to \BB$ be a sequence of holomorphic mappings such that the sequence $\psi_j$ converges to $I$ uniformly on $r\BB$ for every $0<r<1$. Then for each $0<r<1$ there exists a $k\ge 1$ such that $\psi_j$ is injective on $r\BB$ and $\psi_j(\BB)\supset r\BB$ whenever $j\ge k$.
By the Cauchy estimates, we can deduce that $d\psi_j$ converges to $I$ uniformly on each $r\BB$. Thus, for fixed constants $r>0$ and $\epsilon<1$, it holds that $$\|\psi_j(x)-\psi_j(y)-(x-y)\|<\epsilon\|x-y\|$$ for any $x, y\in rB$ and any sufficiently large $j$. Observe that the injectivity of $\psi_j$ on $r\BB$ follows from this inequality immediately.
Now, let $0<r<1$. Choose $0<\epsilon<1/8$ so that $(1+2\epsilon)r<1$. Let $j$ be sufficiently large so that $\psi=\psi_j$ satisfies $\|d\psi-I\|<\epsilon$ on $(1+2\epsilon)r\BB$. Fix $x\in r\BB$. Let $y_0=x$ and let $y_k=x+y_{k-1}-\psi(y_{k-1})$ for $k=1,2,\ldots$. Then we see that $$\begin{aligned}
\|\psi(y_k)-x\| &=& \|\psi(y_k)-\psi(y_{k-1})-(y_k-y_{k-1})\| \\
&\le& \epsilon\|y_k-y_{k-1}\| \\
&=& \epsilon\|\psi(y_{k-1})-x\|.\end{aligned}$$ It follows that $\|y_{k+1}-y_k\|=\|\psi(y_k)-x\|\le
\epsilon^k\|y_1-y_0\|$. Thus $\|\psi(y_k)-x\|\to 0$ and $\{y_k\}$ is a Cauchy sequence. The completeness of $\cH$ implies that $y_k$ converges to a certain $y$. Moreover, it is obvious now that $\psi(y)=x$. The remaining assertion follows immediately.
Proof of Theorem \[T:main\]
---------------------------
By the hypothesis, we are given a point $q\in \Omega$ such that the $Aut(\Omega)$-orbit of $q \in \Omega$ accumulates at a strongly pseudoconvex boundary point $p$.
For $j=2, 3, \dots$ let $b_j=u(1-1/j)=(1/2)\ln(2j-1)$. Recall that the following have been proved in \[BGK\] (Also see Proposition 4.2 of this article):
- $\Omega$ is hyperbolic.
- There exist subdomains $\Omega_j\subset
\Omega$, injective holomorphic mappings $\sigma_j:\Omega_j\to \cH$ ($j=2, 3, \dots$), and a holomorphic mapping $\sigma: \Omega\to
\cH$, satisfying the following.
- $\displaystyle{\bigcup_{j=1}^\infty \Omega_j=\Omega}$;
- $\Omega_j\supset \{x\in \Omega: d_\Omega(x,q)< b_j\}$;
- $\sigma_j(q)=0$, $\sigma(q)=0$;
- $d\sigma_j(q)$ and $d\sigma(q)$ are [*calibrated*]{} in the sense that they map the flag subspace $\Sigma_n=\CC e_1\oplus
\cdots\oplus \CC e_n$ into $\Sigma_n$ for each positive integer $n$;
- $(1-1/j)\BB\subset \sigma_j(\Omega_j)\subset(1+1/j)\BB$ and $\sigma(\Omega)\subset\overline \BB$;
- $\sigma_j$ converges weakly to $\sigma$ at every point of $\Omega$;
and
- $d\sigma_j(q)$ converges to $d\sigma(q)$ on every $\Sigma_n$.
Our present goal is to show that $\sigma$ is a biholomorphic mapping onto the open unit ball $\BB$ of the separable Hilbert space $\cH$.
First we observe that $\sigma(\Omega)\subset \BB$ by the maximum modulus principle. It follows by (b), (e) and Lemma \[L:esti\] that, for every $v\in \cH$, $$\label{eq:est}
(1-1/j)k_\Omega(q, v)\le \|d\sigma_j(q)(v)\|
\le (1+1/j)(1-1/j)^{-1}k_\Omega(q, v).$$ With (g) and the fact that $d\sigma_j(q)$ are uniformly bounded, we see that $\|d\sigma_j(q)(v)\|\to \|d\sigma(q)(v)\|$ for every $v\in \cH$, as $j \to \infty$. It follows now that $$\label{eq:est0}
\|d\sigma(q)(v)\|=k_\Omega(q, v).$$
Consider $\sigma_j^{-1}: (1-1/j)\BB\to \Omega$. By (\[eq:est\]) and (\[eq:est0\]), it follows that $$\|d(\sigma\circ \sigma_j^{-1})(0)(v)\|\ge (1-1/j)(1+1/j)^{-1}\|v\|.$$ Let $d(\sigma\circ \sigma_j^{-1})(0)=P_j U_j$ be the polar decomposition of the invertible operator $d(\sigma\circ
\sigma_j^{-1})(0)$, where $P_j$ is positive and $U_j$ unitary. Define a map $\tau_j: \BB\to\Omega$ by $\tau_j(x)=\sigma_j^{-1}((1-1/j)U_j^{-1}x)$. Then $\sigma\circ
\tau_j: \BB\to \BB$, $\sigma\circ \tau_j(0)=0$. Moreover, the positive operator $d(\sigma\circ \tau_j)(0)=(1-1/j)P_j$ satisfies $$\|d(\sigma\circ \tau_j)(0)(v)\|\ge c_j\|v\|,$$ where $c_j=(1-1/j)^2(1+1/j)^{-1}$. It follows that $d(\sigma\circ
\tau_j)(0)\ge c_j I$, and that $c_j\to 1$. By Lemma \[L:ball\], $\sigma\circ \tau_j$ converges to $I$ uniformly on $r\BB$ for every $0<r<1$.
Fix $0<r<1$. By Lemma \[L:final\], $\sigma\circ\tau_j(\BB)\supset r\BB$ for sufficiently large $j$. Hence $\sigma(\Omega)\supset r\BB$. Since this is true for each $0<r<1$, we see that $\sigma(\Omega)=\BB$.
Fix $a>0$ and consider $Q_a=\{x\in \Omega: d_\Omega(x,q)<a\}$. Let $r=(1+\tanh(a))/2$ and $t_j=\tanh(a/\tanh(b_j-a))$. If $j$ is sufficiently large, then $b_j>a$ and $t_j(1+1/j)(1-1/j)^{-1}<r$. By Lemma \[L:esti\], $Q_a\subset \{x\in \Omega_j: d_{\Omega_j}(x,q)<a/\tanh(b_j-a)\}$. This, together with $\sigma_j(\Omega_j)\subset (1+1/j)\BB$, implies that $\sigma_j(Q_a)\subset
t_j(1+1/j)\BB$. It follows that $$\tau_j(r{\Bbb B})\supset\tau_j(t_j(1+1/j)(1-1/j)^{-1}\BB)
=\sigma_j^{-1}(t_j(1+1/j)\BB)\supset Q_a.$$ For sufficiently large $j$, $\sigma\circ \tau_j$ is injective on $r\BB$, hence $\sigma$ is injective on $\tau_j(r\BB)\supset Q_a$. Since $\sigma$ is injective on $Q_a$ for every $a>0$, it must be injective on $\Omega$. Therefore, $\sigma$ is an injective holomorphic mapping from $\Omega$ onto $\BB$. It follows that $\Omega$ is biholomorphic to $\BB$.
Concluding Remarks
==================
Recall that the localization theorem is obtained from the existence of pluri-subharmonic peak functions. Since the pluri-subharmonic functions are extremely flexible as far as the extension properties are concerned, the whole argument of this article is valid for the domains in a separable Hilbert manifold. More precisely our main theorem extends to the following.
Let $\Omega$ be a domain in a separable Hilbert manifold $X$. If $\Omega$ admits an automorphism orbit accumulating at a strongly pseudoconvex boundary point, then it is biholomorphic to the unit ball in a separable Hilbert space $\cH$.
Notice that this is the infinite dimensional version of the main theorem of \[GKK\].
The Wong-Rosay theorem in $\CC^n$ has been generalized to several other domains that are not necessarily strongly pseudoconvex. Better known theorems include \[BEP\], \[KIM\], \[KIP\] and \[KKS\]. They characterized the Thullen domains and the polydiscs from Wong type conditions on existence of boundary accumulating orbits. However, the authors do not know at this time of writing as how to generalize these theorems to infinite dimensions. Thus we close this article posing the following question.
Formulate and prove the Wong-Rosay type characterization of the unit ball in the space $c_0$ of complex sequences converging to zero, or in the space $\ell^\infty$ of bounded complex sequences.
[999]{}
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[^1]: The first named author’s research is supported in part by The Grant KRF-2002-070-C00005 of The Korea Research Foundation.
|
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---
abstract: 'We deal with minimal surfaces in spheres that are locally isometric to a pseudoholomorphic curve in a totally geodesic $\mathbb{S}^{5}$ in the nearly K[ä]{}hler sphere $\mathbb{S}^6$. Being locally isometric to a pseudoholomorphic curve in $\mathbb{S}^5$ turns out to be equivalent to the Ricci-like condition $\Delta\log(1-K)=6K,$ where $K$ is the Gaussian curvature of the induced metric. Besides flat minimal surfaces in spheres, direct sums of surfaces in the associated family of pseudoholomorphic curves in $\mathbb{S}^5$ do satisfy this Ricci-like condition. Surfaces in both classes are exceptional surfaces. These are minimal surfaces whose all Hopf differentials are holomorphic, or equivalently the curvature ellipses have constant eccentricity up to the last but one. Under appropriate global assumptions, we prove that minimal surfaces in spheres that satisfy this Ricci-like condition are indeed exceptional. Thus, the classification of these surfaces is reduced to the classification of exceptional surfaces that are locally isometric to a pseudoholomorphic curve in $\mathbb{S}^5.$ In fact, we prove, among other results, that such exceptional surfaces in odd dimensional spheres are flat or direct sums of surfaces in the associated family of a pseudoholomorphic curve in $\mathbb{S}^5$.'
address: 'Department of Mathematics, University of Ioannina, 45110 Ioannina, Greece'
author:
- 'Amalia-Sofia Tsouri and Theodoros Vlachos'
title: 'Minimal surfaces in spheres and a Ricci-like condition'
---
[^1]
Introduction
============
A fundamental problem in the study of minimal surfaces is to classify those surfaces isometric to a given one. More precisely, the following question has been addressed by Lawson in [@Law]:
> *Given a minimal surface $f\colon M\to\mathbb{Q}_c^n$ in a $n$-dimensional space form of curvature $c$, what is the moduli space of all noncongruent minimal surfaces $\tilde{f}
> \colon M\to\mathbb{Q}_c^{n+m},$ any $m$, which are isometric to $f.$*
Partial answers to this problem were provided by several authors. For instance, see [@Cal2; @J; @L; @Law; @M1; @N; @S; @S1; @V9; @V08].
A classical result due to Ricci-Curbastro [@Ric] asserts that the Gaussian curvature $K\le0$ of any minimal surface in $\mathbb{R}^3$ satisfies the so-called Ricci condition $$\Delta\log(-K)=4K,$$ away from totally geodesic points, where $\Delta$ is the Laplacian operator of the surface with respect to the induced metric $ds^2.$ This condition is equivalent to the flatness of the metric $d\hat{s}^2=(-K)^{1/2}ds^2.$ Conversely (see [@L0]), a metric on a simply connected $2$-dimensional Riemannian manifold with negative Gaussian curvature is realized on a minimal surface in $\mathbb{R}^3,$ if the Ricci condition is satisfied. Hence, the Ricci condition is a necessary and sufficient condition for a $2$-dimensional Riemannian manifold to be locally isometric to a minimal surface in $\mathbb{R}^3.$
Lawson [@L] studied the above problem for minimal surfaces in a Euclidean space that are isometric to minimal surfaces in $\mathbb{R}^3.$ Using the Ricci condition and the holomorphicity of the Gauss map, he classified all minimal surfaces in $\mathbb{R}^n$ that are isometric to a minimal surface in $\mathbb{R}^3.$ Calabi [@Cal2] obtained a complete description of the moduli space of all noncongruent minimal surfaces in $\mathbb{R}^n$ which are isometric to a given holomorphic curve in $\mathbb{C}^n.$
The aforementioned problem turns out to be more difficult for minimal surfaces in spheres. The difficulty arises from the fact that their Gauss map is merely harmonic, while for minimal surfaces in the Euclidean space it is holomorphic. The classification problem of minimal surfaces in spheres that are isometric to minimal surfaces in the sphere $\mathbb{S}^3$ was raised by Lawson in [@L], where he stated a conjecture that is still open. This conjecture has been only confirmed for certain classes of minimal surfaces in spheres (see [@N; @S; @S1; @V9; @V08]). It is worth noticing that a surface is locally isometric to a minimal surface in $\mathbb{S}^3$ if its Gaussian curvature satisfies the spherical Ricci condition $$\Delta\log(1-K)=4K.$$
A distinguished class of minimal surfaces in spheres is the one of *pseudoholomorphic curves* in the nearly K[ä]{}hler sphere $\mathbb{S}^6,$ that was introduced by Bryant [@Br] and has been widely studied (cf. [@BVW; @H; @EschVl]). The pseudoholomorphic curves in $\mathbb{S}^6$ are nonconstant smooth maps from a Riemann surface into the nearly K[ä]{}hler sphere $\mathbb{S}^6,$ whose differential is complex linear with respect to the almost complex structure of $\mathbb{S}^6$ that is induced from the multiplication of the Cayley numbers.
In analogy with Calabi’s work [@Cal2], we consider the problem of classifying minimal surfaces in spheres that are isometric to pseudoholomorphic curves in the nearly K[ä]{}hler sphere $\mathbb{S}^6.$ More precisely, in the present paper we focus on the following problem: *Classify minimal surfaces in spheres that are locally isometric to pseudoholomorphic curves in a totally geodesic $\mathbb{S}^5\subset\mathbb{S}^6.$* The case of pseudoholomorphic curves that are substantial in $\mathbb{S}^6$ requires a different treatment and will be the subject of a forthcoming paper.
A characterization of Riemannian metrics that arise as induced metrics on pseudoholomorphic curves in $\mathbb{S}^5$ was given in [@H; @EschVl]. In fact, the Gaussian curvature $K\leq1$ of a pseudoholomorphic curve in $\mathbb{S}^5$ satisfies the condition $$\Delta\log(1-K)=6K, \tag{$\ast$}$$ away from totally geodesic points, where $\Delta$ is the Laplacian operator of the induced metric $ds^2.$ This condition is equivalent to the flatness of the metric $d\hat
{s}^2=(1-K)^{1/3}ds^2.$ Conversely, any two-dimensional Riemannian manifold $(M,ds^2),$ with Gaussian curvature $K<1$, that satisfies the Ricci-like condition ($\ast$) can be locally isometrically immersed as a pseudoholomorphic curve in $\mathbb{S}^5.$ Thus the classification of minimal surfaces in spheres that are locally isometric to a pseudoholomorphic curve in $\mathbb{S}^5\subset\mathbb{S}^6$ is equivalent to the classification of those surfaces whose induced metrics satisfy the condition $(\ast).$
Obviously flat minimal surfaces in spheres satisfy the condition $(\ast).$ These surfaces were classified in [@K2] and lie in odd dimensional spheres. Another class of minimal surfaces in odd dimensional spheres that satisfy the condition $(\ast)$ is constructed in the following way. Let $g_{\theta}, 0\leq \theta<\pi$, be the associated family of a simply connected pseudoholomorphic curve $g\colon M\to
\mathbb{S}^{5}.$ We consider the surface $\hat g\colon M\to\mathbb{S}^{6m-1}$ defined by $$\hat g=a_{1}g_{\theta _{1}}\oplus \cdots\oplus a_{m}g_{\theta _{m}}, \tag{$\ast \ast$}$$ where $a_{1},\dots\,,a_{m}$ are any real numbers with $\sum_{j=1}^{m}a_{j}^{2}=1,$ $0\leq \theta _{1}<\cdots<\theta_{m}<\pi,$ and $\oplus $ denotes the orthogonal sum with respect to an orthogonal decomposition of the Euclidean space $\mathbb{R}^{6m}.$ It is easy to see that $\hat g$ is minimal and isometric to $g.$
It would be interesting to know whether there exist other minimal surfaces in spheres whose induced metrics satisfy the condition $(\ast),$ besides the flat ones and surfaces of the type $(\ast \ast).$
We prove that minimal surfaces given by $(\ast\ast)$ belong to the class of exceptional surfaces that were studied in [@V08; @V16]. These are minimal surfaces whose all Hopf differentials are holomorphic, or equivalently all curvature ellipses of any order have constant eccentricity up to the last but one. This class of surfaces contains the superconformal ones.
It is then natural to investigate whether any minimal surface that satisfies the Ricci-like condition $(\ast)$ is an exceptional surface. We are able to prove that minimal surfaces in spheres that satisfy the condition $(\ast)$ are exceptional under appropriate global assumptions. In view of this result, the study of minimal surfaces in spheres that satisfy the condition $(\ast)$ is reduced to the class of exceptional surfaces.
In fact, we prove that besides flat minimal surfaces in odd dimensional spheres, the only simply connected exceptional surfaces that satisfy the condition $(\ast)$ are of the type $(\ast\ast).$ Furthermore, we show that compact minimal surfaces in $\mathbb{S}^n, 4\le n\le7,$ that are not homeomorphic to the torus, cannot be locally isometric to a pseudoholomorphic curve in $\mathbb{S}^5,$ unless $n=5.$ Moreover, we prove that, under certain assumptions, there are no minimal surfaces in even dimensional spheres that satisfy the condition $(\ast).$
It is worth noticing that a necessary and sufficient condition for a $2$-dimensional Riemannian manifold to be locally isometric to a minimal Lagrangian surface in $\mathbb{C}P^2$ (see [@EGT Theorem 3.8]) is that its induced metric satisfies the condition $(\ast).$ Thus our results apply to minimal surfaces in spheres that are locally isometric to minimal Lagrangian surfaces in $\mathbb{C}P^2.$
The paper is organized as follows: In Section 2, we fix the notation and give some preliminaries. In Section 3, we recall the notion of Hopf differentials and some results about exceptional surfaces. In Section 4, we give some basic facts about absolute value type functions, a notion that was introduced in [@EGT; @ET]. In section 5, we discuss pseudoholomorphic curves in $\mathbb{S}^{5}$ and give some useful properties for minimal surfaces that satisfy the Ricci-like condition ($\ast$). In Section 6, we study the class of surfaces of type ($\ast\ast$), we compute their Hopf differentials and show that they are indeed exceptional. Section 7 is devoted to minimal surfaces that are exceptional and satisfy the condition $(\ast).$ There we prove that, besides flat minimal surfaces in odd dimensional spheres, the only simply connected exceptional surfaces that satisfy the condition $(\ast)$ are given by $(\ast\ast).$ In the last section, we prove our global results.
Preliminaries
=============
In this section, we collect several facts and definitions about minimal surfaces in spheres. For more details on these facts we refer to [@DF] and [@DV2k15].
Let $f\colon M\to\mathbb{S}^n$ denote an isometric immersion of a two-dimensional Riemannian manifold. The $k^{th}$*-normal space* of $f$ at $x\in M$ for $k\geq 1$ is defined as $$N^f_k(x)={\rm span}\left\{\alpha^f_{k+1}(X_1,\ldots,X_{k+1}):X_1,\ldots,X_{k+1}\in
T_xM\right\},$$ where $$\alpha^f_s\colon TM\times\cdots\times TM\to N_fM,\;\; s\geq 3,$$ denotes the symmetric tensor called the $s^{th}$*-fundamental form* given inductively by $$\alpha^f_s(X_1,\ldots,X_s)=\left(\nabla^\perp_{X_s}\cdots\nabla^\perp_{X_3}
\alpha^f(X_2,X_1)\right)^\perp$$ and $\alpha^f\colon TM\times TM\to N_fM$ stands for the standard second fundamental form of $f$ with values in the normal bundle. Here, $\nabla^{\perp}$ denotes the induced connection in the normal bundle $N_fM$ of $f$ and $(\,\cdot\,)^\perp$ means taking the projection onto the orthogonal complement of $N^f_1\oplus\cdots\oplus N^f_{s-2}$ in $N_fM.$ If $f$ is minimal, then ${\rm dim}N^f_k(x)\le2$ for all $k\ge1$ and any $x\in M$ (cf. [@DF]).
A surface $f\colon M\to\mathbb{S}^n$ is called *regular* if for each $k$ the subspaces $N^f_k$ have constant dimension and thus form normal subbundles. Notice that regularity is always verified along connected components of an open dense subset of $M.$
Assume that an immersion $f\colon M\to\mathbb{S}^n$ is minimal and substantial. By the latter, we mean that $f(M)$ is not contained in any totally geodesic submanifold of $\mathbb{S}^n.$ In this case, the normal bundle of $f$ splits along an open dense subset of $M$ as $$N_fM=N_1^f\oplus N_2^f\oplus\dots\oplus N_m^f,\;\;\; m=[(n-1)/2],$$ since all higher normal bundles have rank two except possible the last one that has rank one if $n$ is odd; see [@Ch] or [@DF]. Moreover, if $M$ is oriented, then an orientation is induced on each plane subbundle $N_s^f$ given by the ordered base $$\alpha^f_{s+1}(X,\ldots,X),\;\;\;\alpha^f_{s+1}(JX,\ldots,X),$$ where $0\neq X\in TM$.
If $f\colon M\to\mathbb{S}^n$ is a minimal surface, then at $x\in M$ and for each $N_r^f$, $1\leq r\leq m$, the *$r^{th}$-order curvature ellipse* $\mathcal{E}^f_r(x)\subset N^f_r(x)$ is defined by $$\mathcal{E}^f_r(x) = \left\{\alpha^f_{r+1}(Z^{\varphi},\ldots,Z^{\varphi})\colon\,
Z^{\varphi}=\cos\varphi Z+\sin\varphi JZ\;\mbox{and}\;\varphi\in[0,2\pi)\right\},$$ where $Z\in T_xM$ is any vector of unit length.
A substantial regular surface $f\colon M\to\mathbb{S}^{n}$ is called *$s$-isotropic* if it is minimal and at any $x\in M$ the curvature ellipses $\mathcal{E}^f_r(x)$ contained in all two-dimensional $N^f_r$$\,{}^{\prime}$s are circles for any $1\le r\le s.$
The $r$-th *normal curvature* $K_{r}^{\perp}$ of $f$ is defined by $$K_{r}^{\perp}={\frac{2}{\pi}}{\hbox {Area}}(\mathcal{E}^f\sb r).$$ If $\kappa _{r}\geq \mu_{r}\geq 0$ denote the length of the semi-axes of the curvature ellipse $\mathcal{E}^f\sb r,$ then $$\label{elipsi}
K_{r}^{\perp}=2\kappa _{r}\mu _{r}.$$
The *eccentricity* $\varepsilon\sb r$ of the curvature ellipse $\mathcal{E}^f\sb r$ is given by $$\varepsilon\sb r=\frac{\left(\kappa^{2}_{r}-\mu^{2}_{r}\right)^{1/2}}{\kappa_{r}},$$ where $\left(\kappa^{2}_{r}-\mu^{2}_{r}\right)^{1/2}$ is the distance from the center to a focus, and can be thought of as a measure of how far $\mathcal{E}^f\sb r$ deviates from being a circle.
The $a$-*invariants* (see [@V16]) are the functions $$a^{\pm}_{r}= \kappa _{r}{\pm} \mu _{r}= \left(2^{-r} \Vert \alpha^f_{r+1}
\Vert^{2} \pm K_{r}^{\perp}\right)^{1/2}.$$ These functions determine the geometry of the $r$-th curvature ellipse.
Denote by $\tau^o_f$ the index of the last plane bundle, in the orthogonal decomposition of the normal bundle. Let $\{e_1,e_2\}$ be a local tangent orthonormal frame and $\{e_\alpha\}$ be a local orthonormal frame of the normal bundle such that $\{e_{2r+1},e_{2r+2}\}$ span $N_r^f$ for any $1\le r\le\tau^o_f$ and $e_{2m+1}$ spans the line bundle $N^f_{m+1}$ if $n=2m+1.$ For any $\alpha=2r+1$ or $\alpha=2r+2,$ we set $$h_1^{\alpha}=\langle
\alpha^f_{r+1}(e_1,\dots,e_1),e_{\alpha}\rangle,{\ }h_2^{\alpha}=\langle
\alpha^f_{r+1}(e_1,\dots,e_1,e_2),e_{\alpha}\rangle.$$ Introducing the complex valued functions $$H_{\alpha}=h_1^{\alpha}+ih_2^{\alpha}\;\;\text{for any}\;\;\alpha=2r+1\;\;\text{or}\;\;\alpha=2r+2,$$ it is not hard to verify that the $r$-th normal curvature is given by $$\label{prwthsxeshden}
K_r^{\perp}=i\left(H_{2r+1}{\overline{H}_{2r+2}}-{\overline{H}_{2r+1}}
H_{2r+2}\right).$$
The length of the $(r+1)$-th fundamental form $\alpha^f_{r+1}$ is given by $$\label{deuterhsxeshden}
\Vert \alpha^f_{r+1}\Vert ^2=2^r\big(|{H_{2r+1}}|^2+|{H_{2r+2}}|
^2\big),$$ or equivalently (cf. [@A]) $$\label{si}
\Vert \alpha^f_{r+1}\Vert ^2=2^r(\kappa_r^2+\mu_r^2).$$
Each plane subbundle $N_r^f$ inherits a Riemannian connection from that of the normal bundle. Its *intrinsic curvature* $K^*_r$ is given by the following proposition (cf. [@A]).
\[5\] *The intrinsic curvature $K_r^{\ast}$ of each plane subbundle $N_{r}^f$ of a minimal surface* $f\colon M\to \mathbb{S}^{n}$ *is given by* $$K_{1}^{\ast}=K_1^{\perp}-{\frac{\Vert \alpha^f_3\Vert ^2}{2K_1^{\perp}}}
\;\; \text{and}\;\; K_r^{\ast}={\frac{K_r^{\perp}}{(K_{r-1}^{\perp})^2}}{\frac{
\Vert \alpha^f_{r}\Vert ^{2}}{2^{r-2}}}-{\frac{\Vert \alpha^f_{r+2}\Vert ^2}{
2^rK_r^{\perp}}}\;\;\text{for}\;\;2\leq r\leq \tau_f^o.$$
Let $f\colon M\to\mathbb{S}^n$ be a minimal isometric immersion. If $M$ is simply connected, there exists a one-parameter *associated family* of minimal isometric immersions $f_\theta\colon M\to\mathbb{S}^n,$ where $\theta\in\mathbb{S}^1=[0,\pi).$ To see this, for each $\theta\in\mathbb{S}^1$ consider the orthogonal parallel tensor field $$J_{\theta}=\cos\theta I+\sin\theta J,$$ where $I$ is the identity endomorphism of the tangent bundle and $J$ is the complex structure of $M$ induced by the metric and the orientation. Then, the symmetric section $\alpha^f(J_\theta\cdot,
\cdot)$ of the bundle $\text{Hom}(TM\times TM,N_f M)$ satisfies the Gauss, Codazzi and Ricci equations, with respect to the same normal connection; see [@DG2] for details. Therefore, there exists a minimal isometric immersion $f_{\theta}\colon M\to
\mathbb{S}^n$ whose second fundamental form is $$\label{sff}
\alpha^{f_{\theta}}(X,Y)=T_\theta\alpha^f(J_{\theta}X,Y),$$ where $T_\theta\colon N_fM\to N_{f_{\theta}}M$ is the parallel vector bundle isometry that identifies the normal subspaces $N_s^f$ with $N_s^{f_\theta}$, $s\geq 1.$
Hopf differentials and Exceptional surfaces
===========================================
Let $f\colon M\to\mathbb{S}^n$ be a minimal surface. The complexified tangent bundle $TM\otimes \mathbb{C}$ is decomposed into the eigenspaces $T^{\prime}M$ and $T^{\prime \prime}M$ of the complex structure $J$, corresponding to the eigenvalues $i$ and $-i.$ The $(r+1)$-th fundamental form $\alpha^f_{r+1}$, which takes values in the normal subbundle $N_{r}^f$, can be complex linearly extended to $TM\otimes
\mathbb{C}$ with values in the complexified vector bundle $N_{r}^f\otimes \mathbb{C}$ and then decomposed into its $(p,q)$-components, $p+q=r+1,$ which are tensor products of $p$ differential 1-forms vanishing on $T^{\prime \prime}M$ and $q$ differential 1-forms vanishing on $T^{\prime}M.$ The minimality of $f$ is equivalent to the vanishing of the $(1,1)$-part of the second fundamental form. Hence, the $(p,q)$-components of $\alpha^f_{r+1}$ vanish unless $p=r+1$ or $p=0,$ and consequently for a local complex coordinate $z$ on $M$, we have the following decomposition $$\alpha^f_{r+1}=\alpha_{r+1}^{(r+1,0)}dz^{r+1}+\alpha_{r+1}^{(0,r+1)}d\bar{z}^
{r+1},$$ where $$\alpha_{r+1}^{(r+1,0)}=\alpha^f_{r+1}(\partial,\dots,\partial),\;\;\alpha_{r+1}^{(0,r+1)}=\overline{\alpha_{r+1}^{(r+1,0)}}\;\;\;\text{and}\;\;\;\partial ={\frac{1}{2}}\big({\frac{\partial}{\partial x}}-i{\frac{\partial}{\partial y}}\big).$$
The *Hopf differentials* are the differential forms (see [@V]) $$\Phi _{r}=\langle \alpha_{r+1}^{(r+1,0)},\alpha_{r+1}^{(r+1,0)}\rangle dz^{2r+2}$$ of type $(2r+2,0),r=1,\dots,[(n-1)/2],$ where $\langle \cdot,\cdot\rangle$ denotes the extension of the usual Riemannian metric of $\mathbb{S}^n$ to a complex bilinear form. These forms are defined on the open subset where the minimal surface is regular and are independent of the choice of coordinates, while $\Phi _{1}$ is globally well defined.
Let $\{e_1,e_2\}$ be a local orthonormal frame in the tangent bundle. It will be convenient to use complex vectors, and we put $$\text{ }E=e_1-ie_2\;\;
\text{and}\;\; \phi =\omega _{1}+i\omega _2,$$ where $\{\omega_1,\omega_2\}$ is the dual frame. We choose a local complex coordinate $z=x+iy$ such that $\phi =Fdz.$
From the definition of Hopf differentials, we easily obtain $$\Phi _{r}={\frac{1}{4}}\left({\overline{H}_{2r+1}^2}+{\overline{H}_{2r+2}^2}
\right) \phi^{2r+2}.$$
Moreover, using (\[prwthsxeshden\]) and (\[deuterhsxeshden\]), we find that $$\label{what}
\left\vert \langle \alpha_{r+1}^{(r+1,0)},\alpha_{r+1}^{(r+1,0)}\rangle \right\vert
^2=\frac{F^{2r+2}}{2^{2r+4}}\left(\Vert \alpha^f_{r+1}\Vert
^4-4^r(K_r^{\perp})^2\right).$$ Thus, the zeros of $\Phi _r$ are precisely the points where the $r$-th curvature ellipse $
\mathcal{E}^f\sb r$ is a circle. Being $s$-isotropic is equivalent to $\Phi_r=0$ for any $1\le r\le s.$
The Codazzi equation implies that $\Phi _{1}$ is always holomorphic (cf. [@Ch; @ChW]). Besides $\Phi_1$, the rest Hopf differentials are not always holomorphic. The following characterization of the holomorphicity of Hopf differentials was given in [@V08], in terms of the eccentricity of curvature ellipses of higher order.
\[ena\] Let $f\colon M\to \mathbb{S}^n$ be a minimal surface. Its Hopf differentials $\Phi _{2},\dots,\Phi_{r+1}$ are holomorphic if and only if the higher curvature ellipses have constant eccentricity up to order $r.$
A minimal surface in $\mathbb{S}^n$ is called *$r$-exceptional* if all Hopf differentials up to order $r+1$ are holomorphic, or equivalently if all higher curvature ellipses up to order $r$ have constant eccentricity. A minimal surface in $\mathbb{S}^n$ is called *exceptional* if it is $r$-exceptional for $r=[(n-1)/2-1].$ This class of minimal surfaces may be viewed as the next simplest to superconformal ones. In fact, superconformal minimal surfaces are indeed exceptional, for superconformal minimal surfaces are characterized by the fact that all Hopf differentials vanish up to the last but one, which is equivalent to the fact that all higher curvature ellipses are circles up to the last but one. There is an abundance of exceptional surfaces. Besides flat minimal surfaces, we show in Section 6 that minimal surfaces of the type $(\ast\ast)$ are indeed exceptional.
We recall some results for exceptional surfaces proved in [@V08], that will be used in the proofs of our main results.
\[3i\] Let $f\colon M\to\mathbb{S}^n$ be an $(r-1)$-exceptional surface. At regular points the following hold:
\(i) For any $1\leq s\leq r-1,$ we have $$\Delta \log \left\Vert \alpha_{s+1}\right\Vert ^2=2\big((s+1)K-K_s^{\ast}\big),$$ where $\Delta $ is the Laplacian operator with respect to the induced metric $ds^{2}.$
\(ii) If $\Phi _{r}\neq 0$*, then* $$\Delta \log \left(\left\Vert \alpha_{r+1}\right\Vert^2+2^rK_r^{\perp}\right) =2\big((r+1)K-K_r^{\ast}\big)$$ and $$\Delta \log \left(\left\Vert \alpha_{r+1}\right\Vert^2-2^rK_r^{\perp}\right) =2\big((r+1)K+K_r^{\ast}\big).$$
\(iii) If $\Phi _{r}=0$*, then* $$\Delta \log \left\Vert \alpha_{r+1}\right\Vert^2=2\big((r+1)K-K_r^{\ast}\big).$$
\(iv) The intrinsic curvature of the $s$-th normal bundle $N_s^f$* is* $K_{s}^{\ast}=0$ if $1\leq s\leq r-1$ and $\Phi _s\neq 0.$
We will need the following proposition which was proved in [@V08].
\[neoksanaafththfora\] Let $f\colon M\to \mathbb{S}^n$ be an $r$-exceptional surface. Then the set $L_0$, where $f$ fails to be regular, consists of isolated points and all $N_s^f$’s and the Hopf differentials $\Phi_s$’s extend smoothly to $L_0$ for any $1\leq s\leq r.$
Absolute value type functions
=============================
For the proof of our results, we shall use the notion of absolute value type functions introduced in [@EGT; @ET]. A smooth complex valued function $p$ defined on a Riemann surface is called of *holomorphic type* if locally $p=p_0p_1,$ where $p_0$ is holomorphic and $p_1$ is smooth without zeros. A function $u\colon M\to\lbrack 0,+
\infty)$ defined on a Riemann surface $M$ is called of *absolute value type* if there is a function $p$ of holomorphic type on $M$ such that $u=|p|.$
The zero set of such a function on a connected compact oriented surface $M$ is either isolated or the whole of $M$, and outside its zeros the function is smooth. If $u$ is a nonzero absolute value type function, i.e., locally $u=|t_0|u_1$, with $t_0$ holomorphic, the order $k\ge1$ of any point $p\in M$ with $u(p)=0$ is the order of $t_0$ at $p.$ Let $N(u)$ be the sum of the orders for all zeros of $u.$ Then $\Delta\log u$ is bounded on $M\smallsetminus\left\{u=0\right\}$ and its integral is computed in the following lemma that was proved in [@EGT; @ET].
\[forglobal\] Let $(M,ds^2)$ be a compact oriented two-dimensional Riemannian manifold with area element $dA.$
\(i) If $u$ is an absolute value type function on $M,$ then $$\int_{M}\Delta \log udA=-2\pi N(u).$$
\(ii) If $\Phi $ is a holomorphic symmetric $(r,0)$-form on $M,$ then either $\Phi =0$ or $N(\Phi)=-r\chi (M),$ where $\chi (M)$ is the Euler-Poincaré characteristic of $M.$
The following lemma, that was proved in [@N], provides a sufficient condition for a function to be of absolute value type.
\[dena\] Let $D$ be a plane domain containing the origin with coordinate $z$ and $u$ be a real analytic nonnegative function on $D$ such that $u(0)=0.$ If $u$ is not identically zero and $\log u$ is harmonic away from the points where $u=0$, then $u$ is of absolute value type and the order of the zero of $u$ at the origin is even.
Pseudoholomorphic curves in $\mathbb{S}^5$
==========================================
It is known that the multiplicative structure on the Cayley numbers $\mathbb{O}$ can be used to define an almost complex structure $J$ on the sphere $\mathbb{S}^6$ in $\mathbb{R}^7.$ This almost complex structure is not integrable but it is nearly K[ä]{}hler. A *pseudoholomorphic curve* [@Br] is a nonconstant smooth map $g\colon M\to\mathbb{S}^6$ from a Riemann surface $M$ into the nearly K[ä]{}hler sphere $\mathbb{S}^6,$ whose differential is complex linear.
It is known [@Br; @EschVl] that any pseudoholomorphic curve $g\colon M\to\mathbb{S}
^6$ is $1$-isotropic. The nontotally geodesic pseudoholomorphic curves in $\mathbb{S}^6$ are 2-isotropic and substantial in $\mathbb{S}^6$, substantial in $\mathbb{S}^6$ but not 2-isotropic, or substantial in a totally geodesic $\mathbb{S}^5\subset\mathbb{S}^6.$
The following theorem [@EschVl] provides a characterization of Riemannian metrics that arise as induced metrics on pseudoholomorphic curves in $\mathbb{S}^5.$
Let $(M,ds^2)$ be a simply connected Riemann surface, and let $K\leq 1$ be its Gaussian curvature and $\Delta$ its Laplacian operator. Suppose that the function $1-K$ is of absolute value type. Then there exists an isometric pseudoholomorphic curve $g\colon M\to\mathbb{S}^5$ if and only if $$\Delta\log(1-K)=6K.\tag{$\ast$}$$ In fact, up to translations with elements of $G_2$, that is the set $Aut(\mathbb{O})\subset SO_7,$ there is precisely one associated family of such maps.
The above result shows that a minimal surface in a sphere is locally isometric to a pseudoholomorphic curve in $\mathbb{S}^5$ if its Gaussian curvature satisfies the condition $(\ast)$ at points where $K<1$ or equivalently if the metric $d\hat{s}^2=(1-K)^{1/3}ds^2$ is flat.
The following lemma is fundamental for our proofs.
\[avtf\] Let $f\colon (M,ds^2)\to\mathbb{S}^n$ be a nontotally geodesic minimal surface. If $(M,ds^2)$ satisfies the Ricci-like condition $(\ast)$, at points with Gauss curvature $K<1$, then the function $1-K$ is of absolute value type with isolated zeros of even order. Moreover, if $M$ is compact and $p_j,j=1,\dots,m,$ are the isolated zeros of $1-K$ with corresponding order $\mathrm{ord}_{p_j}(1-K)=2k_j,$ then we have $$\label{denginetai}
\sum_{j=1}^{m}k_j=-3\chi (M),$$ where $\chi (M)$ is the Euler-Poincaré characteristic of $M.$ In particular, $M$ cannot be homeomorphic to the sphere $\mathbb{S}^2.$
Let $M_{0}$ be the set of points where $K=1.$ The open subset $M\smallsetminus M_0$ is dense on $M,$ since minimal surfaces in spheres are real analytic. Around each point $p_0\in M_0,$ we choose a local complex coordinate $z$ such that $p_0$ corresponds to $z=0$ and the induced metric is written as $ds^2=F|dz|^2.$ The Gaussian curvature $K$ is given by $$K=-\frac{2}{F}\partial \bar{\partial}\log F.$$
Moreover, the condition $(\ast)$ is equivalent to $$4\partial \bar{\partial}\log (1-K)=6KF.$$ Thus we have $$\partial \bar{\partial}\log \left((1-K)F^3\right) =0.$$
According to Lemma \[dena\], the function $1-K$ is of absolute value type with isolated zeros $p_j,j=1,\dots,m,$ and corresponding order $\mathrm{ord}_{p_j}(1-K)=2k_{j}.$ Then, (\[denginetai\]) follows from Lemma \[forglobal\](i) and the condition $(\ast).$
Let $g\colon M\to\mathbb{S}^5$ be a pseudoholomorphic curve and let $\xi\in N_fM$ be a smooth unit vector field that spans the extended line bundle $N_2^g$ over the isolated set of points where $f$ fails to be regular (see Proposition \[neoksanaafththfora\]). The surface $g^*\colon M\to\mathbb{S}^5$ defined by $g^*=\xi$ is called the *polar surface* of $g.$ It has been proved in [@V16 Corollary 3] that the surfaces $g$ and $g^*$ are congruent.
A class of minimal surfaces that are locally isometric to a pseudoholomorphic curve in $\mathbb{S}^5$
=====================================================================================================
The aim of this section is to study a class of minimal surfaces that are exceptional, nonflat and locally isometric to a pseudoholomorphic curve in $\mathbb{S}^5.$
Let $g\colon M\to\mathbb{S}^5$ be a simply connected pseudoholomorphic curve with Gaussian curvature $K<1,$ with respect to the induced metric $\langle\cdot,\cdot \rangle=ds^2,$ and let $g_\theta,\theta\in\mathbb{S}^1=[0,\pi),$ be its associated family.
Take $$\bold{a}=(a_1,\dots,a_m)\in\mathbb{S}^{m-1}\subset\mathbb{R}^m
\;\; \text{with}\;\; \prod\limits_{j=1}^{m}a_j\neq0$$ and $${\pmb{\theta}}=(\theta_1,\dots,\theta_m)\in\mathbb{S}^1\times\cdots\times
\mathbb{S}^1, \;\; \text{where} \;\; 0\leq \theta _1<\cdots<\theta_m<\pi.$$
We consider the map $\hat{g}=g_{\bold{a},\pmb{\theta}}\colon M\to\mathbb{S}^
{6m-1}\subset\mathbb{R}^{6m}$ defined by $$\hat{g}=g_{\bold{a},{\pmb{\theta}}}=a_1g_{\theta_{1}}\oplus\cdots
\oplus a_mg_{\theta_m},$$ where $\oplus$ denotes the orthogonal sum with respect to an orthogonal decomposition of $\mathbb{R}^{6m}$. Its differential is given by $$d\hat{g}=a_{1}dg_{\theta_1}\oplus\cdots\oplus a_mdg_{\theta_m}.$$ It is obvious that $\hat{g}$ is an isometric immersion. We can easily see that the second fundamental form of the surface $\hat{g}$ is given by $$\hat{\alpha}_2(X,Y)=\sum\limits_{j=1}^ma_j\alpha^{g_{\theta_j}}(X,Y),\,\,
X,Y\in TM,$$ which implies that $\hat{g}$ is minimal.
The following proposition provides several properties for the above class of minimal surfaces.
\[hola\] For any simply connected pseudoholomorphic curve $g\colon M\to\mathbb{S}^5,$ the minimal surface $\hat{g}\colon M\to\mathbb{S}^{6m-1}$ is substantial and isometric to $g.$ Moreover, it is an exceptional surface and the following hold:
\(i) The length of its $(s+1)$-th fundamental form is given by $$\label{alphaf}
\left\Vert \hat{\alpha}_{s+1}\right\Vert^2=
\begin{cases}
\,\hat{b}_s(1-K)^{s/3} &\text{ \,if\, }
s\equiv 0\;\mathrm{mod}\; 3, \\[2mm]
\,\hat{b}_s(1-K)^{(s+2)/3}& \text{ \,if\, }
s\equiv 1\;\mathrm{mod}\; 3, \\[2mm]
\,\hat{b}_s(1-K)^{(s+1)/3} &\text{ \,if\, }
s\equiv 2\;\mathrm{mod}\; 3,
\end{cases}$$ for any $1\le s\le 3m-1,$ where $\hat{b}_s$ are positive numbers.
\(ii) Its $s$-th normal curvature is given by $$\label{curvaturef}
\hat{K}^{\perp}_s=
\begin{cases}
\,\hat{c}_s\,(1-K)^{s/3}&\text{ \,if\, }
s\equiv 0\;\mathrm{mod}\; 3, \\[2mm]
\,\hat{c}_s\,(1-K)^{(s+2)/3}&\text{ \,if\, }
s\equiv 1\;\mathrm{mod}\; 3, \\[2mm]
\,\hat{c}_s\,(1-K)^{(s+1)/3}&\text{ \,if\, }
s\equiv 2\;\mathrm{mod}\; 3,
\end{cases}$$ for any $1\le s< 3m-1,$ where $\hat{c}_s$ are positive numbers.
\(iii) Its $s$-th Hopf differential is given by $$\label{hopff}
\hat{\Phi}_s=
\begin{cases}
\,\hat{d}_s\Phi^{(s+1)/3}&\text{ \,if\, }
s\equiv 2\;\mathrm{mod}\; 3, \\[1mm]
\,0&\text{ \,\,otherwise,}
\end{cases}$$ for any $1\le s\le 3m-1,$ where $\hat{d}_s\in\mathbb{C}$ and $\Phi$ is the second Hopf differential of $g.$
We consider a local orthonormal frame $\{e_1,e_2\}$ in the tangent bundle away from totally geodesic points of $g.$ Moreover, we choose a local orthonormal frame field $\{\xi_1, \xi_2,\xi_3\}$ in the normal bundle of $g$ such that $$\xi_{1}=\frac{\alpha(e_1,e_1)}{\left\Vert \alpha(e_1,e_1)\right\Vert},\,\,\,
\xi_{2}=\frac{\alpha(e_1,e_2)}{\left\Vert \alpha(e_1,e_2)\right\Vert}.$$
From [@V16 Lemma 5] it follows that $h_1^3=\kappa,$ $h_2^3=0,$ $h_1^4=0$ and $h_2^4=\kappa,$ where $\kappa$ is the radius of the first circular curvature ellipse. Hence $H_{3}=\kappa$ and $H_{4}=i\kappa.$ Moreover, we have that $h_2^5=0$ and $h_1^5=\kappa.$ Therefore, it follows that $$\langle\nabla_{e_1}^{\perp}\xi_1,\xi_3\rangle=1,\,\,\,
\langle\nabla_{e_1}^{\perp}\xi_2,\xi_3\rangle=0,$$ $$\langle\nabla_{e_2}^{\perp}\xi_1,\xi_3\rangle=0,\,\,\,
\langle\nabla_{e_2}^{\perp}\xi_2,\xi_3\rangle=-1,$$ or equivalently $$\label{apeiroeths}
\langle\nabla_{\overline{E}}^{\perp}\xi_{3},\xi_{1}-i\xi_{2}\rangle=0,\,\,\,
\langle\nabla_{\overline{E}}^{\perp}\xi_{3},\xi_{1}+i\xi_{2}\rangle=-2,$$ where $E=e_1-ie_2.$
In order to show that the minimal surface $\hat{g}$ is substantial, it is sufficient to prove that $$\sum\limits_{j=1}^m a_j\langle g_{\theta_j},w_j\rangle=0\label{paragwgisi}$$ for $(w_1,\dots,w_m)\in\mathbb{R}^{6m}=\mathbb{R}^6\oplus\cdots\oplus\mathbb{R}^6$ implies that $w_j=0$ for any $j=1,\dots,m.$
Assume to the contrary that $w_j\neq0$ for all $j=1,\dots,m.$ Differentiating (\[paragwgisi\]) we obtain $$\label{dgneo}
\sum\limits_{j=1}^m a_j\langle dg_{\theta_j},w_j\rangle=0,$$ and $$\sum\limits_{j=1}^{m}a_j\left\langle \alpha^{g_{\theta_j}}, w_j\right\rangle=0.$$ Using (\[sff\]), we have that $$\sum\limits_{j=1}^ma_j\left\langle T_{\theta_j}\alpha^g(J_{\theta_j}\overline{E},
\overline{E}), w_j\right\rangle=0,$$ where $T_{\theta_{j}}\colon N_gM\to N_{g_{\theta_j}}M$ is a parallel vector bundle isometry. Since $J_{\theta}\overline{E}=e^{-i\theta}\overline{E}$, it follows that $$\label{Ttheta}
\sum\limits_{j=1}^{m}a_je^{-i\theta_j}\left\langle T_{\theta_j}(\xi_1+i\xi_2),
w_j\right\rangle=0.$$ Differentiating with respect to $\overline{E}$, and using the Weingarten formula, we obtain $$\sum\limits_{j=1}^{m}a_je^{-i\theta_j}\left\langle\nabla^\perp_{\overline{E}}
T_{\theta_j}(\xi_1+i\xi_2), w_j\right\rangle=\sum\limits_{j=1}^{m}a_je^{-i\theta_j}
\left\langle dg_{\theta_j}\circ A_{T_{\theta_j}(\xi_1+i\xi_2)}(\overline{E}),w_j\right
\rangle,$$ where $A_{T_{\theta_j}\eta}$ is the shape operator of ${g_{\theta_j}}$ with respect to its normal direction $T_{\theta_j}\eta.$ It follows from (\[sff\]) that $$A_{T_{\theta_j}(\xi_1+i\xi_2)}=e^{i\theta_j}A_{\xi_1+i\xi_2}.$$ This and yield $$\sum\limits_{j=1}^{m}a_je^{-i\theta_j}\left\langle T_{\theta_j}\left(\nabla^\perp_
{\overline{E}}\left(\xi_1+i\xi_2\right)\right), w_j\right\rangle=0.$$ Using and , the above is written as $$\sum\limits_{j=1}^{m}a_je^{-i\theta_j}\left\langle T_{\theta_j}\xi_3,w_j\right\rangle=0,$$ or equivalently $$\sum\limits_{j=1}^{m}a_je^{-i\theta_j}\langle g^*_{\theta_j},w_j\rangle=0,$$ where $g^*_{\theta_j}=T_{\theta_j}\xi_3$ is the polar surface of $g_{\theta_j}.$ This is equivalent to $$\sum\limits_{j=1}^{m}a_j\cos\theta_j\langle g^*_{\theta_j},w_j\rangle=0
\;\;\text{and}\;\;\sum\limits_{j=1}^{m}a_j\sin\theta_j\langle g^*_{\theta_j},
w_j\rangle=0.$$ Eliminating $\langle g^*_{\theta_m},w_m\rangle$, we can easily see that $$\label{g*w}
\sum\limits_{j=1}^{m-1}a_j\langle g^*_{\theta_j},w_j^{(m)}\rangle=0,$$ where $w_j^{(m)}=\sin(\theta_m-\theta_j)w_j\neq0.$ Using the fact that the polar surface of $g_{\theta_j}$ is congruent to $g_{\theta_j}$ (cf. [@DV Lemma 11] or [@V16 Corollary 3]) and arguing as for , we have that $$\sum\limits_{j=1}^{m-2}a_j\langle g_{\theta_j},w_j^{(m-1)}\rangle=0,$$ where $w_j^{(m-1)}=\sin(\theta_{m-1}-\theta_j)w_j^{(m)}.$ Arguing as before, we inductively obtain that $$a_1\langle g_{\theta_1},w\rangle=0 \text{\,\, or \,\,} a_1\langle g^*_{\theta_1},
w\rangle=0$$ for a vector $w\in\mathbb{R}^6\smallsetminus\{0\},$ which is a contradiction.
*Claim.* We now claim that the higher fundamental forms of $\hat{g}$ are given by $$\label{1}
\hat{\alpha}_s(\overline{E},\dots,\overline{E})=
\begin{cases}
\,\kappa^{s/3}\sum_{j=1}^{m}c_j^sg_{\theta_j}&\text{ \ if \ }s\equiv 0
\;\mathrm{mod}\;6,\\[2mm]
\,\kappa^{(s-1)/3}\sum_{j=1}^{m}c_j^sdg_{\theta_j}(\overline{E})
&\text{ \ if \ }s\equiv 1\;\mathrm{mod}\; 6, \\[2mm]
\,\kappa^{(s+1)/3}\sum_{j=1}^{m}c_j^sT_{\theta_j}(\xi_1+i\xi_2)
&\text{ \ if \ }s\equiv 2\;\mathrm{mod}\; 6, \\[2mm]
\,\kappa^{s/3}\sum_{j=1}^{m}c_j^sT_{\theta_j}\xi_3&\text{ \ if \ }s\equiv
3\;\mathrm{mod}\; 6, \\[2mm]
\,\kappa^{(s-1)/3}\sum_{j=1}^{m}c_j^sT_{\theta_j}(\xi_1-i\xi_2)
&\text{ \ if \ }s\equiv 4\;\mathrm{mod}\; 6, \\[2mm]
\,\kappa^{(s+1)/3}\sum_{j=1}^{m}c_j^sdg_{\theta_j}(E)
&\text{ \ if \ }s\equiv 5\;\mathrm{mod}\; 6,
\end{cases}$$ where the complex vectors $\bold{C}_s=(c_1^s,\dots,c_m^s)\in\mathbb{C}^m\smallsetminus
\{0\}$, $2\le s\le 3m$ satisfy the following orthogonality conditions, with respect to the standard Hermitian product $(\cdot,\cdot)$ on $\mathbb{C}^m$: $$\label{2}
(\bold{C}_{t},\overline{\bold{C}}_{t'})=0 \,\,\, \text{if} \,\,\, t\equiv1\;\mathrm{mod}\; 6
\,\,\text{and}\,\, t'\equiv5\;\mathrm{mod}\; 6, \,\,\,\text{or}\,\,
t\equiv2\;\mathrm{mod}\; 6\,\,\text{and}\,\, t'\equiv4\;\mathrm{mod}\; 6,$$ $$(\bold{C}_{t},\overline{\bold{C}}_{t'})=0=(\bold{C}_{t},\bold{C}_{t'})\,\,\text{if} \,\,
t\neq t' \,\,\text{and} \,\, t,t'\equiv0\;\mathrm{mod}\; 6, \,\,\,\text{or}\,\,t, t'\equiv3\;\mathrm{mod}
\; 6,$$ $$(\bold{C}_{t},\bold{C}_{t'})=0=(\overline{\bold{C}}_{t},\overline{\bold{C}}_{t'})
\text{ if\,\, } t\neq t' \,\,\text{and} \,\, t,t'\equiv1\;\mathrm{mod}\; 6, \,\,\,\text{or} \,\, t, t'\equiv2\;
\mathrm{mod}\; 6, \\$$ $$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{or} \,\, t, t'\equiv4\;\mathrm{mod}\; 6, \,\,\text{or} \,\, t, t'\equiv5\;\mathrm{mod}\; 6$$ and $$\label{6}
(\bold{C}_{t},\bold{a})=0 \text{ if\,\, }t\equiv0,1,5\;\mathrm{mod}\; 6.$$ In particular, these complex vectors are defined inductively by $$\label{celeosksananea}
\bold{C}_{s+1}=
\begin{cases}
\,\bold{C}_{s}-\sum\limits_{t\equiv1\;\mathrm{mod}\; 6}^{s}
\frac{(\bold{C}_{s},
\bold{C}_{t})}{\left\Vert\bold{C}_{t}\right\Vert^{2}}\bold{C}_{t}
-\sum\limits_{t\equiv5\;\mathrm{mod}\; 6}^{s}\frac{(\bold{C}_{s},
\overline{\bold{C}}_{t})}{\left\Vert\bold{C}_{t}\right\Vert^{2}}\overline{\bold{C}}_{t}
&\text{\,\,\,if $s\equiv 0\;\mathrm{mod}\; 6, $}\\[4mm]
\,2T_{\pmb{\theta}}\bold{C}_{s}
&\text{\,\,\,if $s\equiv 1\;\mathrm{mod}\; 6,$}\\[4mm]
\,2\bold{C}_{s}
&\text{\,\,\,if $s\equiv 2\;\mathrm{mod}\; 6,$}\\[3mm]
\,-\bold{C}_{s}+\sum\limits_{t\equiv2\;\mathrm{mod}\; 6}^{s}\frac{(\bold{C}_{s},
\overline{\bold{C}}_{t})}{\left\Vert\bold{C}_{t}\right\Vert^{2}}\overline{\bold{C}}_{t}
+\sum\limits_{t\equiv4\;\mathrm{mod}\; 6}^{s}\frac{(\bold{C}_{s},
\bold{C}_{t})}{\left\Vert\bold{C}_{t}\right\Vert^{2}}\bold{C}_{t}
&\text{\,\,\,if $s\equiv 3\;\mathrm{mod}\; 6,$ }\\[4mm]
\,-2T_{\pmb{\theta}}\bold{C}_{s}
&\text{\,\,\,if $s\equiv 4\;\mathrm{mod}\; 6,$}\\[4mm]
\,-2\bold{C}_{s}
&\text{\,\,\,if $s\equiv 5\;\mathrm{mod}\; 6,$}
\end{cases}$$ where $\bold{C}_1=\bold{a}$ and $T_{\pmb{\sigma}}\colon\mathbb{C}^m\to\mathbb{C}^m$ denotes the unitary transformation given by $$T_{\pmb{\sigma}}\bold{u}=(u_{1}e^{-i\sigma_1},\dots,u_{m}e^{-i
\sigma_{m}}),\;\; \bold{u}=(u_{1},\dots,u_{m})\in\mathbb{C}^{m}$$ for any ${\pmb{\sigma}}=(\sigma_{1},\dots,\sigma_{m})\in\mathbb{R}^{m}.$ It is worth noticing that implies that $\bold{C}_{s}\neq0$ for every $2\le s\le 3m,$ since the surface $\hat{g}$ is substantial.
To prove the claim, we proceed by induction on $s.$ Using that $$d\hat{g}(\overline{E})=\sum\limits_{j=1}^{m}a_{j}dg_{\theta_j}(\overline{E}),$$ the Gauss formula for $g$ and $g_{\theta_j},\,\, j=1,\dots,m,$ yields $$\hat{\alpha}_{2}(\overline{E},\overline{E})=\kappa\sum_{j=1}^{m}c_j^2(\xi_{1}
^{\theta_j}+i\xi_2^{\theta_j}),$$ where $c_j^2=2a_je^{-i\theta_j}.$ Hence, $\bold{C}_2=2T_{\pmb{\theta}}
\bold{C}_1=2T_{\pmb{\theta}}
\bold{a}$ and this proves (\[1\]) for $s=2.$
Let us assume that (\[1\])-(\[celeosksananea\]) hold for any $t\le s.$ We shall prove that it is also true for $t=s+1.$ From the definition of the higher fundamental forms, we have that $$\begin{aligned}
\hat{\alpha}_{s+1}(\overline{E},\dots,\overline{E})&=&\left(\overline{\nabla}_
{\overline{E}}\hat{\alpha}_{s}(\overline{E},\dots,\overline{E})\right)^{N_s^
{\hat{g}}}\notag\\
&=&\kappa^{\lambda_{s}}\left(\tilde{\nabla}_{\overline{E}}\left(\frac{1}{\kappa^
{\lambda_{s}}}\hat{\alpha}_{s}(\overline{E},\dots,\overline{E})\right)\right)^{N_s^
{\hat{g}}},\label{a}\end{aligned}$$ where $\overline{\nabla}$ is the induced connection of $g^\ast(T\mathbb{S}^{6m-1}),$ $\tilde{\nabla}$ is the induced connection of the induced bundle $(i\circ g)^\ast(T
\mathbb{R}^{6m})$, $i\colon \mathbb{S}^{6m-1}\to\mathbb{R}^{6m}$ is the standard inclusion, $\lambda_s$ is the exponent of the function $\kappa$ in (\[1\]) and $(\,\cdot\,)
^{N_s^{\hat{g}}}$ stands for the projection onto the $s$-th normal bundle of $\hat{g}.$ Taking into account, we obtain that $$\label{naii}
\tilde{\nabla}_{\overline{E}}
\left(\frac{1}{\kappa^{\lambda_s}}\hat{\alpha}_{s}(\overline{E},\dots,\overline{E})
\right)=\sum\limits_{j=1}^{m}c^s_jdg_{\theta_{j}}(\overline{E})
\text{\,\,\,if $s\equiv 0\;\mathrm{mod}\; 6$,}$$
$$\begin{aligned}
\tilde{\nabla}_{\overline{E}}\left(\frac{1}{\kappa^{\lambda_{s}}}\hat{\alpha}_{s}
(\overline{E},\dots,\overline{E})\right)
&=\dfrac{1}{2}&\sum\limits_{j=1}^{m}c^s_j\bigg(\langle\nabla_{\overline{E}}
\overline{E},E\rangle dg_{\theta_{j}}(\overline{E})\\ \nonumber
&&+4\kappa e^{-i\theta_{j}}T_{\theta_j}(\xi_{1}+i\xi_{2})\bigg)\text{\,\,if\,\,}s
\equiv 1\;\mathrm{mod}\; 6,\end{aligned}$$
$$\tilde{\nabla}_{\overline{E}}\left(\frac{1}{\kappa^{\lambda_{s}}}\hat{\alpha}_{s}
(\overline{E},\dots,\overline{E})\right)=\sum\limits_{j=1}^{m}c^{s}_{j}\left(-i\langle
\nabla^\perp_{\overline{E}}\xi_{1},\xi_{2}\rangle T_{\theta_j}(\xi_{1}+i\xi_{2})+
2T_{\theta_j}\xi_{3}\right) \text{\,\,if $s\equiv 2\;\mathrm{mod}\; 6$},$$
$$\tilde{\nabla}_{\overline{E}}\left(\frac{1}{\kappa^{\lambda_{s}}}\hat{\alpha}_{s}
(\overline{E},\dots,\overline{E})\right)=-\sum\limits_{j=1}^{m}c^{s}_{j}T_{\theta_j}
(\xi_{1}-i\xi_{2})\text{\,\,\,if $s\equiv 3\;\mathrm{mod}\; 6,$}$$
$$\begin{aligned}
\tilde{\nabla}_{\overline{E}}\left(\frac{1}{\kappa^{\lambda_{s}}}\hat{\alpha}_{s}
(\overline{E},\dots,\overline{E})\right)&=&\sum\limits_{j=1}^{m}c^{s}_{j}\Big(-2\kappa
e^{-i\theta_{j}}dg_{\theta_{j}}(E)\\ \nonumber
&&+i\langle\nabla^\perp_{\overline{E}}\xi_{1},\xi_{2}\rangle T_{\theta_j}(\xi_{1}-i
\xi_{2})\Big)\text{\,\,\,if $s\equiv 4\;\mathrm{mod}\; 6$}\end{aligned}$$
and $$\label{naiii}
\tilde{\nabla}_{\overline{E}}\left(\frac{1}{\kappa^{\lambda_{s}}}\hat{\alpha}_{s}
(\overline{E},\dots,\overline{E})\right)=\dfrac{1}{2}\sum\limits_{j=1}^{m}c^{s}_{j}\left(\langle
\nabla_{\overline{E}}E,\overline{E}\rangle dg_{\theta_{j}}(E)-4g_{\theta_{j}}\right)
\text{\,\,\,if $s\equiv 5\;\mathrm{mod}\; 6$,}$$ where $\nabla$ is the Levi-Civitá connection on $M.$
Using (\[a\]) and (\[naii\])-(\[naiii\]), after some tedious computations, we obtain that (\[1\]) holds for $t=s+1.$ Taking into account (\[1\]) for $t=s+1,$ the orthogonality of the higher normal bundles and (\[2\])-(\[6\]) for $t\le s,$ we obtain that (\[2\])-(\[6\]) are also true for $t=s+1,$ and this completes the proof of the claim.
From (\[si\]) and since the length of the semi-axes $\kappa_s$ and $\mu_s$ of the $s$-th curvature ellipse satisfy $$\kappa^2_s+\mu^2_s=2^{-2s}\left\Vert \hat{\alpha}_{s+1}(\overline{E},\dots,
\overline{E})\right\Vert^2,$$ we have $$\left\Vert \hat{\alpha}_{s+1}\right\Vert^2=2^{-s}\left\Vert \hat{\alpha}_{s+1}
(\overline{E},\dots,\overline{E})\right\Vert^2.$$ Clearly (\[alphaf\]) follows from (\[1\]) with $$\hat{b}_s=
\begin{cases}
\,{2^{(3-4s)/3}}\left\Vert\bold{C}_{s+1}\right\Vert^{2}&\text{ \ if \ }
s\equiv 0\;\mathrm{mod}\; 3,\\[2mm]
\,2^{(1-4s)/3}\left\Vert\bold{C}_{s+1}\right
\Vert^{2}
&\text{ \ if \ }s\equiv 1\;\mathrm{mod}\; 3, \\[2mm]
\,2^{-(1+4s)/3}\left\Vert\bold{C}_{s+1}\right
\Vert^{2}
&\text{ \ if \ }s\equiv 2\;\mathrm{mod}\; 3.
\end{cases}$$
Furthermore, the $s$-th normal curvature is given by $$\hat{K}_{s}^\perp=2^{-2s}\left(\left\Vert \hat{\alpha}_{s+1}(\overline{E},\dots,
\overline{E})\right\Vert^4-|\langle \hat{\alpha}_{s+1}(E,\dots,E),\hat{\alpha}_{s+1}
(E,\dots,E)\rangle|^2\right)^{1/2}.$$ This, combined with (\[1\]) yields (\[curvaturef\]), where $$\hat{c}_s=
\begin{cases}
\,2^{(3-7s)/3}\left\Vert\bold{C}_{s+1}\right\Vert^{2}&\text{ \ if \ }s\equiv 0\;
\mathrm{mod}\;3,\\[3mm]
\,2^{(1-7s)/3}\left\Vert\bold{C}_{s+1}\right\Vert^{2}
&\text{ \ if \ }s\equiv 1\;\mathrm{mod}\; 3, \\[2mm]
\,2^{-(1+7s)/3}\left(\left\Vert \bold{C}_{s+1}\right\Vert^{4}-|(\bold{C}_{s+1},
\overline{\bold{C}}_{s+1})|^2\right)^{1/2} &\text{ \ if \ }s\equiv 2\;\mathrm{mod}\; 3.
\end{cases}$$
Using (\[1\]) and the fact that the $s$-th Hopf differential of $\hat{g}$ is written as $$\hat{\Phi}_{s}=4^{-(s+1)} \left\langle \hat{\alpha}_{s+1}(E,\dots,E),\hat{\alpha}_{s+1}
(E,\dots,E)\right\rangle\phi^{2s+2},$$ we obtain that $$\hat{\Phi}_{s}=4^{-(s+1)}\kappa^{2\frac{s+1}{3}}
\sum\limits_{j=1}^{m}(\overline{c}_j^{s+1})^2\phi^{2s+2} \;\;\text{if}\;\; s\equiv2
\;\mathrm{mod}\; 3$$ and $\hat{\Phi}_{s}=0$ otherwise.
The fact that the second Hopf differential $\Phi$ of $g$ is given by $\Phi=2^{-2}
\kappa^2\phi^6$ completes the proof of part $(iii)$, where $\hat{d}_s=2^{-4(s+1)/3}(\overline{\bold{C}}_{s+1},\bold{C}_{s+1}).$ Obviously, all Hopf differentials are holomorphic and consequently $\hat{g}$ is exceptional according to Theorem \[ena\].
In the subsequent lemma, we determine the associated family of any surface $\hat{g}=g_{\bold{a},{\pmb{\varphi}}}.$
\[asociatefamilygtheta\] The associated family $\hat{g}_\varphi$ of any minimal surface $\hat{g}=g_{\bold{a},
{\pmb{\theta}}}$ is given by $\hat{g}_\varphi=g_{\bold{a},{\pmb{\varphi}}}$, where ${\pmb{\varphi}}=(\theta_1+\varphi,\dots,\theta_m+\varphi).$
Let $f\colon M\to\mathbb{S}^{6m-1}$ be the minimal surface given by $f=g_{\bold{a},{\pmb{\varphi}}}.$ From (\[celeosksananea\]) we can easily see that the complex vectors $\bold{C}^{f}_{s},\,\bold{C}_{s}\in\mathbb{C}^m
\smallsetminus\{0\}$ associated to $f$ and $\hat{g}=g_{\bold{a},{\pmb{\theta}}}$, respectively, satisfy $$\bold{C}^{f}_{s}=e^{-i\varphi}\bold{C}_{s} \;\;\text{for any}\;\; 2\le s\le 3m.$$
Moreover, Proposition \[hola\](iii) implies that the $s$-th Hopf differential of $f$ is given by $$\Phi^f_s=
\begin{cases}
\,d^f_s\Phi^{(s+1)/3}&\text{ \,if\, }s\equiv 2\;\mathrm{mod}\; 3, \\[1mm]
\,0&\text{ \,\,otherwise, }\
\end{cases}$$ where $d^f_s=2^{-4(s+1)/3}(\overline{\bold{C}}^f_{s+1},\bold{C}^f_{s+1}).$ Equivalently, we have $$\label{kanahopf}
\Phi^f_s=
\begin{cases}
\,2^{-4(s+1)/3}e^{2i\varphi}(\overline{\bold{C}}_{s+1},\bold{C}_{s+1})\Phi^{(s+1)/
3}&\text{ \,if\, }s\equiv 2\;\mathrm{mod}\; 3, \\[1mm]
\,0&\text{ \,\,otherwise.}
\end{cases}$$ Thus the Hopf differentials of $f$ and $\hat{g}$ satisfy $$\Phi^f_s= e^{2i\varphi}\hat{\Phi}_s \;\; \text{for any} \;\;1\le s\le 3m-1.$$ According to [@V Theorem 5.2], the associated family of the surface $\hat{g}$ is $g_{\bold{a},{\pmb{\varphi}}}$ and this completes the proof.
Exceptional surfaces and the Ricci-like condition
=================================================
In this section, we study exceptional surfaces that satisfy the Ricci-like condition $(\ast).$ To prove our main results, we need the following proposition.
\[whhat\] Let $f\colon M\to \mathbb{S}^n$ be a nonflat $r$-exceptional surface which satisfies the Ricci-like condition $(\ast).$ Then the following hold:
For any $1\leq s\leq r+1,$ we have: $$\label{aexc}
\left\Vert \alpha_{s+1}\right\Vert^2=
\begin{cases}
\,b_s(1-K)^{s/3} &\text{ \,if\, }s\equiv 0\;\mathrm{mod}\; 3, \\[1mm]
\,b_s(1-K)^{(s+2)/3} &\text{ \,if\, }s\equiv 1\;\mathrm{mod}\; 3, \\[1mm]
\,b_s(1-K)^{(s+1)/3} &\text{ \,if\, }s\equiv 2\;\mathrm{mod}\; 3,
\end{cases}$$ where $\left\{ b_s\right\}, \left\{ \rho_s\right\}$ are sequences of positive numbers such that $b_1=2, b_{s+1}=\rho_s^2b_s, \rho _s\leq1$ and $\rho _s=1$ if $s\equiv0,1\;\mathrm{mod}\; 3.$
Moreover for any $1\leq s\leq r,$ the following hold: $$\label{first}
\Phi_s=0 \text{\,\, if\,\,\,}s\equiv0,1\;\mathrm{mod}\; 3,\\$$ $$\label{olalazoun}
K_s^*=
\begin{cases}
\,K &\text{ \,if\, }s\equiv 0\;\mathrm{mod}\; 3, \\[1mm]
\,-K&\text{ \,if\, }s\equiv 1\;\mathrm{mod}\; 3, \\[1mm]
\,0 &\text{ \,if\, }s\equiv 2\;\mathrm{mod}\; 3,
\end{cases}$$ and $$\label{last}
K_s^\perp=
\begin{cases}
\,c_s(1-K)^{s/3} &\text{ \,if\, }s\equiv 0\;\mathrm{mod}\; 3, \\[1mm]
\,c_s(1-K)^{(s+2)/3} &\text{ \,if\, }s\equiv 1\;\mathrm{mod}\; 3, \\[1mm]
\,c_s(1-K)^{(s+1)/3} &\text{ \,if\, }s\equiv 2\;\mathrm{mod}\; 3,
\end{cases}$$ where $c_s=2^{-s}\rho _sb_s.$
We set $\rho _s=2^sK_s^{\perp}/\left\Vert \alpha_{s+1}\right\Vert^2.$ Since $f$ is $r$-exceptional, the function $\rho _s$ is constant for any $1\leq s\leq r.$ We proceed by induction on $r.$
Assume that $f$ is $1$-exceptional. The Gauss equation implies $\left\Vert
\alpha_2\right\Vert^2=2(1-K).$ Then from Proposition \[3i\](i) for $s=1$ and the Ricci-like condition $(\ast)$, we find $K_1^{\ast}=-K.$ Moreover, we have $K_1^{\perp}=\rho_1(1-K).$ We claim that $\rho_1=1.$ Assume to the contrary that $\rho_1\neq1.$ Then $\Phi_1\neq0$ and Proposition \[3i\](ii) combined with the condition $(\ast)$ yield $K^*_1=K$, which is a contradiction. Hence $\rho_1=1$ and Proposition \[5\] yields (\[aexc\]) for $s=2$ with $b_2=2.$ This settles the case $r=1.$
Suppose now that (\[first\])-(\[last\]) hold if $f$ is $r$-exceptional. We shall prove that (\[first\])-(\[last\]) also hold assuming that $f$ is $(r+1)$-exceptional. By Theorem \[ena\], the Hopf differential $\Phi _{r+1}$ is holomorphic, hence either it is identically zero or its zeros are isolated.
At first we assume that $r\equiv0\;\mathrm{mod}\;3.$ From the inductive assumption, we have $$\left\Vert \alpha_{r+2}\right\Vert^2=b_{r+1}(1-K)^{(r+3)/3}.$$ We claim that $\rho_{r+1}=1.$ Arguing indirectly, we assume that $\Phi _{r+1}\neq 0.$ Then Proposition \[3i\](iv) yields $K_{r+1}^{\ast}=0.$ Taking into account the condition $(\ast)$, Proposition \[3i\](ii) implies that $M$ is flat and this is a contradiction. Thus $\Phi _{r+1}$ is identically zero, or equivalently $\rho _{r+1}=1.$ From Proposition \[3i\](iii) and the condition $(\ast)$, we obtain $K_{r+1}^
{\ast}=-K.$ Furthermore, we have $$K_{r+1}^{\perp}=2^{-(r+1)}\rho _{r+1}\left\Vert \alpha_{r+2}\right\Vert^2,$$ or equivalently $$K_{r+1}^{\perp}=c_{r+1}(1-K)^{(r+3)/3},$$ with $c_{r+1}=2^{-(r+1)}b_{r+1}.$ Then using Proposition \[5\], we obtain $$\left\Vert \alpha_{r+3}\right\Vert^2=b_{r+2}(1-K)^{(r+3)/3},$$ with $b_{r+2}=b_{r+1}.$
Assume now that $r\equiv1\;\mathrm{mod}\;3.$ From the inductive assumption, we have $$\left\Vert \alpha_{r+2}\right\Vert ^{2}=b_{r+1}(1-K)^{(r+2)/3}.$$ If $\Phi _{r+1}\neq 0,$ then Proposition \[3i\](iv) yields $K_{r+1}^{\ast}=0.$ If $\Phi _{r+1}$ is identically zero, or equivalently $\rho _{r+1}=1$, then Proposition \[3i\](iii) and the condition $(\ast)$ imply that $K_{r+1}^{\ast}=0.$ Furthermore, we have $$K_{r+1}^{\perp}=2^{-(r+1)}\rho _{r+1}\left\Vert \alpha_{r+2}\right\Vert^2,$$ or equivalently $$K_{r+1}^{\perp}=c_{r+1}(1-K)^{(r+2)/3},$$ with $c_{r+1}=2^{-(r+1)}\rho_{r+1}b_{r+1}.$ From Proposition \[5\], we obtain $$\left\Vert \alpha_{r+3}\right\Vert^2=b_{r+2}(1-K)^{(r+2)/3},$$ with $b_{r+2}=\rho^2_{r+1}b_{r+1}.$
Finally, we suppose that $r\equiv2\;\mathrm{mod}\;3.$ From the inductive assumption, we have $$\left\Vert \alpha_{r+2}\right\Vert^2=b_{r+1}(1-K)^{(r+1)/3}.$$ We claim that $\rho_{r+1}=1.$ Assume to the contrary that $\rho_{r+1}\neq1$ or equivalently $\Phi _{r+1}\neq 0.$ Then Proposition \[3i\](iv) yields $K_{r+1}^{\ast}
=0.$ Taking into account the Ricci-like condition $(\ast),$ Proposition \[3i\](ii) implies that $M$ is flat, which is a contradiction. Hence $\Phi _{r+1}$ is identically zero, or equivalently $\rho _{r+1}=1.$ From Proposition \[3i\](iii) and the condition $(\ast)$, we obtain $K_{r+1}^{\ast}=K.$ Furthermore, we have $$K_{r+1}^{\perp}=2^{-(r+1)}\rho _{r+1}\left\Vert \alpha_{r+2}\right\Vert^2,$$ or equivalently $$K_{r+1}^{\perp}=c_{r+1}(1-K)^{(r+1)/3},$$ with $c_{r+1}=2^{-(r+1)}\rho_{r+1}b_{r+1}.$ Using Proposition \[5\], it follows that $$\left\Vert \alpha_{r+3}\right\Vert^2=b_{r+2}(1-K)^{(r+4)/3}$$ with $b_{r+2}=b_{r+1}$ and this completes the proof.
We are ready to prove the main result of this section.
\[kanenadenexwlabel\] Let $f\colon M\to\mathbb{S}^n$ be a nonflat simply connected exceptional surface with substantial odd codimension. If $f$ satisfies the Ricci-like condition $(\ast)$ away from the isolated points with Gaussian curvature $K=1$, then $n=6m-1$ and there exists $\bold{a}=(a_1,\dots,a_m)\in\mathbb{S}^{m-1}\subset\mathbb{R}^m$ with $\Pi_{j=1}^{m}a_j\neq0$ and ${\pmb{\theta}}=(\theta_1,\dots,\theta_m)\in\mathbb{S}^1\times\cdots\times\mathbb{S}^1,$ where $0\leq \theta _1<\cdots<\theta_m<\pi$ such that $f=g_{\bold{a},\pmb{\theta}}.$
We claim that $n\equiv 5\;\mathrm{mod}\;6.$ Arguing indirectly, we suppose at first that $n=6l+1.$ Since $f$ is $(3l-1)$-exceptional, (\[aexc\]) yields $\left\Vert\alpha_{3l+1}
\right\Vert ^{2}=b_{3l}(1-K)^{l}.$ Moreover, viewing $f$ as a minimal surface in $\mathbb{S}^{6l+2},$ we obviously have $K_{3l}^{\perp}=K_{3l}^{\ast}=0.$ Then from Proposition \[3i\](ii), we obtain $$\Delta \log\left\Vert \alpha_{3l+1}\right\Vert ^2=2(3l+1)K.$$ Thus $K=0,$ which is a contradiction.
Suppose that $n=6l+3.$ Since $f$ is $3l$-exceptional, (\[aexc\]) yields $$\left\Vert\alpha_{3l+2}\right\Vert^2=b_{3l+1}(1-K)^{l+1}.$$ Moreover, viewing $f$ as a minimal surface in $\mathbb{S}^{6l+4},$ we obviously have $K_{3l+1}^{\perp}=K_{3l+1}^{\ast}=0.$ Then from Proposition \[3i\](ii), it follows that $$\Delta \log
\left\Vert \alpha_{3l+2}\right\Vert^2=2(3l+2)K.$$ Thus $K=0,$ which is a contradiction.
Hence $n\equiv 5\;\mathrm{mod}\;6$ and we may set $n=6m-1.$ According to (\[first\]), we have that $\Phi _{r}=0$ if $r\equiv0,1\;\mathrm{mod}\;3.$ Let $$r_{0}=\min \left\{ r:2\leq r\leq 3m-1\;\; \text{with}\;\; \Phi _{r}\neq0\right\}.$$ Obviously $r_0\equiv2\;\mathrm{mod}\;3.$ Let $z$ be a local complex coordinate such that the induced metric is given by $ds^2=F|dz|^2.$ From the definition of Hopf differentials we know that $\Phi _r=f_rdz^{2r+2},$ where $f_r=\langle\alpha_
{r+1}^{(r+1,0)}, \alpha_{r+1}^{(r+1,0)}\rangle.$
For any $r\equiv2\;\mathrm{mod}\;3$ and $r\geq r_0$ such that $\Phi_r\neq0,$ we may write $\Phi _r=|f_r|e^{i\sigma _r}dz^{2r+2}.$ Using (\[aexc\]), (\[last\]) and (\[what\]), we obtain $$\Phi _r=2^{-(r+2)}b_rF^{r+1}e^{i\sigma _r}(1-K)^{(r+1)/3}
\left(1-\rho _r^2\right)^{1/2}dz^{2r+2}.$$
We pick a branch $h$ of $f_{r_0}^{3/(r_0+1)}$ and define the form $\Phi =c_0hdz^6,$ where $c_0$ is given by $$c_0=
\begin{cases}
\,\left(\frac{2}{b_{r_0}\left(1-\rho^2_{r_0}\right)^{1/2}}\right)^{3/(r_0+1)} &\text{if $r_0<3m-1,$}\\[3mm]
\,\left(\frac{2}{b_{r_0}}\right)^{3/(r_0+1)} &\text{if $r_0=3m-1.$}
\end{cases}$$ It is obvious that $\Phi $ is well defined and holomorphic. It follows that $$\Phi _{r}=\frac{1}{2}b_r\left(1-\rho _r^2\right)^{1/2}e^{i\big(
\sigma _r-\frac{r+1}{r_0+1}\sigma _{r_0}\big)}\Phi ^{(r+1)/3}.$$ From the holomorphicity of $\Phi _r$ and $\Phi,$ we deduce that $\sigma
_r-\frac{r+1}{r_0+1}\sigma _{r_0}$ is constant. Moreover, we easily see that $$|c_0h|^2=\left(\frac{F}{2}\right)^6\left(1-K\right)^2.$$
Using , we infer that there exists a pseudoholomorphic curve $g\colon M\to\mathbb{S}^5$ whose second Hopf differential is $\Phi.$
We consider a surface $\hat{g}=g_{\bold{a},\pmb{\theta}}$ which according to Proposition \[hola\] is exceptional for any $\bold{a}\in\mathbb{S}^{m-1}$ and $\pmb{\theta}.$ Setting $\hat{\rho}_s=2^s\hat{K}_s^\perp/\left\Vert\hat{\alpha}_{s+1}\right\Vert^2,$ it follows from Proposition \[hola\] that $$\hat{\rho}_s=
\begin{cases}
\,\left(1-\frac{\left|\left(\bold{C}_{s+1},\overline{\bold{C}}_{s+1}\right)\right|^2}
{\left\Vert \bold{C}_{s+1}\right\Vert^4}\right)^{1/2}
&\text{ \ if \ }s\equiv 2\;\mathrm{mod}\; 3,\\[4mm]
\,\,\,1&\text{ \ otherwise. \ }
\end{cases}$$
We now claim that we can choose $\bold{a}\in\mathbb{S}^{m-1}$ and $\pmb{\theta}$ such that $$b_r=\hat{b}_r,\,\, c_r=\hat{c}_r \;\;\text{and}\;\; \rho_r=\hat{\rho}_r \;\;\text{for every}\;\;
1\leq r\leq 6m-3,$$ where $\hat{b}_r, \hat{c}_r, b_r, c_r $ and $\rho_r$ are the sequences in Proposition \[hola\] and Proposition \[whhat\], respectively. Obviously, Proposition \[whhat\] gives that $$b_r=\hat{b}_r=2 \;\;\text{for}\;\; r=1,2,\;\; c_1=\hat{c}_1=1 \;\;\text{and}\;\; \rho_1=
\hat{\rho}_1=1.$$ We choose $\bold{a}$ and $\pmb{\theta}$ such that the unitary transformation $T_{2\pmb{\theta}}$ satisfies $$\left|\left(T_{2\pmb{\theta}}\bold{a},\bold{a}\right)\right|^2=1-\rho^2_2.$$
According to (\[celeosksananea\]), the above is equivalent to $\rho_2=\hat{\rho}_2.$ Then using Proposition \[whhat\], we obtain that $$b_{r+1}=\hat{b}_{r+1},\, c_r=\hat{c}_r \;\;\text{and}\;\; \rho_r=\hat{\rho}_r
\;\;\text{for}\;\; 1\le r\le4.$$ Similarly, we may choose $\bold{a}$ and $\pmb{\theta}$ such that $$\frac{\left|\left(T_{2\pmb{\theta}}\bold{C}_4,\bold{C}_4\right)\right|^2}{\|\bold{C}_4
\|^4}=1-\rho_5^2,$$ or equivalently $\rho_5=\hat{\rho}_5,$ according to (\[celeosksananea\]). Repeating this argument, and choosing $\bold{a}$ and $\pmb{\theta}$ such that $\rho_r=\hat{\rho}_r$ for any $r\equiv 2\;\mathrm{mod}\; 3$, the claim follows inductively.
Thus, Proposition \[whhat\] implies that the $a$-invariants of the minimal surface $f$ coincide with those of $\hat{g}=g_{\bold{a}, \pmb{\theta}}$ for appropriate $\bold{a}$ and $\pmb{\theta}.$
It follows from [@V Theorem 5.2] that $f$ is a member of the associate family of $\hat{g}$, which in view of Lemma \[asociatefamilygtheta\] completes the proof.
For the proof of Theorem \[artiadimension\] below, we recall the following well known lemma.
\[QPgrad\] Let $M$ be a two-dimensional Riemannian manifold and let $f\colon M\to
\mathbb{R}$ be a smooth function such that $\Delta f=P(f)$ and $\left\Vert\nabla
f\right\Vert^{2}=Q(f)$ for smooth functions $P,Q\colon\mathbb{R}\to \mathbb{R},$ where $\nabla f\ $denotes the gradient of $f.$ Then on $\left\{ p\in
M:\nabla f(p)\neq 0\right\},$ the Gaussian curvature $K$ satisfies $$2KQ+(2P-Q^{\prime})(P-Q^{\prime })+Q(2P^{\prime }-Q^{\prime \prime })=0.$$
For minimal surfaces in substantial even codimension, we prove the following result.
\[artiadimension\] (i) Substantial exceptional surfaces in $\mathbb{S}^{6m}$ cannot satisfy the Ricci-like condition $(\ast).$
\(ii) Substantial $\left[\frac{n-1}{2}\right]$-exceptional surfaces in an even dimensional sphere $\mathbb{S}^n$ cannot satisfy the Ricci-like condition $(\ast).$
\(i) Assume to the contrary that $f\colon M \to \mathbb{S}^{6m}$ is a substantial exceptional surface that satisfies the condition $(\ast).$ Since $f$ is $(3m-2)$-exceptional, Proposition \[whhat\] yields $\left\Vert\alpha_{3m}\right\Vert ^2=b_{3m-1}(1-K)^m$ and $K_{3m-2}^{\ast}
=-K.$ Moreover, combining Proposition \[5\] with Proposition \[whhat\], we find that $$K_{3m-1}^{\ast}=\frac{2^{3m-1}K_{3m-1}^{\perp}}{b_{3m-1}(1-K)^m}.$$
By Theorem \[ena\], $\Phi _{3m-1}$ is holomorphic. Hence either it is identically zero or its zeros are isolated. If $\Phi _{3m-1}$ is identically zero, then $
f$ is $(3m-1)$-exceptional, and (\[olalazoun\]) yields $K_{3m-1}^{\ast}=0.$ Then the above equation implies $K_{3m-1}^{\perp}=0.$ This means that $f$ lies in a totally geodesic $\mathbb{S}^{6m-1}$ of $\mathbb{S}^{6m}$ (cf. [@O p. 96]), which is a contradiction. Suppose now that $\Phi _{3m-1}\neq 0.$ By virtue of Proposition \[3i\](ii), we have $$\begin{aligned}
\Delta \log \left(\left\Vert \alpha_{3m}\right\Vert^2
+2^{3m-1}K_{3m-1}^{\perp}\right) &=&2\big(3mK-K_{3m-1}^{\ast}\big), \\
\Delta \log \left(\left\Vert \alpha_{3m}\right\Vert
^2-2^{3m-1}K_{3m-1}^{\perp}\right) &=&2\big(3mK+K_{3m-1}^{\ast}\big).\end{aligned}$$
Using the condition $(\ast)$ and setting $\rho=2^{3m-1}K_{3m-1}^{\perp}/\left\Vert \alpha_{3m}\right\Vert ^2,$ the above equations are equivalent to $$\label{paliapothnarxh}
\Delta \log \left(1+\rho \right) =-2K_{3m-1}^{\ast} \;\; \text{and} \;\; \Delta
\log \left(1-\rho \right) =2K_{3m-1}^{\ast}.$$ Since $\rho = 2^{3m-1}K_{3m-1}^{\perp}/\left\Vert \alpha_{3m}\right\Vert^2,$ we obtain $K_{3m-1}^{\ast}=\rho.$
Then equations (\[paliapothnarxh\]) are written equivalently $$\Delta \rho =-2\rho (1+\rho^2)
\;\; \text{and} \;\; \left\Vert\nabla \rho \right\Vert^2=2\rho^2(1-\rho^2).$$ If the function $\rho$ is constant, then $\rho =0$ and consequently $K_{3m-1}^{\perp}=0,$ which contradicts the fact that $f$ is substantial. If $\rho$ is not constant, then Lemma \[QPgrad\] yields $K=-8,$ which contradicts the Ricci-like condition $(\ast).$
\(ii) Assume that $f\colon M\to \mathbb{S}^n$ is a substantial $[(n-1)/2]$-exceptional surface which satisfies the condition $(\ast),$ where $n$ is even. It suffices to consider the case $n=6m+2$ and $n=6m+4,$ since the case $n=6m$ was settled in (i).
At first let us suppose that $n=6m+2.$ Since $f$ is $3m$-exceptional, (\[first\]) and (\[olalazoun\]) yield $\Phi _{3m}=0$ and $K_{3m}^{\ast}=K.$ By virtue of Proposition \[5\], we obtain $$K_{3m}^{\ast}=\frac{K_{3m}^{\perp}\left\Vert \alpha_{3m}\right\Vert^2}{
2^{3m-2}\left(K_{3m-1}^{\perp}\right)^2}.$$ Then, using (\[aexc\]) and (\[last\]), we have that $K_{3m}^{\ast}=1$, which is a contradiction.
We suppose now that $n=6m+4.$ Since $f$ is $(3m+1)$-exceptional, (\[first\]) and (\[olalazoun\]) yield $\Phi _{3m+1}=0$ and $K_{3m+1}^{\ast}=-K.$ From Proposition \[5\] it follows that $$K_{3m+1}^{\ast}=\frac{K_{3m+1}^{\perp}\left\Vert \alpha_{3m+1}\right\Vert^2}
{2^{3m-1}\left(K_{3m}^{\perp}\right)^2}.$$ Using (\[aexc\]) and (\[last\]), we find that $K_{3m+1}^{\ast}=1-K$, which is a contradiction, and this completes the proof.
Global results
==============
In this section, we prove results for compact minimal surfaces that satisfy the condition $(\ast)$ and are not homeomorphic to the torus. We recall from Lemma \[avtf\], that such surfaces cannot be homeomorphic to the sphere $\mathbb S^2.$
\[pamegiallaksana\] Let $f\colon M\to \mathbb{S}^n$ be a compact substantial minimal surface with genus $g\geq 2$ which satisfies the Ricci-like condition $(\ast)$ away from isolated points where the Gaussian curvature satisfies $K=1.$ If the eccentricity $\varepsilon _r$ of the higher curvature ellipses of order $r\equiv 0\;\mathrm{mod}\; 3$ for any $1\leq r\leq s$ satisfies the condition $$\int_{M}\frac{\varepsilon_r}{\left(1-K\right)^{\gamma}}dA<\infty$$ for some constant $\gamma \geq 4/3,$ then $f$ is $s$-exceptional.
According to Lemma \[avtf\], the function $1-K$ is of absolute value type with nonempty zero set $M_0=\left\{ p_1,\dots,p_m\right\} $ and corresponding order $\mathrm{ord}_{p_j}(1-K)=2k_j.$ For each point $p_j, j=1,\dots,m,$ we choose a local complex coordinate $z$ such that $p_j$ corresponds to $z=0$ and the induced metric is written as $ds^2=F|dz|^2.$ Around $p_j,$ we have that $$\label{den}
1-K=|z|^{2k_j}u_0,$$ where $u_0$ is a smooth positive function.
We shall prove that $f$ is $s$-exceptional by induction. At first we show that $f$ is $1$-exceptional. In fact, we can prove that $f$ is $1$-isotropic. We know that the first Hopf differential $\Phi _1=f_1dz^4$ is holomorphic. Hence either $\Phi _1$ is identically zero, or its zeros are isolated. Assume now that $\Phi _1$ is not identically zero. Obviously, $\Phi _1$ vanishes at each $p_j.$ Thus we may write $f_1=z^{l_1(p_j)}\psi _1$ around $p_j,$ where $l_1(p_j)$ is the order of $\Phi_1$ at $p_j,$ and $\psi _1$ is a nonzero holomorphic function. Bearing in mind (\[what\]), we obtain $$\frac{1}{4}\left\Vert \alpha_2\right\Vert^4-(K_1^{\perp})^2=2^4F^{-4}|\psi _1|^2|z|^{2l_{1}(p_j)}$$ around $p_j.$ In view of (\[den\]) and the fact that $\left\Vert
\alpha_2\right\Vert^2=2(1-K),$ we find that the function $u_1\colon M
\smallsetminus M_0\to \mathbb{R}$ defined by $$u_1=\frac{\left((1-K)^2-(K_1^{\perp})^2\right)^3}{(1-K)^4},$$ around $p_j,$ is written as $$\label{tora}
u_1=2^{12}F^{-12}u_0^{-4}|\psi _1|^6|z|^{6l_{1}(p_j)-8k_j}.$$
Since $
u_1\leq (1-K)^2,$ from (\[tora\]) we deduce that $l_1(p_j)\geq 2k_j$ and we can extend $u_1$ to a smooth function on $M.$ It follows from Proposition \[3i\](ii) and the condition $(\ast)$ that $\log u_1$ is harmonic away from the isolated zeros of $u_1$. By continuity, the function $u_1$ is subharmonic everywhere on $M.$ Using the maximum principle, we deduce that $u_1$ is a positive constant. This contradicts the fact that $K=1$ on $M_0.$
Suppose now that $f$ is $(r-1)$-exceptional for $r\ge2.$ We note that $M$ cannot be flat due to our assumption on the genus. We shall prove that $f$ is also $r$ -exceptional. From [@V08 Proposition 4], we know that $\Phi _r=f_rdz^{2r+2}$ is globally defined and holomorphic. Hence either $\Phi _r=0$ or its zeros are isolated. In the former case, $f$ is $r$-exceptional.
Assume now that $\Phi _r$ is not identically zero. Obviously, $\Phi _r$ vanishes at $p_j.$ Hence we may write $f_r=z^{l_r(p_j)}\psi _r$ around $p_j,$ where $l_r(p_j)$ is the order of $\Phi _r$ at $p_j,$ and $\psi _r$ is a nonzero holomorphic function. Bearing in mind (\[what\]), we obtain $$\label{aque}
\left\Vert \alpha_{r+1}\right\Vert^4-4^r(K_r^{\perp})^2=4^{r+2}F^{-2(r+1)}|\psi _r|^2|z|^{2l_r(p_j)}$$ around $p_j.$ In view of (\[den\]), we find that $$\label{utora}
u_r=4^{3(r+2)}F^{-6(r+1)}u_0^{-2(r+1)}|\psi
_r|^6|z|^{6l_r(p_j)-4k_j(r+1)},$$ where $u_{r}\colon M\smallsetminus M_0\to \mathbb{R}$ is the smooth function (see Proposition \[neoksanaafththfora\]) given by $$u_r=\frac{\left(\left\Vert \alpha_{r+1}\right\Vert^4-4^r(K_r^{\perp})^2\right)^3}{(1-K)^{2(r+1)}}.$$
We claim that $r\equiv 2\;\mathrm{mod}\; 3.$ Arguing indirectly, we at first assume that $r\equiv 0\;\mathrm{mod}\; 3.$ Since $\varepsilon_r^2/(2-
\varepsilon_r^2)\leq \varepsilon_r,$ our assumption implies $$\int_{M}\frac{\varepsilon_r^2}{(2-\varepsilon_r^2)\left(1-K\right)^{\gamma}}
dA<\infty,$$ or equivalently, bearing in mind (\[elipsi\]) and (\[si\]), $$\int_{M}\frac{\left(\left\Vert \alpha_{r+1}\right\Vert^4-4^r(K_r^{\perp})^2\right)^{1/2}}{\left(1-K\right)^{\gamma}\left\Vert \alpha_{r+1}\right\Vert^2}dA<\infty.$$ Taking into account (\[aexc\]), the above becomes $$\int_{M}\frac{\left(\left\Vert \alpha_{r+1}\right\Vert^4-4^r(K_r^{\perp})^2\right)^{1/2}}{\left(1-K\right)^{\gamma +\frac{r}{3}}}dA<\infty.$$ We consider the subset $$U_{\delta}(p_j)=\left\{p\in M: |z(p)|<\delta \right\}, j=1,\dots,m.$$ Using (\[den\]) and (\[aque\]), the above inequality implies that $$\int_{U_{\delta _0}(p_j)\smallsetminus U_{\delta}(p_j)}|z|^{l_r(p_j)-2k_j(\gamma+\frac{r}{3})}dA<c$$ for any $\delta <\delta _0,$ where $c$ is a positive constant and $\delta_0$ is small enough. We set $z=\rho e^{i\theta}.$ Since $dA=F\rho d\rho\wedge d\theta,$ we deduce that $$\int_0^{\delta _0}\rho^{l_r(p_j)-2k_j(\gamma+\frac{r}{3})+1}d\rho <\infty.$$ This implies that $$l_r(p_j)>2k_j(\gamma+\frac{r}{3})-2.$$ Summing up, we obtain $$N(\Phi_r)+2m>2(\gamma+\frac{r}{3})\sum_{j=1}^{m}k_j.$$ Using Lemma \[forglobal\](ii) and in Lemma \[avtf\], it follows that $$\chi (M)(3\gamma-1)+m>0.$$ On the other hand, implies that $m\leq -3\chi (M)$, which contradicts the above and the hypothesis that $\chi (M)<0.$
Now assume that $r\equiv 1\;\mathrm{mod}\;3.$ Bearing in mind (\[aexc\]), we deduce that $u_r\leq
b_r^6(1-K)^2.$ Using (\[den\]) and (\[utora\]), we obtain $3l_r(p_j)
\ge2k_j(r+2).$ Then from Lemma \[avtf\], we conclude that $$N(\Phi_r)\ge-2(r+2)\chi (M).$$ Due to Lemma \[forglobal\](ii), the above contradicts our hypothesis on the genus.
Therefore, we conclude that $r\equiv2\;\mathrm{mod}\; 3.$ By virtue of (\[aexc\]), we obtain $u_r\leq b_r^6.$ Then (\[utora\]) implies $3l_r(p_j)\geq 2k_j(r+1),$ and we can extend $u_r$ to a smooth function on $M.$ It follows from Proposition \[3i\](ii) and the Ricci-like condition $(\ast)$ that $\log u_r$ is harmonic away from the zeros which are isolated, and consequently by continuity $u_r$ is subharmonic everywhere on $M.$ By the maximum principle, we deduce that the function $u_r$ is a positive constant. This shows that the $r$-th curvature ellipse has constant eccentricity, i.e., the surface $f$ is $r$-exceptional. This completes the proof.
For compact minimal submanifolds in spheres with low codimension, we prove the following result.
Let $f\colon M\to\mathbb{S}^n$ be a substantial minimal surface with $4\le n\le7.$ If $M$ is compact and not homeomorphic to the torus, then it cannot be locally isometric to a pseudoholomorphic curve in $\mathbb{S}^5$, unless $n=5.$
From Lemma \[avtf\], we have that the genus of $M$ satisfies $g\geq2.$ We assume that $n\neq5.$ For $n=4$ and $n=6,$ the result follows immediately from Theorem \[pamegiallaksana\] and Theorem \[artiadimension\](ii). In the case where $n=7$, Theorem \[pamegiallaksana\] implies that the surface is exceptional and the result follows from Theorem \[kanenadenexwlabel\].
The assumption in Theorem \[pamegiallaksana\] on the eccentricity of curvature ellipses of order $r\equiv 0\;\mathrm{mod}\; 3$ could be replaced by the condition $$\varepsilon_r\leq (1-K)^\beta$$ for positive constants $c$ and $\beta>1/3.$ Both conditions claim that the curvature ellipses of order $r\equiv 0\;\mathrm{mod}\; 3$ tend to be circles close to totally geodesic points. We don’t know whether Theorem \[pamegiallaksana\] holds without this assumption in any codimension.
The following global result is complementary to Theorem \[artiadimension\].
Let $f\colon M\to\mathbb{S}^{6m+4}, m\geq1,$ be a substantial exceptional surface. If $M$ is compact with genus $g\geq 2,$ then it cannot be locally isometric to a pseudoholomorphic curve in $\mathbb{S}^5.$
We assume to the contrary that the surface satisfies the Ricci-like condition ($\ast$). Since $f$ is $3m$-exceptional, from Proposition \[neoksanaafththfora\] we know that the Hopf differential $\Phi _{3m+1}=f_{3m+1}dz^{6m+4}$ is globally defined and holomorphic. Hence either $\Phi _{3m+1}=0$ or its zeros are isolated.
Theorem \[artiadimension\](ii) implies that the Hopf differential $\Phi _{3m+1}$ cannot vanish identically.
According to Lemma \[avtf\], the function $1-K$ is of absolute value type with nonempty zero set $M_0=\left\{ p_1,\dots,p_m\right\} $ and corresponding order $\mathrm{ord}_{p_j}(1-K)=2k_j.$ For each point $p_j, j=1,\dots, m,$ we choose a local complex coordinate $z$ such that $p_j$ corresponds to $z=0$ and the induced metric is written as $ds^2=F|dz|^2.$ Around $p_j,$ we have that $$\label{ksanaidio}
1-K=|z|^{2k_j}u_0,$$ where $u_0$ is a smooth positive function.
Obviously, $\Phi _{3m+1}$ vanishes at $p_{j}.$ Hence we may write $f_{3m+1}=
z^{l(p_{j})}\psi$ around $p_{j},$ where $l(p_{j})$ is the order of $\Phi _{3m+1}$at $p_{j},$ and $\psi$ is a nonzero holomorphic function. Bearing in mind (\[what\]),we obtain $$\label{aque2}
\left\Vert \alpha_{3m+2}\right\Vert^4-4^{3m+1}(K_{3m+1}^{\perp})^2=2^{6(m+1)}F^{-2(3m+2)}|\psi|^2|z|^{2l(p_j)}$$ around $p_j.$ In view of (\[ksanaidio\]), we find that $$\label{utoratora}
u=2^{18(m+1)}F^{-6(3m+2)}u_0^{-2(3m+2)}|\psi
|^6|z|^{6l(p_j)-4k_j(3m+2)},$$ where $u\colon M\smallsetminus M_0\to \mathbb{R}$ is the smooth function (see Proposition \[neoksanaafththfora\]) given by $$u=\frac{\left(\left\Vert \alpha_{3m+2}\right\Vert^4-4^{3m+1}(K_{3m+1}^{\perp})^2\right)^3}{(1-K)^{2(3m+2)}}.$$ Using (\[aexc\]), it follows that $u\leq
b_{3m+1}^6(1-K)^2.$ Then (\[ksanaidio\]) and (\[utoratora\]) imply that $l(p_j)\ge2k_j(m+1).$ By Lemma \[avtf\], we deduce that $$N(\Phi_{3m+1})\ge-6(m+1)\chi (M).$$ It follows from Lemma \[forglobal\](ii) that the above contradicts our hypothesis on the genus and the theorem is proved.
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M. Sakaki, *Rigidity of superconformal minimal surfaces lying fully in odd-dimensional unit spheres*, Math. Proc. Camb. Phil. Soc. **117** (1995), 251–257.
Th. Vlachos, *Minimal surfaces in a sphere and the Ricci condition*, Ann. Global Anal. Geom. **17** (1999), 129–150.
Th. Vlachos, *Congruence of minimal surfaces and higher fundamental forms*, Manusucripta Math. **110** (2003), 77–91.
Th. Vlachos, *Minimal surfaces, Hopf differentials and the Ricci condition*, Manusucripta Math. **126** (2008), 201–230.
Th. Vlachos, *Exceptional minimal surfaces in spheres*, Manusucripta Math. **150** (2016), 73–98.
[^1]: The first named author would like to acknowledge financial support by the General Secretariat for Research and Technology (GSRT) and the Hellenic Foundation for Research and Innovation (HFRI) Grant No: 133.
|
{
"pile_set_name": "ArXiv"
}
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---
abstract: 'This is an overview of nuclear star cluster observations, covering their structure, stellar populations, kinematics and possible connection to black holes at the centers of galaxies.'
date: '?? and in revised form ??'
---
Nucleation Fraction, Stellar Populations, and Kinematics
========================================================
Nuclear Star Clusters (NCs) are a very common structural component at the centers of galaxies. They are found in $77\%$ of late type galaxies ([@boeker02]), $55\%$ of spirals ([@carollo98]), and at least $66\%$ of (dwarf) ellipticals and S0s ([@cote06]). These studies find that the half-light radii of NCs are typically $5pc$, and due to these small sizes, their detection requires very high spatial resolution observations, making the HST crucial for systematic searches. NCs are on average 4mag brighter than GCs ([@boeker04]), i.e. they are more massive but have similar half-light radii. This makes NCs the densest stellar systems in the universe ([@walcher05; @misgeld11]). They lie at the high-mass end of the star cluster mass function, and are structurally very different from bulges.
Nuclear star clusters truly occupy the centers of galaxies, both photometrically but also kinematically ([@boeker02; @neumayer11], respectively), and it may be this special location at the bottom of the potential well of the galaxies, that causes the star formation history of NCs to be rather complex. Several studies have shown that NCs have multiple stellar populations both in late type ([@walcher06; @seth06; @rossa06]) and also early type galaxies ([@seth10]). The NC seems to be typically more metal-rich and younger than the surrounding galaxy ([@koleva11]), and the abundance ratios \[$\alpha$/Fe\] show that NCs are more metal enriched than globular clusters (GCs) ([@evstigneeva07]). This finding suggests that NCs cannot solely be the merger product of GCs, but need some gas for recurrent star formation. This finding is also supported by recent kinematical studies ([@hartmann11; @delorenzi12]), where cluster infall alone cannot explain the dynamical state of the NC.
Recent studies of the kinematics of NCs with integral-field spectroscopy show that the cluster as a whole rotates (Seth et al. 2008, 2010). Combined with the superb spatial resolution of adaptive-optics, the 2D velocity maps resolve stellar and gas kinematics down to a few parsecs on physical scales. In addition, due to the extremely high central stellar density in NCs, it becomes possible to pick-up kinematic signatures for intermediate-mass black hole inside NCs ([@seth10], Neumayer et al. in prep).
Connection to Black Holes
=========================
Unlike black holes, NCs provide a visible record of the accretion of stars and gas into the center of a galaxy, and studying their stellar populations, structure and kinematics allow us to disentangle their formation history. NCs do co-exist with black holes. The best studied example is the NC in our own Galaxy (see R. Schödel this edition). But there are also NCs with very tight upper limits on the mass of a central black hole (see [@neumayer12] for an overview), and it is not yet clear under what conditions galaxies make NCs and/or BHs. Figure \[fig:MBH\_MNC\] shows a compilation of mass measurements of BHs and NCs. For the lowest mass NCs BHs are very hard to detect (if present), while for the highest mass BHs, the surrounding NCs seem to have been destroyed already. The underlying physical connection remains unclear for now, but it may well be that BHs grow inside NCs, that may thus be the precursors of massive BHs at the nuclei of galaxies.
![ [The mass of the BH mass vs. the NC mass. The two full lines indicate a NC mass of $3 \times 10^6 M_{\odot}$ and a MBH / MNC mass ratio of 100. These lines separate NC dominated galaxy nuclei (lower left of both lines) from BH dominated galaxy nuclei (upper left of both lines) and a transition region (to the right of both lines) (Figure 5 from [@neumayer12]). ]{} \[fig:MBH\_MNC\]](Mbh_MNC.pdf){width="\textwidth"}
I acknowledge the support by the DFG cluster of excellence ‘Origin and Structure of the Universe’, and thank the IAU for financial support.
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Evstigneeva, E. A., Gregg, M. D., Drinkwater, M. J., & Hilker, M. 2007, *AJ*j, 133, 1722
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Neumayer, N., & Walcher, C. J. 2012, *Advances in Astronomy*, 2012,
, J., [van der Marel]{}, R. P., [B[ö]{}ker]{}, T., et al. 2006, *AJ*, 132, 1074
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|
{
"pile_set_name": "ArXiv"
}
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---
abstract: |
The LOPES experiment, a digital radio interferometer located at KIT (Karlsruhe Institute of Technology), obtained remarkable results for the detection of radio emission from extensive air showers at MHz frequencies.\
Features of the radio lateral distribution function (LDF) measured by LOPES are explored in this work for a precise reconstruction of two fundamental air shower parameters: the primary energy and the shower X$_{\mathrm{max}}$.\
The method presented here has been developed on (REAS3-)simulations, and is applied to LOPES measurements. Despite the high human-made noise at the LOPES site, it is possible to reconstruct both the energy and X$_{\mathrm{max}}$ for individual events. On the one hand, the energy resolution is promising and comparable to the one of the co-located KASCADE-Grande experiment. On the other hand, X$_{\mathrm{max}}$ values are reconstructed with the LOPES measurements with a resolution of 90g/cm$^{2}$. A precision on X$_{\mathrm{max}}$ better than 30g/cm$^{2}$ is predicted and achievable in a region with a lower human-made noise level.
address:
- 'Institut für Kernphysik, Karlsruhe Institute of Technology (KIT), Germany'
- 'Institut für Experimentelle Kernphysik, KIT - Karlsruher Institut für Technologie, Germany'
- 'Radboud University Nijmegen, Department of Astrophysics, The Netherlands'
- 'Dipartimento di Fisica Generale dell’ Università di Torino, Italy'
- 'Max-Planck-Institut für Radioastronomie Bonn, Germany'
- 'National Institute of Physics and Nuclear Engineering, Bucharest, Romania'
- 'Fachbereich Physik, Universität Siegen, Germany'
- 'Istituto di Fisica dello Spazio Interplanetario, INAF Torino, Italy'
- 'ASTRON, Dwingeloo, The Netherlands'
- 'Fachbereich Physik, Universität Wuppertal, Germany'
- 'Institut für Prozessdatenverarbeitung und Elektronik, KIT - Karlsruher Institut für Technologie, Germany'
- 'Soltan Institute for Nuclear Studies, Lodz, Poland'
- 'Department of Physics, University of Bucharest, Bucharest, Romania'
- 'now at: Univ Michoacana, Morelia, Mexico'
- 'now at: Univ S$\tilde{a}$o Paulo, Inst. de Física de São Carlos, Brasil'
- 'now at: Inst. Space Sciences, Bucharest, Romania '
author:
- 'W.D. Apel'
- 'J.C. Arteaga'
- 'L. Bähren'
- 'K. Bekk'
- 'M. Bertaina'
- 'P.L. Biermann'
- 'J. Blümer'
- 'H. Bozdog'
- 'I.M. Brancus'
- 'A. Chiavassa'
- 'K. Daumiller'
- 'V. de Souza'
- 'F. Di Pierro'
- 'P. Doll'
- 'R. Engel'
- 'H. Falcke'
- 'B. Fuchs'
- 'D. Fuhrmann'
- 'H. Gemmeke'
- 'C. Grupen'
- 'A. Haungs'
- 'D. Heck'
- 'J.R. Hörandel'
- 'A. Horneffer'
- 'D. Huber'
- 'T. Huege'
- 'P.G. Isar'
- 'K.H. Kampert'
- 'D. Kang'
- 'O. Krömer'
- 'J. Kuijpers'
- 'K. Link'
- 'P. [Ł]{}uczak'
- 'M. Ludwig'
- 'H.J. Mathes'
- 'M. Melissas'
- 'C. Morello'
- 'J. Oehlschläger'
- 'N. Palmieri'
- 'T. Pierog'
- 'J. Rautenberg'
- 'H. Rebel'
- 'M. Roth'
- 'C. Rühle'
- 'A. Saftoiu'
- 'H. Schieler'
- 'A. Schmid'
- 'F.G. Schröder'
- 'O. Sima'
- 'G. Toma'
- 'G.C. Trinchero'
- 'A. Weindl'
- 'J. Wochele'
- 'M. Wommer'
- 'J. Zabierowski'
- 'J.A. Zensus'
title: 'Reconstructing energy and X$_{\mathrm{max}}$ of cosmic ray air showers using the radio lateral distribution measured with LOPES'
---
radio detection,cosmic rays,air showers,LOPES ,Xmax ,primary energy 96.50.sd,95.55.Jz
Introduction
============
Knowing the type of cosmic rays which interact in the atmosphere with an energy larger than 10$^{14}$eV still remains a fundamental goal in cosmic ray physics. A precise knowledge of the mass composition for the complete energy spectrum will help in distinguishing between several models for cosmic ray origin and propagation.\
The detection of radio emission from cosmic ray air showers in the MHz regime as well as the understanding of its emission mechanisms made impressive progress in the recent years.\
The LOPES experiment [@lopes] is one of the pioneering in radio detection and still supplies us with remarkable results [@frank3]. One of its main advantage is the co-location with the particle detector KASCADE-Grande [@kg] at KIT, Germany.\
Recently, two independent methods have been separately investigated and tested on the LOPES data in order to extract X$_{\mathrm{max}}$, i.e. atmospheric depth of the shower maximum, from radio measurements and thus, indirectly, to achieve information on the type of primary cosmic rays. One method considers the shape of the radio shower front [@frank2; @frank3], the other method is presented in the following (slope method) and uses the slope of the radio lateral distribution function (LDF), i.e. the radio amplitudes at several distances from the shower axis [^1].\
The correlation between the slope of the radio LDF and the shower maximum depth has been predicted since years from simulations [@REAS3; @MGMRc]. This dependence can easily be referred to a geometrical effect: iron nuclei interact earlier in the atmosphere and develop faster compared to proton primaries; the radio source for an iron event is, thus, further away from an observer at ground, and, as a consequence, the lateral distribution function slope is flatter compared to proton events.\
Taking the analysis in [@REAS2c] as guideline, a simulation-based method (slope method) which uses the LDF slope to extract two fundamental shower parameters (primary energy and X$_{\mathrm{max}}$) is developed and applied to LOPES data. The main results are presented in the following.
LOPES events
============
The LOPES experiment [@lopes] is a digital interferometric radio antenna array placed at KIT, Germany. The events selected for the slope method analysis have been measured with the LOPES30 and LOPESpol setups [@timArena]. The first consisted of 30 calibrated dipole antennas, all oriented in the east-west direction, while the second used only 15 antennas aligned in the east-west direction. Due to a higher statistics of events expected in the east-west aligned antennas compared to the north-south direction [@geomagnEffect], only the analysis on east-west detected events is shown. All the antennas operate in the effective frequency range of 43-74MHz.\
LOPES profits from the reconstruction of air shower parameters, such as primary energy, shower core and incoming direction, from the particle detector array KASCADE-Grande [@kg].\
In the selected events the primary energy is around $\sim 10^{17}$eV and the zenith angle is less then 40deg. The core position of the shower is required to be at a distance of at most 90m from the center of the LOPES array, in order to avoid events with amplitudes in the tail of the LDF, thus affected by larger fluctuations. High signal-to-noise and high coherency for the radio signal in the antennas is required as well. Further qualitative cuts demand a good fit for the lateral distribution function, i.e. small $\chi^{2}$. Over 200 events are selected in this way.\
The radio lateral distribution for each individual event is fit with an exponential function, which is considered a good approximation for the radio lateral behaviour at the distances probed by LOPES [@Nunzia_thesis].\
Improvements are constantly made in modeling the radio emission from extensive air showers. The results shown in the following are based on REAS3 simulations (REAS3.11 and CoREAS are the most recent versions [@coreas; @coreas2]), which already showed a good agreement with the LOPES measured LDFs for almost all the events [@reas3].\
The slope method
================
![Normalized REAS3 lateral distribution function for the events with zenith angle smaller than 20deg., simulated once as proton (black), once as iron (gray). The points are fitted with an exponential function.[]{data-label="LDF"}](ldf2mod2.eps){width="100.00000%"}
The slope method has been developed with REAS3 simulations, and it is then applied to LOPES data. For each LOPES measured event, one proton-generated air shower and one iron-generated air shower is produced with CORSIKA [@corsika] (QGSJetII is used as hadronic interaction model). Afterwards, REAS3 [@reas3] generates the radio emission for the CORSIKA simulated air showers.\
For the purpose of this analysis, the simulations are created in order to reproduce a realistic case: on the one hand, information about the LOPES selected events, such as core position, primary energy, the number of muons (N$\mu$), and incoming direction reconstructed by KASCADE(-Grande) are used as input parameters for the CORSIKA showers [@reas3].\
On the other hand, shower-to-shower fluctuation are included, since not a typical shower, i.e. with a typical X$_{\mathrm{max}}$, is selected. In order to represent at best the recorded event, a further step is made in the pre-selection of the CORSIKA showers: the N$\mu$ measured by KASCADE(-Grande) is used as discriminator for this purpose. 200 CONEX showers for proton and 100 for iron are simulated with QGSJetII and UrQMD respectively for high and low energy interaction[@corsika; @reas3]. Among all, the CONEX shower which can best reproduce the measured N$\mu$ is chosen. In this way a specific shower similar to what has been detected by KASCADE is used.\
Moreover, the observer positions for the REAS3-simulated LDF represent the real arrangement of the LOPES antennas in the field with respect to the core of the shower.\
As for the LOPES events, an exponential fit is applied to each REAS3 lateral distribution functions.\
The inclination of the air shower, as well as the type of the primary particle, considerably affect the slope of the radio LDF. Again this is related to a geometrical effect: for larger zenith angle, the radio source is further away from the observer at ground, thus a flatter slope for the LDFs is expected compared to vertical air showers. Thereby, several zenith angle bins are separately considered, and only the plots for the first zenith bin (up to 20deg.) are shown here.\
The slope method aims to compare LDFs of events (in the same zenith bin) with different primary energies and arrival directions, thus normalizations for the radio amplitudes are required. The primary energy reconstructed by KASCADE(-Grande) is used for the energy normalization, while the corrections for the arrival direction involve the Lorentz force vector (precisely the east-west component of the vector Pew). This last implies the reasonable assumption that the predominant contribution to the radio emission in air showers has geomagnetic origin [@geomagnEffect; @Nunzia_thesis].\
The LDFs for 54 events (first zenith bin) are shown in Fig. \[LDF\] and the difference between irons and protons is clearly visible by eye.\
![The correlation between the reconstructed primary energy and the REAS3 radio pulse in the flat region (d$_{0}$) is fit with a linear function, with k$_{l}$ the free parameter. The radio amplitude, in the east-west direction, is normalized by the single component of the Lorentz vector. The RMS-spread is of $\sim$6$\%$.[]{data-label="REAS_energy_lin"}](energy_2sel_E_2bin_fitlinear_poster_col.eps){width="100.00000%"}
Primary energy correlation
--------------------------
A specific region (d$_{0}$), so called flat region, where all the LDFs profiles intersect and where the radio amplitude does not carry any information about the primary type is expected [@REAS2c]. This region is identified by looking at the RMS spread of the LDF fits at 12 distances from the shower core and picking the smallest RMS value. For the LOPES events d$_{\mathrm{0}}$ happens in the distance range between 70 and 90m from the shower axis [^2] depending on the zenith angle bin, and it is marked by the dashed line in Fig. \[LDF\] (60m).\
From previous analyses [@REAS2c], a direct correlation is expected between the radio amplitude in d$_{\mathrm{0}}$ - normalized for the arrival direction with Pew (as explained above)- and the energy, in particular with the fraction of the energy from the electromagnetic component of the air shower. This is due to the intrinsic nature of the radio emission.\
For the LOPES experiment, only the total primary energy reconstructed by KASCADE(-Grande) is available (Fig. \[REAS\_energy\_lin\]). A linear fit is used for this correlation. The spread around the fit is an indicator of the precision one acquires in determining the energy with this method. For this LOPES selection, an RMS spread of maximum 8$\%$ is found for the complete LOPES selection ($\sim$6$\%$ in Fig. \[REAS\_energy\_lin\] for zenith angle $<$20deg.) for the total primary energy correlation. Much higher precision - RMS spread $<4\%$ - is predicted for the electromagnetic energy (not shown here) [@Nunzia_thesis].
![Correlation between the true CORSIKA X$_{\mathrm{max}}$ and the radio simulated LDF slope, for proton (blue) and iron (red) simulated primaries. The RMS-spread is of $\sim$29 gcm$^{-2}$.[]{data-label="REAS_Xmax"}](Xmfit_1bin_1sel_qg_poster_col.eps){width="100.00000%"}
X$_{\mathrm{max}}$ correlation
------------------------------
The slope of the radio LDFs is sensitive to the depth of the shower maximum (X$_{\mathrm{max}}$). Independently of the function used to fit the LDF, the ratio ($\epsilon_{\mathrm{ratio}}$) of the radio amplitudes at two different distances is an indicator of the LDF slope. In this analysis the amplitudes at d$_{\mathrm{0}}$ and at d$_{\mathrm{0}}$+170m are considered.\
The function Eq. \[xm\_eq\] [@REAS2c] is used to fit the correlation $$X_{max} = a \left[\mathrm{ln} \left(b \epsilon_{\mathrm{ratio}}\right)\right]^{c}
\label{xm_eq}$$ between REAS3 $\epsilon_{\mathrm{ratio}}$ and X$_{\mathrm{max}}$ from CORSIKA simulations, separately for each zenith angle bin (Fig. \[REAS\_Xmax\]). The RMS spread is an indicator of the uncertainty on X$_{\mathrm{max}}$ reconstruction with the slope method; for the complete zenith angle range an uncertainty of 20-40g/cm$^{2}$ is predicted, with the larger values due to the larger zenith angles.\
The values for the three fitting parameters (*a, b* and *c*) will be used in the following to reconstruct X$_{\mathrm{max}}$ with the LOPES measurements.
![TOP: Linear correlation of the KASCADE-Grande reconstructed primary energy and the LOPES measured radio pulse in d$_{0}$, normalized for Pew. The RMS-spread if of $\sim23\%$ BOTTOM: Comparison of the RMS-spread for the linear energy fit computed at 4 distances from the shower axis, for zenith angle $<$20deg. In d$_{0}$ the RMS-spread has the smallest value. The LOPES measurements confirm the distinctive feature of d$_{0}$. []{data-label="LOPES_en"}](rmsLinearFit_energy2_poster.eps){width="100.00000%"}
LOPES measurements
==================
One of the main results already achieved with the LOPES experiment is the clear dependence of the recorded radio pulse (CC-beam amplitude) with the energy of the primary cosmic ray [@geomagnEffect]. In the following another approach for the primary energy identification by using the radio signal at a well specific distance from the shower axis is presented.\
The same distances d$_{\mathrm{0}}$, previously identified for each zenith angle bin with the REAS3 simulations, are used for the LOPES measurements as well. For each individual event, the LOPES $\epsilon_{\mathrm{d_{0}}, Pew}$ is the value of the fit of the LOPES lateral distribution function - corrected for the arrival direction - at the predicted distance d$_{\mathrm{0}}$. As for the REAS3 simulations, the linear correlation with the primary energy reconstructed by KASCADE(-Grande) is shown (Fig. \[LOPES\_en\] - top side). The uncertainty on the energy reconstruction is again identified by the RMS spread, and it is of about 20$\%$ averaging over the complete selection, i.e. zenith angle between 0 and 40deg. This value is comparable with the statistical uncertainty of KASCADE [@kg].\
As a cross-check of the real existence in the measurements of this preferable distance for the energy reconstruction, the energy correlation is investigated considering the radio amplitudes at three further distances from the shower axis: 0, 100, d$_{\mathrm{0}}$+170m (Fig. \[LOPES\_en\] - bottom side). The RMS spread to the linear fit results to be lowest when the radio amplitude is taken at d$_{\mathrm{0}}$, confirming the expectations.
![X$_{\mathrm{max}}$ reconstructed with the slope method for the LOPES measurements (black) and for REAS3 simulations (proton (blue) and iron (red)), for the complete zenith angle range- 0-40deg. []{data-label="LOPES_xm"}](X_col.eps){width="100.00000%"}
The X$_{\mathrm{max, LOPES}}$ are reconstructed for the complete LOPES selection using the Eq. \[xm\_eq\] and the *a, b* and *c* identified with the REAS3 simulations. In Figure \[LOPES\_xm\], X$_{\mathrm{max, LOPES}}$ of 600 $\pm$ 90g/cm$^{2}$, i.e. mean and standard deviation values. The LOPES reconstructed X$_{\mathrm{max}}$ values are almost compatible with the expectations from the cosmic ray nuclei. However, they are shifted to X$_{\mathrm{max}}$ smaller than the REAS3 iron-like predictions. This shift is surely influenced by the still existing systematic divergence between the slope of the simulated (REAS3) and the measured (LOPES) lateral distribution functions. The most recent REAS3.11 and CoREAS simulations have already shown a better agreements with the measurements, therefore improvements are expected by the upcoming application of the slope method to the newest simulations.
conclusion
==========
The slope method has been successfully applied to REAS3 simulations and measurements of LOPES data.\
This method reveals itself to be a powerful tool for both energy and mass composition investigations with the radio data.\
A well specific distance from the shower axis is confirmed even in the LOPES measurements to be the best place for the primary energy reconstruction. A precision of almost 20$\%$ in energy reconstruction is found for the LOPES data and it is comparable to the statistical uncertainty of the KASCADE experiment.\
X$_{\mathrm{max}}$ values obtained for the LOPES measured data are almost comparable with expectations. Systematic divergences are also discussed and improvements are foreseen with REAS3.11 and CoREAS simulations.\
An upper-limit precision of 90g/cm$^{2}$ is found with the LOPES measurements, being the highest precision on X$_{\mathrm{max}}$ sensitivity to date achievable with the radio data.\
Despite the high environmental noise which limits the performance of the LOPES experiment, important results are found.\
Higher precision in both energy and X$_{\mathrm{max}}$ reconstruction predicted by the slope method may be achieved with a negligible noise background. Especially in the case of AERA (Auger Engineering Radio Array [@aera]), located at the Pierre Auger Observatory, which could cross-check the reconstructed X$_{\mathrm{max}}$ with the experimental values obtained by the Fluorescence Detector (FD).
**Acknowledgments** LOPES and KASCADE-Grande have been supported by the German Federal Ministry of Education and Research. KASCADE-Grande is partly supported by the MIUR and INAF of Italy, the Polish Ministry of Science and Higher Education and by the Romanian Authority for Scientific Research UEFISCDI (PNII-IDEI grant 271/2011). This research has been supported by grant number VH-NG-413 of the Helmholtz Association.
H. Falcke et al. (LOPES collaboration) *Nature*, [**435**]{} (2005) 313 F. G. Schröder et al. (LOPES collaboration), “Cosmic Ray Measurements with LOPES: Status and Recent Results", *Proceedings of the ARENA 2012 workshop (Erlangen, Germany)*, AIP Conference Proceedings, to be published. G. Navarra et al. (KASCADE-Grande Collaboration), *NIM A* [**518**]{} (204) 207. F. G. Schröder et al. (LOPES collaboration), “Investigation of the Radio Wavefront of Air Showers with LOPES and REAS3", *32nd ICRC Proceedings*, 2011. M. Ludwig, T. Huege, *Astrop. Phys.* [**34**]{} (2010) 438. Krijn D. de Vries et al., *Astropart. Phys.*, [**34**]{} (2010) 267 T. Huege et al., *Astropart. Phys.*,[**30**]{} (2008) 96. T. Huege et al. (LOPES collaboration), *NIM A* (in press), DOI:*10.1016/j.nima.2011.11.081* Horneffer et al. (LOPES collaboration), *30th ICRC Proceedings*, V4, 83-86 N. Palmieri, Ph.D. thesis (2012), in preparation. T. Huege, “Simulating radio emission from air showers with CoREAS", *Proceedings of the ARENA 2012 workshop (Erlangen, Germany)*, AIP Conference Proceedings, to be published. M. Ludwig, “Comparison of LOPES measurements with CoREAS and REAS 3.1 simulations", *Proceedings of the ARENA 2012 workshop (Erlangen, Germany)*, AIP Conference Proceedings, to be published. M. Ludwig , Ph.D. thesis (2011). D. Heck et al., *Report FZKA* 6019, Forschungszentrum Karlsruhe (1998). M. Melissas, “Recent developments at the Auger Engineering Radio Array", *Proceedings of the ARENA 2012 workshop (Erlangen, Germany)*, AIP Conference Proceedings, to be published.
[^1]: Hence the name “slope method"
[^2]: in shower plane coordinate system
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We studied the background dwarf nova of KIC 11412044 in the Kepler public data and identified it with GALEX J194419.33$+$491257.0. This object turned out to be a very active SU UMa-type dwarf nova having a mean supercycle of about 150 d and frequent normal outbursts having intervals of 4–10 d. The object showed strong persistent signal of the orbital variation with a period of 0.0528164(4) d (76.06 min) and superhumps with a typical period of 0.0548 d during superoutbursts. Most of the superoutbursts were accompanied by a precursor outburst. All these features are unusual for this very short orbital period. We succeeded in detecting the evolving stage of superhumps (stage A superhumps) and obtained a mass ratio of 0.141(2), which is unusually high for this orbital period. We suggest that the unusual outburst properties are a result of this high mass ratio. We suspect that this object is a member of the recently recognized class of cataclysmic variables (CVs) with a stripped core evolved secondary which are evolving toward AM CVn-type CVs. The present determination of the mass ratio using stage A superhumps makes the first case in such systems.'
author:
- 'Taichi <span style="font-variant:small-caps;">Kato</span>'
- 'Yoji <span style="font-variant:small-caps;">Osaki</span>'
title: 'GALEX J194419.33$+$491257.0: An Unusually Active SU UMa-Type Dwarf Nova with a Very Short Orbital Period in the Kepler Data'
---
Introduction
============
The Kepler mission ([@bor10Keplerfirst]; [@Kepler]), which was aimed to detect extrasolar planets, has provided unprecedentedly sampled data on several cataclysmic variables (CVs). This satellite also recorded previously unknown CVs as by-products of the main target stars. The best documented example has been the background dwarf nova of KIC 4378554 ([@bar12j1939]; [@kat13j1939v585lyrv516lyr]). In addition to this object, the group Planet Hunters [@fis12PlanetHunters] detected several candidate background CVs.[^1]
We studied one of these background dwarf novae, the one in the field of KIC 11412044 (hereafter J1944). This object was discovered by the Planet Hunters group as a background SU UMa-type dwarf nova of KIC 11412044, in which superoutbursts and frequent normal outbursts were recognized. [^2] Since it was bright enough and it was frequently included in the aperture mask of KIC 11412044, the outburst behavior can be immediately recognized in Kepler [SAP\_FLUX]{} light curve of KIC 11412044.
Data Analysis
=============
(160mm,230mm)[fig1.eps]{}
[ (160mm,120mm)[fig2\_s.eps]{} ]{}
[ (88mm,88mm)[fig3.eps]{} ]{}
We used Kepler public long cadence (LC) data (Q1–Q17) for analysis. Since the outbursts were immediately recognizable in each light curve of the Kepler target pixel images, we used a custom aperture consisting of 4–6 pixels showing outbursts as we did in the background dwarf nova of KIC 4378554 [@kat13j1939v585lyrv516lyr]. We used surrounding pixels to subtract the background from KIC 11412044. We further corrected small long-term baseline variations by subtracting a locally-weighted polynomial regression (LOWESS: [@LOWESS]) and spline functions. Since the quiescent magnitude is difficult to determine, we artificially set the level to be 22.0 mag.
Characterization and Identification of Object
=============================================
Outburst Properties
-------------------
The resultant light curve indicates that this object is an SU UMa-type dwarf nova with frequent outbursts (figure \[fig:j1944lc\]). There were eight observed superoutbursts, and from the regular pattern, another superoutburst most likely occurred between BJD 2455553 and 2455568 (a data gap in Q8) and we numbered the superoutburst and supercycle assuming that there is a superoutburst in this gap. The intervals between successive superoutbursts (supercycles) were in a range of 120–160 d. We determined the mean supercycle of 147(1) d. Most of superoutbursts were associated with a precursor outburst with a various degree of separation from the main superoutburst. The typical duration of the superoutburst is $\sim$8 d including the precursor part. This duration is shorter than those of many other SU UMa-type dwarf novae.
The number of normal outbursts in one supercycle ranged from 11 to 21. The intervals of normal outbursts were 4–10 d, one of the shortest known except ER UMa stars [@kat99erumareview]. The amplitudes of normal outbursts increased as the supercycle phase progresses. Some of the normal outbursts were “failed”, i.e. they decayed before reaching the full maximum.
Frequency Analysis and Source Identification
--------------------------------------------
As shown in figure \[fig:j1944spec2d\], a two-dimensional Fourier analysis (using the Hann window function) of the light curve of this object yielded two periods. There was a signal of a constant frequency (18.93 cycle d$^{-1}$) with the almost constant strength. Using all the data segment, we determined the period to be 0.0528164(4) d (18.934 cycle d$^{-1}$). We refer this signal to “0.0528 d” signal. Based on the high stability of the 0.0528 d signal during the entire Kepler observations, we identified this period to be the orbital period ($P_{\rm orb}$) of this object. During superoutbursts, there were transient signals of superhumps at frequencies around 18.1–18.5 cycle d$^{-1}$ as expected.
Let us now examine the source position of the background dwarf nova in figure \[fig:kepmap\]. We checked the pixels which showed the dwarf nova-type variation. The peak of signal of dwarf nova-type outbursts was found two pixels away from the center of KIC 11412044 (star 1 on the DSS image). At this location, there is a GALEX [@GALEX] ultraviolet source GALEX J194419.33$+$491257.0 \[NUV magnitude 21.3(3)\] and we identified this source as the UV counterpart of this dwarf nova (figure \[fig:kepmap\], Q16), confirming the suggestion in the Planet Hunters’ page. The superhump component and 0.0528 d component were also confirmed at the location of this object (figure \[fig:kepmap\], Q14), and we consider that the 0.0528 d signal indeed comes from this dwarf nova. This has also been confirmed by the non-detection of the 0.0528 d signal in the [SAP\_FLUX]{} of KIC 11412044 when this dwarf nova was outside the aperture of KIC 11412044.
Variation of Superhump Period
-----------------------------
Since the superhump period is less than three LC exposures, it is difficult to determine the times of superhump maxima by the conventional method. We employed the Markov-chain Monte Carlo (MCMC) modeling used in @kat13j1939v585lyrv516lyr. Although we only show the result of SO3 (figure \[fig:j1944humpall\]), the pattern is similar in other superoutbursts. In the $O-C$ diagram, stages A–C (for an explanation of these stages, see [@Pdot]) can be recognized. Long-period superhumps (stage A superhumps) with growing superhumps were recorded during the late part of the precursor outburst to the maximum of the superoutburst. The overall pattern is very similar to those of other SU UMa-type dwarf novae, including V1504 Cyg and V344 Lyr ([@osa13v344lyrv1504cyg]; [@osa14v1504cygv344lyrpaper3]). The object makes the fourth case (after V1504 Cyg, V344 Lyr and V516 Lyr) in the Kepler field in which the growing superhumps lead smoothly from the precursor to the main superoutburst and thus gives further support to the thermal-tidal instability (TTI) model [@osa89suuma] as the explanation for the superoutburst.
(88mm,110mm)[fig4.eps]{}
System Properties
-----------------
The inferred fractional superhump period excess $\varepsilon \equiv P_{\rm SH}/P_{\rm orb}-1$, where $P_{\rm SH}$ is the superhump period, of $\sim$3.8% is, however, unusually large for this $P_{\rm orb}$ (cf. figure 15 in [@Pdot]).
@kat13qfromstageA recently proposed that stage A superhumps can be used to determine the mass ratio ($q=M_2/M_1$) and the resultant mass ratios are as accurate as those obtained from eclipse modeling. We have succeeded in measuring the period of stage A superhump during the three superoutbursts: 0.0555(2) d (SO3), 0.05546(5) d (SO6), 0.05552(6) d (SO7). The corresponding fractional superhump excesses in the frequency unit $\varepsilon^* \equiv 1-P_{\rm orb}/P_{\rm SH}$ are 4.8%, 4.77% and 4.88%. These values correspond to the $q$ value of 0.14, 0.139 and 0.143, respectively. We therefore adopted $q$=0.141(2). This mass ratio implies a massive (approximately two times more massive) secondary for this very short orbital period comparable to most WZ Sge-type dwarf novae (figure \[fig:evolloc\]). This result may alternatively suggest the possibility of an unusually low-mass white dwarf. If we assume that the secondary of J1944 has a normal mass for this orbital period, such as 0.066$M_\odot$ in WZ Sge [@kat13qfromstageA], the mass of the white dwarf must be $\sim$0.47$M_\odot$. According to @zor11SDSSCVWDmass, the fraction of CVs having white dwarf lighter than 0.5$M_\odot$ is only 7$\pm$3 %, even including suspicions measurements. Furthermore, there is evidence from modern eclipse observations that mass of the white dwarf in short-$P_{\rm orb}$ CVs is not diverse [@sav11CVeclmass]. We therefore consider the interpretation of a massive secondary more likely.
[ (88mm,70mm)[fig5.eps]{} ]{}
The presence of precursor outburst and the high frequency of normal outbursts are usual features of longer-$P_{\rm orb}$ systems such as V1504 Cyg and V344 Lyr. Systems with $P_{\rm orb}$ like J1944 are usually WZ Sge-type dwarf novae with very rare (super)outbursts (e.g. [@kat01hvvir]) or ER UMa-type dwarf novae, a rare subgroup with very frequent outbursts and short supercycles (e.g. [@kat95eruma]; [@rob95eruma]). J1944 does not match the properties of either group. This can be understood if the outburst properties are a reflection of the mass ratio rather than the orbital period since the $q$ value of J1944 is closer to those of longer-$P_{\rm orb}$ SU UMa-type dwarf novae.
The presence of such a system would pose a problem in terms of the CV evolution since the secondary loses its mass during the CV evolution and $q$ value is expected to be as low as $\sim$0.08 around the orbital period of J1944. In recent years, some objects showing hydrogen lines in their spectra (this excludes the possibility of double-degenerate AM CVn-type systems) have been discovered around this period or even in shorter period. These objects include EI Psc ([@uem02j2329letter]; [@tho02j2329]), V485 Cen [@aug96v485cen] and GZ Cet [@ima06j0137]. These objects are considered to be CVs whose secondary had an evolved core at the time of the contact, and are considered to be progenitors of AM CVn-type double white dwarfs ([@pod03amcvn]; [@nel04amcvn]; [@uem02j2329letter]; [@tho02j2329]).
None of these objects have been reported for $q$ determination directly from radial-velocity studies, and $q$ values have only been inferred from the traditional $\varepsilon$, which has an unknown uncertainty [@kat13qfromstageA]. The detection of stage A superhumps in J1944 allowed the first reliable determination of $q$ in such stripped-core ultracompact binaries. There is, however, a marked difference of the outburst frequency between J1944 and these known objects since the frequency of outbursts in such systems have been reported to be low [@tho13j1340]. This suggests that J1944 has an anomalously high mass-transfer rate among these objects. The object may be in a phase analogous to ER UMa-type dwarf novae, whose high mass-transfer rates may be a result of a recent classical nova explosion (cf. [@kat95eruma]; [@pat13bklyn]). Since the object can be within the reach of the ground-based telescopes, the exact optical identification and the search for the feature of the secondary star are encouraged to solve the mystery.
We thank the Kepler Mission team and the data calibration engineers for making Kepler data available to the public. We also thank the Planet Hunters group for making their information on the background dwarf novae public which enabled us to study this interesting object. This work was supported by the Grant-in-Aid “Initiative for High-Dimensional Data-Driven Science through Deepening of Sparse Modeling” from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan.
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[^1]: $<$http://talk.planethunters.org/objects/APH51255246/\
discussions/DPH101e5xe$>$.
[^2]: $<$http://keplerlightcurves.blogspot.jp/2012/07/\
dwarf-novae-candidates-at-planet.html$>$.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'This note gives an example of closed smooth manifolds $M$ and $N$ for which the rank of $M\times N$ is strictly greater than $rank M + rank N$.'
address:
- ' Geometr[í]{}a y Topolog[í]{}a, Facultad de Ciencias, Campus de Teatinos, s/n, 29071-M[á]{}laga, Spain'
- ' University of Michigan, Ann Arbor, MI 48109-1003, USA'
author:
- 'Francisco-Javier Turiel'
- 'Arthur G. Wasserman'
title: On the rank of a product of manifolds
---
Milnor defined the [*rank*]{} of a smooth manifold $M$ as the maximal number of commuting vector fields on $M$ that are linearly independent at each point.
One of the questions raised by Milnor at the Seattle Topology Conference of 1963, and echoed by Novikov [@No], was $$is\ rank(M\zpor N)=rank(M)+rank(N)$$ whenever $M$ and $N$ are smooth closed manifolds?
[*In this note we give a negative answer to this question.*]{}
We need a simple result about mapping tori.
Let $f\zdp X\zfl X$ be a diffeomorphism of a manifold X and let $$M(f)={\frac {I\zpor X} {(0,x)_{\widetilde~} (1,f(x))}}$$ be the mapping torus of f where $I=[0,1]$.
Equivalently, $M(f)=\zgran {\frac {{\mathbb R}\zpor X} {{\mathbb Z}}}$ where the action of ${\mathbb Z}$ on ${\mathbb R}\zpor X$ is given by $\za(k)(t,x)=(t+k,f^k (x))$. $M(f)$ is a fibre bundle over $S^1$ with fiber $X$. We note that $\zp_1 (M(f))=\zp_1 (X)\ast_{f}{\mathbb Z}$ where $\ast$ denotes the semi-direct product and $f_\ast\zdp\zp_1 (X)\zfl\zp_1 (X)$.
\[P1\] Consider two periodic diffeomorphisms $f\zdp X\zfl X$ and $g\zdp Y\zfl Y$ with periods $m$ and $n$ respectively. Assume $m$ and $n$ are relatively prime, i.e., there are integers $c,d$ such that $mc+nd=1$.
Then $M(f) \zpor M(g)$ is diffeomorphic to $ M(h)$ where $h\zdp S^1 \zpor X\zpor Y\zfl S^1 \zpor X\zpor Y$ is defined by $h(\zh,x ,y)=(\zh,f^{-d}(x ),g^c (y ))$. Moreover $h^{m-n}=(id,f,g)$.
$M(f) \zpor M(g)$ can be identified with the quotient of ${\mathbb R}^2 \zpor X\zpor Y$ under the action of ${\mathbb Z}^2$ given by $\zb(z)(u,x,y)=(u+z, f^{z_1}(x), g^{z_2}(y ))$, where $z=(z_1 ,z_2 )\zpe{\mathbb Z}^2$, $u=(u_1 ,u_2 )\zpe{\mathbb R}^2$ and $(x ,y)\zpe X\zpor Y$.
Set $\zl=(m,n)$ and $\zm=(-d,c)$. Since $mc+nd=1$, ${\mathcal B}=\{\zl,\zm\}$ is at the same time a basis of ${\mathbb Z}^2$ as a ${\mathbb Z}$-module and a basis of ${\mathbb R}^2$ as a vector space. On the other hand $$\zb(\zl)(u,x,y)=(u+\zl,x,y)\quad \text{and}\quad \zb(\zm)(u,x,y)=(u+\zm,f^{-d} (x ),g^c (y )).$$
Therefore the action $\zb$ referred to the new basis ${\mathcal B}$ of ${\mathbb Z}^2$ and ${\mathbb R}^2$ is written now: $$\zb(k,r)(a,b,x ,y)=(a+k,b+r,{\zf}^r (x ),{\zg}^r (y ))$$ where $\zf=f^{-d}$ and $\zg=g^c$.
As the action of the first factor of ${\mathbb Z}^2$ on $X\zpor Y$ is trivial, identifying $S^1$ with $\zgran{\frac {{\mathbb R}} {{\mathbb Z}}}$ shows that $M(f) \zpor M(g)$ is diffeomorphic to $M(h)$.
Finally from $(-n)(-d)=1-cm$ and $cm=1-dn$ follows that $h^{m-n}=(id,f,g)$.
On the other hand:
\[L1\] Let $f\zdp N\zfl N$ be a diffeomorphism and let $X_1 ,\dots,X_k$ be a family of commuting vector fields on $N$ that are linearly independent everywhere. Assume $f_\ast X_i =\zsu_{j=1}^k a_{ij}X_j$, $i=1,\zps,k$, where the matrix $(a_{ij})\zpe GL(k,{\mathbb R})$. Then $rank(M(f))\zmai k$.
[**Proof.**]{} It suffices to construct $k$ commuting vector fields ${\widetilde X}_1 ,\zps,{\widetilde X}_k$ on $I\zpor N$ that are linearly independent at each point and such that every ${\widetilde X}_i (t,x)$ equals $X_i (x)$ if $t$ is close to zero and $f_\ast X_i (x)$ when $t$ is close to 1 ($X_1 ,\dots,X_k$ are considered vector fields on $I\zpor N$ in the obvious way).
If $\zbv a_{ij}\zbv>0$ consider an interval $[a,b]\zco (0,1)$ and a (differentiable) map $(\zf_{ij} )\zdp I\zfl GL(k,{\mathbb R})$ such that $\zf_{ij}([0,a])=\zd_{ij}$ and $\zf_{ij}([b,1])=a_{ij}$, and set ${\widetilde X}_i (t,x) =\zsu_{j=1}^k\zf_{ij}(t)X_j (x)$.
When $\zbv a_{ij}\zbv<0$ first take an interval $[c,d]\zco (0,1/2)$ and a function $\zr\zdp [0,1/2]\zfl {\mathbb R}$ such that $\zr([0,c])=1$, $\zr([d,1/2])=-1$, and on $[0,1/2]\zpor N$ set ${\widetilde X}_1 (t,x)=\zr(t)X_1 (x)+(1-\zr^2 (t)){\frac{\zpar} {\zpar t}}$ and ${\widetilde X}_i (t,x)=X_i (x)$, $i=2,\zps,k$.
The matrix of coordinates of $f_\ast X_1 ,\zps,f_\ast X_k$ with respect to the basis $\{-X_1 ,X_2 ,\zps,X_k \}$ has positive determinant, so by doing as before we can extend ${\widetilde X}_1 ,\zps,{\widetilde X}_k$ to $[1/2,1]\zpor N$ by means of an interval $[a,b]\zco (1/2,1)$ and a suitable map $(\zf_{ij} )\zdp [1/2,1]\zfl GL(k,{\mathbb R})$. $\quad\square$
$\,$
Proposition \[P1\] and Lemma \[L1\] quickly yield a counterexample.
$\,$
Assume $X$ is a torus $\zgran{\mathbb T}^k ={\frac {{\mathbb R}^k} {{\mathbb Z}^k}}$ and $f$ is the map induced by a nontrivial element of $GL(k,{\mathbb Z})$. Then by the above lemma applied to ${\frac {\zpar} {\zpar\zh_j}}$, $j=1,\dots,k$, $rank(M(f))\zmai k$. But $M(f)$ has non-abelian fundamental group, so it is not a torus and $rank(M(f))=k$. (If M is a closed connected n-manifold of rank n then M is diffeomorphic to the n-torus.)
For the same reason if $Y={\mathbb T}^r$ and $g$ is induced by a nontrivial element of $GL(r,{\mathbb Z})$ then $rank(M(g))=r$.
If $f$ and $g$ are periodic with relatively prime periods $m$ and $n$ respectively then by Proposition \[P1\] $M(f) \zpor M(g)$ =$M(h)$ where $h\zdp{\mathbb T}^{k+r+1}\zfl{\mathbb T}^{k+r+1}$ is induced by a nontrivial element of $GL(k+r+1,{\mathbb Z})$. Moreover $rank(M(h))=k+r+1$ . Therefore: $$rank(M(f)\zpor M(g))>rank(M(f))+rank(M(g)).$$
For instance, set $k=r=2$ and consider $f,g$ induced by the elements in $SL(2,{\mathbb Z})\zco GL(2,{\mathbb Z})$ $$\begin{pmatrix} -1 & 0 \\ 0&-1 \\ \end{pmatrix}
\quad\text{and}\quad
\begin{pmatrix} 0&1\\ -1&-1\\ \end{pmatrix}$$ respectively, so $M(f)$ and $M(g)$ are orientable. Then the period of $f$ is 2 and that of $g$ equals 3 .
An even simpler but non-orientable counterexample can be constructed as follows. Take $r$ and $g$ as before, $k=1$ and $f$ induced by $(-1)$. Then $M(f)$ is the Klein bottle which has rank 1 and M(g) has rank 2; however, $M(f) \zpor M(g)$ is diffeomorphic to $ M(h)$ and hence has rank 4.
$\,$
\[BB\]
The [*file*]{} of a manifold $M$ was defined by Rosenberg [@Ro] to be the largest integer $k$ such that ${\mathbb R}^k$ acts locally free on $M$. When $M$ is closed $file(M)$ equals $rank(M)$ but $file({\mathbb R}\zpor S^2 )=1$, [@Ro], while $rank({\mathbb R}\zpor S^2 )=3$.
The analog of Milnor’s question for the file of a product of noncompact manifolds also fails. Indeed, let ${\mathbb R}_{e}^4$ be any exotic ${\mathbb R}^4$. Then $file({\mathbb R}_{e}^4 )\zmei 3$ otherwise ${\mathbb R}_{e}^4 ={\mathbb R}^4$. But ${\mathbb R}_{e}^4 \zpor{\mathbb R}={\mathbb R}^5$ because there in no exotic ${\mathbb R}^5$, so $file({\mathbb R}_{e}^4 \zpor{\mathbb R})=5>file({\mathbb R}_{e}^4 )+file({\mathbb R})$.
Orientable closed connected $n$-manifolds of rank $n-1$ are completely described in [@RRW; @CR; @Ti].
[99]{}
G. Chatelet and H. Rosenberg, [*Manifolds which admit ${\mathbb R}^n$ actions*]{}, Inst. Hautes Études Sci. Publ. Math. [**43**]{} (1974), 245–260.
S. P. Novikov, [*The Topology Summer Institute (Seattle, USA, 1963)*]{}, Russian Math. Surveys [**20**]{} (1965), 145–167. http://www.mi.ras.ru/$_{\widetilde~}$snovikov/16.pdf.
H. Rosenberg, [*Singularities of ${\mathbb R}^2$ actions*]{}, Topology [**7**]{} (1968), 143–145.
H. Rosenberg, R. Roussarie and D. Weil, [*A classification of closed oriented 3-manifold of rank two*]{}, Ann. of Math. [**91**]{} (1970), 449–464.
D. Tischler, [*Manifolds $M^n$ of rank $n-1$*]{}, Proc. Amer. Math. Soc. [**94**]{} (1985),158–160.
|
{
"pile_set_name": "ArXiv"
}
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---
abstract: 'We consider the setting of a sensor that consists of a speed-scalable processor, a battery, and a solar cell that harvests energy from its environment at a time-invariant recharge rate. The processor must process a collection of jobs of various sizes. Jobs arrive at different times and have different deadlines. The objective is to minimize the *recharge rate*, which is the rate at which the device has to harvest energy in order to feasibly schedule all jobs. The main result is a polynomial-time combinatorial algorithm for processors with a natural set of discrete speed/power pairs.'
author:
- Neal Barcelo
- 'Peter Kling[^1]'
- Michael Nugent
- 'Kirk Pruhs[^2]'
bibliography:
- 'references.bib'
title: Optimal Speed Scaling with a Solar Cell
---
[^1]: Supported by fellowships of the Postdoc-Programme of the German Academic Exchange Service (DAAD) and the Pacific Institute of Mathematical Sciences (PIMS). Work done while at the University of Pittsburgh.
[^2]: Supported, in part, by NSF grants CCF-1115575, CNS-1253218, CCF-1421508, CCF-1535755, and an IBM Faculty Award.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Near-infrared (IR) imaging polarimetry in the $J$, $H$, and $K_s$ bands has been carried out for the protostellar cluster region around NGC 2264 IRS 2 in the Monoceros OB1 molecular cloud. Various infrared reflection nebulae clusters (IRNCs) associated with NGC 2264 IRS 2 and IRAS 12 S1 core were detected as well as local infrared reflection nebulae (IRNe). The illuminating sources of the IRNe were identified with known or new near- and mid-IR sources. In addition, 314 point-like sources were detected in all three bands and their aperture polarimetry was studied. Using a color–color diagram, reddened field stars and diskless pre-main sequence stars were selected to trace the magnetic field (MF) structure of the molecular cloud. The mean polarization position angle of the point-like sources is 81$\arcdeg$ $\pm$ 29$\arcdeg$ in the cluster core, and 58$\arcdeg$ $\pm$ 24$\arcdeg$ in the perimeter of the cluster core, which is interpreted as the projected direction on the sky of the MF in the observed region of the cloud. The Chandrasekhar–Fermi method gives a rough estimate of the MF strength to be about 100 $\mu$G. A comparison with recent numerical simulations of the cluster formation implies that the cloud dynamics is controlled by the relatively strong MF. The local MF direction is well associated with that of CO outflow for IRAS 12 S1 and consistent with that inferred from submillimeter polarimetry. In contrast, the local MF direction runs roughly perpendicular to the Galactic MF direction.'
author:
- 'Jungmi Kwon, Motohide Tamura , Ryo Kandori , Nobuhiko Kusakabe, Jun Hashimoto, Yasushi Nakajima, Fumitaka Nakamura, Takahiro Nagayama, Tetsuya Nagata, James H. Hough, Michael W. Werner, and Paula S. Teixeira'
title: |
Complex Scattered Radiation Fields and Multiple Magnetic Fields\
in the Protostellar Cluster in NGC 2264
---
INTRODUCTION
============
NGC 2264, including the Cone Nebula, Fox Fur Nebula and Christmas Tree Cluster in the Monoceros OB1 molecular cloud, is a site of active star formation (Walker 1954, 1956) and is a cornerstone for the study of the formation and time evolution of young stellar objects (YSOs). The median age of NGC 2264 is 1–3 Myr with an age dispersion of $\sim$5 Myr (Dahm & Simon 2005). It is located at a distance of $\sim$760 pc (Sung et al. 1997, but see also Baxter et al. 2009, Sung et al. 2010) and lies 37 pc above the galactic plane (Pérez 1991) in front of a dark molecular cloud complex. The mass of the cloud complex is estimated to be at least $\sim$3.7 $\times$ 10$^4$ M$_{\sun}$ (Dahm 2008). NGC 2264 has an intermediate stellar density and its stellar population, which is known down to very low masses, is dominated by the O7V multiple star S Monocerotis (Schwartz et al. 1985).
The NGC 2264 region has been exceptionally well studied at various wavelengths from centimeters to X-rays. Previous studies have revealed that active star formation is ongoing in this region, as evidenced by the presence of numerous embedded clusters of protostars, molecular outflows, and Herbig–Haro objects (e.g., Adams et al. 1979; Margulis et al. 1989; Walsh et al. 1992; Hodapp 1994; Reipurth et al. 2004; Young et al. 2006, hereafter YTL). There are two prominent sites of star formation activity centered near NGC 2264, identified by IR and millimeter observations, IRAS 06384+0932 (hereafter IRS 1) and IRAS 06382+0939 (hereafter IRS 2). NGC 2264 IRS 1, known as Allen’s source (Allen 1972), is a deeply embedded early-type (B2-B5) star. NGC 2264 IRS 2, discovered by Sargent et al. (1984), was identified as a Class I source. NGC 2264 IRS 2 was designated as IRAS 12 (06$^h$ 41$^m$ 02$\fs$7 +09$\arcdeg$ 36$\arcmin$ 10$\arcsec$) by Margulis et al. (1989), and Castelaz & Grasdalen (1988) showed that NGC 2264 IRS 2 is a binary source (RNO-E and RNO-W) separated by 2$\farcs$8.
Krügel et al. (1987) presented an extended ammonia map of the central region of the molecular cloud associated with NGC 2264 including both IRS 1 and IRS 2. Williams & Garland (2002) also observed the dust and gas near the two young stellar clusters IRS 1 and IRS 2 in NGC 2264 and presented an 870 $\mu$m continuum emission map. The elongated shape of IRS 1 shows signs of substructure, while IRS 2 is more fragmented, indicating a more evolved cluster of protostars. IRAS 12 S1 located south-west of NGC 2264 IRS 2 was found by Cohen et al. (1985), and it is located precisely at the peak of the IRAS 12 S1 in the submillimeter maps of Wolf-Chase et al. (2003). It is also associated with the core C of Williams & Garland (2002), the NGC 2264 D-MM1 of Peretto et al. (2006), and the dense microcluster of Class 0 protostars of Teixeira et al. (2007). Wolf-Chase et al. (2003) derived an envelope mass of 17.6 M$_{\sun}$ within a 29$\arcsec$ diameter, and Peretto et al. (2006) estimated a mass infall rate of M$_{D-MM1}$ $\sim$1.1 $\times$ 10$^{-4}$ M$_{\sun}$yr$^{-1}$ from millimeter observations, toward the rotating IRAS 12 S1. However, YTL and Teixeira et al. (2007) showed that IRAS 12 S1 is in fact a multiple source, composed of at least 7 protostars.
Teixeira et al. (2006) identified bright 24 $\mu$m protostars that are radially distributed around NGC 2264 IRS 2, and embedded within dusty filaments. The cluster of protostars was therefore named the Spokes cluster. The projected separation of the 24 $\mu$m protostars is 20 $\pm$ 5$\arcsec$, similar to the Jeans length of the cloud, 27, implying the filaments had undergone thermal fragmentation. In addition, the evolutionary stages of protostars in the NGC 2264 IRS 2 region were studied through detailed spectral energy distributions (SEDs) (Forbrich et al. 2010, hereafter FTR), and the outflow activity was derived in the vicinity of IRAS 12 S1. Bourke et al. (2001), from OH Zeeman observations using the NRAO Green Bank 43-m telescope (FWHM beam size $\sim$18$\arcmin$), estimated the magnetic field strength for NGC 2264 to be 21 $\pm$ 16 $\mu$G.
The outflows and the Herbig–Haro objects are considered as indicators of recent star formation activity. The characteristics of outflows and jets associated with YSOs are related to the magnetic field and the rotation of the protostar and circumstellar disk. The outflows and jets commonly tend to be aligned with each other and with the cloud-scale magnetic field, as determined from polarization observations (e.g., Cohen et al. 1984; Strom et al. 1986; Vrba et al. 1986, 1988; Tamura & Sato 1989; Jones & Amini 2003). Herbig (1974) performed early surveys of the NGC 2264 region and identified several candidate Herbig–Haro objects, as well as a survey of emission line stars (Herbig 1954).
Multi-color near-IR polarimetry is useful for understanding magnetic fields and the properties of dust grains that cause scattering and absorption in various environments (e.g., Tamura et al. 2007). Polarimetric studies of the NGC 2264 region have been reported previously by several authors (e.g., Breger & Hardorp 1973; Kobayashi et al. 1978; Dyck & Lonsdale 1979; Heckert & Zeilik 1981, 1984; Schreyer et al. 2003; Dotson et al. 2010) but almost all of these studies only covered the NGC 2264 IRS 1 region. The polarization position angle of NGC 2264 IRS 1 was found to be $\sim$106$\arcdeg$–117$\arcdeg$ in the near-IR (Breger & Hardorp 1973; Kobayashi et al. 1978; Heckert and Zeilik 1981). Dotson et al. (2010) recently reported 350 $\mu$m polarimetry in the direction of IRS 1 and IRAS 12 S1 near IRS 2 in NGC 2264.
In this paper, we first present wide-field near-IR imaging polarimetry of the IRS 2 region, as part of our ongoing survey project of $JHK_s$ polarimetry for star forming regions. In Section 2, we describe the observations. In Section 3, we present the results of the imaging and aperture polarimetry. In Section 4, we discuss the illuminating sources of infrared reflection nebulae and the magnetic field structure related to the star forming activity in the NGC 2264 IRS 2 region. A summary is given in Section 5.
OBSERVATIONS
============
The observations in the direction of the IRAS 12 S1 region were carried out using the SIRPOL imaging polarimeter on the Infrared Survey Facility (IRSF) 1.4-m telescope at the South African Astronomical Observatory. SIRPOL consists of a single-beam polarimeter (an achromatic half-wave plate rotator unit and a polarizer) and an imaging camera (Nagayama et al. 2003). The camera, SIRIUS, has three 1024 $\times$ 1024 HgCdTe infrared detectors. IRSF/SIRPOL enables deep and wide-field (77 $\times$ 77 with a scale of 045 pixel$^{-1}$) imaging polarimetry in the $J$, $H$, and $K_s$ bands simultaneously (Kandori et al. 2006).
The observations were made on the night of 2007 February 19. We performed 10-s exposures at four wave-plate angles (in the sequence 0, 45, 22$\fdg$5, and 67$\fdg$5) at 10 dithered positions for each set. The same observation sets were repeated 10 times toward the target object and for the sky background to increase the signal-to-noise ratio. The total integration time was 1000 s per wave plate angle. The typical seeing size during the observations was $\sim$13 (2.9 pixels) in the $J$ band. SIRPOL has been routinely used since 2006 and the instrumental polarization is negligible (Kandori et al. 2006). The polarization efficiencies are 95.5 %, 96.3 %, and 98.5 % in the $J$, $H$, and $K_s$ bands, respectively.
The data were processed using IRAF in the standard manner, which included dark-field subtraction, flat-field correction, median sky subtraction, and frame registration. The remaining artificial stripe pattern was then removed using IDL. The pixel coordinates of point-like sources found on the reduced images were matched with the celestial coordinates of their counterparts in the Two Micron All Sky Survey (2MASS) point source catalog. The IRAF IMCOORDS package was applied to the matched list to obtain plate transform parameters. The rms uncertainty in the coordinate transformation was $\sim$01.
Figure 1 shows the $J$-$H$-$K_s$ color composite intensity image of the 8$\farcm$0 $\times$ 8$\farcm$0 region around NGC 2264 IRS 2 (hereafter the IRS 2 field) including IRAS 12 S1 core. The field is larger than the field-of-view of the camera because of dithering.
Mid-IR data obtained from Spitzer IRAC and MIPS were used for data analysis as well as our near-IR data. The IRAC 5.8 $\mu$m data were acquired in two epochs 7 months apart (2004 March 6 and October 8), with two dithers at each epoch to allow easy removal of asteroids and other transients, as part of Spitzer Guaranteed Time Observation program 37. Basic data reduction and calibration were done with the Spitzer Science Center (SSC) pipeline, version S14.0. The MIPS 24 $\mu$m data were obtained on 2004 March 16 using the scan-map mode, as part of Spitzer Guaranteed Time Observation program 58, and the data were reduced using the MIPS Data Analysis Tool, version S9.5. Since the spatial resolution of the MIPS 24 $\mu$m data is low, we paid considerable attention to finding the appropriate IRAC 5.8 $\mu$m and SIRPOL counterparts.
Figure 2 shows the $J$-$H$-$K_s$–5.8 $\mu$m–24 $\mu$m color composite intensity image of the IRS 2 region. Since protostars have significant circumstellar disks and envelopes, the 24 $\mu$m band may be particularly useful for identifying the Class I and Class 0 objects.
In this paper, we designate individual infrared reflection nebula(e) as IRN(e) and clusters of those infrared reflection nebulae as IRNC(s). We also designate the IRN associated with an IR source X as IRN (X) such as IRN (AR 6), IRN (IRS 2), and IRN (D-MM15), rather than numbering each.
RESULTS
=======
Polarimetry of extended sources
-------------------------------
Polarimetry of extended sources was carried out on the combined intensity images for each exposure cycle (a set of exposures at four wave plate angles at the same dithered position). The Stokes parameters $I$, $Q$, and $U$ were calculated by $$I = {1 \over 2} (I_{0} + I_{22.5} + I_{45} + I_{67.5}),$$ $$Q = I_{0} - I_{45},$$ and $$U = I_{22.5} - I_{67.5},$$ where $I_a$ is the intensity with the half wave plate oriented at $a\arcdeg$. The polarization degree $P$ and the polarization position angle $\theta$ were calculated by $$P = {{\sqrt{Q^2 + U^2}} \over {I}}$$ and $$\theta = {1\over2} \arctan {U \over Q}.$$ The degree of polarization $P$ was corrected using the polarization efficiencies of SIRPOL.
Figure 3 shows the $H$ polarization vector map of the whole IRS 2 region superposed on the intensity $I$ image combined with SIRPOL$+$Spitzer image data. This figure shows prominent and extended polarization nebulosities over the IRS 2 region for the first time. The polarimetry of extended sources can reveal the locations of illuminating sources within nebulae because the observed polarization is perpendicular to ray from illuminating source to scatterer.
Polarimetry of point-like sources
---------------------------------
### Photometry
The IRAF DAOPHOT package was used for source detection (Stetson 1987). The DAOPHOT program automatically detected point-like sources with peak intensities greater than 10$\sigma$ above the local sky background, where $\sigma$ is the rms uncertainty. The automatic detection procedure misidentified some spurious sources, which were removed by careful visual inspection, and missed a few real sources such as AR 6B. We included AR 6B because we were interested in AR 6B (see Section 4.1.1). Then the IDL photometry package adapted from DAOPHOT was used to perform aperture photometry. The aperture radius was 3 pixels and the sky annulus radius was set to 10 pixels with a 5-pixel width. The pixel coordinates of the detected sources were matched with the celestial coordinates of their counterparts in the 2MASS point source catalog.
The Stokes $I$ intensity of each point-like source was calculated using equation (1). The magnitude and color of the photometry were transformed into the 2MASS system by $$\rm MAG_{2MASS} = MAG_{IRSF} + \alpha_1 \times COLOR_{IRSF} + \beta_1$$ and $$\rm COLOR_{2MASS} = \alpha_2 \times COLOR_{IRSF} + \beta_2,$$ where MAG$_{\rm IRSF}$ is the instrumental magnitude from the IRSF images and MAG$_{\rm 2MASS}$ is the magnitude from the 2MASS Point Source Catalog. The parameters were determined by fitting the data using a robust least absolute deviation method, and the mean absolute deviation for each data are 0.11393877, 0.072393811, and 0.084156421 for $J$, $H$, and $K_s$, respectively. For the magnitudes, $\alpha_1$ = 0.0484596, $-$0.0307835, and 8.31423e$-$05, and $\beta_1$ = $-$4.75512, $-$4.50592, and $-$5.22532 for $J$, $H$, and $K_s$, respectively. For the colors, $\alpha_2$ = 1.066 and 0.989 and $\beta_2$ = $-$0.283 and 0.732 for $J - H$ and $H - K_s$, respectively. The coefficients $\beta_1$ and $\beta_2$ include both the zero-point correction and aperture correction. The derived magnitudes are listed in Table 1. The 10$\sigma$ limiting magnitudes were 19.2, 18.8, and 16.8 for $J$, $H$, and $K_s$, respectively.
The resulting list contains 314 sources whose photometric uncertainties are less than 0.1 mag in all three bands (Table 1).
### Aperture Polarimetry
Polarimetry of point-like sources (hereafter aperture polarimetry) was carried out on the combined intensity images for each wave plate angle at 10 dithered positions for each set, instead of using the Stokes $Q$ and $U$ images. This is because the center of the sources cannot be determined satisfactorily in the $Q$ and $U$ images. From the aperture photometry for each wave plate angle image, the Stokes parameters of each point-like source were derived by equations (2) and (3). The aperture and sky radii were the same as those used in the aperture photometry of the $I$ images. The degree of polarization $P$ and the polarization position angle $\theta$ can be calculated by $$P_0 = {{\sqrt{Q^2 + U^2}} \over {I}},$$ $$P = \sqrt {P_0^2 - \delta P^2},$$ and $$\theta = {1\over2} \arctan {U \over Q},$$ where $\delta P$ is the uncertainty in $P_0$. Equation (9) is necessary to de-bias the polarization degree (Wardle & Kronberg 1974). The degree of polarization $P$ was corrected using the polarization efficiencies of SIRPOL.
Table 2 shows the derived source parameters. The uncertainties given in Table 2 (and elsewhere in this paper) are 1$\sigma$ values. The aperture polarimetry can indicate the structure of magnetic fields by measuring dichroic polarization of background stars which is produced by aligned interstellar dust grains.
DISCUSSION
==========
Scattered Radiation Field in the NGC 2264 IRS 2 Region
------------------------------------------------------
There are at least three IRNCs as well as several local IRNe in the outer parts of three IRNCs in Figure 3, characterized by both a high degree of polarization and a centrosymmetric pattern of polarization vectors, albeit often only partial, around an illuminating source whose location can often be identified. Some IRNe overlap with ill-defined boundaries. The polarization vector patterns around illuminating sources are often complex, and rarely entirely centrosymmetric. For example, illuminating sources may be close together, circumstellar disks and envelopes can involve multiple scattering and dichroic extinction, and foreground dust in the molecular cloud can also produce dichroic polarization. That significant regions of centrosymmetric patterns are observed implies that the amount of foreground dichroic extinction is relatively small, and/or patchy. In Figure 3, it is clear that each of the protostar cluster members indicated by 24 $\mu$m sources (FTR sources), is associated with their own local nebulosity, indicating that each protostar is self-luminous and associated with the circumstellar matter such as envelopes or disks. We discuss the illuminating sources of IRNe in each IRNC in the following subsections.
### IRNC 1
Figure 4 shows a close-up of the $H$ polarization vector map of IRNC 1 shown in Figure 3. It is clear that the IRNC 1 region is dominated by various highly polarized local IRNe with illuminating sources, while the outer parts are roughly centrosymmetric around the central stellar sources (Figure 4).
The main illuminating source showing a large, centrosymmetric nebulosity extending to the south-west (more than 1$\arcmin$) and also probably to the north-east (up to the middle point between AR 6 and IRS 2) is most likely to be AR 6. AR 6 consists of three stellar components, the brightest star AR 6A, the southern source 6B, and the faintest companion 6C. Unfortunately, our data could not resolve AR 6C because the distance between AR 6A and AR 6C is only 084, smaller than our seeing size. AR 6A$+$6C and AR 6B correspond to our source 218 and 314, respectively. The polarization of source 218 is 5–7% at $\sim$74$\arcdeg$, and the polarization of source 314 is 2–5% at 42$\arcdeg$–74$\arcdeg$.
The large IRN clearly shows a centrosymmetric pattern, indicating that the nebulosity is most likely associated with the central source AR 6. Therefore, we call it IRN (AR 6). This nebula corresponds to the blue region surrounding AR 6 in Figure 1, suggesting the central source is “evolved". It is consistent with the illuminating source AR 6 being a visible YSO.
As shown in Figure 5, an enlargement of IRN (AR 6), AR 6 itself shows a compact ($\sim$20$\arcsec$ $\times$ 10$\arcsec$) cometary nebulosity, extending from the west of the stars and curving around them to the south (Aspin & Reipurth 2003). To the east of AR 6, polarizations are partly offset and not always centrosymmetric around AR 6 but are directing south-east of AR 6. There may be a polarization disk (Bastien & Mènard 1988, 1990; Tamura et al. 1991) around AR 6. However, it is rare that a polarization disk is observed around a visible YSO because the polarization disk is believed to be caused by multiple scattering in the dense disk. Aspin & Reipurth (2003) suggested that AR 6A is a young star within the NGC 2264, as an FUor-like pre-main-sequence object, and all of the nebulosity seen may be connected to AR 6A. AR 6B is also in an elevated FUor state as an FUor-like object, but it does not appear intimately associated with the nebulosity.
IRS 2 also belongs to the infrared reflection nebula cluster IRNC 1. It shows a generally centrosymmetric pattern, even though there are slight changes in the vector pattern at several parts of the nebula. The IRN (IRS 2) is approximately centrosymmetric out to $\sim$30$\arcsec$, except for around point-sources in Figure 4. In the outer part of up to 15 to the north-west, the nebula also seems to be illuminated by IRS 2 rather than AR 6. Note that near the middle point between IRS 2 and AR 6, the polarization pattern is a mixture of the two polarization patterns.
IRS 2 is resolved into two sources in our images (RNO-E and RNO-W; Castelaz & Grasdalen 1988). RNO-E and RNO-W correspond to our sources 241 and 240, respectively (see Table 1). It is noteworthy that the polarization pattern very close to these sources (a few arcsec) is not centrosymmetric but rather aligned in IRN (IRS 2); this can trace the magnetic fields rather than the local nebula.
There are other notable reflection nebulae and their illuminating sources in this field besides those associated with AR 6 and IRS 2 as described below.
The most interesting feature of Figure 4 is the small nebula, IRN (D-MM15), associated with a faint source or a cluster of faint sources $\sim$30$\arcsec$ to the east of IRS 2. This does not follow the centrosymmetric pattern around IRS 2 or the nearest source 246, suggesting that this faint source is an independent self-luminous source. This source is not point-like and is only visible in the $K_s$ band. It appears that this source corresponds to a millimeter/submillimeter source D-MM15 of Peretto et al. (2006). The position of D-MM15 is $\alpha$ = 06$^h$ 41$^m$ 04$\fs$6, $\delta$ = +09$\arcdeg$ 36$\arcmin$ 19$\arcsec$ \[J2000.0\]. Hashimoto et al. (2008) proposed an evolutionary sequence of massive YSOs based on the morphology of the associated IRNe. The Type A IRNe defined by Hashimoto et al. (2008) have a non-point-like morphology and extend in a direction relatively perpendicular to the polarization vectors, and is similar to IRN (D-MM15) as shown in Figure 4. They suggested that the monopolar or bipolar IRN of aligned vectors corresponds to the earlier phase because multiple scattering is dominant in the circumstellar matter around the youngest YSOs.
Source 200 is within the red nebula around IRS 2, and the vector pattern is partly centrosymmetric in the outer part of source 200. However, the vectors in the inner part are well aligned in one direction that is associated with the magnetic field direction in the IRS 2 region. It is highly polarized (P = 4–9%) in all three bands (Table 2).
Source 169 shows similar vector patterns to source 200. The vector pattern is partly centrosymmetric in the outer part of source 169, and it is also highly polarized (P = 5–10%) in all three bands (Table 2).
### IRNC 2
A faint red nebula around FTR 51 and FTR 53 (sources 51 and 53 of FTR) extends to the east of source 200 in Figure 4. The vector pattern of this nebula is not centrosymmetric around source 198, suggesting that the scattered radiation field here is dominated by the sources in IRNC 1. However, there is another IR cluster to the east whose nebulae are distinct from IRNC 1.
Figure 6 shows the polarization vector maps around FTR 56, as a close-up of the $H$ polarization vector map of IRNC 2 shown in Figure 3. This source is very red and in fact not point-like. The polarization map shows that the source itself is a reflection nebula and it extends significantly to the east (up to $\sim$20) but is partly interrupted by the blue point source 201. A small ($\sim$10) reflection nebula extending to the west is also clearly recognized. There is an independent nebula around the north-east blue sources 216 and 213 but the polarization pattern is not centrosymmetric. The north-west source 223 is also associated with a nebula but again its polarization pattern is not simple. The bright 24 $\mu$m sources FTR 51 and FTR 53 are totally invisible in the near-IR.
### IRNC 3
Figure 7 shows a close-up of the $H$ polarization vector map of IRNC 3 shown in Figure 3, including the IRAS 12 S1. As shown in Figure 7, the vector pattern around IRAS 12 S1 itself is not highly polarized ($\sim$10 % or less) compared with those in IRNC 1 and IRNC 2 ($\sim$10 % or more), but our data suggest a clear association of a medium-size reflection nebula with a centrosymmetric pattern around this “microcluster” of protostars (Teixeira et al. 2006; YTL). In the outer part (up to $\sim$20$\arcsec$), the nebula is roughly centrosymmetric, in particular to the south, south-west, and north-west in the $H$ band. There is a more simple, interesting pattern in the $K_s$ band. However, the central part and the region to the north-east are very complex.
Figure 8 shows enlargements of the $H$ and $K_s$ polarization vector maps for the IRAS 12 S1, i.e., in the right lower quadrant of Figure 7. These vector maps show very complex vector patterns slightly different from each other because of the sources only detected in the $K_s$ band, or the difference between the amount of foreground dichroic extinction by different wavelengths. It also indicates that there might be other hidden sources as well as infrared sources and submillimeter sources in Figure 8.
In the $H$ polarization vector map, the positions of the sources detected in the $K_s$ band with Baade/PANIC (YTL) are marked with circles and their source numbers. YTL reported several knots of molecular hydrogen in this “microcluster”, for example, between YTL 47 and YTL 49 and between YTL 40 and YTL 37. They also showed that YTL 44, associated with molecular hydrogen emission, has a distinctly bipolar appearance with an opaque dust lane, most likely caused by a circumstellar disk. Of particular interest is YTL 44, which shows a monopolar IRN of aligned vectors (Type A; Hashimoto et al. 2008). In the $K_s$ polarization vector map, the positions of the seven SMA continuum data using the Submillimeter Array whose synthesized beam had dimensions of 1$\farcs$4 $\times$ 1$\farcs$3 (Teixeira et al. 2007) are marked with plus signs on the basis of the source positions from their $K_s$ band image (0$\farcs$125 pixel$^{-1}$). Teixeira et al. (2007) reported that SMA 1 and SMA 3 have mid-IR counterparts and SMA 2, SMA 6, and SMA 7 are associated with diffuse mid-IR emission. SMA 4 and SMA 5 do not appear to have any near- and/or mid-IR emission, even though there is near- and mid-IR diffuse emission located between these sources. In Figure 8 (bottom panel), the position of the SMA sources are slightly shifted (by $\sim$1$\arcsec$), which is within the positional errors of the submillimeter array. We found that the shifted position show a better correlation between the near-IR nebula and the submillimeter sources.
FTR 50 and FTR 44 was classified as “genuine protostars” based on detailed SEDs (FTR). FTR 41 and FTR 43 that are multiple sources in IRAS 12 S1 also clearly showed protostar SEDs (FTR). They all do not show a centrosymmetric pattern of polarization vectors in Figure 7.
### Local IRNe
We found that several 24 $\mu$m sources such as FTR 39, FTR 38, FTR 37, and FTR 35 are associated with a local IRNe outside of three IRNCs in the observed region. The vector patterns around these local IRNe except for FTR 39 were not highly polarized compared with those in three IRNCs. IRN (FTR 39) is considered as one of members of IRNC 1.
Multiple Magnetic Fields in the IRS 2 Region
--------------------------------------------
Figures 9–11 show the polarization maps of point-like sources superposed on the Stokes $I$ images. The aperture polarimetry of the 314 point-like sources resulted in the positive detection ($P/\delta P \ge$ 3) of 181, 212, and 97 sources in the $J$, $H$, and $K_s$ bands, respectively, and 241 sources in at least one of the three bands.
The relation between the polarimetric and spectral data may be useful in understanding the nature of the polarization. The degree of polarization appears to be correlated with near-IR colors (Figure 12). The empirical relation for the upper limit of interstellar polarization suggested by Jones (1989) is $$P_{K, \rm max} = \tanh \left\{ 1.5 E(H-K) {{1-\eta}\over{1+\eta}} \right\},$$ where $\eta$ = 0.875 and $E(H-K)$ is the reddening arising from extinction. As shown in Figure 12, most of the sources are within this limit. A few sources are above the $P_{\rm max}$ limit, but their uncertainties are large. The near-IR polarization-to-extinction efficiency of the point-like sources in the IRS 2 field is consistent with that caused by aligned dust grains in the dense interstellar medium. Therefore, the near-IR polarizations of these sources are most likely dominated by dichroic extinction from aligned dust grains by a magnetic field, and the intrinsic polarization, if any, does not significantly affect the observed degree of polarization. However, this result does not completely exclude the possibility that some of the sources have intrinsic polarization because depolarization is also possible.
Aperture polarimetry in the $K$ band using a 4$\arcsec$ aperture is reported by Castelaz & Grasdalen (1988), for both RNO-E and RNO-W. The polarization degree of RNO-E was 3.3 $\pm$ 0.3 % with a position angle of 30$\arcdeg$ $\pm$ 3$\arcdeg$, and the polarization degree of RNO-W was 3.5 $\pm$ 0.3 % with a position angle of 7$\arcdeg$ $\pm$ 3$\arcdeg$. In our near-IR data, the polarization degree of RNO-E is 4–8% with a position angle of 91$\arcdeg$–98$\arcdeg$ with smaller errors. The polarization position angle of Castelaz & Grasdalen (1988) is somewhat smaller than our near-IR measurements. It is not clear what caused the difference in these sources.
### Source Classification
The magnetic field structure of molecular clouds can be inferred from the polarization produced by dichroic extinction, provided sources are selected with no intrinsic polarization. YSOs in the cloud can exhibit a substantial degree of intrinsic polarization caused by circumstellar material. Such sources may show a large IR excess which can be identified from multiwavelength photometry.
Figure 13 shows a color–color diagram for all sources detected in all three bands. The diagram is divided into several domains. Based on the location in this diagram, sources can be classified into a few groups (Lada & Adams 1992). The area near the locus of main-sequence/giant stars is called domain A0. Sources in domain A0 are either field stars (dwarfs and giants) or pre-main-sequence (PMS) stars with little IR excess (weak-lined T Tauri stars and some classical T Tauri stars) and with little reddening. There is a clear gap just above domain A0, and the area above this gap in the direction of the reddening vector is called domain Ar. Sources in domain Ar are either field stars or PMS stars with little IR excess and with substantial reddening. Domain B is the area next to domain Ar in the direction of higher $H$ – $K_s$ (to the right) and above the locus of classical T Tauri stars. Sources in domain B are PMS stars with IR excess emission from disks. Domain C is the area next to and to the right of domain B. Sources in domain C are IR protostars or Class I sources. Herbig AeBe stars tend to occupy lower parts of domains B and C. This classification based on the color–color diagram, however, is far from perfect. A certain fraction of classical T Tauri stars may reside in domains A0 or Ar, some protostars can be found in domain B, and some extremely reddened AeBe stars may be found among protostars (Lada & Adams 1992). This “contamination” will eventually contribute to the uncertainty in statistical quantities derived from the classification, but the estimation of this uncertainty is beyond the scope of this paper.
There are 125 sources in domain A0, and they are collectively called group A0. They are either foreground stars or those seen along the line of sight with little extinction. One hundred fifty one sources were found in domain Ar. These sources (group Ar) are either background stars or PMS stars in the Monoceros OB1 cloud. They are the most useful sources for the study of the magnetic fields in the cloud (see Section 4.4 of Kwon et al. 2010).
Forty-five sources were found in domain B. These sources (group B) may be PMS stars associated with the Monoceros OB1 cloud.
Three sources (sources 255, 307, and 314) are located in domain C. AR 6B (source 314) is considered a variable star of FU Ori type (Aspin & Reipurth 2003).
### Magnetic Field Strength
The sources in group Ar (Figure 13) are best for studying the magnetic fields in the molecular cloud because they are likely to have little intrinsic polarization, and the high reddening can lead to polarization through dichroic extinction that is selective attenuation of different components of the electric vector when light passes through a medium in which the grains are aligned by a magnetic field. Since there is significant difference between the vectors of the cluster core and outer parts in Figures 9–11, only the sources in the cluster core were chosen to estimate the magnetic field strength of the IRS 2 region, among group Ar sources. In Figure 10, contours of 870 $\mu$m continuum emission (Williams & Garland 2002) were superposed on the $H$ polarization vector map. In the following discussion, we designate the sources within blue contours (dense cluster of IRS 2 region) as that of cluster core, and the sources outside blue and red contours as that of the perimeter of the cluster core. Note that the sources within red contours (dense cluster of IRS 1 region) are excluded from following discussions for magnetic fields.
Figure 14 shows the histogram for the $H$ polarization position angles of the group Ar sources in both the cluster core and perimeter. The dispersion of the polarization position angles toward the IRAS 12 S1 core, covering a region of about a 4$\arcmin$ $\times$ 4$\arcmin$, is larger than that of the point-like sources in the perimeter of the cluster core. For the point-like sources of the dense cluster, the mean angle is 81$\arcdeg$ with a standard deviation of 29$\arcdeg$. For the sources in the perimeter of the cluster core, the mean angle is 58$\arcdeg$ with a standard deviation of 24$\arcdeg$. The difference between the cluster core and the perimeter is 23$\arcdeg$, which is smaller than the standard deviation of each of them. Note that there is a systematic gradient in the magnetic field orientation over the imaged field, even though there is relatively large dispersion of polarization angles in the cluster core.
Although the polarization measurement does not provide a direct estimate of the magnetic field strength at each data point in the image, a rough estimate over a large region is possible by statistically comparing the dispersion of the polarization orientation with the degree of turbulence in the cloud (Chandrasekhar & Fermi 1953). Assuming that velocity perturbations are isotropic, the strength of the magnetic field projected on the plane of the sky can be calculated by $$B_p = {\cal Q} \sqrt{4\pi\rho}\ {{\delta v_{\rm los}}\over{\delta\theta}},$$ where ${\cal Q}$ is a factor to account for various averaging effects, $\rho$ is the mean density of the cloud, $\delta v_{\rm los}$ is the rms line-of-sight velocity, and $\delta\theta$ is the dispersion of the polarization angles. Ostriker et al. (2001) suggested that ${\cal Q}$ $\sim$0.5 is a good approximation when the angle dispersion is small ($\delta\theta \lesssim$ 25) from numerical simulations, and Houde (2004) also suggested that a correction factor of $\sim$0.5 is appropriate in most cases when the magnetic field is not too weak. Since the magnetic field seems to be ordered over the IRS 2 region, the magnetic field is expected to be strong and we adopt a correction factor of 0.5 to estimate the magnetic field strength, even though the angle dispersion (29) in the cluster core is a little larger than the 25$\arcdeg$ suggested by Ostriker et al. (2001). From observations of cluster forming clumps in the N$_2$H$^+$ (1–0), Peretto et al. (2006) estimated a mean H$_2$ density of $6.0 \times 10^4$ cm$^{-3}$ assuming a spherical volume and a mean clump diameter of 0.9 pc. They also estimated a 3D velocity dispersion of 1.4 km s$^{-1}$, assuming isotropic motion. Then, from equation (12), the strength of the magnetic field projected on the plane of the sky is derived to be $B_p \approx$ 100 $\mu$G. The uncertainty in this estimate is rather large because the observed IRS 2 field is only a part of the Monoceros OB1 cloud and because the mean density $\rho$ used in above equation is small due to imperfect coupling between ions and neutrals. It has been suggested that the gas densities are too low toward the edge of the CO outflow in the IRS 2 region (Wolf-Chase & Walker 1995). Thus, it should be taken as an order-of-magnitude estimate. The estimated magnetic field strength of the IRS 2 region is similar to that of other molecular clouds (20–200 $\mu$G) derived using the Chandrasekhar–Fermi method (e.g., Andersson & Potter 2005; Poidevin & Bastien 2006; Alves et al. 2008; Kwon et al. 2010; Sugitani et al. 2010).
### The Role of Magnetic Fields in Cluster Formation
Previous observations suggest that the local magnetic field associated with molecular clouds correlates with the Galactic magnetic field (Kobayashi et al. 1978; Dyck & Lonsdale 1979). This indicates that the Galactic magnetic field is likely to play an important role in the formation and evolution of molecular clouds. For example, Dyck & Lonsdale (1979) compared infrared aperture polarization for 31 protostellar sources buried in compact HII regions and molecular clouds with the interstellar polarization in the vicinity of the sources as determined from field stars. There was a strong tendency for the infrared and interstellar polarization vectors to be approximately parallel, with sixty-five percent of the sample (29 sources) having position angles of polarization lying within 30$\arcdeg$ of the average local interstellar polarization direction. NGC 2264 IRS 1 was one of the exceptions, with the infrared polarization vector closer to the orientation of the galactic plane than to the average interstellar polarization direction, but is not close to either. In the following, we compare near-infrared and optical polarization data to reveal the relationship between the local and Galactic magnetic fields in the IRS 2 region.
Interstellar polarizations in a (20$\arcdeg$ $\times$ 20$\arcdeg$) region centered around IRS 2, were taken from the optical polarization data in the Heiles catalogue (2000). [^1]. Figure 15 shows the relations between both polarization degrees and polarization angles against distances for the optical data. It shows larger polarization degrees at larger distances, as expected, as well as larger dispersion. For the position angles there is a peak in the angles beyond the distance of 760 pc in contrast to the sources in front of that. Figure 16 shows histograms of the polarization position angles for the local (our near-IR data) and Galactic magnetic field, and Figure 17 shows the interstellar polarization vector map of the Galactic magnetic field from optical polarization data. The direction of the Galactic magnetic field runs roughly perpendicular to the direction of the local magnetic field. In other words, the local magnetic field is not aligned with the Galactic magnetic field. Therefore, some other factors other than the Galactic magnetic field may have played a role in the formation and evolution of the cluster core in this region. It is interesting to note that the IRAS 12 S1 core may be rotating (Wolf-Chase & Walker 1995) and that the suggested axis of rotation in NGC 2264 coincides with the axis of the Galaxy (Dyck & Lonsdale 1979).
Very recently, Nakamura & Li (2010) performed numerical simulations of clustered star formation in parsec-scale dense clumps with different magnetic field strength and discussed the role of magnetic fields in cluster formation. The initial cloud was a centrally condensed spherical clump with a simulation box length of 2 pc. The central density was 5.0 $\times$ 10$^{-20}$ g cm$^{-3}$ (1.5 $\times$ 10$^4$ cm$^{-3}$), yielding a total clump mass of 884 M$_{\sun}$. These physical quantities appear to be in a reasonable agreement with those of the observed IRS 2 field. They also assumed an isothermal equation of state with a sound speed of 0.23 km s$^{-1}$ for a mean molecular weight of 2.33 and a gas temperature of 15 K. In addition, the magnetic field strength was specified by the plasma beta $\beta$, the ratio of thermal pressure to magnetic pressure at the clump center, through $B_0 = 25.8 \beta^{-1/2}$ $\mu$G. In their numerical models it was found that the filamentary clumps are almost perpendicular to the global magnetic fields when the cluster forming clumps are created out of a strongly-magnetized cloud. In contrast, when the magnetic field associated with the parent molecular clouds is dynamically weak, the random component parallel to the filamentary clumps tends to be stronger preferentially in the dense parts when the turbulent flows play a role in the formation of the dense clumps. The observed, spatially ordered magnetic field seems consistent with their strongly-magnetized model for which the cloud dynamics is regulated by the strong magnetic field. Therefore, our comparison between local and global magnetic fields may indicate that the cluster forming clumps of the IRS 2 region including the protostars in the distribution corresponding to the Jeans length may be created out of such a cloud, in the presence of strong magnetic fields. However, even in a magnetically subcritical cloud, partly distorted field components exist. This could explain the difference in the polarization position angles between the cluster core and the perimeter.
### Protostellar Outflows and Magnetic Fields
Most cloud cores are magnetically supercritical (Nakano 1998, 1999; Crutcher 1999; Shu et al. 1999), and outflows are closely related to the magnetic field in the cloud as well as the accretion process, angular momentum transportation.
Molecular outflows in the Monoceros OB1 cloud were reported previously by Margulis & Lada (1988). The direction ($\sim$60$\arcdeg$) of the CO molecular outflow measured by connecting the peaks of the red- and blue-shifted components is well aligned with that of our observed local magnetic field in the IRS 2 region ($\sim$60$\arcdeg$). Based on Wolf-Chase et al. (2003), if IRAS 12 S1, a multiple source, is the main contributor to this outflow, then the outflow momentum flux is roughly an order of magnitude greater than expected for a Class I object of comparable bolometric luminosity. It means that IRAS 12 S1 consists of young Class 0 objects (Bontemps et al. 1996).
Outflow formation has been simulated by many researchers, as has protostellar collapse (e.g., Tomisaka 1998, 2002; Machida et al. 2005a, 2005b, 2006; Matsumoto & Tomisaka 2004; Matsumoto et al. 2006; Ziegler 2005; Banerjee & Pudritz 2006; Fromang et al. 2006). According to these simulations, outflow axes tend to be aligned parallel to the local magnetic fields and perpendicular to rotating disks, when the magnetic fields associated with parent cloud cores are not weak. This implies that the magnetic field strength in the IRS 2 region may be strong enough to align the outflows in the direction of the local-scale magnetic field.
In addition, the NGC 2264 cluster contains stars at different stages of formation, and the star formation process appears to have continued for a few Myrs (Dahm & Simon 2005), corresponding to more than 10 free-fall times for a gas clump with a few 10$^4$ cm$^{-3}$. In other words, the star formation in NGC 2264 is likely to be relatively slow. Such slow star formation could be explained by protostellar outflow feedback that can impede the global gravitational collapse by regenerating supersonic turbulence. Recent numerical simulations have shown that the star formation rate is significantly reduced to an observed low level particularly in the presence of moderately-strong magnetic field. (e.g., Li & Nakamura 2006; Nakamura & Li 2007, 2010; Wang et al. 2010).
### Comparison with Submillimeter Polarimetry
An interesting issue is how useful near-IR polarimetry is in tracing the magnetic field structure of dense clouds. Goodman et al. (1995) suggested that the polarizing power of dust grains may drop in the dense interior of some dark clouds and that near-IR polarization maps of background sources may be unreliable. However, the relevant physics is surprisingly complex (Lazarian 2007) and there are various pieces of observational evidence and theoretical explanations for aligned grains in dense cloud cores (Ward-Thompson et al. 2000; Cho & Lazarian 2005; Hough et al. 2008). To compare the magnetic field direction derived from dichroic polarization at near-IR wavelengths with that from thermal emission polarization at submillimeter wavelengths, it may be a worthwhile experiment to confirm the potential inefficiency of grain alignment in a dense region, even though previous observations have already shown agreement in the magnetic field structures at various wavelengths such as near-, far-IR, or submillimeter wavelengths (Tamura et al. 1996, 2007; Houde et al. 2004; Kandori et al. 2007). Figure 18 shows a comparison with submillimeter polarization vector map (Dotson et al. 2010), and Figure 14 shows histograms of polarization position angles at $H$ and at a wavelength of 350 $\mu$m (Dotson et al. 2010). Interestingly, the distribution of the polarization position angles at 350 $\mu$m is in better agreement with that of the $H$ band in the perimeter of the cluster core than in the cluster core. The mean angle of the polarization position angles at 350 $\mu$m is 60$\arcdeg$.
The general trend derived from the above two wavelengths represents a good agreement with each other, although their spatial scale is somewhat different. Therefore, our results indicate that both near-IR dichroic polarization and submillimeter emission polarization may trace the magnetic field structures associated with the IRS 2 region.
### Wavelength Dependence of Interstellar Polarization
Though the near-IR polarizations of selected point-like sources are most likely dominated by dichroic extinction with no nebulosity, we cannot rule out the possibility of the presence of unresolved reflection nebulae in some of the sources. To discriminate between the contributions from the dichroic extinction and from the scattering, the wavelength dependence of polarization can be measured by calculating the ratio of polarization degrees. In the IR wavelength range, the polarization by dichroic extinction decreases with wavelength, while the polarization by scattering is not a strong function of wavelength (Whittet et al. 1992; Casali 1995). Figure 19 shows the $P_J/P_{K_s}$ and $P_H/P_{K_s}$ ratios for the sources in the IRS 2 region. It is very clear that group A0 and Ar show very different behavior. Within group Ar, the ratios are reasonably constant, and the weighted average values are $P_J/P_{K_s}$ = 2.22 and $P_H/P_{K_s}$ = 1.54 with standard deviations of 0.80 and 0.50, respectively. These values are consistent with the empirical relation $P \propto \lambda ^{-\beta}$ = 1.6–2.0 (Whittet 1992). Therefore, the polarization of group Ar sources can be very well explained by dichroic extinction. Source 11 shows an unusually high ratio in Figure 19 (top panel). The $J$ polarization degree of source 11 is too high with large errors, while the $P_H/P_{K_s}$ ratio is near 1 as expected. Since source 11 is located around boundary, it may be possible that it was caused by a bad pixel in the $J$ band.
In contrast, the $P_J/P_{K_s}$ and $P_H/P_{K_s}$ ratios of group A0 sources are near unity, which is significantly different from those of group Ar sources. Therefore, the polarization mechanism for low extinction sources may be dominated by the circumstellar scattering (see more discussions in Section 5.4 of Casali 1995).
The $P_J/P_{K_s}$ and $P_H/P_{K_s}$ ratios of group B are similar to that of group Ar, and Source 8 also shows an unusually high ratio in Figure 19 (top panel). The $J$ polarization degree is too high with large errors, while the $P_H/P_{K_s}$ ratio is near 1 as expected. It is the same as source 11.
SUMMARY
=======
We conducted deep and wide-field $J$-$H$-$K_s$ imaging polarimetry In the direction of a 8$\arcmin$ $\times$ 8$\arcmin$ region around NGC 2264 IRS 2 in the Monoceros OB1 cloud. The main results in this study are summarized as follows.
1. Our polarization data revealed three IRNCs associated with IRS 2, and several local IRNe. In addition, the illuminating sources of the IRNe were identified with near- and mid-IR sources.
2. Aperture photometry of point-like sources, with 314 detected in all three bands, was used to classify sources using a color–color diagram.
3. Aperture polarimetry of the point-like sources allowed positive detection of 241 sources in at least one of the three bands. Most of the near-IR polarizations of the point-like sources can be explained by dichroic polarization.
4. Sources in group Ar are expected to be either background stars or PMS stars with little intrinsic polarization. The histogram of polarization position angles of group Ar sources in the cluster core has a well-defined peak at $\sim$81$\arcdeg$, which we interpret as the projected direction, on the sky, of the magnetic fields in the IRS 2 region. From the 29$\arcdeg$ standard deviation of the polarization position angles, the strength of the magnetic field projected on the plane of the sky is roughly estimated at $\sim$100 $\mu$G using the Chandrasekhar–Fermi method.
5. By comparing the observed magnetic field lines and those derived from recent numerical simulations, we suggest that the clump including the IRS 2 region is likely to have formed under the influence of a strong magnetic field. The local magnetic field runs roughly perpendicular to the Galactic magnetic field. The significant difference between the local and Galactic magnetic field directions may imply that the local magnetic field is strong enough to control the cloud dynamics. In fact, the direction of the powerful molecular outflow from IRAS 12 S1 is along the local magnetic field, instead of the Galactic magnetic field.
6. The magnetic field direction inferred from our observations appears to be consistent with that inferred from 350 $\mu$m thermal continuum emission polarimetry centered around IRAS 12 S1. The 350 $\mu$m polarization position angles were especially consistent with the group Ar sources in the outer parts of the cluster core.
7. For the group Ar sources, the wavelength dependence of polarization is consistent with the dichroic extinction. Sources in group A0 have a small amount of extinction, and their polarization seems to be caused by the circumstellar scattering. Sources in group B show similar behavior to the group Ar sources in our data.
This work was supported by the Korean Scholarship Foundation. M. T. has been supported by the MEXT, Grants-in-Aid 19204018 and 22000005. This work is based on observations made at the South African Astronomical Observatory (Department of Astronomy, Nagoya University). This work is based in part on observations made with the Spitzer Space Telescope, which is operated by the Jet Propulsion Laboratory, California Institute of Technology under a contract with NASA. This publication makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation. This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France, as well as IRAF and the IDL Astronomy Library.
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[^1]: An agglomeration of stellar polarization catalogs with results for 9286 stars
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{
"pile_set_name": "ArXiv"
}
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---
abstract: 'The Galaxy And Mass Assembly (GAMA) survey furnishes a deep redshift catalog that, when combined with the Wide-field Infrared Explorer ([*WISE*]{}), allows us to explore for the first time the mid-infrared properties of $> 110, 000$ galaxies over 120deg$^2$ to $z\simeq0.5$. In this paper we detail the procedure for producing the matched GAMA-[*WISE*]{} catalog for the G12 and G15 fields, in particular characterising and measuring resolved sources; the complete catalogs for all three GAMA equatorial fields will be made available through the GAMA public releases. The wealth of multiwavelength photometry and optical spectroscopy allows us to explore empirical relations between optically determined stellar mass (derived from synthetic stellar population models) and 3.4 and 4.6 [*WISE*]{} measurements. Similarly dust-corrected H$\alpha$-derived star formation rates can be compared to 12 and 22 luminosities to quantify correlations that can be applied to large samples to $z<0.5$. To illustrate the applications of these relations, we use the 12 star formation prescription to investigate the behaviour of specific star formation within the GAMA-[*WISE*]{} sample and underscore the ability of [*WISE*]{} to detect star-forming systems at $z\sim 0.5$. Within galaxy groups (determined by a sophisticated friends-of-friends scheme), results suggest that galaxies with a neighbor within $100\,h^{-1}$kpc have, on average, lower specific star formation rates than typical GAMA galaxies with the same stellar mass.'
author:
- 'M.E. Cluver, T.H. Jarrett, A.M. Hopkins, S.P. Driver, J. Liske, M.L.P. Gunawardhana, E.N. Taylor, A.S.G. Robotham, M. Alpaslan, I. Baldry, M.J.I. Brown, J.A. Peacock, C.C. Popescu, R.J. Tuffs, A.E. Bauer, J. Bland-Hawthorn, M. Colless, B. Holwerda, M.A. Lara-L[ó]{}pez, K. Leschinski, A.R. L[ó]{}pez-S[á]{}nchez, P. Norberg, M. Owers, L. Wang, S.M. Wilkins'
title: 'Galaxy And Mass Assembly (GAMA): Mid-Infrared Properties and Empirical Relations from [*WISE*]{}'
---
Introduction
============
Spanning almost six decades and counting, large area surveys have revolutionised our view of the structure and evolution of the universe, for example, the Palomar Observatory Sky Survey (POSS II, Reid et al. 1991), the IIIa-J SRC Southern Sky Survey, the Infrared Astronomy Satellite (IRAS) Sky Survey Atlas (Wheelock et al. 1994), 2dF Galaxy Redshift Survey [2dFGRS, @Coll01], the Sloan Digital Sky Survey (SDSS, Abazajian et al. 2003), the [H[i]{}]{} Parkes All Sky Survey (HIPASS, Meyer et al. 2004), the 6dF Galaxy Survey [6dFGS, @Jon04], the 2 Micron All Sky Survey (2MASS, Skrutskie et al. 2006) and the UKIRT Infrared Deep Sky Survey [UKIDSS, @Law07]. Uncovering the properties of large galaxy populations has proved essential to our understanding of how galaxies develop and transform. Increasing sensitivity and scale in the time domain, as with Skymapper and the planned LSST (Large Synoptic Survey Telescope) holds the promise of enormous future scientific returns. In the near future, leading-edge radio telescopes and planned strategic surveys will add a key ingredient to the mix – [H[i]{}]{} and continuum measurements will trace neutral gas reservoirs and activity to unprecedented depth, sky coverage and resolution with WALLABY and EMU on ASKAP (Australian Square Kilometer Array Pathfinder), Apertif-WNSHS, LADUMA [@Hol10] on MeerKAT, and the JVLA.
In order to better understand the key physics at work in star-forming galaxies, and crucially the efficacy of different physical mechanisms, one requires the combination of extended area and sufficient depth in order to capture a cosmologically and evolutionarily significant volume. The GAMA [Galaxy and Mass Assembly, @Driv09; @Driv11] survey provides exactly this laboratory. At its heart, GAMA is an optical spectroscopic survey of up to $\sim$ 300 000 galaxies [@Hop13] obtained at the Anglo-Australian Telescope situated at Siding Spring Observatory. Three equatorial fields (G09, G12 and G15) covering 180deg$^2$ sample large-scale structure to a redshift of z$\simeq 0.5$, with overall median redshift of z$\simeq 0.3$, and two southern fields (G02 and G23). Multiwavelength ancillary data from ultraviolet to far-infrared wavelengths provides comprehensive SED (Spectral Energy Distribution) wavelength real estate. At mid-infrared wavelengths it falls to [*WISE*]{}, the Wide-field Infrared Survey Explorer, to span the near-infrared transition from stellar to dust emission.
Launched in December 2009, [*WISE*]{} surveyed the entire sky in four mid-infrared bands: 3.4 4.6, 12 and 22 [@Wr10]. In the local Universe, the 3.4 (W1) and 4.6 (W2) bands chiefly trace the continuum emission from evolved stars [see, for example, @Meidt12]. The W1 band is the most sensitive to stellar light, typically reaching $L^{\ast}$ depths to $z\simeq 0.5$. The W2 band is additionally sensitive to hot dust; hence, this makes the 3.4 $-$ 4.6 color very sensitive to galaxies dominated globally by active galactic nucleus (AGN) emission [see e.g., @Jar11; @Ster12]. The 12 (W3) band is broad [see @Wr10 for relative system response curves] and can trace both the 9.7 silicate absorption feature, as well as the 11.3 PAH (polycyclic aromatic hydrocarbon) and $[$Ne[ii]{}$]$ emission lines. Lastly, the W4 band traces the warm dust continuum at 22, sensitive to reprocessed radiation from star formation and AGN activity. [*WISE*]{} is thus optimally suited to study the diverse emission mechanisms of galaxies.
[*WISE*]{} is confusion-noise-limited – the structure of the background (e.g. distant galaxies and scattered light) increases noise by contributing flux – and under these conditions point spread function (PSF) profile-fitted photometry performs robustly [@Mar12]. For unresolved sources, this is the adopted method of obtaining photometry and accordingly the [*WISE*]{} Source Catalog is optimised and calibrated for point sources. The angular resolution of [*WISE*]{} frames, used for determining the profile photometry measurement (keyword [*w$\star$mpro*]{} in the [*WISE*]{} All-Sky Data Release), is 6.1, 6.4, 6.5and 12.0 for W1, W2, W3 and W4, respectively. The [*WISE*]{} public-release data products also includes an image Atlas of tiled mosaics; however, the angular resolution is slightly degraded in the Atlas images because they are convolved with a matched filter to optimize point-source detection. The crucial point here is that the [*WISE*]{} catalogs and images are not well-suited to characterization of resolved sources. Resolved sources will either be detected and measured as conglomerations of several point sources, or have their flux grossly underestimated by the PSF profile. As a consequence, it is left to the community to properly measure sources that are resolved by [*WISE*]{} (i.e., nearby galaxies). In this work we have produced new, better-optimized images to extract resolved galaxy measurements from the GAMA fields to complement the point-source measurements obtained from the public-release catalogs.
A detailed [*WISE*]{} study of several nearby ($<60$ Mpc) galaxies [@Jar13] has highlighted how [*WISE*]{} data can be used in multiwavelength studies of star formation and interstellar medium (ISM) conditions, as well as tracing global properties such as stellar mass and star formation. The work of @Don12 showed the power of combining [*WISE*]{} with a large-area survey (SDSS); they investigated the effect of star formation- and AGN-activity on the [*WISE*]{} properties of $>95, 000$ galaxies, including calibrations using the 12 and 22 luminosities. These empirical calibrations have also been investigated in 22-selected galaxies in SDSS galaxies with $z<0.3$ by @Lee13. [*WISE*]{} colours have proved to be an excellent AGN selection tool [@Jar11; @Ster12; @Ass13], and as a diagnostic for determining the accretion modes amongst radio-loud AGN [@Gur13]. @Yan13 combine [*WISE*]{} and SDSS to provide a phenomenological characterization for [*WISE*]{} extragalactic sources.
In this paper we harness the power of the GAMA survey, and its value-added data products, cross matching GAMA galaxies in two completed GAMA fields (the G12 and G15 regions) to their [*WISE*]{} counterparts. The G09 region is not considered in this analysis, but will be included as part of the GAMA-[*WISE*]{} data release.
GAMA counterparts at low redshift will be predominantly resolved, particularly in the W1 and W2 bands. Unlike previous studies, we include robust measurements of resolved sources in these fields and distinguish between resolved and unresolved systems to determine the photometry most appropriate for a given source. Since GAMA is highly spectroscopically complete to $r_{\rm AB}<19.8$ within the two GAMA regions, we are able to push [*WISE*]{} to higher redshifts than previously possible for wide-field surveys. We explore empirical relationships for stellar mass and star formation rates, the color distribution of [*WISE*]{} sources in GAMA and their behaviour in dense environments. The paper is arranged as follows: in §2 we describe the data used in this analysis, §3 discusses how the GAMA-[*WISE*]{} matched catalog is constructed, §4 contains results derived from the combined surveys and §5 illustrates scientific applications of the catalog. Conclusions are presented in §6.
The cosmology adopted throughout this paper is $H_0 = 70 $kms$^{-1}$ Mpc$^{-1}$, $h={\rm H}_0/100$, $\Omega_M=0.3$ and $\Omega_{\Lambda} = 0.7$. All magnitudes are in the Vega system ([*WISE*]{} calibration described in Jarrett et al. 2011) unless indicated explicitly by an AB subscript.
Data
====
GAMA
----
Our sample is drawn from the G12 and G15 equatorial fields of the GAMA II survey [@Driv09; @Driv11] combining high spectroscopic completeness ($\simeq$ 97%) to a limiting magnitude of $r_{\rm AB}=19.8$, with a wealth of ancillary photometric data. Details of target selection for the survey are outlined in @Bal10 and optimal tiling for the survey in @Robot10. Spectra were obtained primarily with the 2dF instrument mounted on the 3.9-m Anglo-Australian Telescope, and additionally from the Sloan Digital Sky Survey [SDSS DR7; @Ab09]. Reduction and analysis of the spectra is discussed in detail in @Hop13 and Liske et al. (in prep.).
Photometric data for galaxies within the G12 and G15 volumes (ApMatchedCatv05) is drawn from SDSS imaging ($u,g,r,i,z$) as outlined in @Hill11 and VISTA VIKING ($ZYJHK$) as detailed in Driver et al. (in prep.). Photometry is corrected for Galactic foreground dust extinction using @Sch98. Stellar mass estimates (StellarMassesv15) are derived from matched-aperture photometry using synthetic stellar population models from @BC03 as detailed in @Tay11. We take advantage of the completed observations in the equatorial fields and updated redshifts (Baldry et al., in prep.) and stellar masses. Throughout our analysis we only use sources with reliable redshifts i.e. $nQ \ge 3$ [@Driv11].
Star formation rates (SFRs) can be derived from the H$\alpha$ equivalent width, combined with the $r$-band absolute magnitude, and are available for the GAMA phase-I survey ($r_{\rm AB}<19.4$ in G15 and $r_{\rm AB}<19.8$ in G12) as determined in @Gun11 [@Gun13]. Corrections are applied for the underlying Balmer stellar absorption, dust obscuration and fibre aperture effects. To maintain consistency we only use SFRs from @Gun13 derived for galaxies with redshifts matching in both GAMA I and GAMA II ($\simeq 10\%$ were excluded because of this criterion).
[*WISE*]{} Image Construction
-----------------------------
The [*WISE*]{} mission provides ‘Atlas’ Images via its data repository, which are co-added and interpolated from the multiple single exposure frames taken during the survey [@Cut12]. These 1.56$\arcdeg \times 1.56\arcdeg$ mosaics have a 1.375 pixel scale, but the resampling and co-addition method is optimised for point source detection; Atlas images have beam sizes of 8.1, 8.8, 11.0 and 17.5 for the W1, W2, W3 and W4 bands, respectively. To better preserve the native resolution of the single frames we employ a mosaic construction that uses the ‘drizzle’ resampling technique. Variable-Pixel Linear Reconstruction or drizzling co-addition algorithm using a tophat PRF (point response function) can be used to improve the spatial resolution compared to the nominal Atlas Images [@Jar12]. Drizzled image mosaics, 1.3$\times$1.3 degrees in size, were created using the software package ICORE [@Mas13] achieving a resolution of 5.9, 6.5, 7.0 and 12.4 in the 3.4, 4.6, 12 and 22, respectively. Within ICORE, background level offset-matching, flagging and outlier rejection and co-addition using overlap area-weighted interpolation ensures optimal background stability. The number of images that are combined is dependent on the depth of coverage and additional orbits, a function of the field’s location relative to the ecliptic. For the GAMA fields the coverage is typically 12 and 24 frames for the G12 and G15 fields, respectively, where G15 benefits from additional scans in that region arising from multiple epoch [*WISE*]{} orbits.
{width="15cm"}
Cross Match to GAMA G12 and G15
-------------------------------
Galaxies were extracted from the G12 and G15 GAMA II catalogs of observed sources (i.e. TilingCatv41 with redshifts as in SpecAllv21 and $z>0.001$) – 69,693 and 64,822 for G12 and G15, respectively, and cross-matched to the [*WISE*]{} All-Sky Catalog using the NASA/IPAC Infrared Science Archive (IRSA) and a 3 cone search radius. The number of unique objects in each field were 60,645 GAMA matches for G12 and 58,199 for G15, i.e., a 87.0% and 89.8% match rate for G12 and G15, respectively, with a higher match rate in G15 due to the increased [*WISE*]{} depth (from additional scans) and therefore sensitivity. Further details are provided in Table \[match\] and a region of G15 shown in Figure \[fig1\] to illustrate the difference between the optical and mid-infrared sky.
For this analysis, we rely on several parameters output by the [*WISE*]{} All-Sky Catalog; these are listed in Table \[param\] with a brief explanation.
[l l l l l]{}\
\
Band & Detections & SNR$^{1}$ $>2$ & SNR$ >5$ & Upper Limits$^{\dagger}$\
W1 & 100% & 100% & 99.9% & 0%\
W2 & 100% & 96% & 77% & 4%\
W3 & 99% & 47% & 17% & 52%\
W4 & 98% & 19% & 2% & 79%\
\
\
\
Band & Detections & SNR$>2$ &SNR$ >5$ & Upper Limits$^{\dagger}$\
W1 & 100% & 100% & 99.9% & 0.01%\
W2 & 100% & 98% & 88% & 2%\
W3 & 99% & 58% & 29% & 42%\
W4 & 98% & 19% & 3% & 78%\
\
\
\
\
[l l]{}\
[*designation*]{} & unique [*WISE*]{} source designation\
[*ra*]{} (deg) & Right Ascension (J2000)\
[*dec*]{} (deg) & Declination (J2000)\
[*sigra*]{} (arcsec) & uncertainty in RA\
[*sigdec*]{} (arcsec) & uncertainty in DEC\
[*w$\star$mpro*]{} (mag) & instrumental profile-fit photometry magnitude\
[*w$\star$sigmpro*]{} (mag) & instrumental profile-fit photometry uncertainty\
[*w$\star$rchi2*]{} & instrumental profile-fit photometry reduced $\chi^2$
\
[*nb*]{} & number of blend components used in each fit\
[*na*]{} & active deblend flag (=1 if actively deblended)\
[*xscprox*]{} (arcsec) & distance between source center and XSC$^{\dagger}$ galaxy\
[*w$\star$rsemi*]{} & (scaled) semi-major axis of galaxy from XSC$^{\dagger}$\
[*w$\star$gmag*]{} & elliptical aperture mag of extracted galaxy\
\
\
\
Sources Resolved by [*WISE*]{}
==============================
A key feature of a catalog containing [*WISE*]{} photometry is determining which of the galaxies are resolved and then ensuring their fluxes are measured robustly. A broad indication is given by the reduced $\chi^2$ ([*w$\star$rchi2*]{}) of the profile-fit solution, where high values of the [*w$\star$rchi2*]{} parameter indicate that the wpro algorithm measurement is underestimating the flux of the source. Sources with [*w$\star$rchi2*]{} $>2$ are often resolved, with [*w$\star$rchi2*]{} $>5$ usually indicating well-resolved systems.
Unfortunately [*w$\star$rchi*]{} is not by itself a robust measure of ‘resolvedness’, particularly where sources have a low signal to noise. The ‘reduced’ metric tends to unity as noise begins to dominate the measurement. Moreover, the [*w$\star$mpro*]{} fitting process can be fooled for sources with relatively bright cores that are just being resolved by [*WISE*]{} – this happens with 2MASS compact extended sources which have low [*w$\star$rchi2*]{}, but the [*w$\star$mpro*]{} and isophotal photometry can be systematically offset. Even though 2MASS has superior (2$\times$) spatial resolution compared to [*WISE*]{}, with its greater sensitivity [*WISE*]{} is able to resolve many nearby, relatively small galaxies. Therefore, all 2MASS Extended Source Catalog [XSC, @Jar00] sources should be tested to determine if they are resolved by [*WISE*]{} (see §\[rfuzzy\]). For galaxies not in the 2MASS XSC, values of [*w$\star$rchi2*]{} $\ge 2$ should be used as an initial selection for resolved objects. Caution is required, however, since blended objects will also satisfy this criterion and can be a source of false positives, notably when the stellar confusion is significant.
Source Characterization of Potentially Resolved Sources
-------------------------------------------------------
Sources are measured using custom software adapting tools and algorithms developed for the 2MASS XSC [@Jar00] and [*WISE*]{} photometry pipelines [@Jar11; @Cut12; @Jar13]. The process is semi-automated in that photometry measurements are automated, but each result is assessed by visual inspection with intervention where necessary.
The first step is to remove point sources by PSF subtraction which preserves the structure of the background. If necessary surrounding contaminating sources are masked (e.g., bright stars). The local background is estimated from pixel value (trimmed average) distribution that lies within an elliptical annulus located just outside of an ‘active region’ which represents the image area that contains measurable light from the galaxy. The active region is not initially known, but it determined through successive iteration of the characterisation process until convergence is reached.
The galaxy is modeled using an ellipsoid built from azimuthally averaging the (background-subtracted) surface brightness; any masked pixels are recovered using a weighted combination of the local background (to the masked source) and the galaxy model. The best fit axis ratio and ellipticity are determined using the 3$\sigma$ isophote and the galaxy shape is defined by this isophote and is assumed to be fixed at all radii. The primary ‘isophotal’ photometry (W$\star$iso) is then measured from the 1$\sigma$ (of the background RMS) elliptical isophote, capturing over 90% of the total light (see Jarrett et al 2013). Other measurements include a double Sérsic fit, to the inner galaxy region (i.e., the bulge) and the outer region (i.e., the disk), thus allowing estimation of the total integrated flux that extends beyond the 1$\sigma$ isophote. Since [*WISE*]{} is confusion-noise limited, we track the photometry using a curve-of-growth table. Where a large mismatch occurs between the isophotal radius and the radius where the change in flux is less than 2% (the ‘convergence point’), the size of the active region is automatically decreased. This usually occurs where background levels are elevated due to a neighboring bright source which contaminates the background level. The process is iterated to adjust the active region until the source measurements have converged.
Once all the sources in a field are measured and modeled accordingly, the process is rerun to effectively perform galaxy-galaxy deblending. Where a measurement exists for a galaxy when measuring an adjacent source, the galaxy model is subtracted in addition to neighboring point sources. This mitigates contamination from nearby resolved sources.
The Resolved Parameter [*Rfuzzy*]{} {#rfuzzy}
-----------------------------------
The [*WISE*]{} bands at 3.4 and 4.6 achieve the best spatial resolution, at 6.1 and 6.4, respectively, and capture extended light arising from evolved stars in the bulge and disk regions. Since [*w1rchi2*]{} cannot discriminate well between faint, resolved and unresolved systems, we use the 2nd order intensity-weighted moment to describe the shape (major and minor axis) of the object and measure the intensity-weighted moment of the object on either side of the main axis.
If the object is symmetric and resolved on both sides of the major axis (as with a nominal resolved source), then each moment will have a resolved signature (i.e., the moment is large compared to a point source). If the object is, for example, a star blended with a galaxy, then their moments will be different (one half will have a moment that is point-like, the other resolved). Thus, taking the minimum moment value between the two halves, determines whether the source is resolved. Fuzzy objects will have a large moment regardless of which major axis half that is measured; stars or double stars will have a minimum moment that appears to be unresolved. The performance of the Rfuzzy parameter is illustrated in §\[rfuzz\_sec\] of the appendix.
Catalog of Resolved Sources
---------------------------
Isolating the resolved sources using the Rfuzzy parameter (in W1) gives 1,390 and 1,368 sources in G12 and G15, respectively, i.e. $2-3$% of the total [*WISE*]{} matched sources in these fields. It is important to bear in mind that despite the resolution of the W1 and W2 bands, [*WISE*]{} is far more sensitive compared to, for example, 2MASS and will detect the extended light profiles from nearby galaxies. The typical W1 1$\sigma$ isophotal radius is more than a factor of $\simeq$2 in scale compared to the equivalent 2MASS K$_{\rm s}$-band isophotal radius. An illustration of this is shown in Figure \[res\] where the 2MASS XSC-derived scaled aperture photometry ([*w$\star$gmag*]{}) are compared to the isophotal photometry of resolved sources in the GAMA G12 and G15 fields. The 2MASS XSC-derived magnitudes underestimate the inferred flux due to the increased sensitivity of [*WISE*]{}. Note that no star subtraction or deblending has been attempted when measuring the [*w$\star$gmag*]{}s and contamination may be present; as a consequence, these measurements should only be used as a last resort (employing the offset shown in Figure \[res\] to obtain the correct galaxy flux). The behavior of resolved sources is further explored in §\[resphot\] of the Appendix.
GAMA-[*WISE*]{} Catalog Photometry
----------------------------------
For point sources, the primary photometry are the profile fit measurements ([*w$\star$mpro*]{}), and for well-resolved sources the isophotal photometry (§3.1). When the source is not well resolved, or the S/N (signal-to-noise) is low, the following steps are used to choose the best photometry:
- [*w$\star$mpro*]{} photometry is used when the S/N is low, as measured by the isophotal aperture process for a given band: S/N of 10, 10, 15 and 15 for W1, W2, W3 and W4, respectively.
- If the Rfuzzy parameter is false (measured in the W1 band), classify source as unresolved and use [*w$\star$mpro*]{} photometry for bands W1 and W2.
- If [*w3rchi2*]{} $<$ 2, classify source as unresolved and revert to the [*w3mpro*]{} photometry.
- If [*w4rchi2*]{} $<$ 2, classify source as unresolved and revert to the [*w4mpro*]{} photometry.
- W1 is the most sensitive [*WISE*]{} band. To obtain an accurate W1$-$W2 color for galaxies resolved in W1 and W2, the following steps are employed. Firstly, an accurate W1$-$W2 color is determined using a matched aperture derived from the smaller of the two isophotal radii (which is usually W2). Then, from the isophotal magnitude of the W1 band, the corresponding W2 magnitude is determined, thereby reflecting the sensitivity of the W1 band. This is also done for the W4 band using W3 in the same way as W1, if sources are resolved at these wavelengths, which in GAMA is rare (only 16 sources in G12 and G15). We note that the W1 and W3 bands cannot be matched in this way since the flux at these wavelengths is produced by different physical processes (evolved stars vs. PAH features).
Aperture corrections are applied to isophotal measurements as detailed in the Explanatory Supplement to the [*WISE*]{} All-Sky Data Release Products[^1].
In Figure \[sens\] we plot the signal-to-noise ratio (SNR) for the G12 and G15 fields; due to the additional $\simeq$2$\times$ frame coverage the G15 field has higher signal-to-noise detections compared to G12 at a given W1 magnitude. The magnitude sensitivity limit in the W1 band for the G12 field is 16.6 (10-$\sigma$), 17.3 (5-$\sigma$) and for G15, 17.0 (10-$\sigma$) and 17.7 (5-$\sigma$). The combined fields have a 10-sigma magnitude sensitivity limit of 16.6 or 71 $\mu$Jy. The redshift distribution of sources in the two fields (excluding upper limits) is shown in Figure \[cat1\]. Since the G15 field is more sensitive, the peak of the distribution is shifted to a slightly higher redshift compared to G12. The two fields show very similar redshift distributions for $z<1$ (Figure \[cat1\]b), although the G12 matches have relatively more high-$z$ sources.
![The [*WISE*]{} W1 band (3.4) signal-to-noise in the W1 band as a function of magnitude for G12 and G15 sources (resolved and unresolved sources). Note that G15 has better overall coverage and thus achieves greater sensitivity in luminosity.[]{data-label="sens"}](fig3.eps){width="8cm"}
Blending {#blend}
--------
Point sources are both passively and actively deblended by the [*WISE*]{} pipeline and the [*WISE*]{} All-Sky catalog provides the [*na*]{} and [*nb*]{} blending flags where a value of “0" indicates no active deblending has been performed, and “1" if sources have been deblended; deblending introduces an additional uncertainty on the measurement provided (nb is the number of components that were deblended).
However, an additional flag is needed to indicate where [*WISE*]{} cannot distinguish between GAMA sources (one [*WISE*]{} source for multiple GAMA sources) and where contamination is expected to be high since more than one GAMA source lies within the [*WISE*]{} beam. As such, we use an additional blending flag as determined by the proximity of neighboring GAMA-[*WISE*]{} sources. Sources within 5 are viewed as a catastrophic blend due to the size of the [*WISE*]{} beam and multiple GAMA sources will probably have the same [*WISE*]{} source matched to it and/or have a highly uncertain match. A source within 15 indicates a potential blend or contamination and is also flagged. For the analysis in this study we remove all flagged sources from our sample ($\approx$ 6% of G12 and G15) to ensure the cleanest [*WISE*]{} photometry, but may bias the sample against the densest environments.
Rest Frame Colours
------------------
Rest-frame colours are determined using the GAMA redshift by building an SED (Spectral Energy Distribution) combining the optical, near-infrared and mid-infrared flux densities and fitting to an empirical template library, consisting of 126 galaxy templates, of local well-studied and morphologically diverse galaxies [e.g. SINGS, the [*Spitzer*]{} Infrared Nearby Galaxy Survey @Ken03] from @Br13. The templates are constructed from optical and [*Spitzer*]{} spectroscopy, with matched aperture photometry from [*GALEX*]{} [@Mar05], [*Swift*]{} [UVOT; @Rom05], SDSS, 2MASS, [*Spitzer*]{} [@Wer04] and [*WISE*]{}, combined with MAGPHYS [@dacun08]. The [*WISE*]{} relative spectral response curves are from @Jar11 and are available as part of the [*WISE*]{} Explanatory Supplement.
The best template fit is determined by minimising the function that describes the difference between the flux density in the measured band, and the model-template flux density. Each resulting best-fit is assigned a normalised score derived from the reduced $\chi^2$ minimisation, with the most consistent fits having values of $<2$ and relatively poor fits with scores of $>3$ . In the sections that follow we use only those sources with a score $\le2$, although the fitting accuracy is minimally important for nearby galaxies (i.e. with low rest-frame color corrections).
Results
=======
[*WISE*]{} Colours of the G12 and G15 Fields
--------------------------------------------
As shown by @Jar11, the 3.4, 4.5 and 12 bands of [*WISE*]{} can be combined in a mid-infrared color-color diagram (Figure \[col\]); the shortest bands are sensitive to the evolved stellar population and hot dust, therefore their color indicates increased activity (AGN or starburst). The 12 band is dominated by the 11.3 PAH, as well as the dust continuum, sensitive to star formation. This color-color diagnostic is therefore useful to separate galaxy populations, particularly old stellar population-dominated, star-forming and systems dominated by AGN-activity [@Jar11; @Ster12]. In Figure \[col\]a the observed colors of the G12 and G15 are shown, and the k-corrected version, Figure \[col\]b, shows the distribution of the rest-frame colors. Both fields have similar distributions, biased towards star-forming systems. This is not surprising given the optical selection of the [*GAMA*]{} sample. Heavily obscured galaxies, notably Ultra-luminous Infrared Galaxies (ULIRGS) are absent due to the insensitivity (selective extinction) of SDSS and GAMA optical catalogs. Systems globally dominated by AGN-activity also appear sparse within the sample; this is consistent with the findings of @Gun13 where spectroscopically-identified AGN make up $<20\%$ of the GAMA I emission-line catalog [see also @Lar13].
Aggregate Stellar Mass Estimation
---------------------------------
The 3.4 band of [*WISE*]{} is dominated by the light from old stars and can be used as an effective measure of stellar mass [@Jar13]. To explore this further, we calculate the ‘in-band’ luminosity for W1, i.e. the luminosity of the source as measured relative to the Sun in the W1 band, and use the stellar masses for GAMA as determined by @Tay11 to determine a mass-to-light (M/L) ratio. These stellar masses are best constrained for $z< 0.15$ [@Tay11] and we apply this cut to our sample. As shown by @Jar13, the stellar mass-to-light has a linear trend with [*WISE*]{} W1$-$W2 and W2$-$W3 color, reflecting systematic M/L differences between passive and star-forming systems.
In order to empirically calibrate a relation we require high signal-to-noise measurements and additionally use only the rest-frame W1$-$W2 color since the detection rate is much higher in W2 compared to W3. We apply a S/N cut of 13.5 in W1 and W2 and remove known AGN [from the GAMA I spectroscopy measurements of @Gun13] and also systems with [*WISE*]{} colors consistent with AGN activity dominating their global colors (W1$-$W2$\ge 0.8$) as discussed by @Ster12.
The first population we investigate are resolved, low-$z$ galaxies, shown in Figure \[sm1\]a. We illustrate the danger of a least-squares minimisation fit compared to a bivariate-Gaussian, maximum-likelihood (or ‘best’ fit) line [@Tay13; @Hogg10]. Throughout the paper we plot both the least-squares and maximum-likelihood lines to show the large discrepancies that may exist; we provide relations for the maximum-likelihood (or best fit) only. We find very good agreement with the M$_{\rm stellar}$/L$_{\rm W1}$ relation of @Jar13 based on their relatively small sample of 17 galaxies and with stellar masses effectively derived from 2$\micron$ $K_s$ band photometry. Our relation for resolved low-$z$ sources is:
$${\rm log_{10}}\, M_{\rm stellar}/L_{\rm W1} = -2.54({\rm W}_{3.4\mu m} - {\rm W}_{4.6\mu m}) - 0.17, \\$$
with $$L_{\rm W1}\ ( L_{\odot}) = 10^{ -0.4 (M - M_{\rm Sun})}$$\
where M is the absolute magnitude of the source in W1, M$_{\rm Sun} = 3.24$, and ${\rm W}_{3.4\mu m} - {\rm W}_{4.6\mu m}$ reflects the rest-frame color of the source.
For comparison to the resolved sample, in Figure \[sm1\]b we show all sources that meet our S/N selection criterion and redshift cut, which shows the distribution to be shifted to “warmer" W1$-$W2 colors i.e. signifying systems with increased activity such as star-formation or low-power AGN. In addition, the contours suggest that the distribution is made up of two populations, most probably due to passive galaxies having a larger mass-to-light ratio than disk-dominated and dwarf galaxies. We investigate this further in Figure \[sm2\] where we make a color-distinction based on the W3 measurement. By selecting galaxies with W2$-$W3$\ge 1.5$ (see Figure \[col\]) we choose systems that are most likely dominated by star formation. These selected types show a clear trend with mass-to-light (Figure \[sm2\]a), but lie parallel and offset to the relation of @Jar13. But now including the passive galaxies, Figure \[sm2\]b, we see a clear clustering around a fixed mass-to-light ratio of $\simeq 0.7$.
We conclude that the best-fit for our entire sample is:
$${\rm log_{10}}\, M_{\rm stellar}/L_{\rm W1} = -1.96({\rm W}_{3.4\mu m} - {\rm W}_{4.6\mu m}) - 0.03 \\$$
\
and for star-forming (lower mass-to-light) systems only:
$${\rm log_{10}}\, M_{\rm stellar}/L_{\rm W1} = -1.93({\rm W}_{3.4\mu m} - {\rm W}_{4.6\mu m}) - 0.04 \\$$
This likely explains the shift observed in the relation for resolved sources and the entire field sample. The resolved sources are relatively nearby and will be a mix of passive and star-forming systems. However, at higher redshifts we add relatively more star-forming galaxies (higher infrared luminosity) causing a larger spread in W1$-$W2 color. Notably galaxies with higher star formation rates will have a larger W1$-$W2 color due to more hot dust emission giving a brighter W2 measurement. Also, at higher redshifts we will preferentially detect galaxies with higher star formation rates (see next section). It should be noted that AGN activity would have a similar effect (as shown by the upward trend in Figure \[col\]b) and some contamination from nuclear activity is inevitable.
[{width="8cm"}]{}
In Figure \[sm3\] we plot the residuals of the GAMA stellar masses (used to calibrate the WISE relation in equation 2) and the WISE-derived values themselves. The mass estimates agree within a factor of 1.2 at $10^{10}$M$_\odot$ (with a standard deviation of 0.5), within a factor of 0.98 at $5\times10^{10}$M$_\odot$ (with a standard deviation of 0.4) and a factor of 0.96 at $10^{11}$M$_\odot$ (with a standard deviation of 0.4). For stellar masses $>10^{10}$M$_\odot$ the WISE-derived masses appear overall lower than the GAMA stellar masses. This is probably unsurprising given that the sample is dominated by star-forming galaxies (see Figure \[col\]) and the WISE 3.4 band is sensitive to the light from evolved stars in passively evolving galaxies.
Star Formation Rate Comparisons
-------------------------------
### 22 Warm Dust Continuum
[*IRAS*]{}, [*ISO*]{} and [*Spitzer*]{} revolutionised our understanding of dust emission as a tracer of star formation. In the mid-infrared, the [*Spitzer*]{} MIPS 24 band measures the warm dust continuum excited by hot, young stars and is therefore sensitive to recent star formation, as well as AGN-activity. Numerous studies have investigated its stability as a measure of SFR [see e.g. @Al06; @Cal07; @Riek09] and we are able to transfer this understanding to [*WISE*]{} and its 22 band.
We determine the luminosity density ($\nu$L$_{\nu}$) for the 22 band and normalise by the total solar luminosity (L$_{\odot}$= 3.839$\times$10$^{26}$W). The W4 band is the least sensitive [*WISE*]{} band, but we impose a S/N cut of 7 to ensure high quality photometry. Sources flagged as AGN based on optical spectroscopy diagnostics are removed from the sample. Cross-matching these sources with star formation rates available for GAMA I, using sources with SFR between 0.1 and 100 M$_{\odot}$yr$^{-1}$ [see @Gun13], yields the best-fit relation shown in Figure \[sf2\]. The optical star formation rates are determined from H$\alpha$ equivalent widths applying an extinction-correction based on the Balmer decrement [full details can be found in @Gun13] and are sampled to $z<0.35$ (i.e. beyond which H$\alpha$ is redshifted out of the observed spectral range). Galaxies where the H$\alpha$ line is contaminated by atmospheric O$_2$ (A band) absorption, in the redshift range 0.155$< z < $ 0.170 are not included in the sample as recommended in @Gun13. The points in Figure \[sf2\] are color-coded according to Balmer decrement and on average higher infrared luminosities correspond to higher Balmer decrements (i.e., increased dust obscuration). However, without detailed knowledge of the dust geometry this is merely a crude comparison [see also @Wij11a; @Wij11b], and as we discuss below, biases due to extinction are probably in play at higher redshifts. Balmer decrement has also been shown to be correlated with stellar mass, stellar mass surface density and metallicity [see for example, @Bos13]
![H$\alpha$-derived star formation rates as a function of $\nu$L$_{22\mu m}$ luminosity color-coded by Balmer Decrement (H$\alpha$/H$\beta$) from @Gun13.[]{data-label="sf2"}](fig9.eps){width="8cm"}
Figure \[sf2\] also shows the linear fit of @Jar13, where the [*WISE*]{} 22 -based SFR relation was calibrated using [*Spitzer*]{} 24 photometry and the relation of @Riek09. This fit is somewhat steeper, likely explained by the distinction made by @Riek09 based on total infrared luminosity of the source and the relatively small number of sources in the @Jar13 sample (but which also included Local Group dwarf galaxies). The best fit to the GAMA sample distribution is:
$${\rm log_{10}\, SFR}_{{\rm H}\alpha} (M_\odot/{\rm yr}) = 0.82\, {\rm log_{10}}\, \nu L_{22\mu \rm m} (L_\odot) -7.3 \\$$
### 12 ISM Tracer
Infrared-luminous sources will be detected by [*WISE*]{} 22 out to moderate redshifts ($z\sim$ 2 to 3), but most typical galaxies would not. Hence there is a strong bias for W4 detections to be luminous galaxies (LIRGS and ULIRGS), which are highly obscured at optical wavelengths. The [*WISE*]{} 12 band, on the other hand, has greater sensitivity by comparison (see Table \[match\]) and also probes the ISM, sensitive to a larger (representative) sampling of galaxies, and thus making it potentially the primary star formation indicator for [*WISE*]{}. The dominant feature within this W3 band is the 11.3 PAH (polycyclic aromatic hydrocarbon), and to a lesser extent the $[$Ne[ii]{}$]$ emission line. This PAH is large, neutral and excited by ultraviolet radiation from young stars, as well as radiation from older, evolved stars [see, for example, @Kan08]. $[$Ne[ii]{}$]$ is associated with emission from [H[ii]{}]{} regions. Passive disks generally lack the 7.7 PAH tracing current star formation, but still show prominent 11.3 PAH features [see, for example, @Clu13]. As first demonstrated with [*Spitzer*]{} measurements [see for example @Hou07; @Farr07], PAHs can be used to estimate star formation, although there is larger scatter (and potential biases) relative to the superior 24 mid-infrared or 70 far-infrared tracers. The [*WISE*]{} 12 band is thus a powerful, yet also more problematic, tracer of recent star formation.
We proceed as with W4, but with a S/N cut of 10 in the W3 (12 band), similarly color-coding to reflect Balmer decrement (Figure \[sf1\]a). A contour plot of the distribution is shown in Figure \[sf1\]b and the trend appears very tight for SFR $>5\, M_\odot\,{\rm yr}^{-1}$. At low SFR and $\nu$L$_{\nu}$, however, the distribution appears to flatten and probably accounts for the differences in slope between our relation and the least squares fit of @Don12.The relation of @Jar13 lies offset below our best fit, most likely as a result of relatively low SFR within their small sample (SFR $<5\, M_\odot\,{\rm yr}^{-1}$) of nearby galaxies. The best fit relation for the 12 band is:
$${\rm log_{10}\, SFR}_{{\rm H}\alpha} (M_\odot/{\rm yr}) = 1.13\, {\rm log_{10}}\, \nu L_{12\mu \rm m} (L_\odot) -10.24 \\$$
This flattening of the distribution in Figure \[sf1\] at low SFR is most likely a feature of the 11.3 PAH tracer (since we do not observe it at 22) and is also observed by @Lee13 in their study.
Given that the 12 micron band emission appears too low for the given H$\alpha$ SFR, a possible explanation is that the relative abundance of PAH molecules to big grain emission is diminished, due to low metalicity in these galaxies. If this is the case, we do not expect to see a similar effect at 22 micron, since this band is dominated by big grains in equilibrium with the strong radiation fields inside star formation regions. Indeed, the 22 sample of @Lee13 does not show such a flattening.
An alternative explanation would be that in a low SFR regime, the preponderance of the diffuse medium increases with respect to the star-formation component of the interstellar medium (galaxies are more quiescent). Bear in mind that the 24 warm dust emission is powered by the UV radiation fields from the massive stars inside the star forming regions, while the 12 emission is mainly powered by the diffuse interstellar radiation fields [@Pop11]. Thus, the PAH and small grains emission seen in the 12 band increases with increasing SFR, but this increase is mediated by the propagation of the UV photons in the diffuse ISM, while the 24 emission directly traces star formation regions, as does the Balmer line corrected H$\alpha$ emission from which the SFRs are derived.
To investigate this further we show individual points for contiguous redshift slices overplotted on the full sample in Figure \[sf3\]; this serves as an indication of how the distribution is built up. The flattening at low SFR appears to be a feature of nearby sources alone (Figure \[sf3\]a) supporting the hypothesis that we are observing the PAH features and small grain emission of relatively quiescent galaxies, dominated by the diffuse ISM. The flattening in the slope is not detected at higher redshifts ($z>0.05$) as these galaxies have more star formation activity (i.e. more clumpy). In Figures \[sf3\]b-d we see how as we move to higher redshifts, we sample systems with higher $\nu L_{12\mu \rm m}$ and, on average, higher SFRs. The relatively low number of sources in Figure \[sf3\]d reflects the atmospheric contamination affecting sources at redshifts of $z\simeq0.16$.
The higher infrared luminosity in these sources, however, also indicate greater obscuration in the optical bands; this explains the sharp drop in H$\alpha$ luminosities relative to W3, apparent in Figs. 10c and d with SFR(H$\alpha$) $<$ 3 M$_{\odot}$yr$^{-1}$. Corrections using the Balmer decrement become ineffective when the extinctions are high, (Av $>>1$), which is why infrared tracers of obscured star formation and UV/optical tracers of unobscured star formation are best combined to estimate the total star formation rate.
Science with the GAMA-WISE Catalog
==================================
The GAMA-[*WISE*]{} catalog and empirically-derived relations can be used any number of ways to probe the behaviour of large populations of galaxies. To illustrate, we highlight below how [*WISE*]{} can be used to study the specific star formation of galaxies.
Specific Star Formation
-----------------------
![Specific star formation with redshift. The star formation derived from the $\nu$L$_{12\mu m}$ luminosity, for the entire sample $\nu$L$_{12\mu m}$ as a function of stellar mass, giving the sSFR, color-coded by redshift. Lines of constant SFR (0.1, 1, 10 and 20 M$_{\odot}$yr$^{-1}$) are shown as dotted, solid, dashed and dash-dot lines, respectively.[]{data-label="ssfr"}](fig12.eps){width="8cm"}
We can investigate the specific star formation rate (sSFR) of the entire sample by using $L_{12\mu \rm m}$-derived star formation rates (equation 5) and the GAMA stellar masses [@Tay11]. In Figure \[ssfr\] we show specific star formation as a function of stellar mass, color-coded by redshift. Two clear trends are seen: (1) lower mass galaxies are actively building their disks while massive galaxies have expended their gas reservoirs rendering mostly passive evolution (the “SFR-M$_\star$" relation) and (2) with increasing redshift, a shift to higher SFR for a given stellar mass as we capture these infrared-luminous systems. The behavior of star formation and specific star formation, as a function of stellar mass and redshift, within the GAMA sample is explored in detail in @Bau13 and @Lar13. The observed increase in mean sSFR of star-forming galaxies with increasing redshift is well-established [see for example @Noes07; @Elb07; @Rod10] and we illustrate here that the GAMA-[*WISE*]{} sample is sufficiently large and diverse to explore galaxy evolution between the local universe and $z<$ 0.5.
From the relation of @Bou10, the “main-sequence" of galaxies with stellar mass of 10$^{11}$M$_\odot$ at $z\simeq0.5$ have typical SFRs of $\simeq 20$ M$_\odot$yr$^{-1}$. Converting to luminosity density by way of equation (5) gives log$_{10}L_{12\mu \rm m} \simeq 10.2$L$_\odot$. As illustrated in Figure \[lumw3\], [*WISE*]{} can detect these systems that are within the GAMA-[*WISE*]{} sample. A luminosity density of log$_{10}L_{12\mu \rm m} \simeq 10.2$L$_\odot$ corresponds to a flux density of 0.24 mJy or 12.8mag; using the S/N detection statistics of the G15 sources, [*WISE*]{} will detect this source with S/N$\simeq 20$. The magnitude sensitivity limit in the W3 band for the G12 field is 13.2 (10-$\sigma$), 14.0 (5-$\sigma$) and for G15, 13.8 (10-$\sigma$) and 14.5 (5-$\sigma$).
Specific Star Formation of Galaxies in Pairs
--------------------------------------------
The 12-derived star formation rates, computed from equation (5), can also be used to investigate sSFR trends within populations of GAMA galaxies. For example, we use the updated GAMA II Galaxy Group Catalogue (G$^3$Cv6), as detailed in @Rob11, to isolate galaxies that reside within a pair. In G$^3$C a galaxy pair is defined as two galaxies within 100$h^{-1}{\rm kpc}$, in physical separation, and 1000 kms$^{-1}$ in velocity separation. Figure \[pairs\] is a normalised histogram of all galaxies in the sample with stellar mass $>$ 10$^{10}$M$_{\odot}$ and the subset that reside in a pair (17,475 galaxies) and a close pair (i.e. separation $<20\,h^{-1}{\rm kpc}$). We note that potential blends (as outlined in §\[blend\]) are removed.
The striking feature of Figure \[pairs\]a is that galaxies that reside within a pair appear to have, on average, lower sSFR than a typical GAMA galaxy of the same mass, suggesting that instead of star formation being enhanced in these systems, it is suppressed. Although initially counter-intuitive, and contrary to studies of local galaxy pairs compared to field control samples [see, for example, @Ell13], it should be noted that $\sim40\%$ of GAMA galaxies reside within a group, with numerous mechanisms at play. Within the context, therefore, of the environment of a typical galaxy this highlights the complexity of interacting and merging systems and is discussed further in a detailed study of merging and interacting galaxies within GAMA (Robotham et al., in prep.). We note that a lack of SFR enhancement has emerged from studies of galaxy pairs probing higher redshift samples [see, for example, @Xu12], with one suggested explanation that higher gas fractions at higher redshift reduce the efficiency of torque-driven gas infall.
Further we include in Figure \[pairs\]b a histogram of galaxies within 20kpc/h of their neighbor, as compared to the distribution from Figure \[pairs\]a. It appears that this subset shows a broader range of sSFRs compared to the larger sample of galaxies in pairs, and may even be bimodal, consisting of suppressed systems and enhanced systems. Adding this kinematic trigger to galaxy evolution is clearly a complex process, but crucial to understand how galaxies evolve in the group environment. Finally, we note that blending limitations of the [*WISE*]{} data within pairs that have a smaller separation make this parameter space uncertain, but optical tracers will be exploited here in forthcoming GAMA papers.
Conclusions
===========
We have detailed the procedure for creating a source-matched catalog between galaxies in the GAMA G12 and G15 fields, and [*WISE*]{} photometry. In particular we have outlined how best to extract photometry for low signal to noise, unresolved and resolved sources in the [*WISE*]{} All-Sky Catalog. Complete GAMA-[*WISE*]{} catalogs for the G09, G12 and G15 fields will be made available through the GAMA Public Releases (see http://www.gama-survey.org/).
Using the [*WISE*]{} measurements and matched GAMA galaxies in the G12 and G15 fields, we have investigated the following:
- The [*WISE*]{} color distribution for the sample shows most systems are globally dominated by star formation, with few passive and AGN-dominated systems; this is consistent with the GAMA sample selection and spectroscopic sensitivity to higher redshift, star-forming galaxies.
- Empirical relations of stellar mass as a function of 3.4$-$4.6 color for resolved sources, our entire sample and star formation-dominated galaxies only. We provide relations that can be applied to large samples for $z<0.5$.
- Star formation rate relations can be derived using 22 and 12 luminosities. The 12-derived SFR relation (equation 5) is a complex tracer of the ISM and should, however, be used with caution. We show that the distribution of galaxies in the 12 luminosity and H$\alpha$ SFR plane as a function of redshift are affected by selection effects and most probably dust geometry.
- Specific star formation (using 12-derived SFRs) for the sample illustrates how the GAMA-[*WISE*]{} catalog detects only the most massive, highest star-forming systems at the highest redshifts. [*WISE*]{} is, however, able to detect star-forming main-sequence systems, of stellar mass $\sim$10$^{11}$M$_\odot$, to $z\simeq 0.5$ with S/N $>10$.
- Controlling for stellar mass, galaxies with an associated neighbor appear to experience, on average, a shift to lower specific star formation. Extracting pairs with relatively small separations ($<$ 20$h^{-1}$kpc) suggests a broader behaviour consistent with populations experiencing either star formation suppression or enhancement.
Acknowledgements {#acknowledgements .unnumbered}
================
MEC and THJ acknowledge support from the National Research Foundation (South Africa). MEC, MLL, MO, AEB and MJIB acknowledge support from the Australian Research Council (FS110200023, FS100100065, FT100100280). MLPG acknowledges support from a European Research Council Starting Grant (DEGAS-259586).
This publication makes use of data products from the Wide-field Infrared Survey Explorer, which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, funded by the National Aeronautics and Space Administration.
GAMA is a joint European-Australasian project based around a spectroscopic campaign using the Anglo-Australian Telescope. The GAMA input catalog is based on data taken from the Sloan Digital Sky Survey and the UKIRT Infrared Deep Sky Survey. Complementary imaging of the GAMA regions is being obtained by a number of independent survey programs including GALEX MIS, VST KiDS, VISTA VIKING, WISE, Herschel- ATLAS, GMRT and ASKAP providing UV to radio coverage. GAMA is funded by the STFC (UK), the ARC (Australia), the AAO, and the participating institutions. The GAMA website is http://www.gama-survey.org/.
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Analysis of [*WISE*]{} Photometry for Nominally Resolved Sources
================================================================
In order to understand the photometry of resolved and partially resolved sources in [*WISE*]{} we use several diagnostics. Since [*WISE*]{} is calibrated in the Vega system, images and all-sky catalogs, all comparisons are done as such.
Rfuzzy Defined {#rfuzz_sec}
--------------
The Rfuzzy parameter is used to determine whether a source is resolved in the 3.4 band of [*WISE*]{} (see §\[rfuzzy\]). The Rfuzzy parameter is measured in the following way:
1. The source is rotated to determine the optimal 2nd moment parameters (Rmajor, Rminor, position angle of major axis)
2. The source is divided along the minor axis into two halves: positive x, and negative x, where the central x position is the nominal position of the source
3. Compute the 2nd order intensity-weighted moment for each half and derive the Rmajor for each half: Rmajor$_{\rm a}$, Rmajor$_{\rm b}$
4. Rfuzzy $=$ minimum(Rmajor$_{\rm a}$, Rmajor$_{\rm b}$).
If Rfuzzy $<$ Rfuzzy$_{\rm limit}$, then the source is unresolved. The Rfuzzy$_{\rm limit}$ is determined by measuring sources that are expected to be unresolved. For example, Figure \[rfuzz\] shows the Rfuzzy values for [*WISE*]{} sources in the GAMA G12 region with [*w1rchi2*]{}$<$2 which preferentially selects unresolved sources. The typical value for Rfuzzy is $\simeq$10.0 for high signal to noise point sources (e.g. stars), but the distribution can be used to identify the limit where resolvedness can be determined. A power series fit to the 2$\sigma$ mean of the distribution yields the Rfuzzy$_{\rm limit}$ curve – points that lie above this curve are classified as resolved.
Rfuzzy can be defined in terms of isophotal signal to noise or isophotal magnitude. Given the differing exposure coverage (number of [*WISE*]{} frames that samples a given patch of sky) for the entire sky, the sensitivity limits vary depending on location on the sky. For this reason, a magnitude-dependent Rfuzzy$_{\rm limit}$ curve will be sensitive to the [*WISE*]{} coverage and would need to be derived for each region. However, a signal to noise function is robust against this and therefore the curve shown in Figure \[rfuzz\]a is used.
As a test of the performance of the Rfuzzy parameter, Figure \[rfuzz\_perf\] plots Rfuzzy for a sample from GAMA G12 expected to be dominated by resolved systems (based on [*w1rchi2*]{}). The majority of sources with [*w1rchi2*]{}$>$5 lie above the curve defined by the point sources in Figure \[rfuzz\]a.
![The performance of Rfuzzy is shown for sources in G15 where the majority are expected to be resolved i.e. [*w1$\star$rchi2*]{}$>$5.[]{data-label="rfuzz_perf"}](fig16.eps){width="8cm"}
In Figure \[delmag\_rfuzz\_sn\] we test 2590 sources in G15 chosen using the prescription of §3, i.e. most of which are expected to be resolved in W1. This shows that the largest offset between the W1iso and [*w1mpro*]{} photometry measurements occurs for sources deemed resolved by the Rfuzzy parameter. This behavior is consistent with sources whose flux is underestimated by profile fitting i.e. resolved. Unresolved sources show the expected scatter around 0. In Figure \[delmag\_rfuzz\_sn\]c we see that few sources are resolved in the W3 band.
Behaviour of Resolved Photometry {#resphot}
--------------------------------
In order to establish the reliablity of isophotal photometry, we use our test sample (2590 potentially resolved sources from the G15 GAMA field). In Figure \[delmag\_snr\] we compare the difference between W$\star$iso and [*w$\star$mpro*]{} photometry as a function of S/N ratio. At low S/N the isophotal measurement becomes unreliable – in some cases the fluxes are inflated, notably for W3 and W4 – due to contamination from background sources and [*w$\star$mpro*]{} provide the most robust measurement.
In Figure \[delmag\_rchi\_sn\] we again use the difference between W$\star$iso and [*w$\star$mpro*]{} photometry to illustrate that the sensitivity of the W1 and W2 bands prevents the [*w1rchi2*]{} and [*w2rchi*]{} values from acting as a reliable discriminator of resolvedness. For W3 and W4, however, it can be used and the limits derived from the plots are indicated.
We explore the relationship between the isophotal photometry and the [*w$\star$gmag*]{} photometry for sources in the 2MASS XSC. Since the [*w$\star$gmag*]{}s are measured using elliptical apertures with radii scaled to twice the 2MASS radii, this provides an indication of the additional sensitivity. We note that [*w$\star$gmag*]{}s can be contaminated by nearby objects since no attempt is made to remove neighboring sources. In Figure \[resplots\] we show the difference between [*w$\star$gmag*]{} and W$\star$iso photometry for resolved galaxies in G12 and G15. This shows that for fainter sources [*w$\star$gmag*]{} is brighter than W$\star$iso, likely due to contamination.
Finally, in Figure \[radplots\] the [*WISE*]{} isophotal and 2MASS isophotal radii of resolved sources which clearly shows that the 1$\sigma$ isophotal radii are systematically larger than the 2MASS radii by a factor of 2 to 2.5. The largest offset occurs for the most compact 2MASS sources probably due to the increased sensitivity of WISE in the W1 and W2 bands resolving more of the galaxy relative to 2MASS.
[^1]: [http://{\\it WISE}2.ipac.caltech.edu/docs/release/allsky/expsup/ sec4\$\_\$4c.html\$\#\$apcor](http://{\it WISE}2.ipac.caltech.edu/docs/release/allsky/expsup/ sec4$_$4c.html$#$apcor)
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'It is shown that an arbitrary singular Lagrangian theory (with first and second class constraints up to $N$-th stage presented in the Hamiltonian formulation) can be reformulated as a theory with at most third-stage constraints. The corresponding Lagrangian $\tilde L$ can be obtained by pure algebraic methods, its manifest form in terms of quantities of the initial formulation is find. Local symmetries of $\tilde L$ are obtained in closed form. All the first class constraints of the initial Lagrangian turn out to be gauge symmetry generators for $\tilde L$.'
author:
- 'A. A. Deriglazov[^1]'
date: |
Dept. de Matematica, ICE, Universidade Federal de Juiz de Fora,\
MG, Brazil;\
and\
LAFEX - CBPF/MCT, Rio de Janeiro, RJ, Brazil.
title: Formulation of a constrained system in terms of extended Lagrangian and its local symmetries
---
Introduction
============
Conventional way to describe a relativistic theory is to formulate it in terms of a singular Lagrangian. In turn, analysis of the singular theory can be carried out in a Hamiltonian formalism. In this framework, possible motions of the singular system are restricted to lie on some surface of a phase space. Algebraic equations of the surface (constraints) can be revealed in the course of a Dirac procedure, the latter in general case requires a number of stages. According to the order of appearance, the constraints are called primary, second-stage, ... , N-th stage constraints. All the constraints, beside the primary ones are called higher-stage constraints below. Whenever are appeared, the higher-stage constraints represent perceptible problems for analysis of the theory. In particular, search for local symmetries of the Lagrangian action, which is main subject of the present work, turns out to be rather nontrivial issue in a general case \[1-5\]. So, it may be reasonable to adopt a different approach to the problem. Namely, instead of looking for properties of the initial Lagrangian $L$ (provided all its constraints are known), we work out an equivalent Lagrangian $\tilde L$, the latter implies more transparent structure of constraints (in fact, all the higher-stage constraints of the original formulation enter into $\tilde L$ in a manifest form, see the last term in Eq. (\[13\]) below). It allows one to find infinitesimal local symmetries of the formulation in closed form, in terms of the constraints of the initial formulation. For some particular examples, such a kind possibility has been tested in the recent work \[6\]. Here we develop the formalism for an arbitrary theory, with first and second class constraints up to $N$-th stage presented in the original formulation $L$. $\tilde L$ is called an extended Lagrangian, since the corresponding complete Hamiltonian turns out to be closely related with an extended Hamiltonian of the original formulation[^2]. So, in this work we also clarify a relation among the complete and the extended Hamiltonian formulations of a given theory.
The work is organized as follows. With the aim to fix our notations, we outline in Section 2 the Hamiltonization procedure for an arbitrary singular Lagrangian theory. In Section 3 we formulate pure algebraic recipe for construction of the extended Lagrangian. All the higher-stage constraints of $L$ appear as secondary constraints for $\tilde L$. Besides, we demonstrate that $\tilde L$ is a theory with at most third-stage constraints. Then it is proved that $\tilde L$ and $L$ are equivalent[^3]. Since the original and the reconstructed formulations are equivalent, it is matter of convenience to use one or another of them for description of a theory under investigation[^4]. In Section 4 we demonstrate one of advantages of the extended formulation by finding its complete irreducible set of local symmetries. Properties of the extended formulation for some particular cases of original gauge algebra are discussed in the Conclusion.
Initial formulation with higher-stage first and second class constraints
========================================================================
Let $L(q^A, \dot q^B)$ be Lagrangian of singular theory: $rank\frac{\partial^2 L}{\partial\dot q^A\partial\dot q^B}=[i]<[A]$, defined on configuration space $q^A, A=1, 2, \ldots , [A]$. From the beginning, it is convenient to rearrange the initial variables in such a way that the rank minor is placed in the upper left corner. Then one has $q^A=(q^i, q^\alpha)$, $i=1, 2, \ldots , [i]$, $\alpha=1, 2, \ldots , [\alpha]=[A]-[i]$, where $\det\frac{\partial^2 L}{\partial\dot q^i\partial\dot q^j}\ne 0$.
Let us construct the Hamiltonian formulation for the theory. To fix our notations, we carry out the Hamiltonization procedure in some details. One introduces conjugate momenta according to the equations $p_i$ $=$ $\frac{\partial L}{\partial\dot q^i}$, $p_\alpha$ $=$ $\frac{\partial L}{\partial\dot q^\alpha}$. They are considered as algebraic equations for determining velocities $\dot q^A$. According to the rank condition, the first $[i]$ equations can be resolved with respect to $\dot q^i$, let us denote the solution as $$\begin{aligned}
\label{2}
\dot q^i=v^i(q^A, p_j, \dot q^\alpha).\end{aligned}$$ It can be substituted into remaining $[\alpha]$ equations for the momenta. By construction, the resulting expressions do not depend on $\dot q^A$ and are called primary constraints $\Phi_\alpha(q, p)$ of the Hamiltonian formulation. One finds $$\begin{aligned}
\label{3}
\Phi_\alpha\equiv p_\alpha-f_\alpha(q^A, p_j)=0,\end{aligned}$$ where $$\begin{aligned}
\label{4}
f_\alpha(q^A, p_j)\equiv\left.\frac{\partial L}{\partial\dot q^\alpha}
\right|_{\dot q^i=v^i(q^A, p_j, \dot q^\alpha)}.\end{aligned}$$ The original equations for the momenta are thus equivalent to the system (\[2\]), (\[3\]). By construction, one has the identities $$\begin{aligned}
\label{1}
\left.\frac{\partial L(q, \dot q)}{\partial\dot q^i}\right|_{\dot q^i\rightarrow v^i(q^A, p_j, \dot q^\alpha)}\equiv p_i, \qquad
\left.v^i(q^A, p_j, \dot q^\alpha)\right|_{p_j\rightarrow\frac{\partial L}{\partial\dot q^j}}\equiv\dot q^i.\end{aligned}$$
Next step of the Hamiltonian procedure is to introduce an extended phase space parameterized by the coordinates $q^A, p_A, v^\alpha$, and to define a complete Hamiltonian $H$ according to the rule $$\begin{aligned}
\label{5}
H(q^A, p_A, v^\alpha)=H_0(q^A, p_j)+v^\alpha\Phi_\alpha(q^A, p_B),\end{aligned}$$ where $$\begin{aligned}
\label{6}
H_0=\left.(p_i\dot q^i-L+ \dot q^\alpha\frac{\partial L}{\partial \dot q^\alpha})\right|
_{\dot q^i\rightarrow v^i(q^A, p_j, \dot q^\alpha)}.\end{aligned}$$ Then the following system of equations on this space $$\begin{aligned}
\label{7}
\dot q^A=\{q^A, H\}, \qquad \dot p_A=\{p_A, H\}, \qquad
\Phi_\alpha(q^A, p_B)=0,\end{aligned}$$ is equivalent to the Lagrangian equations following from $L$, see \[8\]. Here $\{ , \}$ denotes the Poisson bracket.
From Eq. (\[7\]) it follows that all the solutions are confined to lie on a surface of the extended phase space defined by the algebraic equations $\Phi_\alpha=0$. It may happen, that the system (\[7\]) contains in reality more then $[\alpha]$ algebraic equations. Actually, derivative of the primary constraints with respect to time implies, as algebraic consequences of the system (\[7\]), the so called second stage equations: $\{\Phi_\alpha, H\}$ $\equiv$ $\{\Phi_{\alpha}, \Phi_\beta\}v^\beta+\{\Phi_{\alpha}, H_0\}$ $=$ $0$. They can be added to Eq. (\[7\]), which gives an equivalent system. Let on-shell one has $rank\{\Phi_{\alpha}, \Phi_\beta\}=[\alpha ']\leq [\alpha]$. Then $[\alpha ']$ equations of the second-stage system can be used to represent some $v^{\alpha '}$ through other variables. It can be substituted into the remaining $[\alpha '']\equiv [\alpha]-[\alpha ']$ equations, the resulting expressions do not contain $v^\alpha$ at all. Thus the second-stage system can be presented in the equivalent form $$\begin{aligned}
\label{7.0}
v^{\alpha '}=v^{\alpha '}(q^A, p_j, v^{\alpha ''}), \qquad T_{\alpha ''}(q^A, p_j)=0.\end{aligned}$$ Functionally independent equations among $T_{\alpha ''}=0$, if any, represent secondary Dirac constraints. Thus all the solutions of the system (\[7\]) are confined to the surface defined by $\Phi_\alpha=0$ and by the equations (\[7.0\]).
The secondary constraints may imply third-stage constraints, and so on. We suppose that the theory has constraints up to $N$-th stage, $N\ge 2$. Higher stage constraints are denoted by $T_a(q^A, p_j)=0$. Then the complete constraint system is $G_I\equiv(\Phi_\alpha, T_a)$, while all the solutions of Eq. (\[7\]) are confined to the surface defined by the equations $\Phi_\alpha=0$ as well as by[^5] $$\begin{aligned}
\label{7.1}
\{ G_I, H\}=0.\end{aligned}$$ By construction, after substitution of the velocities determined during the Dirac procedure, these equations vanish on the complete constraint surface $G_J$.
Suppose that $\{G_I, G_J\}=\triangle_{IJ}(q^A, p_j)$, where $\left. rank\triangle_{IJ}\right|_{G_I=0}=[I_2]<[I]$, that is both first and second class constraints are presented. It will be convenient to separate them. According to the rank condition, there exist $[I_1]={I}-[I_2]$ independent null-vectors $\vec K_{I_1}$ of the matrix $\triangle$ on the surface $G_I=0$, with the components $K_{I_1}{}^J(q^A, p_j)$. Then bracket of the constraints $G_{I_1}\equiv K_{I_1}{}^JG_J$ with any $G_I$ vanishes, hence $G_{I_1}$ represent first class subset. Let $K_{I_2}{}^J(q^A, p_j)$ be any completion of the set $K_{I_1}{}^J$ up to a basis of $[I]$-dimensional vector space. By construction, the matrix $$\begin{aligned}
\label{7.2}
K_{I}{}^J\equiv
\left(K_{I_1}{}^J\atop K_{I_2}{}^J\right),\end{aligned}$$ is invertible. So the system $\tilde G_I$ $\equiv$ $(G_{I_1}\equiv K_{I_1}{}^JG_J$, $G_{I_2}\equiv K_{I_2}{}^JG_J)$ is equivalent to the initial system of constraints $G_I$. The constraints $G_{I_2}$ form the second class subset of the complete set. In arbitrary theory, the constraints obey the following Poisson bracket algebra: $$\begin{aligned}
\label{8}
\{\tilde G_I, \tilde G_J\}=\triangle_{IJ}(q^A, p_B), \qquad \quad \qquad \qquad \quad \cr
\{G_{I_1}, G_J\}=c_{I_1 J}{}^K(q^A, p_B)G_K, \quad \{G_{I_1}, H_0\}=b_{I_1}{}^J(q^A, p_B)G_J, \cr
\{G_{I_2}, G_{J_2}\}=\triangle_{I_2 J_2}(q^A, p_B), \qquad \qquad \quad \qquad\end{aligned}$$ where $$\begin{aligned}
\label{8.1}
\left. rank\triangle_{IJ}\right|_{G_I=0}=[I_2], \qquad
\left. \det\triangle_{I_2 J_2}\right|_{G_I=0}\ne 0.\end{aligned}$$
Construction of the extended Lagrangian and its properties
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Starting from the theory described above, we construct here a Lagrangian $\tilde L(q^A, \dot q^A, s^a)$ defined on the configuration space $q^A, s^a$. In the Hamiltonian formalism, it leads to the Hamiltonian[^6] $H_0+s^aT_a$, and to the primary constraints $\Phi_\alpha=0, ~ \pi_a=0$, where $\pi_a$ represent conjugate momenta for $s^a$. Due to special form of the Hamiltonian, preservation in time of the primary constraints implies, that all the higher stage constraints $T_a$ of initial theory appear as secondary constraints for the theory $\tilde L$. Moreover, the Dirac procedure stops on third stage: $\tilde L$ turns out to be a theory with at most third-stage constraints presented. Besides, we demonstrate that the formulations $L$ and $\tilde L$ are equivalent.
To construct the extended Lagrangian for $L$, let us consider the following equations for the variables $q^A,\omega_j, s^a$: $$\begin{aligned}
\label{9}
\dot q^i-v^i(q^A, \omega_j, \dot q^\alpha)-s^a\frac{\partial T_a(q^A, \omega_j)}{\partial\omega_i}=0.\end{aligned}$$ Here the functions $v^i(q^A, \omega_j, \dot q^\alpha)$, $T_a(q^A, \omega_j)$ are taken from the initial formulation. The equations can be resolved algebraically with respect to $\omega_i$ in a neighboard of the point $s^a=0$. Actually, Eq. (\[9\]) with $s^a=0$ coincides with Eq. (\[2\]) of the initial formulation, the latter can be resolved, see Eq. (\[1\]). Hence $\det\frac{\partial (Eq.(\ref{9}))^i}{\partial\omega_j}\ne 0$ at the point $s^a=0$. Then the same is true in some vicinity of this point, and Eq. (\[9\]) thus can be resolved. Let us denote the solution as $$\begin{aligned}
\label{10}
\omega_i=\omega_i(q^A, \dot q^A, s^a).\end{aligned}$$ By construction, one has the identities $$\begin{aligned}
\label{11}
\left.\omega_i(q, \dot q, s)\right|_{\dot q^i\rightarrow
v^i(q^A, \omega_j, \dot q^\alpha)+s^a\frac{\partial T_a(q^A, \omega_j)}{\partial\omega_i}}\equiv\omega_i,
\qquad \cr
\left.\left(v^i(q^A, \omega_j, \dot q^\alpha)+s^a\frac{\partial T_a(q^A, \omega_j)}{\partial\omega_i}\right)
\right|_{\omega_i(q, \dot q, s)}\equiv\dot q^i,\end{aligned}$$ as well as the following property of the function $\omega$ $$\begin{aligned}
\label{12}
\left.\omega_i(q^A, \dot q^A, s^a)\right|_{s^a=0}=\frac{\partial L}{\partial\dot q^i}.\end{aligned}$$ Now, the extended Lagrangian for $L$ is defined according to the expression $$\begin{aligned}
\label{13}
\tilde L(q^A, \dot q^A, s^a)=L(q^A, v^i(q^A, \omega_j, \dot q^\alpha), \dot q^\alpha)+ \cr
\omega_i(\dot q^i-v^i(q^A, \omega_j, \dot q^\alpha))-s^aT_a(q^A, \omega_j), \quad\end{aligned}$$ where the functions $v^i, \omega_i$ are given by Eqs. (\[2\]), (\[10\]). As compare with the initial Lagrangian, $\tilde L$ involves the new variables $s^a$, in a number equal to the number of higher stage constraints $T_a$. Let us enumerate some properties of $\tilde L$ $$\begin{aligned}
\label{14}
\tilde L(s^a=0)=L,\end{aligned}$$ $$\begin{aligned}
\label{15}
\left.\frac{\partial\tilde L}{\partial\omega_i}
\right|_{\omega(q, \dot q, s)}=0,\end{aligned}$$ $$\begin{aligned}
\label{16}
\frac{\partial\tilde L}{\partial\dot q^\alpha}=
\left.\left.\frac{\partial L(q^A, v^i, \dot q^\alpha)}{\partial\dot q^\alpha}
\right|_{v^i(q, \omega, \dot q^\alpha)}\right|_{\omega(q, \dot q, s)}=
f_\alpha(q^A, \omega_j(q, \dot q, s)).\end{aligned}$$ Eq. (\[14\]) follows from Eqs. (\[12\]), (\[1\]). Eq. (\[15\]) is a consequence of the identities (\[1\]), (\[11\]). Eq. (\[15\]) will be crucial for discussion of local symmetries in the next section. At last, Eq. (\[16\]) is a consequence of Eqs. (\[15\]), (\[1\]).
Following to the standard prescription \[7, 8\], let us construct the Hamiltonian formulation for $\tilde L$. By using of Eqs. (\[15\]), (\[16\]), one finds the conjugate momenta for $q^A, s^a$ $$\begin{aligned}
\label{17}
\tilde p_i=\frac{\partial\tilde L}{\partial\dot q^i}=\omega_i(q^A, \dot q^A, s^a), \qquad
\tilde p_\alpha=\frac{\partial\tilde L}{\partial\dot q^\alpha}=f_\alpha(q^A, \omega_j), \cr
\pi_a=\frac{\partial\tilde L}{\partial\dot s^a}=0. \qquad \qquad \qquad \quad\end{aligned}$$ Due to the identities (\[11\]), these expressions can be rewritten in the equivalent form $$\begin{aligned}
\label{18}
\dot q^i=v^i(q^A, \tilde p_j, \dot q^\alpha)+s^a\frac{\partial T_a(q^A, \tilde p_j)}{\partial\tilde p_i}, ~
\tilde p_\alpha-f_\alpha(q^A, \tilde p_j)=0, ~ \pi_a=0.\end{aligned}$$ Thus the velocities $\dot q^i$ have been determined. There are presented trivial constraints $\pi_a=0$, in a number equal to the number of all the higher stage constraints of the initial formulation, as well as all the primary constraints $\Phi_\alpha=0$ of the initial theory. Using the definition (\[6\]), one obtains the Hamiltonian $\tilde H_0$ $=$ $H_0+s^aT_a$, so the complete Hamiltonian for $\tilde L$ is given by the expression $$\begin{aligned}
\label{19}
\tilde H%(q^A, \tilde p_A, s^a, \pi_a, v^\alpha, v^a)
=H_0(q^A, \tilde p_j)+
s^aT_a(q^A, \tilde p_j)
+v^\alpha\Phi_\alpha(q^A, \tilde p_B)+v^a\pi_a, \quad\end{aligned}$$ where $v^\alpha, v^a$ are multipliers corresponding to the primary constraints. Note that, if one discards the constraints $\pi_a=0$, $\tilde H$ coincides with the extended Hamiltonian for $L$ after identification of configuration space variables $s^a$ with the Lagrangian multipliers for higher stage constraints of the original formulation.
Further, preservation in time of the primary constraints $\pi_a$ implies the equations $T_a=0$. Hence all the higher stage constraints of the initial formulation appear now as the secondary constraints. Preservation in time of the primary constraints $\Phi_\alpha$ leads to the equations $\{\Phi_\alpha, \tilde H\}$ $=$ $\{\Phi_\alpha, H_0\}$ $+$ $\{\Phi_\alpha, \Phi_\beta\}v^\beta$ $+$ $\{\Phi_\alpha, T_b\}s^b$ $=$ $0$. In turn, preservation of the secondary constraints $T_a$ leads to the similar equations $\{T_a, \tilde H\}$ $=$ $\{T_a, H_0\}$ $+$ $\{T_a, \Phi_\beta\}v^\beta$ $+$ $\{T_a, T_b\}s^b$ $=$ $0$. To continue the analysis, it is convenient to unify them as follows: $$\begin{aligned}
\label{20}
\{G_I, H_0\}+\{G_I, G_J\}S^J=0.\end{aligned}$$ Here $G_I$ are all the constraints of the initial formulation and it was denoted $S^J\equiv(v^\alpha, s^a)$. Using the matrix (\[7.2\]), the system (\[20\]) can be rewritten in the equivalent form $$\begin{aligned}
\label{21}
\{G_{I_1}, H_0\}+O(G_I)=0,\end{aligned}$$ $$\begin{aligned}
\label{22}
\{G_{I_2}, H_0\}+\{G_{I_2}, G_J\}S^J=O(G_I).\end{aligned}$$ Eq. (\[21\]) does not contain any new information, since the first class constraints commute with the Hamiltonian, see Eq. (\[8\]). So, let us analyze the system (\[22\]). First, one notes that due to the rank condition $\left.rank\{ G_{I_2}, G_J\}\right|_{G_I}$ $=$ $[I_2]=max$, exactly $[I_2]$ variables among $S^I$ can be found from the equations. According to the Dirac prescription, one needs to find maximal number of $v^\alpha$. To make this, let us restore $v$-dependence in Eq. (\[22\]): $\{G_{I_2}, \Phi_\alpha\}v^\alpha$ $+$ $\{G_{I_2}, H_0\}+\{G_{I_2}, T_b\}s^b$ $=$ $0$. Since the matrix $\{G_{I_2}, \Phi_\alpha\}$ is the same as in the initial formulation, from these equations one determines some group of variables $v^{\alpha_2}$ through the remaining variables $v^{\alpha_1}$, where $[\alpha_2]$ is number of primary second-class constraints among $\Phi_\alpha$. After substitution of the result into the remaining equations of the system (\[22\]), the latter acquires the form $$\begin{aligned}
\label{23}
v^{\alpha_2}=v^{\alpha_2}(q, p, s^a, v^{\alpha_1}), \qquad
Q_{a_2 b}(q, p)s^b+P_{a_2}(q, p)=0,\end{aligned}$$ where $[a_2]$ is the number of higher-stage second class constraints of the initial theory. It must be $P\approx 0$, since for $s^b=0$ the system (\[22\]) is a subsystem of (\[7.1\]), but the latter vanish after substitution of the multipliers determined during the procedure, see discussion after Eq. (\[7.1\]). Besides, one notes that $rank Q=[a_2]=max$. Actually, suppose that $rank Q=[a']<[a_2]$. Then from Eq. (\[22\]) only $[\alpha_2]+[a']<[I_2]$ variables among $S^I$ can be determined, in contradiction with the conclusion made before. In resume, the system (\[20\]) for determining the second-stage and the third-stage constraints and multipliers is equivalent to the following one $$\begin{aligned}
\label{24}
v^{\alpha_2}=v^{\alpha_2}(q, p, s^{a_1}, v^{\alpha_1}),\end{aligned}$$ $$\begin{aligned}
\label{25}
s^{a_2}=Q^{a_2}{}_{b_1}(q, p)s^{b_1}.\end{aligned}$$ Conservation in time of the constraints (\[25\]) does not produce new constraints, giving equations for determining the multipliers $$\begin{aligned}
\label{25.1}
v^{a_2}=\{ Q^{a_2}{}_{b_1}(q, p)s^{b_1}, \tilde H\},\end{aligned}$$ The Dirac procedure for $\tilde L$ stops on this stage. All the constraints of the theory have been revealed after completing the third stage.
Now we are ready to compare the theories $\tilde L$ and $L$. Dynamics of the theory $\tilde L$ is governed by the Hamiltonian equations $$\begin{aligned}
\label{26}
\dot q^A=\{q^A, H\}+s^a\{q^A, T_a\}, \qquad \dot{\tilde p}_A=\{\tilde p_A, H\}+s^a\{\tilde p_A, T_a\}, \cr
\dot s^a=v^a, \qquad \qquad \qquad \qquad \qquad \dot\pi_a=0, \qquad \qquad \qquad \qquad \quad\end{aligned}$$ as well as by the constraints $$\begin{aligned}
\label{27}
\Phi_\alpha=0, \qquad T_a=0,\end{aligned}$$ $$\begin{aligned}
\label{28}
\pi_{a_1}=0,\end{aligned}$$ $$\begin{aligned}
\label{29}
\pi_{a_2}=0, \qquad s^{a_2}=Q^{a_2}{}_{b_1}(q, p)s^{b_1}.\end{aligned}$$ Here $H$ is complete Hamiltonian of the initial theory (\[5\]), and the Poisson bracket is defined on the phase space $q^A, s^a, p_A, \pi_a$. The constraints $\pi_{a_1}=0$ can be replaced by the combinations $\pi_{a_1}-\pi_{a_2}Q^{a_2}{}_{a_1}(q, p)=0$, the latter represent first class subset. Let us make partial fixation of a gauge by imposing the equations $s^{a_1}=0$ as a gauge conditions for the subset. Then $(s^a, \pi_a)$-sector of the theory disappears, whereas the equations (\[26\]), (\[27\]) coincide exactly with those of the initial theory[^7] $L$. Let us remind that $\tilde L$ has been constructed in some vicinity of the point $s^a=0$. The gauge $s^{a_1}=0$ implies $s^a=0$ due to the homogeneity of Eq. (\[25\]). It guarantees a self consistency of the construction. Thus $L$ represents one of the gauges for $\tilde L$, which proves equivalence of the two formulations.
Local symmetries of the extended Lagrangian
===========================================
Since the initial Lagrangian is one of gauges for $\tilde L$, physical system under consideration can be equally analyzed by using of the extended Lagrangian. In contrast to $L$, the extended Lagrangian contains the higher-stage constraints $T_a$ of $L$ in the manifest form, see Eq. (\[13\]). Moreover, while $T_a$ appear as the secondary constraints of the formulation $\tilde L$, they are also presented in the manifest form in the complete Hamiltonian $\tilde H$. Here we demonstrate one of consequences of this property: all the infinitesimal local symmetries of $\tilde L$ can be found in closed form.
According to the analysis made in the previous section, the primary constraints of the extended formulation are $\Phi_\alpha=0$, $\pi_a=0$. Among $\Phi_\alpha=0$ there are presented first class constraints, in a number equal to the number of primary first class constraints of $L$. Among $\pi_a=0$, we have find the first class constraints $\pi_{a_1}-\pi_{a_2}Q^{a_2}{}_{a_1}(q, p)=0$, in a number equal to the number of all the higher-stage first class constraints of $L$. Thus the number of primary first class constraints of $\tilde L$ coincide with the number $[I_1]$ of all the first class constraints of $L$. Hence one expects $[I_1]$ local symmetries presented in the formulation $\tilde L$. Now we demonstrate that the action $S_{\tilde L}=\int d\tau\tilde L$ is invariant (modulo to a surface term) under the following infinitesimal transformations: $$\begin{aligned}
\label{30}
\delta_{I_1} q^A=\epsilon^{I_1}\left.\{q^A, G_{I_1}(q^A, p_B)\}
\right|_{p_i\rightarrow\omega_i(q, \dot q, s), p_\alpha\rightarrow f_\alpha(q, \omega(q, \dot q, s))}, \qquad \qquad ~ ~\cr
\delta_{I_1} s^a= \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \cr
\left.\left[\dot\epsilon^{I_1}K_{I_1}{}^a+\epsilon^{I_1}\left( b_{I_1}{}^a+s^bc_{I_1 b}{}^a+
\dot q^\beta c_{I_1 \beta}{}^a\right)\right]
\right|_{p_i\rightarrow\omega_i(q, \dot q, s), p_\alpha\rightarrow f_\alpha(q, \omega(q, \dot q, s))}.\end{aligned}$$ Here $\epsilon^{I_1}(\tau)$, $I_1=1, 2, \ldots , [I_1]$ are the local parameters, and $K$ is the conversion matrix, see Eq. (\[7.2\]). Eq. (\[30\]) gives the symmetries of $\tilde L$ in closed form in terms of the first class constraints $G_{I_1}$ of the initial formulation. One should note that the transformations of $q^A$ represent Lagrangian version of canonical transformations with the generators being $G_{I_1}$.
In the subsequent computations we omit all the terms which are total derivatives. Besides, the notation $\left. A\right|$ implies the substitution indicated in Eq. (\[30\]).
To make a proof, it is convenient to represent the extended Lagrangian (\[13\]) in terms of the initial Hamiltonian $H_0$, instead of the initial Lagrangian $L$. With help of Eq. (\[6\]) one writes $$\begin{aligned}
\label{31}
\tilde L(q^A, \dot q^A, s^a)=
\omega_i\dot q^i+f_\alpha(q^A, \omega_j)\dot q^\alpha-
H_0(q^A, \omega_j)-s^aT_a(q^A, \omega_j),\end{aligned}$$ where the functions $\omega_i(q, \dot q, s)$, $f_\alpha(q, \omega)$ are defined by Eqs. (\[10\]), (\[4\]). Using the identity (\[15\]), variation of this expression under the transformation (\[30\]) can be presented in the form $$\begin{aligned}
\label{32}
\delta\tilde L=-\dot\omega_i(q, \dot q, s)\left.\frac{\partial G_{I_1}}{\partial p_i}\right|\epsilon^{I_1}
-\dot f_\alpha(q, \omega(q, \dot q, s)\left.\frac{\partial G_{I_1}}{\partial p_\alpha}\right|\epsilon^{I_1}
\qquad \qquad \quad ~ \cr
-\left. \left(\frac{\partial H_0(q^A, p_j)}{\partial q^A}+
\dot q^\alpha\frac{\partial\Phi_\alpha(q^A, p_B)}{\partial q^A}+
s^a\frac{\partial T_a(q^A, p_j)}{\partial q^A}\right)\right|\left.\{q^A, G_{I_1}\}\right|\epsilon^{I_1} \cr
-\delta_{I_1}s^aT_a(q^A, \omega_j).\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad\end{aligned}$$ To see that $\delta\tilde L$ is total derivative, we add the following zero $$\begin{aligned}
\label{33}
0\equiv\left.\left[\left.\frac{\partial\tilde L}{\partial\omega_i}\right|_{\omega_i}\{p_i, G_{I_1}\}\right.\right.
\qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad ~\cr
\left.\left.
-\left(\frac{\partial H_0}{\partial p_\beta}+
\dot q^\alpha\frac{\partial\Phi_\alpha}{\partial p_\beta}+
s^a\frac{\partial T_a}{\partial p_\beta}\right)\{p_\beta, G_{I_1}\}+
\dot q^\alpha\{p_\alpha, G_{I_1}\}\right]\right|\epsilon^{I_1},\end{aligned}$$ to r.h.s. of Eq. (\[32\]). It gives the expression $$\begin{aligned}
\label{34}
\delta\tilde L=
\left.\left[\dot\epsilon^{I_1}G_{I_1}-\epsilon^{I_1}\left(\{H_0, G_{I_1}\}+
\dot q^\alpha\{\Phi_\alpha, G_{I_1}\}+s^a\{T_a, G_{I_1}\}\right)\right]\right| \cr
-\delta_{I_1}s^aT_a(q^A, \omega_j)= \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \cr
\left.\left[\dot\epsilon^{I_1}G_{I_1}+\epsilon^{I_1}\left(b_{I_1}{}^I+
\dot q^\alpha c_{I_1 \alpha}{}^I+s^bc_{I_1 b}{}^I\right)G_I\right]\right|-\delta_{I_1}s^aT_a(q^A, \omega_j),\end{aligned}$$ where $b, c$ are coefficient functions of the constraint algebra (\[8\]). Using the equalities $\left. G_{I}\right|=(0, ~ T_a(q^A, \omega_j))$, $\left. G_{I_1}\right|=K_{I_1}{}^a T_a(q^A, \omega_j)$, one finally obtains $$\begin{aligned}
\label{35}
\delta\tilde L= \qquad \qquad \qquad \qquad \qquad \qquad \qquad \cr
\left.\left[\dot\epsilon^{I_1}K_{I_1}{}^a+\epsilon^{I_1}\left(b_{I_1}{}^a+
\dot q^\alpha c_{I_1 \alpha}{}^a+s^bc_{I_1 b}{}^a\right)-\delta_{I_1}s^a\right]\right|_
{p_i\rightarrow\omega_i}T_a.\end{aligned}$$ Then the variation of $s^a$ given in Eq. (\[30\]) implies $\delta\tilde L=div$, as it has been stated.
Conclusion
==========
In this work we have presented a relatively simple way for finding the local symmetries in a singular theory of a general form. Instead of looking for the symmetries of initial Lagrangian, one can construct an equivalent Lagrangian $\tilde L$ given by Eq. (\[13\]), the latter implies at most third-stage constraints in the Hamiltonian formulation[^8]. Due to special structure of $\tilde L$ (all the higher-stage constraints $T_a$ of the original formulation enter into $\tilde L$ in a manifest form, see the last term in Eq. (\[13\])), local symmetries of $\tilde L$ can be immediately written according to Eq. (\[30\]). The latter gives the symmetries in terms of the first class constraints $G_{I_1}$ of the initial formulation, moreover, transformations of $q^A$ represent Lagrangian version of canonical transformations with the generators being $G_{I_1}$. In contrast to a situation with symmetries of $L$ \[2-5\], the transformations (\[30\]) do not involve the second class constraints.
The extended formulation can be appropriate tool for development of a general formalism for conversion of second class constraints into the first class ones according to the ideas of the work \[10\]. To apply the method proposed in \[10\], it is desirable to have a formulation with some configuration space variables entering into the Lagrangian without derivatives. It is exactly what happens in the extended formulation.
To conclude with, we discuss properties of the extended formulation for some particular cases of the original gauge algebra (\[8\]).
Suppose that all the original constraints $G_I$ are first class. It implies the extended formulation with at most secondary constraints. One obtains the primary constraints $\Phi_\alpha=0$, $\pi_a=0$ and the secondary constraints $T_a=0$, all of them being the first class. An appropriate gauge for $\pi_a=0$ is $s^a=0$. For the case, Eq. (\[30\]) reduces to the result obtained in \[6\].
Suppose that all the original constraints $G_I$ are second class (that is there are no of local symmetries in the theory). It implies the extended formulation with at most third-stage constraints, all of them being the second class: $\Phi_\alpha=0$, $T_a=0$, $\pi_a=0$, $s^a=0$.
Suppose that the original $L$ represents a formulation with at most second-stage first and second class constraints. It implies the extended formulation with at most third-stage constraints. Nevertheless, namely for the extended formulation the local symmetries can be find in a manifest form according to Eq. (\[30\]).
Acknowledgments
===============
Author would like to thank the Brazilian foundations CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico - Brasil) and FAPERJ for financial support.
[nn]{} J.L. Anderson and P.G. Bergmann, Phys. Rev. [**83**]{} (1951) 1018; P.G. Bergmann and I. Goldberg, Phys. Rev. [**98**]{} (1955) 531. M. Henneaux, C. Teitelboim and J. Zanelli, Nucl. Phys. [**B 332**]{} (1990) 169; M. Henneaux and C. Teitelboim, Quantization of Gauge Systems (Princeton: Princeton Univ. Press, 1992). V. A. Borokhov and I. V. Tyutin, Physics of Atomic Nuclei [**61**]{} (1998) 1603; Physics of Atomic Nuclei [**62**]{} (1999) 1070. A. A. Deriglazov, and K.E. Evdokimov, Int. J. Mod. Phys. [**A 15**]{} (2000) 4045 \[hep-th/9912179\]; A. A. Deriglazov, Int. J. Mod. Phys. [**A 22**]{} (2007) 2105; A. A. Deriglazov, Search for gauge symmetry generators of singular Lagrangian theory, hep-th/0509222. D. M. Gitman and I. V. Tyutin, Int. J. Mod. Phys. [**A 21**]{} (2006) 327. A. A. Deriglazov, J. Phys. A: Math. Theor. [**40**]{} (2007) 11083. P.A.M. Dirac, Can. J. Math. [**2**]{} (1950) 129; Lectures on Quantum Mechanics (Yeshiva Univ., New York, 1964). D. M. Gitman and I. V. Tyutin, Quantization of Fields with Constraints (Berlin: Springer-Verlag, 1990). A. A. Deriglazov, Phys. Lett. [**B 626**]{} (2005) 243. A. A. Deriglazov and Z. Kuznetsova, Phys. Lett. [**B 646**]{} (2007) 47.
[^1]: [email protected] On leave of absence from Dept. Math. Phys., Tomsk Polytechnical University, Tomsk, Russia.
[^2]: By definition, the extended Hamiltonian is obtained from the complete one by addition of the higher-stage constraints with corresponding Lagrangian multipliers. It is known \[8\] that the two formulations are equivalent.
[^3]: Popular physical theories usually do not involve more than third-stage constraints (example of a theory with third-stage constraints is the membrane, in the formulation with world-volume metric). Our result can be considered as an explanation of this fact.
[^4]: Let us point out that the higher stage constraints usually appear in a covariant form. So one expects manifest covariance of the extended formulation.
[^5]: It is known \[8\], that the procedure reveals all the algebraic equations presented in the system (\[7\]). Besides, surface of solutions of Eq. (\[7\]) coincides with the surface $\Phi_\alpha=0$, $\{ G_I, H\}=0$.
[^6]: Let us stress once again, that in our formulation the variables $s^a$ represent a part of the configuration-space variables.
[^7]: In more rigorous treatment, one writes Dirac bracket corresponding to the equations $\pi_{a_1}-\pi_{a_2}Q^{a_2}{}_{a_1}=0$, $s^{a_1}=0$, and to the second class constraints (\[29\]). After that, the equations used in construction of the Dirac bracket can be used as strong equalities. For the case, they reduce to the equations $s^a=0, \pi_a=0$. For the remaining phase-space variables $q^A, p_A$, the Dirac bracket coincides with the Poisson one.
[^8]: In the recent work \[9\] it was demonstrated that the primary constraints, while are convenient, turn out to be not necessary for the Hamiltonization procedure. So, one can said that for any theory there exists a formulation with secondary and tertiary constraints.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We prove that the known formulae for computing the optimal number of maximally entangled pairs required for entanglement-assisted quantum error-correcting codes (EAQECCs) over the binary field hold for codes over arbitrary finite fields as well. We also give a Gilbert-Varshamov bound for EAQECCs and constructions of EAQECCs coming from punctured self-orthogonal linear codes which are valid for any finite field.'
author:
- 'C. Galindo, F. Hernando, R. Matsumoto and D. Ruano'
title: 'Entanglement-assisted quantum error-correcting codes over arbitrary finite fields'
---
Introduction
============
The Shor’s proposal of using quantum error correction for reducing decoherence in quantum computation [@shor95] and his polynomial-time algorithms for prime factorization and discrete logarithms on quantum computers [@22RBC] clearly illustrate the feasibility and importance of quantum computation and quantum error-correcting.
Most of the quantum error-correcting codes (QECCs) come from classical codes. The first known stabilizer quantum codes were binary [@calderbank98; @gott]. Later, stabilizer codes over any finite field were introduced and studied, they are of particular interest because their utility in fault-tolerant computation. Following [@ketkar06], one can obtain QECCs of length $n$ over a finite field $\mathbb{F}_q$ from additive codes included in $\mathbb{F}_q^{2n}$ which are self-orthogonal with respect to a trace symplectic form. Working on this construction, QECCs of length $n$ over $\mathbb{F}_q$ can be derived from classical self-orthogonal codes with respect to the Hermitian inner product included in $\mathbb{F}_{q^2}^{n}$, and also from codes in $\mathbb{F}_{q}^{n}$ which are self-orthogonal with respect to the Euclidean inner product.
The previously mentioned self-orthogonality conditions (or some similar requirements of inclusion of codes in the dual of others) prevent the usage of many common classical codes for providing quantum codes. Brun, Dvetak and Hsieh in [@brun1] proposed to share entanglement between encoder and decoder to simplify the theory of quantum error-correction and increase the communication capacity. With this new formalism, entanglement-assisted quantum stabilizer codes can be constructed from any classical linear code giving rise to entanglement-assisted quantum error-correcting codes (EAQECCs). A formula to obtain the optimal number of ebits required for a binary entanglement-assisted code of Calderbank-Shor-Steane (CSS) type was showed in [@hsieh] and formulae for more general constructions, including the consideration of duality with respect to symplectic forms were given in [@wilde08]. In fact, [@wilde08] proves that the optimal number $c$ of ebits required for a binary entanglement-assisted quantum error-correcting code with generator matrix $(H_X|H_Z)$ is $\mathrm{rank}(H_X H_Z^T - H_Z H_X^T)/2$, where the superindex $T$ means transpose. Remark 1 in that paper states, without a proof, that the same formula holds when considering codes over finite fields $\mathbb{F}_p$, $p$ being a prime number, a proof can be found in [@schin].
Recently, one can find in the literature some papers where the above formula (or formulae derived from it) are used for determining the entanglement corresponding to EAQECCs over arbitrary finite fields (see, for instance [@chen; @Guenda18; @Liu18; @qian]). Although it holds for any finite field, we have found no proof in the literature and, thus, this work fills this gap. Therefore, this paper is devoted to prove formulae for the minimum required number $c$ of pairs of maximally entangled quantum states, corresponding to EAQECCs codes obtained from linear codes $C$ over any finite field, by using symplectic forms, or Hermitian or Euclidean inner products. We also show (see Subsection \[geometric\]) that in the Hermitian and Euclidean cases, $c$ is easy to compute when one chooses, as a basis of the linear code $C$ of length $n$, a subset of those vectors giving rise to a geometric decomposition of the coordinate space of dimension $n$ that contains $C$ [@MetricStructure].
In [@lai12], a Gilbert-Varshamov type formula for the existence of binary EAQECCs was presented. Still with the idea of extending the binary case to the general one and with the help of our study of entanglement-assisted codes, we give a Gilbert-Varshamov type formula which is valid for any finite field. Furthermore, we will also provide conditions of existence and parameters of EAQECCs coming from classical self-orthogonal codes (say $C$) over any finite field. Since fewer qudits should be transmitted through a noisy channel, they perform better. Constructions of this type have been considered in the binary case for giving a coding scheme with imperfect e-bits [@lai12].
Theorems \[thm:fp\], \[thm:hermitian\] and \[thm:euclidean\] contain our results about the entanglement required for EAQECCs over arbitrary finite fields. Section \[sect2\] also explains how, in the Hermitian and Euclidean cases, nice bases of the vector spaces that contain the supporting linear codes allow us to get the corresponding required number $c$. Section \[sect3\] is devoted to state the mentioned Gilbert-Varshamov type bound and Section \[sec:wip\] contains our results about EAQECCs coming from QECC by considering symplectic, Hermitian or Euclidean duality.
EAQECCs over $\mathbb{F}_q$ {#sect2}
===========================
The first three subsections of this section are devoted to prove formulae for computing the optimal entanglement corresponding to EAQECCs over arbitrary finite fields when considering symplectic forms, or Hermitian or Euclidean inner products.
The symplectic case {#symplectic}
-------------------
Let $p$ be a prime number and $q$ a positive power $q=p^m$. Denote by $\mathbb{F}_q$ the finite field with $q$ elements. We also write $\mathbb{C}$ the field of complex numbers and $\mathbb{C}^r$, $r$ a positive integer, the $r$-coordinate space over $\mathbb{C}$.
Let $n$ be a positive integer, it is known (see for instance [@ketkar06 Theorem 13]) that an $((n, K,d))_q$ stabilizer quantum code over $\mathbb{F}_q$ can be obtained from an additive code $C \subseteq \mathbb{F}_q^{2n}$ of size $q^n/K$ such that $C \subseteq C^{\perp_{ts}}$, and $\mathrm{swt} (C^{\perp_{ts}} \setminus C)= d$ when $K \geq 1$ and $d= \mathrm{swt} (C)$ otherwise. In the above result, we have considered the following notation which will be used in this paper as well. The symbol $\perp_{ts}$ means dual with respect to the [*trace-symplectic form*]{} on $\mathbb{F}_q^{2n}$: $$\left(\vec{a}|\vec{b}\right) \cdot_{ts} \left(\vec{a'}|\vec{b'}\right) = {\mathrm{tr}_{q/p}}\left(\vec{a} \cdot \vec{b'} - \vec{a'} \cdot \vec{b}\right) \in \mathbb{F}_p,$$ where $\left(\vec{a}|\vec{b}\right), \left(\vec{a'}|\vec{b'}\right) \in \mathbb{F}_q^{2n}$, $\vec{a} \cdot \vec{b'}$ and $\vec{a'} \cdot \vec{b}$ are Euclidean products, and ${\mathrm{tr}_{q/p}}: \mathbb{F}_q \rightarrow \mathbb{F}_p$, $${\mathrm{tr}_{q/p}}(x)= x + x^p + \cdots + x^{p^{m-1}},$$ is the standard trace map. Also the symplectic weight is defined as $$\mathrm{swt} \left(\vec{a}|\vec{b}\right) = {{\rm card}}\left\{i \;| \; (a_i,b_i) \neq (0,0), 1 \leq i \leq n \right\},$$ where $\vec{a}=(a_1,a_2, \ldots, a_n)$ and $\vec{b}=(b_1,b_2, \ldots, b_n)$.
We will also use the symplectic form on $\mathbb{F}_q^{2n}$ defined as $$\left(\vec{a}|\vec{b}\right) \cdot_{s} \left(\vec{a'}|\vec{b'}\right) = \left(\vec{a} \cdot \vec{b'} - \vec{a'} \cdot \vec{b}\right) \in \mathbb{F}_q,$$ and the corresponding dual space for an $\mathbb{F}_q$-linear code $C \subseteq \mathbb{F}_q^{2n}$ will be denoted by $C^{\perp_s}$.
For the first part of this paper, we fix a trace orthogonal basis of $\mathbb{F}_q$ over $\mathbb{F}_p$, $B=\{\gamma_1, \gamma_2, \ldots, \gamma_m\}$. Recall that $B$ is a basis of $\mathbb{F}_q$ as a $\mathbb{F}_p$-linear space satisfying that the matrix $$M= \left({\mathrm{tr}_{q/p}}(\gamma_i \gamma_j)\right)_{1\leq i \leq m;\;1\leq j \leq m}$$ is an invertible and diagonal matrix of size $m$ with coefficients in $\mathbb{F}_p$. The existence of a basis as $B$ is proved in [@sero]. We choose a basis as $B$ by convenience, but our results also hold if one considers any other basis. Now consider the $\mathbb{F}_p$-linear map $$h: \mathbb{F}_p^m \rightarrow \mathbb{F}_q, \;\; h(x_1,x_2, \ldots,x_m)= \sum_{i=1}^m x_i \gamma_i:= x.$$ The map $h$ is an isomorphism of $\mathbb{F}_p$-linear spaces and for $x \in \mathbb{F}_q$, $h^{-1}(x)$ gives the coordinates of $x$ in the basis $B$.
Denote by $\Omega$ the inverse matrix of $M$, $\Omega$ is a size $m$ diagonal invertible matrix with entries in $\mathbb{F}_p$. Let $\omega_1, \omega_2, \ldots, \omega_m$ be its diagonal and define the map: $$\phi: \mathbb{F}_p^{2m}=\mathbb{F}_p^m \times \mathbb{F}_p^m \longrightarrow \mathbb{F}_q^2,$$ given by $$\begin{aligned}
\phi\left((x_1,x_2, \ldots,x_m)|(y_1,y_2, \ldots,y_m)\right) &=& \left(\sum_{i=1}^m x_i \gamma_i, \sum_{i=1}^m y_i \omega_i \gamma_i\right)\\
&=& \left( h(x_1,x_2, \ldots,x_m), h[(y_1,y_2, \ldots,y_m)\Omega]\right).\end{aligned}$$ Taking into account that $\omega_i \in \mathbb{F}_p$, $B'=\{\omega_i \gamma_i\}_{i=1}^m$ is also a trace orthogonal basis of $\mathbb{F}_q$ over $\mathbb{F}_p$ whose matrix $\left({\mathrm{tr}_{q/p}}(\omega_i \gamma_i \omega_j \gamma_j)\right)_{1\leq i \leq m;\;1\leq j \leq m}$ is $\Omega$.
In sum, $\phi$ is an isomorphism of $\mathbb{F}_p$-linear spaces and for $(x,y) \in \mathbb{F}_q^2$, $$\phi^{-1} (x,y) = \left(\phi^{-1}_1 (x,y), \phi^{-1}_2 (x,y)\right) \in \mathbb{F}_p^{2m},$$where $\phi^{-1}_{1}$ (respectively, $\phi^{-1}_{2}$) is the first (respectively, second) projection of $\phi^{-1}$ over the first (respectively, second) component of the Cartesian product $\mathbb{F}_p^m \times \mathbb{F}_p^m$. One has that $\phi^{-1}(x,y)$ simply gives a pair whose first components are the coordinates of $x$ in the basis $B$ and the second ones are those of $y$ in the basis $B'$.
The above map can be extended to products of $n$ copies giving rise to the map $$\phi^E: \mathbb{F}_p^{2mn}=(\mathbb{F}_p^m)^n \times (\mathbb{F}_p^m)^n \longrightarrow \mathbb{F}_q^{n} \times \mathbb{F}_q^{n} =\mathbb{F}_q^{2n},$$ defined by $$\begin{gathered}
\phi^E \left[\left( (a_{11}, \ldots a_{1m}), \ldots, (a_{n1}, \ldots a_{nm})|(b_{11}, \ldots b_{1m}), \ldots, (b_{n1}, \ldots b_{nm}) \right) \right] =\\
\left( h(a_{11}, \ldots a_{1m}), \ldots, h(a_{n1}, \ldots a_{nm})
| h[(b_{11}, \ldots b_{1m})\Omega], \ldots, h[(b_{n1}, \ldots, b_{nm})\Omega]
\right).\end{gathered}$$
Notice that $\phi^E$ is again an isomorphism of $\mathbb{F}_p$-linear spaces and $$\left(\phi^E\right)^{-1} \left(\vec{a}|\vec{b}\right)= \left( \left(\phi^E\right)^{-1}_1 \left(\vec{a}|\vec{b}\right)|\left(\phi^E\right)^{-1}_2 \left(\vec{a}|\vec{b}\right)\right),$$where $\left(\phi^E\right)^{-1}_1$ (respectively, $\left(\phi^E\right)^{-1}_2$) is the first (respectively, second) projection of $\left(\phi^E\right)^{-1}_1$ over the first (respectively, second) component of the Cartesian product $(\mathbb{F}_p^m)^n \times (\mathbb{F}_p^m)^ n$. One has that $\left(\phi^E\right)^{-1}\left(\vec{a}|\vec{b}\right)$ equals the vector of coordinates of the element $\left(\vec{a}|\vec{b}\right) \in \mathbb{F}_q^{2n}$ in the basis of $\mathbb{F}_q^{2n}$ over $\mathbb{F}_p$ given by $\oplus_{n\; \mbox{\footnotesize{times}}} B \bigoplus \oplus_{n\; \mbox{\footnotesize{times}}} B'$.
Keeping the above notation, it is easy to deduce the following result in [@ashikhmin00].
\[prop:convert\] The following statements hold:
a)
: Let $x, y \in \mathbb{F}_q$, then $${\mathrm{tr}_{q/p}}(xy) = \left(\phi^{-1}_1 (x,y)\right) \cdot \left(\phi^{-1}_2 (x,y)\right),$$ where $\cdot$ denotes the Euclidean product in $\mathbb{F}_p^m$.
b)
: Let $\left(\vec{a}|\vec{b}\right), \left(\vec{a'}|\vec{b'}\right) \in \mathbb{F}_q^{2n}$, then $$\left(\vec{a}|\vec{b}\right) \cdot_{st} \left(\vec{a'}|\vec{b'}\right)
=
\left[
\left(\phi^E\right)^{-1}_1 \left(\vec{a}|\vec{b}\right) | \left(\phi^E\right)^{-1}_2 \left(\vec{a}|\vec{b}\right)
\right] \cdot_s \left[
\left(\phi^E\right)^{-1}_1 \left(\vec{a'}|\vec{b'}\right) | \left(\phi^E\right)^{-1}_2 \left(\vec{a'}|\vec{b'}\right)
\right],$$ where $\cdot_s$ denotes the symplectic form in $\mathbb{F}_p^{2mn}$.
Our purpose in this section is to determine the optimal required number of pairs of maximally entangled states of the EAQECC over an arbitrary finite field $\mathbb{F}_q$ that can be constructed from an $\mathbb{F}_q$-linear code $C \subseteq \mathbb{F}_q^{2n}$ with dimension $n-k$. Assume that $(H_X|H_Z)$ is an $(n-k)\times 2n$ generator matrix of $C$. The case when $m=1$ (i.e., $q$ is prime) is known (see [@wilde08; @schin]) and the corresponding result is the following:
\[thm:fp\] Let $C \subseteq \mathbb{F}_p^{2n}$ be an $(n-k)$-dimensional $\mathbb{F}_p$-linear space and $H = (H_X|H_Z)$ an $(n-k) \times 2n$ matrix whose row space is $C$. Let $C' \subseteq \mathbb{F}_p^{2(n+c)}$ be an $\mathbb{F}_p$-linear space such that the projection of $C'$ to the $1, 2, \ldots, n, n+c+1, n+c+2, \ldots, 2n+c$-th coordinates is equal to $C$ and $C' \subseteq (C')^{{\perp_s}}$, where $c$ is the minimum required number of maximally entangled quantum states in $\mathbb{C}^p \otimes \mathbb{C}^p$. Then, $$2c = \mathrm{rank}\left(H_X H_Z^T - H_Z H_X^T\right).$$ The encoding quantum circuit is constructed from $C'$, and it encodes $k+c$ logical qudits in $\mathbb{C}^p \otimes \cdots (k+c\; \mbox{times}) \cdots \otimes \mathbb{C}^p$ into $n$ physical qudits using $c$ maximally entangled pairs. The minimum distance is $d:= d_s \left(C^{{\perp_s}}\setminus (C\cap C^{{\perp_s}}) \right)$, where $$d_s\left(C^{{\perp_s}}\setminus (C\cap C^{{\perp_s}})\right) = \min\left\{ \mathrm{swt}\left(\vec{a}|\vec{b}\right) \mid \left(\vec{a}|\vec{b}\right) \in C^{{\perp_s}}\setminus (C\cap C^{{\perp_s}})\right\}.$$ In sum, $C$ provides an $[[n,k+c,d;c]]_p$ EAQECC over the field $\mathbb{F}_p$.
Theorem \[thm:fp\] states that the required number of maximally entangled quantum states is given by the rank of the matrix $H_X H_Z^T - H_Z H_X^T$. Our next result shows that, even in the case of codes over an arbitrary finite field $\mathbb{F}_q$, the above number depends only on the code $C$ and its symplectic dual.
\[propdim\] Let $C \subseteq \mathbb{F}_q^{2n}$ be a linear code over $\mathbb{F}_q$ and $(H_X|H_Z)$ its $(n-k)\times 2n$ generator matrix. Then, $$\mathrm{rank}\left(H_X H_Z^T - H_Z H_X^T\right) =
\dim_{\mathbb{F}_q} C - \dim_{\mathbb{F}_q} (C \cap C^{{\perp_s}}).$$
Consider the $\mathbb{F}_q$-linear map $f : \mathbb{F}_q^{2n} \rightarrow \mathbb{F}_q^{n-k}$ defined by $f\left(\vec{a}|\vec{b}\right) = \vec{a} H_Z^T - \vec{b} H_X^T$. Set $\mathrm{row}(H_X|H_Z)$ the row space of the matrix $(H_X|H_Z)$. Then we have $$\begin{aligned}
\mathrm{rank}\left(H_X H_Z^T - H_Z H_X^T\right) &=& \dim_{\mathbb{F}_q} f\left(\mathrm{row}(H_X|H_Z)\right)\\
&=& \dim_{\mathbb{F}_q} C - \dim_{\mathbb{F}_q} C \cap \ker(f)\\ &=& \dim_{\mathbb{F}_q} C - \dim_{\mathbb{F}_q} (C \cap C^{{\perp_s}}),\end{aligned}$$ which concludes the proof.
Next, with the help of the above proposition, we prove that Theorem \[thm:fp\] can be extended to codes over any finite field $\mathbb{F}_q$.
\[thm:fq\] Let $C \subseteq \mathbb{F}_q^{2n}$ be an $(n-k)$-dimensional $\mathbb{F}_q$-linear space and $H = (H_X|H_Z)$ a matrix whose row space is $C$. Let $C' \subseteq \mathbb{F}_q^{2(n+c)}$ be an $\mathbb{F}_q$-linear space such that its projection to the coordinates $1, 2, \ldots, n, n+c+1, n+c+2, \ldots, 2n+c$ equals $C$ and $C' \subseteq (C')^{{\perp_s}}$, where $c$ is the minimum required number of maximally entangled quantum states in $\mathbb{C}^q \otimes \mathbb{C}^q$. Then, $$2c = \mathrm{rank}\left(H_X H_Z^T - H_Z H_X^T\right) = \dim_{\mathbb{F}_q} C - \dim_{\mathbb{F}_q} \left(C \cap C^{{\perp_s}}\right).$$ The encoding quantum circuit is constructed from $C'$, and it encodes $k+c$ logical qudits in $\mathbb{C}^q \otimes \cdots (k+c\; \mbox{times}) \cdots \otimes \mathbb{C}^q$ into $n$ physical qudits using $c$ maximally entangled pairs. The minimum distance is $d:= d_s \left(C^{{\perp_s}}\setminus (C\cap C^{{\perp_s}}) \right)$, where $d_s$ is defined as in Theorem \[thm:fp\]. In sum, $C$ provides an $[[n,k+c,d;c]]_q$ EAQECC over the field $\mathbb{F}_q$.
One has that the inclusion $C^{{\perp_s}}\subseteq C^{{\perp_{ts}}}$ holds since ${\mathrm{tr}_{q/p}}(0)=0$. In addition, $C^{{\perp_{ts}}}\subseteq C^{{\perp_s}}$. Indeed, following [@ashikhmin00], if $\left(\vec{a}|\vec{b}\right) \in C^{{\perp_{ts}}}$, then $\left(\vec{a}|\vec{b}\right) \cdot_{ts} \left(\vec{x}|\vec{y}\right) =0$ for all $\left(\vec{x}|\vec{y}\right) \in C$. Taking into account that $\alpha \left(\vec{x}|\vec{y}\right) \in C$ for any $\alpha \in \mathbb{F}_q$, then ${\mathrm{tr}_{q/p}}\left((\vec{a}|\vec{b}) \cdot_{s} \alpha (\vec{x}|\vec{y})\right) =0$ for all $\alpha$. This means that ${\mathrm{tr}_{q/p}}\left(\alpha\left( (\vec{a}|\vec{b}) \cdot_{s} (\vec{x}|\vec{y})\right)\right)=0$ for all $\alpha$, which proves $\left(\vec{a}|\vec{b}\right) \cdot_{s} \left(\vec{x}|\vec{y}\right)=0$ and, therefore $\left(\vec{a}|\vec{b}\right) \in C^{{\perp_s}}$.
Now, using the same notation as at the beginning of this section, consider the code over the field $\mathbb{F}_p$, $C_0:= (\phi^E)^{-1} (C)$. It is clear that $\dim_{\mathbb{F}_p} (C_0) = m(n-k)$, and by Proposition \[prop:convert\] and the equality $C^{{\perp_s}}= C^{{\perp_{ts}}}$, we have $$\dim_{\mathbb{F}_p} C_0 = m(n-k) = m \dim_{\mathbb{F}_q} C.$$ Thus, $$\dim_{\mathbb{F}_p} C_0 - \dim_{\mathbb{F}_p} C_0 \cap C_0^{{\perp_s}}\\
= m \left( \dim_{\mathbb{F}_q} C - \dim_{\mathbb{F}_q} (C \cap C^{{\perp_s}}) \right).$$ This shows that, by Theorem \[thm:fp\], we have an entanglement-assisted quantum code encoding $m(k+c)$ qudits in $\mathbb{C}^p$ and consuming $mc$ maximally entangled states in $\mathbb{C}^p \otimes \mathbb{C}^p$. Using the map $\phi^E$ and the fact that $C^{\perp_s} = C^{\perp_{ts}}$, we have an entanglement-assisted quantum code encoding $(k+c)$ qudits in $\mathbb{C}^q$ and consuming $c$ maximally entangled states in $\mathbb{C}^q \otimes \mathbb{C}^q$. In fact, one can construct $C'_0 \subseteq \mathbb{F}_p^{2m(n+c)}$ in the same way as constructed $C'$ from $C$ in Theorem \[thm:fp\]. Applying $\phi^E$ to the code $C'_0$, we get the code $C'$ in the statement with the claimed properties. The minimum distance follows from [@ketkar06 Section III].
The Hermitian case {#Hermitian}
------------------
In this subsection, we specialize the results in Subsection \[symplectic\] by considering the Hermitian inner product instead of a symplectic form. With the above notation, consider the finite field $\mathbb{F}_{q^2}$ and a normal basis $\{w, w^q\}$ of $\mathbb{F}_{q^2}$ over $\mathbb{F}_{q}$. Fix a positive integer $n$ and, following [@ketkar06], define a trace-alternating form over $\mathbb{F}_{q^2}^n$ as $$\vec{x} \cdot_a \vec{y} = {\mathrm{tr}_{q/p}}\left( \frac{\vec{x} \cdot \vec{y}^q - \vec{x}^q \cdot \vec{y}}{w^{2q}-w^2} \right),$$ where $\vec{z}^q$, $\vec{z} \in \mathbb{F}_{q^2}^n$, means the componentwise $q$-power of $\vec{z}$. The map $\varphi: \mathbb{F}_{q}^{2n} \rightarrow \mathbb{F}_{q^2}^{n}$ given by $\varphi \left(\vec{a}|\vec{b}\right) = w \vec{a} + w^q \vec{b}$ is bijective and isometric because the symplectic and the Hamming weights of $\left(\vec{a}|\vec{b}\right)$ and $\varphi\left(\vec{a}|\vec{b}\right)$ coincide. In addition, for $\left(\vec{a}|\vec{b}\right), \left(\vec{a'}|\vec{b'}\right) \in \mathbb{F}_{q}^{2n}$, it holds that $$\left(\vec{a}|\vec{b}\right) \cdot_{ts} \left(\vec{a'}|\vec{b'}\right) = \varphi\left(\vec{a}|\vec{b}\right) \cdot_{a} \varphi\left(\vec{a'}|\vec{b'}\right).$$
Recall that the Hermitian inner product of two vectors $\vec{x}, \vec{y} \in \mathbb{F}_{q^2}^{n}$ is defined to be $\vec{x} \cdot_h \vec{y} = \vec{x}^q \cdot \vec{y}$, where $\cdot$ means Euclidean product, and that, in [@ketkar06], it is proved that for a $\mathbb{F}_{q^2}$-linear code $D$, the dual codes with respect to the products $\cdot_a$ and $\cdot_h$ coincide. With the above ingredients, we are ready to prove the next proposition which will allow us to state and prove our theorem on EAQECCs over arbitrary finite fields by considering Hermitian inner product.
\[propdimh\] Let $C \subseteq \mathbb{F}_{q^2}^n$ be a code over $\mathbb{F}_{q^2}$ of dimension $(n-k)/2$ for some positive integer $k$. Let $H$ be its generator matrix. Then $$\mathrm{rank}(HH^*) =
\dim_{\mathbb{F}_{q^2}} C - \dim_{\mathbb{F}_{q^2}} (C \cap C^{{\perp_h}}),$$ where $H^*$ is the $q$th power of the transpose matrix of $H$.
Define the $\mathbb{F}_{q^2}$-linear map $f : \mathbb{F}_{q^2}^{n} \rightarrow \mathbb{F}_{q^2}^{(n-k)/2}$, given by $f(\vec{a}) = \vec{a} H^*$. Then, $$\begin{aligned}
\mathrm{rank}(H H^*) &=& \dim_{\mathbb{F}_{q^2}} f(\mathrm{row}(H))\\
&=& \dim_{\mathbb{F}_{q^2}} C - \dim_{\mathbb{F}_{q^2}} (C \cap \ker(f))\\ &=& \dim_{\mathbb{F}_{q^2}} C - \dim_{\mathbb{F}_{q^2}} (C \cap C^{{\perp_h}}).\end{aligned}$$
\[thm:hermitian\] Let $C \subseteq \mathbb{F}_{q^2}^{n}$ be an $(n-k)/2$-dimensional code over $\mathbb{F}_{q^2}$, for suitable integers $n$ and $k$. Denote by $H$ its generator matrix. Let $C' \subseteq \mathbb{F}_{q^2}^{(n+c)}$ be an $\mathbb{F}_{q^2}$-linear space whose projection to the coordinates $1, 2, \ldots, n$ equals $C$ and satisfies $C' \subseteq (C')^{\perp_h}$, where $c$ is the minimum required number of maximally entangled quantum states in $\mathbb{C}^q \otimes \mathbb{C}^q$. Then, $$c = \mathrm{rank} \left(H H^* \right) = \dim_{\mathbb{F}_{q^2}} C - \dim_{\mathbb{F}_{q^2}} \left(C \cap C^{\perp_h}\right).$$ The encoding quantum circuit is constructed from $C'$, and it encodes $k+c$ logical qudits in $\mathbb{C}^q \otimes \cdots (k+c\; \mbox{times}) \cdots \otimes \mathbb{C}^q$ into $n$ physical qudits using $c$ maximally entangled pairs. The minimum distance is $d:= d_H \left(C^{\perp_h} \setminus (C\cap C^{\perp_h}) \right)$, where $d_H$ is defined as the minimum Hamming weight of the vectors in the set $C^{\perp_h} \setminus \left(C\cap C^{\perp_h} \right)$. In sum, $C$ provides an $[[n,k+c,d;c]]_q$ EAQECC over the field $\mathbb{F}_q$.
With the above notation, consider the code $C'$ in $\mathbb{F}_{q^2}^n$ of dimension $n-k$ whose generator matrix is $$\mathcal{H} = \left(
\begin{array}{c}
\omega H \\
\omega^q H \\
\end{array}
\right)$$ and set $C_0 = \varphi^{-1} (C')$ the corresponding code in $\mathbb{F}_{q}^{2n}$. Since $\varphi$ is an isometry, to obtain the value $2c$ corresponding to $C_0$, it suffices to compute the rank of the matrix given by the form $\cdot_a$ which is $\mathcal{J}= \mathrm{tr}_{q^2/q} \left((H H^* - H^q H^T)/ \lambda \right)$, where $\lambda = \omega^{2q} - \omega^2$ and $\mathrm{tr}_{q^2/q}$ the trace map from $\mathbb{F}_{q^2}$ to $\mathbb{F}_{q}$. Now, setting $$\mathcal{Z}= \left(
\begin{array}{cc}
\omega^{q+1} & \omega^2 \\
\omega^{2q} & \omega^{q+1} \\
\end{array}
\right),$$ it holds that $\mathcal{J}= (2/\lambda) \left(ZH H^*- Z^T H^q H^T \right)$. Performing elementary operations, we get that $ \mathrm{rank} (\mathcal{J}) = 2 \: \mathrm{rank} \left( H H^*\right)$. Finally, by our previous considerations, $\dim_{\mathbb{F}_q} C_0 = n-k$, $\dim_{\mathbb{F}_q} (C_0 \cap C_0^{{\perp_s}}) = 2c$, and $$d_H \left(C^{{\perp_h}}\setminus C^{{\perp_h}}\cap C\right) = d_s \left(C_0^{{\perp_s}}\setminus (C^{{\perp_s}}\cap C_0) \right),$$ which proves our statement by Theorem \[thm:fq\].
The following corollary is an immediate consequence of the above result.
Let $C$ be an $[n,k,d]_{q^2}$ linear code over $\mathbb{F}_{q^2}$ and set $H$ a parity check matrix of $C$. Then, there exist an $[[n,2k-n+c,d;c]]_q$ EAQECC where $c= \mathrm{rank}(H H^*)$, $H^*$ being the $q$th power of the transpose matrix $H^T$.
The Euclidean case
------------------
In this section we will show that EAQECCs over any finite field $\mathbb{F}_{q}$ can be obtained through a CSS construction, where the Euclidean inner product is considered, and carried out with two $\mathbb{F}_{q}$-linear codes $C_1$ and $C_2$ of length $n$. Assume that $C_1$ (respectively, $C_2$) has dimension $k_1$ and generator matrix $H_1$ (respectively, $k_2$ and $H_2$). Before stating our result, we give the following proposition which will be used in its proof.
\[propdimE\] With the above notations, it holds that $$\label{eq1}
\mathrm{rank}(H_1H_2^T) = \dim_{\mathbb{F}_{q}} C_1 - \dim_{\mathbb{F}_{q}} (C_1 \cap C_2^\perp),$$ and $$\label{eq2}
\mathrm{rank}(H_2H_1^T) = \dim_{\mathbb{F}_{q}} C_2 - \dim_{\mathbb{F}_{q}} (C_2 \cap C_1^\perp),$$ where $\perp$ means Euclidean dual.
To prove Equality (\[eq1\]), consider the $\mathbb{F}_q$-linear map $f : \mathbb{F}_{q}^{n} \rightarrow \mathbb{F}_{q}^{k_2}$ defined by the matrix $H_2^T$, that is $f(\vec{a}) = \vec{a}H_2^T$. Then $$\begin{aligned}
\mathrm{rank}(H_1 H_2^T) &=& \dim_{\mathbb{F}_{q}} f(\mathrm{row}(H_1))\\
&=& \dim_{\mathbb{F}_{q}} C_1 - \dim_{\mathbb{F}_{q}} (C_1 \cap \ker(f))\\ &=& \dim_{\mathbb{F}_{q}} C_1 - \dim_{\mathbb{F}_{q}} (C_1 \cap C_2^\perp).\end{aligned}$$ Equality (\[eq2\]) follows analogously from the map given by $H_1^T$.
Next we state the main result in this section.
\[thm:euclidean\] Let $C_1$ and $C_2$ be two linear codes over $\mathbb{F}_{q}$ included in $\mathbb{F}_q^{n}$ with respective dimensions $k_1$ and $k_2$ and generator matrices $H_1$ and $H_2$. Then, the code $C_0 = C_1 \times C_2 \subseteq \mathbb{F}_q^{2n}$ gives rise to an EAQECC which encodes $n-k_1-k_2 + c$ logical qudits into $n$ physical qudits using the minimum required of maximally entangled pairs $c$, which is $$c = \mathrm{rank}(H_1H_2^T) = \dim_{\mathbb{F}_{q}} C_1 - \dim_{\mathbb{F}_{q}} (C_1 \cap C_2^\perp).$$ The minimum distance of the entanglement-assisted quantum code is larger than or equal to $$d:= \min \left\{ d_H\left(C_1^\perp \setminus (C_2 \cap C_1^\perp)\right), d_H\left(C_2^\perp \setminus (C_1 \cap C_2^\perp)\right) \right\}.$$ In sum, one gets an $[[n, n-k_1-k_2+c,d;c]]_q$ EAQECC.
It suffices to notice that $\dim_{\mathbb{F}_{q}} C_0 = k_1 + k_2$, $C_0^{{\perp_s}}= C_2^\perp \times C_1^\perp$, and $$\begin{aligned}
&& \dim_{\mathbb{F}_{q}} C_0 - \dim_{\mathbb{F}_{q}} (C_0^{{\perp_s}}\cap C_0) \\
&=& \dim_{\mathbb{F}_{q}} \left(C_1 \times C_2\right) -\dim_{\mathbb{F}_{q}} \left((C_2^\perp \cap C_1) \times (C_1^\perp \cap C_2)\right)\\
&=& \left( \dim_{\mathbb{F}_{q}} C_1 - \dim_{\mathbb{F}_{q}} (C_2^\perp \cap C_1) \right) + \left(\dim_{\mathbb{F}_{q}} C_2 - \dim_{\mathbb{F}_{q}} (C_1^\perp \cap C_2) \right)\\
&=& 2c.
\end{aligned}$$ By construction, we have that $$d_s(C_0 \setminus C_0^{{\perp_s}})
\geq \min \left\{ d_H\left(C_1^\perp \setminus (C_2 \cap C_1^\perp)\right),
d_H\left(C_2^\perp \setminus (C_1 \cap C_2^\perp)\right)\right\},$$ and then our statement follows from Theorem \[thm:fq\].
Geometric decomposition of the coordinate space {#geometric}
-----------------------------------------------
In this subsection we consider only the Hermitian and Euclidean cases, and we will explain that the required number of maximally entangled pairs is easy to compute when the generators of the supporting $\mathbb{F}_q$-linear code $C$ in $\mathbb{F}_q^{n}$ are a subset of a basis of $\mathbb{F}_q^{n}$ with a special metric structure and which is said to be [*compatible with a geometric decomposition*]{} of $\mathbb{F}_q^{n}$ (see [@MetricStructure]). Notice that, in the Hermitian case, $q$ should be $q^2$, however, for simplicity’s sake and only in this subsection, we will use $q$ as a generic symbol which means a power of a prime in the Euclidean case or an even power of a prime in the Hermitian case. For avoiding to repeat notation, again only in this subsection, $\langle \vec{a} , \vec{b} \rangle$ will mean either the Hermitian inner product $\vec{a} \cdot_h \vec{b}$ or the Euclidean one $\vec{a} \cdot \vec{b}$.
Let us introduce some notation, we say that $\{\vec{v_1}, \vec{v_2}\}$ are [*geometric generators of a hyperbolic plane*]{} if $\langle \vec{v_1} , \vec{v_1} \rangle = \langle \vec{v_2} , \vec{v_2} \rangle =0$ and $\langle \vec{v_1} , \vec{v_2} \rangle =1$. We say that $\{\vec{v_1}, \vec{v_2}\}$ are [*geometric generators of an elliptic plane*]{} if $\langle \vec{v_1} , \vec{v_1} \rangle =0$ and $\langle \vec{v_2} , \vec{v_2} \rangle = \langle \vec{v_1} , \vec{v_2} \rangle =1$. Finally, we say that $\vec{v}$ [*generates a non-singular space*]{} if $\langle \vec{v}, \vec{v} \rangle \neq 0$.
Let $C \subseteq \mathbb{F}_q^n$ and set $\left\{ \vec{v_1}, \vec{v_2}, \ldots , \vec{v_n} \right\}$ a basis of $\mathbb{F}_q^n$ such that $C$ is generated by $\left\{\vec{v_i}\right\}_{i \in I}$ for $I \subseteq \{1, 2, \ldots , n\}$. We say that $C$ is [*compatible with a geometric decomposition*]{} of $\mathbb{F}_q^n$ if $$\mathbb{F}_q^n = H_1 \oplus \cdots \oplus H_r \oplus L_1 \oplus \cdots \oplus L_s,$$ where the linear spaces from $H_1$, generated by $\left\{\vec{v_1}, \vec{v_2}\right\}$, to $H_r $, generated by $\left\{\vec{v}_{2r-1} , \vec{v_r} \right\}$, are hyperbolic planes, being the $\vec{v_i}$ geometric generators, and from $L_1$, generated by $\vec{v}_{2r+1}$, to $L_s$, generated by $\vec{v}_{2r+s} = \vec{v_n}$, are non-singular spaces. Then, we say that the vectors $\vec{v_1}, \vec{v_2}, \ldots, \vec{v_r}$ (and the indexes $1, 2, \ldots r$) are asymmetric and the vectors $\vec{v}_{r+1}, \vec{v}_{r+2}, \ldots, \vec{v}_n$ (and the indexes $r+1, r+2, \ldots, n$) are symmetric. Moreover, we also say that $(1,2)$, $\ldots$, $(r-1,r)$ are symmetric pairs.
In [@MetricStructure], for the Euclidean inner product, it was proved that for characteristic different from 2, we can always obtain a basis $\left\{\vec{v_1}, \vec{v_2}, \ldots , \vec{v_n}\right\}$ of $\mathbb{F}_q^n$ such that $$\mathbb{F}_q^n = H_1 \oplus \cdots \oplus H_r \oplus L_1 \oplus \cdots \oplus L_s,$$ with $s \le 4$. For characteristic equal to 2, we may have a decomposition as in the case with characteristic different from 2, excepting when the vector $(1, 1, \ldots ,1 )$ belongs to the [*radical*]{} (or [*hull*]{}) of $C$, $C\cap C^\bot$ . In that particular case, it was given in [@MetricStructure] the following decomposition $$\mathbb{F}_q^n = H_1 \oplus \cdots \oplus H_r \oplus L_1 \oplus \cdots \oplus L_s \oplus E,$$ with $s \le 2$ and where $E$ is an elliptic plane.
Let $M = (\langle \vec{v_i} , \vec{v_j} \rangle )_{1\le i,j \le n}$, one has that $M$ has the form
$$M= \left(
\begin{array}{l@{\hspace{0.4cm}}l@{\hspace{0.4cm}}l@{\hspace{0.4cm}}l@{\hspace{0.4cm}}l@{\hspace{0.4cm}}l@{\hspace{0.4cm}}l@{\hspace{0.4cm}}l}
0 & 1 & \nonumber & \nonumber & \nonumber & \nonumber & \nonumber & \nonumber \\
1 & 0 & \nonumber & \nonumber & \nonumber & \nonumber & \nonumber & \nonumber \\
\nonumber & \nonumber & \ddots & \nonumber & \nonumber & \nonumber & \nonumber & \nonumber \\
\nonumber & \nonumber & \nonumber & 0 & 1 & \nonumber & \nonumber & \nonumber \\
\nonumber & \nonumber & \nonumber & 1 & 0 & \nonumber & \nonumber & \nonumber \\
\nonumber & \nonumber & \nonumber & \nonumber & \nonumber & g_1 & \nonumber & \nonumber \\
\nonumber & \nonumber & \nonumber & \nonumber & \nonumber & \nonumber & \ddots & \nonumber \\
\nonumber & \nonumber & \nonumber & \nonumber & \nonumber & \nonumber & \nonumber & g_s \\
\end{array} \right),$$ where $g_1, \ldots, g_s$ are non-zero, except for the case when the characteristic is 2 and $(1, 1, \ldots, 1)$ belongs to the radical of $C$; then we have that $M$ is equal to
$$M= \left(
\begin{array}{l@{\hspace{0.4cm}}l@{\hspace{0.4cm}}l@{\hspace{0.4cm}}l@{\hspace{0.4cm}}l@{\hspace{0.4cm}}l@{\hspace{0.4cm}}l@{\hspace{0.4cm}}l@{\hspace{0.4cm}}l}
0 & 1 & \nonumber & \nonumber & \nonumber & \nonumber & \nonumber & \nonumber & \nonumber \\
1 & 0 & \nonumber & \nonumber & \nonumber & \nonumber & \nonumber & \nonumber & \nonumber \\
\nonumber & \nonumber & \ddots & \nonumber & \nonumber & \nonumber & \nonumber & \nonumber & \nonumber \\
\nonumber & \nonumber & \nonumber & 0 & 1 & \nonumber & \nonumber & \nonumber & \nonumber \\
\nonumber & \nonumber & \nonumber & 1 & 0 & \nonumber & \nonumber & \nonumber & \nonumber \\
\nonumber & \nonumber & \nonumber & \nonumber & \nonumber & g_1 & \nonumber & \nonumber & \nonumber \\
\nonumber & \nonumber & \nonumber & \nonumber & \nonumber & \nonumber & g_s & \nonumber & \nonumber \\
\nonumber & \nonumber & \nonumber & \nonumber & \nonumber & \nonumber & \nonumber & 0 & 1 \\
\nonumber & \nonumber & \nonumber & \nonumber & \nonumber & \nonumber & \nonumber & 1 & 1 \\
\end{array} \right).$$
Now, let $i \in \{ 1 , 2, \ldots , n \}$. We define $i'$ as
- $i+1$ if $\vec{v_i}$ is the first generator of a hyperbolic plane $H$,
- $i-1$ if $\vec{v_i}$ is the second generator of a hyperbolic plane $H$,
- $i$ if $\vec{v_i}$ generates a one-dimensional linear space $L$,
- $i+1$ if $\vec{v_i}$ is the first generator of an elliptic plane $E$.
Notice that we do not define $i'$ when $\vec{v_i}$ is the second geometric generator of an elliptic plane, because in this case, $(1, 1, \ldots, 1)$ is not in the radical of $C$ [@MetricStructure]. For $I \subseteq \{1, 2, \ldots, n \}$ we set $I'=\{ i' : i \in I \}$ and $I^\bot = \{1, 2, \ldots, n \} \setminus I'$. In this way we can compute the dual code $C^\bot$ of a linear code $C$ generated by $\left\{\vec{v_i}\right\}_{i\in I}$ easily since it is generated by $\left\{\vec{v_i}\right\}_{i \in I^\bot}$. Moreover, it can also be used to construct QECCs using the CSS construction since $C \subseteq C^\bot$ if and only if $I \subseteq I^\bot$. These kinds of decomposition arise naturally (i.e., for the usual generators) in some families of evaluation codes as BCH codes, toric codes, $J$-affine variety codes, negacyclic codes, constacylic codes, etc. and the previous approach has been exploited for constructing stabilizer quantum codes, EAQECCs and LCD codes (see [@qc4; @LCD; @Guenda18; @lag2; @Liu18] for instance).
The above paragraphs allow us to give a practical procedure for computing the value $c$ given in Theorem \[thm:hermitian\], for the Hermitian product, and in Theorem \[thm:euclidean\], for the Euclidean product ($C_1=C_2=C)$. Assume that $C$ is a code generated by $\left\{\vec{v_i}\right\}_{i\in I}$ compatible with a geometric decomposition of the corresponding coordinate space and write $I = I_R \sqcup I_L$ (i.e. $I = I_R \cup I_L$ and $I_R \cap I_L = \emptyset$), where the radical of $C$, $C \cap C^\bot$, $\bot$ meaning dual with respect to the inner product $\langle ~ , ~ \rangle$, is generated by $\left\{\vec{v_i}\right\}_{i\in I_R}$. The radical of $C$ can be easily computed in this case: Indeed, given $i \in I$, one has that $i \in I_R$ if it holds that: $\vec{v_i}$ is the first generator of a hyperbolic plane $H$ and $i+1 \not\in I$, $\vec{v_i}$ is the second generator of a hyperbolic plane $H$ and $i-1 \not\in I$, or $\vec{v_i}$ is the first generator of an elliptic plane $E$. Otherwise, $i \in I_L$. An equivalent way to characterize $I_R$ is the following: $I_R$ consists of asymmetric indexes whose pair does not belong to $I$, and $I_L$ consists of symmetric indexes and pairs of asymmetric indexes that belong to $I$. Summarizing, one has that when one considers a suitable basis as above, then $$\begin{aligned}
c &=& \dim_{\mathbb{F}_q} C - \dim_{\mathbb{F}_q} \left(C \cap C^\bot\right)
= \mathrm{card} (I) - \mathrm{card} (I \cap I^\bot)\\
&=& \mathrm{card} (I) - \mathrm{card} (I_R) = \mathrm{card} (I_L).\end{aligned}$$
Note that we have an EAQECC with maximal entanglement when $I=I_L$, i.e., when $I_R = \emptyset$. This fact was used, for instance, in [@Guenda18].
Gilbert-Varshamov-type sufficient condition of existence of entanglement-assisted codes {#sect3}
=======================================================================================
In this section we give a Gilbert-Varshamov-type bound which is valid for EAQECCs over arbitrary finite fields. A similar bound was stated in [@lai14] for the binary case.
\[thmgv\] Assume the existence of positive integers $n$, $k\leq n$, $\delta$, $c\leq (n-k)/2$ such that $$\frac{q^{n+k}-q^{n-k-2c}}{q^{2n}-1}\sum_{i=1}^{\delta - 1}
{n \choose i}(q^2-1)^i < 1. \label{eqgv}$$ Then there exists an $\mathbb{F}_q$-linear code $C \subseteq
\mathbb{F}_q^{2n}$ such that $\dim_{\mathbb{F}_q} C = n-k$, $d_s(C^{{\perp_s}}\setminus (C^{{\perp_s}}\cap C)) \geq \delta$ and $\dim_{\mathbb{F}_q} C - \dim_{\mathbb{F}_q} (C^{{\perp_s}}\cap C)= 2c$.
We will use a close argument to the proof of the Gilbert-Varshamov bound for stabilizer codes [@ketkar06]. Let $\mathrm{Sp}(q,n)$ be the symplectic group over $\mathbb{F}_q^{2n}$ [@grove02 Section 3] and $A(k,c)$ the set of $\mathbb{F}_q$-linear spaces $V \subseteq \mathbb{F}_q^{2n}$ such that $\dim_{\mathbb{F}_q} V = n-k$ and $$\dim_{\mathbb{F}_q} V - \dim_{\mathbb{F}_q} \left(V^{{\perp_s}}\cap V \right)= 2c.$$ For $\vec{0} \neq \vec{e} \in \mathbb{F}_q^{2n}$, define $$B(k,c,\vec{e})
= \left\{ V\in A(k,c) \mid \vec{e} \in V^{{\perp_s}}\setminus (V^{{\perp_s}}\cap V) \right\}.$$ Taking into account that the symplectic group acts transitively on $\mathbb{F}_q^{2n} \setminus \{\vec{0}\}$ [@aschbacher00; @grove02], it holds that for nonzero $\vec{e}_1$, $\vec{e}_2 \in \mathbb{F}_q^{2n}$, there exists $M \in \mathrm{Sp}(q,n)$ such that $\vec{e}_1 M= \vec{e}_2$, and, for $V_1, V_2 \in A(k,c)$, there exists $M \in \mathrm{Sp}(q,n)$ such that $V_1 M= V_2$.
Therefore for nonzero elements $\vec{e}_1, \vec{e}_2 \in \mathbb{F}_q^{2n}$ with $\vec{e}_1 M_1 = \vec{e}_2 \; \left(M_1 \in \mathrm{Sp}(q,n)\right)$ and some fixed linear space $V_1 \in A(k,c)$, we have the following chain of equalities: $$\begin{aligned}
&& \mathrm{card} \left(B(k, c, \vec{e}_1)\right) \\
&=& \mathrm{card}\left(\{ V \in A(k,c) \mid \vec{e}_1 \in V^{{\perp_s}}\setminus (V^{{\perp_s}}\cap V) \}\right) \\
&=& \mathrm{card}\left(\{ V_1 M \mid \vec{e}_1 \in V^{{\perp_s}}M \setminus (V^{{\perp_s}}M \cap V M), M \in \mathrm{Sp}(q,n) \}\right)\\
&=& \mathrm{card} \left(\{V_1 M M_1^{-1} \mid \vec{e}_1 \in V^{{\perp_s}}M M_1^{-1} \setminus (V^{{\perp_s}}M M_1^{-1}\cap V M M_1^{-1}), M \in \mathrm{Sp}(q,n) \}\right)\\
&=& \mathrm{card}\left(\{ V_1 M \mid \vec{e}_1 M_1 \in V^{{\perp_s}}M \setminus (V^{{\perp_s}}M \cap V M), M \in \mathrm{Sp}(q,n) \} \right)\\
&=& \mathrm{card}\left(\{ V_1 M \mid \vec{e}_2 \in V^{{\perp_s}}M \setminus (V^{{\perp_s}}M\cap V M), M \in \mathrm{Sp}(q,n) \} \right)\\
&=& \mathrm{card}\left(\{ V \in A(k,c) \mid \vec{e}_2 \in V^{{\perp_s}}\setminus (V^{{\perp_s}}\cap V) \} \right) \\
&=& \mathrm{card}\left(B(k, c, \vec{e}_2)\right).\end{aligned}$$
For each $V \in A(k,c)$, the number of vectors $\vec{e}$ in $\mathbb{F}_q^{2n}$ such that $\vec{e} \in V^{{\perp_s}}\setminus (V^{{\perp_s}}\cap V)$ is $$\mathrm{card}(V^{{\perp_s}})-\mathrm{card}\left(V^{{\perp_s}}\cap V\right)=q^{n+k}-q^{n-k-2c}.$$ The number of pairs $(\vec{e}$, $V)$ such that $\vec{0} \neq
\vec{e} \in V^{{\perp_s}}\setminus (V^{{\perp_s}}\cap V)$ is $$\sum_{\vec{0} \neq \vec{e}\in \mathbb{F}_q^{2n}} \mathrm{card}\left(B(k,c,\vec{e})\right) = \mathrm{card}\left(A(k,c)\right) \left(q^{n+k}-q^{n-k-2c}\right),$$ which implies $$\frac{\mathrm{card}\left(B(k,c,\vec{e})\right)}{\mathrm{card}\left(A(k,c)\right)} = \frac{q^{n+k}-q^{n-k-2c}}{q^{2n}-1}.
\label{eq101}$$ If there exists $V \in A(k,c)$ such that $V \notin B(k,c,\vec{e})$ for all $1 \leq \mathrm{swt}(\vec{e}) \leq \delta-1$, then there will exist $V$ with the desired properties. The number of vectors $\vec{e}$ such that $1 \leq \mathrm{swt}(\vec{e}) \leq \delta-1$ is given by $$\sum_{i=1}^{\delta - 1} {n \choose i}(q^2-1)^i. \label{eq103}$$ By combining Equalities (\[eq101\]) and (\[eq103\]), we see that Inequality (\[eqgv\]) is a sufficient condition for ensuring the existence of a code $C$ as in our statement. This ends the proof.
To finish this section, we derive an asymptotic form of Theorem \[thmgv\].
\[thmasymp\] Let $R$, $\epsilon$ and $\lambda$ be nonnegative real numbers such that $R \leq 1$, $\epsilon < 1/2$, and $\lambda \leq (1-R)/2$. Let $h(x) := -x \log_q x -(1- x) \log_q (x-1)$ be the $q$-ary entropy function. For $n$ sufficiently large, the inequality $$h(\epsilon) + \epsilon \log_q (q^2-1) < 1-R, \label{eq104}\\$$ implies the existence of a code $C \subseteq
\mathbb{F}_q^{2n}$ over $\mathbb{F}_q$ such that $$\dim_{\mathbb{F}_q} C = \lceil n(1-R) \rceil, \;\;
d_s(C^{{\perp_s}}\setminus (C^{{\perp_s}}\cap C)) \geq \lfloor n\epsilon\rfloor$$ and $$\dim_{\mathbb{F}_q} C - \dim_{\mathbb{F}_q} (C^{{\perp_s}}\cap C) = \lfloor 2n\lambda\rfloor.$$
It follows from Theorem \[thmgv\] using a similar reasoning to that in [@matsumotouematsu01 Section III.C].
EAQECCs coming from punctured QECCs {#sec:wip}
===================================
Our final section gives parameters of EAQECCs obtained from punctured codes coming from self-orthogonal codes with respect to symplectic forms, or Hermitian or Euclidean inner products. Since fewer qudits should be transmitted through a noisy channel, they perform better. Let us start with the symplectic case.
Symplectic form {#sec:wips}
---------------
Let $C \subseteq \mathbb{F}_q^{2n}$ be an $\mathbb{F}_q$-linear code. The [*puncturing*]{} of $C$ to the coordinate set $\{1$, …, $n-c\}$ is defined as the code of length $2(n-c)$ given by $$\begin{gathered}
P(C) = \big\{ (a_1, \ldots, a_{n-c}| b_1, \ldots, b_{n-c}) \mid
(a_1, \ldots, a_n| b_1, \ldots, b_n) \in C \\
\mbox{for some} \;\; a_{n-c+1}, \ldots, a_n, b_{n-c+1}, \ldots, b_n\in \mathbb{F}_q \big\}.\end{gathered}$$
In addition, the [*shortening*]{} of $C$ to the coordinate set $\{1$, …, $n-c\}$ is defined as the code $$S(C) = \{ (a_1, \ldots, a_{n-c}| b_1, \ldots, b_{n-c}) \mid \\
(a_1, \ldots, a_{n-c}, 0, \ldots, 0| b_1, \ldots, b_{n-c},
0, \ldots, 0)\in C \}.$$
When we have a stabilizer code given by an $\mathbb{F}_q$-linear code $C$ such that $C \subseteq C^{{\perp_s}}\subseteq
\mathbb{F}_q^{2n}$, we can construct an entanglement-assisted code from $P(C) \subseteq \mathbb{F}_q^{2(n-c)}$. By [@pless98], $P(C)^{{\perp_s}}= S(C^{{\perp_s}})$ and we deduce $$P(C) \cap P(C)^{{\perp_s}}= P(C)\cap S(C^{{\perp_s}})
= S(C \cap C^{{\perp_s}}) = S(C).$$
The minimum distance of the constructed entanglement-assisted code is $d_s(S(C^{{\perp_s}}) \setminus S(C))$ which is larger than or equal to $d_s(C^{{\perp_s}}\setminus C)$. Following [@pless98], one can prove the following result.
Assume that a positive integer $c$ satisfies $2c \leq d_H(C \setminus \{\vec{0}\})-1$, then $$\begin{array}{l}
\dim_{\mathbb{F}_q} P(C) = \dim_{\mathbb{F}_q} C, \;\;\mbox{and}\\ \\
\dim_{\mathbb{F}_q} P(C)\cap P(C)^{{\perp_s}}= \dim_{\mathbb{F}_q} S(C) = \dim_{\mathbb{F}_q} C - 2c.
\end{array}$$
Summarizing these observations, we have the following theorem. Notice that a close result has been given in [@lai12] for binary codes.
\[thm:newcode\] Let $C \subseteq \mathbb{F}_q^{2n}$ be an $\mathbb{F}_q$-linear code with $\dim_{\mathbb{F}_q} C = n-k$ and $C \subseteq C^{{\perp_s}}$. Assume that a positive integer $c$ satisfies $2c \leq d_H(C \setminus \{\vec{0}\})-1$, then the punctured code $P(C)$ provides an $$[[n-c, k+c, \geq d_s(C^{{\perp_s}}\setminus C); c]]_q$$ entanglement-assisted code.
Our next two sections are devoted to give similar results but considering Hermitian or Euclidean inner product.
Hermitian inner product
-----------------------
Let $C \subseteq \mathbb{F}_{q^2}^n$ an $\mathbb{F}_q$-linear code. The [*$h$-puncturing*]{} of $C$ to the coordinate set $\{1, 2, \ldots,
n-c\}$ is the code of length $n-c$ defined as $$P_h(C) = \\ \left\{ (a_1, a_2, \ldots, a_{n-c}) \mid
(a_1, a_2, \ldots, a_n) \in C \;
\mbox{for some} \; a_{n-c+1}, \ldots, a_n \in \mathbb{F}_{q^2} \right\}.$$ The [*$h$-shortening*]{} of $C$ to the coordinate set $\{1, 2, \ldots, n-c\}$ is the code of length $n-c$ defined as $$S_h(C) = \left\{ (a_1, a_2, \ldots, a_{n-c}) \mid (a_1, a_2, \ldots, a_{n-c}, 0, \ldots, 0) \in C \right\}.$$
The above concepts allows us to state the following theorem.
\[thm:newcodeh\] Let $C \subseteq \mathbb{F}_{q^2}^n$ be an $\mathbb{F}_{q^2}$-linear code with $\dim_{\mathbb{F}_{q^2}} C = (n-k)/2$ and suppose that $C \subseteq C^{{\perp_h}}$. Let $c$ be a positive integer such that $c \leq d_H(C \setminus \{\vec{0}\})-1$, then the punctured code $P_h(C)$ provides an $$[[n-c, k+c, \geq d_H(C^{{\perp_h}}\setminus C); c]]_q$$ entanglement-assisted code.
By the assumption, $\dim_{\mathbb{F}_{q^2}} P_h(C) = \dim_{\mathbb{F}_{q^2}} C$. By a similar argument to that used in Subsection \[sec:wips\], we also see that $P_h(C) \cap P_h(C)^{{\perp_h}}= S_h(C)$. Now we have that $c \leq d_H(C \setminus \{\vec{0}\})-1$, so $\dim_{\mathbb{F}_{q^2}} P_h(C) = \dim_{\mathbb{F}_{q^2}} C$ and $\dim_{\mathbb{F}_{q^2}} S_h(C) = \dim_{\mathbb{F}_{q^2}} C - c$ [@pless98]. It also holds that $d_H(P_h(C)^{{\perp_h}}\setminus S_h(C))
\geq d_H(C^{{\perp_h}}\setminus C)$ and this concludes the proof by Theorem \[thm:hermitian\].
Euclidean inner product
-----------------------
Our result concerning Euclidean duality is the following:
\[thm:newcodee\] Let $C_2 \subseteq C_1 \subseteq \mathbb{F}_{q}^n$ be two $\mathbb{F}_{q}$-linear codes such that $\dim C_i = k_i$, $1 \leq i \leq 2$. The standard construction of CSS codes uses $C_2 \times C_1^\perp$ as the stabilizer. Assume that $c$ is a positive integer such that $$c \leq \min\left\{d_H(C_2 \setminus \{\vec{0}\}),
d_H(C_1^\perp \setminus \{0\})\right\}-1,$$ then the punctured code $P_h(C_2)\times P_h(C_1^\perp)$ provides an $$[[n-c, k_1-k_2+c, \geq \min \left\{ d_H(C_1\setminus C_2), d_H(C_2^\perp\setminus C_1^\perp) \right\}; c]]_q$$ entanglement-assisted code.
The assumption $c \leq \min\{d_H(C_2 \setminus \{\vec{0}\})$, $d_H(C_1^\perp \setminus \{0\})\}-1$ implies the following two equalities: $\dim_{\mathbb{F}_{q}} P_h(C_2) = \dim_{\mathbb{F}_{q}} C_2$ and $\dim_{\mathbb{F}_{q}} P_h(C_1^\perp) = \dim_{\mathbb{F}_{q}} C_1^\perp$. Therefore $$\dim_{\mathbb{F}_{q}} P(C_2 \times C_1^\perp) = \dim_{\mathbb{F}_{q}} P_h(C_2) + \dim_{\mathbb{F}_{q}} P_h(C_1^\perp) = n-(k_1-k_2).$$ Furthermore it holds that $$\begin{gathered}
\dim_{\mathbb{F}_{q}} P (C_2 \times C_1^\perp ) \cap P(C_2 \times C_1^\perp)^{{\perp_s}}= \dim_{\mathbb{F}_{q}} S(C_2 \times C_1^\perp)\\
= \dim_{\mathbb{F}_{q}} [S_h(C_2) \times S_h(C_1^\perp)]
= \dim_{\mathbb{F}_{q}} S_h(C_2) + \dim_{\mathbb{F}_{q}} S_h(C_1^\perp )\\ = \left(\dim_{\mathbb{F}_{q}} S_h(C_2)-c \right) + \left(\dim_{\mathbb{F}_{q}} S_h(C_1^\perp) \right)
= n-(k_1-k_2)-2c.
\end{gathered}$$ Applying Theorem \[thm:newcode\] to the code $C_2 \times C_1^\perp$, the proof is completed.
**Acknowledgments:** We thank Ruud Pellikaan and Francisco R. Fernandes for pointing out a mistake in Theorem \[thmasymp\] on an earlier version of this article.
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|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- |
\
Center for Computational Science, University of Tsukuba,\
Tsukuba, Ibaraki 305-8577, Japan\
E-mail:
title: ' Derivation of Lüscher’s finite size formula for $N\pi$ and $NN$ system '
---
Introduction
==============
Calculation of the scattering phase shift represents an important step for expanding our understanding of the strong interaction based on lattice QCD to dynamical aspects of hadrons. Since Lüscher derived a finite size formula for the two-meson system on 1986 [@fm_LU], which give us a relation between the phase shift and the energy eigenvalue on the finite volume, many lattice calculations of the scatting length and the phase shift of the two-meson systems have been carried with his formula. Recently his formula was extended to that for the $N\pi$ system by Bernard [*et al.*]{} by using the non-relativistic effective theory [@fm_BLMR]. QCDSF collaboration calculated the phase shift of this system with this extended formula and study $\Delta(1232)$ resonance [@QCDSF_Delta].
The extension of formula is necessary to extend our study to many systems. In the present work I consider a derivation of the formula for the elastic $NN$ scattering system, where the formula only for spin singlet state in the non-relativistic limit, which is same as that for the two-meson system given by Lüscher, has been known. My derivation is based only on the relativistic quantum field theory and any effective theories for the two-nucleon interaction are not assumed. Further the extension to the $N\pi$ system can be easily done as discussed latter.
Wave function in infinite volume
==================================
First we consider the wave function in the infinite volume defined by $$\begin{aligned}
\phi^{\infty}_{\alpha\beta}({\bf x};{\bf k})
=
\langle 0 |
\ n_{\alpha}({\bf x}/2)
\ p_{\beta }(-{\bf x}/2)
\ | {\bf k} , \lambda_n , \lambda_p \rangle
\ ,
\label{eq:WF_infV_def}\end{aligned}$$ where $n_{\alpha}({\bf x})$ and $p_{\beta}({\bf x})$ are interpolating operators of the nucleons and $|{\bf k},\lambda_n,\lambda_p\rangle$ is the asymptotic $NN$ state with momentum ${\bf k}$, $-{\bf k}$ and the helicities $\lambda_n$, $\lambda_p$ . Using LSZ reduction formula, the wave function can be written by $$\begin{aligned}
\phi^{\infty}_{\alpha\beta}({\bf x};{\bf k})
=
U_{\alpha\beta}({\bf k},{\lambda}_n,{\lambda}_p)
{\rm e}^{i{\bf x}\cdot{\bf k} }
+
\int
\frac{{\rm d}^3 p}{(2\pi)^3}
\sum_{\xi_n \xi_p }
U_{\alpha\beta}({\bf p},\xi_n,\xi_p)
{\rm e}^{i{\bf x}\cdot{\bf p} }
\frac{ T({\bf p},\xi_n,\xi_p;{\bf k},\lambda_n,\lambda_p) }
{ p^2 - k^2 - i\epsilon }
\ ,
\label{eq:WF_infV_1}\end{aligned}$$ where $U_{\alpha\beta}({\bf k},\lambda_n,\lambda_p)$ is a spinor for two free nucleons given by $U_{\alpha\beta}({\bf k},\lambda_n,\lambda_p)=
u_{\alpha}( {\bf k},\lambda_n)
u_{\beta }(-{\bf k},\lambda_p)$ with the one nucleon spinor $u({\bf k},\lambda)$. $T({\bf p},\xi_n,\xi_p;{\bf k},\lambda_n,\lambda_p)$ is the off-shell scattering amplitude for a process $
n( {\bf k},\lambda_n)
p(-{\bf k},\lambda_p) \to \
n( {\bf p}, \xi_n)
p(-{\bf p}, \xi_p)
$.
We can estimate (\[eq:WF\_infV\_1\]) in the region $|{\bf x}| > R$ for the two-nucleon interaction range $R$, by using a integral formula $$\int \frac{{\rm d}^3 p}{(2\pi)^3}
\frac{ j_l(px) }{ p^2 - k^2 - i\epsilon }
f(p)
=
\frac{k}{4\pi}
\left( i \cdot j_l(kx) + n_l(kx) \right) f(k)
%
\qquad \mbox{ for $F(x)$=0 }
\label{eq:int_formula_NI}$$ where $F(x)$ is the inverse Fourier transformation of $f(p)$. $j_l(x)$ is the spherical Bessel and $n_l(x)$ is the Neumann function, whose conventions agree with those in [@MESSIAH:book] as adopted in [@fm_LU]. This formula is a extension of (A.11) in Appendix A in Ref. [@CP-PACS_wf] to that for arbitrary value of $l$ and can be derived by similar calculations of that paper.
From (\[eq:int\_formula\_NI\]) we know that all values in the numerator of the integrand in (\[eq:WF\_infV\_1\]) can be replaced by the value at on-shell $p=k$. The off-shell scattering amplitude $T({\bf p},\xi_n,\xi_p;{\bf k},\lambda_n,\lambda_p)$ is replaced by the on-shell amplitude, which can be expanded as [@Hamp_JW] $$T(k{\bf e}_p,\xi_n,\xi_p;{\bf k},\lambda_n,\lambda_p)
=
16\pi^2 \frac{\sqrt{s}}{k}
\sum_{JM}
{T}_{\xi_n\xi_p , \lambda_n \lambda_p}^{(J)}(k)
\cdot
N_J{}^2
\ D_{M\xi }^{(J)}{}^{*}({\Omega}_p)
\ D_{M\lambda}^{(J)} ({\Omega}_k)
\ ,
\label{eq:Mphys}$$ where $N_J=\sqrt{(2J+1)/(4\pi)}$, $\lambda = \lambda_n - \lambda_p$, $ \xi = \xi_n - \xi_p$ and $\sqrt{s}=2\sqrt{m^2+k^2}$. In (\[eq:Mphys\]) the helicity amplitude in the subspace of the total energy $\sqrt{s}$ and the total angular momentum $J$ is defined by ${T}_{\xi_n\xi_p,\lambda_n\lambda_p}^{(J)}(k)
=\langle \xi_n \xi_p | \hat{T}^{(J)}(k) | \lambda_n\lambda_p\rangle$. The function $D_{MM'}^{(J)} ({\Omega}_p)
=\langle JM |
\exp(-i\alpha J_z)
\exp(-i\beta J_y) $ $
\exp(-i\gamma J_z)
|JM'\rangle$ is the Wigner’s D-function with the Euler angle $(\alpha,\beta,\gamma)=(\phi_p,\theta_p,-\phi_p)$ for momentum ${\bf p}=
( p \sin\theta_p \cos\phi_p
, p \sin\theta_p \cos\phi_p
, p \cos\theta_p )$.
Using (\[eq:int\_formula\_NI\]) and (\[eq:Mphys\]), we know that the wave function (\[eq:WF\_infV\_1\]) in the region $|{\bf x}| > R$ is written by $$\begin{aligned}
&&
\phi^{\infty}({\bf x};{\bf k})
=
\sum_{JM}
N_J D_{M\lambda}({\Omega}_k) \cdot
\phi^{\infty}_{JM\lambda_n\lambda_p} ({\bf x};k)
\qquad ( \lambda = \lambda_n - \lambda_p )
\ ,
\label{eq:WF_infV_h1}
\\
&&
\phi^{\infty}_{JM\lambda_n\lambda_p}({\bf x};k)
=
\sum_{\xi_n \xi_p }
\left[
{J}_{JM\xi_n\xi_p}({\bf x};k) \cdot \alpha_{\xi_n\xi_p,\lambda_n\lambda_p}^{(J)}(k) +
{N}_{JM\xi_n\xi_p}({\bf x};k) \cdot \beta_{\xi_n\xi_p,\lambda_n\lambda_p}^{(J)}(k)
\right]
\ .
\label{eq:WF_infV_h2}\end{aligned}$$ where $\alpha_{\xi_n\xi_p,\lambda_n\lambda_p}^{(J)}(k)
=\langle \xi_n \xi_p |
\hat{I} + i \hat{T}^{(J)}/2
| \lambda_n\lambda_p\rangle$ and $\beta_{\xi_n\xi_p,\lambda_n\lambda_p}^{(J)}(k)
=\langle \xi_n \xi_p | \hat{T}^{(J)} /2 | \lambda_n\lambda_p\rangle$, which correspond to $\alpha^{(l)} = \cos\delta_l \cdot {\rm exp}(i\delta_l)$ and $ \beta^{(l)} = \sin\delta_l \cdot {\rm exp}(i\delta_l)$ for the two-meson system with the scattering phase shift $\delta_l$. In (\[eq:WF\_infV\_h2\]) the function ${J}_{JM\lambda_n\lambda_p}({\bf x};k)$ is the wave function of two free nucleons with the total energy $\sqrt{s}$, the total angular momentum $JM$ and the helicity $\lambda_n\lambda_p$. Its explicit form is given by $${J}_{JM\lambda_n\lambda_p}({\bf x};k)
=
\hat{L}(\nabla)
\ {J}_{JM\lambda_n\lambda_p}^{\rm NR}({\bf x};k)
\ \hat{R}(\stackrel{ \leftarrow}{\nabla})
\equiv
{J}_{JM\lambda_n\lambda_p}^{\rm NR}({\bf x};k) \bigl|_{\rm R-EX}
\ \ ,
\label{eq:J_JMhh_def}$$ where differential operators $\hat{L}(\nabla)$ and $\hat{R}(\nabla)$ are defined by $$\hat{L}(\nabla)
=
\left(
\begin{array}{c}
\displaystyle
I \\
\displaystyle
\frac{(\sigma\cdot\nabla/i)}{E+m} \\
\end{array}
\right)
\ , \quad
%
\hat{R}(\nabla)
=
\left(
I \ , \
\frac{ - (\sigma^T \cdot \nabla/i)}{E+m}
\right)
%
\ ,$$ with $E=\sqrt{k^2+m^2}$. In (\[eq:J\_JMhh\_def\]) the function ${J}_{JM\lambda_n\lambda_p}^{\rm NR}({\bf x};k)$ is $2\times 2$ non-relativistic spinor defined by $$\begin{aligned}
&&
{J}_{JM\lambda_n\lambda_p}^{\rm NR}({\bf x};k)
= \sum_{ls}
{J}_{JMls}^{\rm NR}({\bf x};k)
\cdot \langle JMls | JM\lambda_n\lambda_p \rangle
\ ,
\label{eq:J_JMls_NR_def_1}
\\
&&
{J}_{JMls}^{\rm NR}({\bf x};k)
= j_l(kx) Y_{JM}^{ls}(\Omega_x) / b_{l}(k)
\ ,
\quad
%
Y_{JM}^{ls}(\Omega_x)
= \sum_{m \mu}
Y_{lm}(\Omega_x)
\phi(s,\mu) \cdot C( lm ; s \mu ; JM )
\ , \ \ \
\label{eq:J_JMls_NR_def_2}\end{aligned}$$ where the coefficient $\langle JMls | JM\lambda_n\lambda_p \rangle$ is the transformation coefficient from the helicity base to the orbit-spin base ($(JMls)$-base) with the angular momentum $l$ and the spin $s$ [@Hamp_JW], and $C(lm;s\mu;JM)$ is the Clebsch-Gordan coefficient for angular momentum state $|lm\rangle\otimes |s \mu\rangle$ and $|JM\rangle$. $\Omega_x$ is the spherical coordinate for ${\bf x}$. $b_l(k)$ is the normalization constant of the state, which takes $1/b_l(k) = (4\pi)i^l \cdot (k^2+m^2)$ for the usual relativistic normalization ($u^\dagger u = 2 E$). $\phi(s,\mu)$ is $2\times 2$ spin wave function for two spin $1/2$ particles with total spin $s\mu$. The function ${N}_{JM\lambda_n\lambda_p}({\bf x};k)$ in (\[eq:WF\_infV\_h2\]) is given by replacing $j_l(kx)$ by $n_l(kx)$ in (\[eq:J\_JMhh\_def\]). We can regard (\[eq:J\_JMhh\_def\]) as a relativistic extension of the non-relativistic spinor ${J}_{JM\lambda_n\lambda_p}^{\rm NR}({\bf x};k)$ to the relativistic one ${J}_{JM\lambda_n\lambda_p}({\bf x};k)$, so that the spinor satisfies the Dirac equation. We use a notation $J^{\rm NR}|_{\rm R-EX}$ for this relativistic extension like as (\[eq:J\_JMhh\_def\]) in the follow.
Next we rewrite (\[eq:WF\_infV\_h1\]) and (\[eq:WF\_infV\_h2\]) by the $(JMls)$-base as $$\begin{aligned}
&&
\phi^{\infty}({\bf x};{\bf k})
=
\sum_{JMls}
C_{JMls}({\bf k}) \cdot
\phi^{\infty}_{JMls}({\bf x};k)
\ ,
\label{eq:WF_infV_ls1}
\\
&&
\phi^{\infty}_{JMls}({\bf x};k)
= \sum_{l's'}
\Bigl[
{J}_{JMl's'}({\bf x};k) \cdot \alpha_{l's',ls}^{(J)}(k) +
{N}_{JMl's'}({\bf x};k) \cdot \beta_{l's',ls}^{(J)}(k)
\Big]
%
\ ,
\label{eq:WF_infV_ls2}\end{aligned}$$ with some constant $C_{JMls}({\bf k})$, where functions of the $(JMls)$-base are defined by $$\begin{aligned}
&&
{J}_{JMls}({\bf x};k)
= \sum_{\lambda_n\lambda_p}
{J}_{JM\lambda_n\lambda_p}({\bf x};k)
\cdot \langle JMls | JM\lambda_n\lambda_p \rangle
\ ,
\label{eq:J_JMls_def}
\\
&&
\alpha_{l's',ls}^{(J)}(k)
=
\sum_{\xi_n \xi_p \lambda_n \lambda_p }
\alpha_{\xi_n\xi_p,\lambda_n\lambda_p }^{(J)}(k) \cdot
\langle JMl's' | JM \xi_n \xi_p \rangle
\langle JMl s | JM \lambda_n \lambda_p \rangle
\ ,
\label{eq:alpha_beta_JMls_def}\end{aligned}$$ and ${N}_{JM\lambda_n\lambda_p}({\bf x};k)$ and $\beta_{l's',ls}^{(J)}(k)$ are similarly defined. Here we should note that ${J}_{JMls}({\bf x};k)$ and ${N}_{JMls}({\bf x};k)$ are not eigenstates of the orbital angular momentum and the spin with $l$ and $s$. These functions satisfy the Dirac equation, thus the upper and the lower components have different orbital angular momenta.
Wave function on the finite volume
====================================
Next we consider the wave function on the finite periodic box of volume $L^3$ defined by $$\phi^{L}_{\alpha\beta}({\bf x};k)
=
\langle 0 |
\ n_{\alpha}( {\bf x}/2)
\ p_{\beta }(-{\bf x}/2) \ | k \rangle
\ ,
\label{eq:WF_finV_def}$$ where $| k \rangle$ is the energy eigenstate with $\sqrt{s}=2\sqrt{m^2+k^2}$ on the finite volume. Here we assume the condition $R<L/2$ for the two-nucleon interaction range $R$ and the lattice size $L$, so that the boundary condition does not distort the shape of the two-nucleon interaction. In the region $R<|{\bf x}|<L$, the wave function satisfies following two equations and the boundary condition. $$\begin{aligned}
&&
\left[\ i ({\bf \gamma}\cdot{\bf \nabla} )
+ \gamma^0 E - m \ \right]
\phi^{L}({\bf x};k)
= 0
\ ,
\quad
\phi^{L}({\bf x};k)
\left[\ -i ({\bf \gamma}\cdot\stackrel{\leftarrow}{\bf \nabla} )
+ \gamma^0 E - m \ \right]^{\rm T}
= 0
\ ,
\label{eq:WF_finV_DEQ}
\\
&&
\phi^{L}({\bf x}+{\bf n}L ;k) =
\phi^{L}({\bf x} ;k)
\qquad ( \ {\bf n} \in \mathbb{Z}^3 \ )
\ ,
\label{eq:WF_finV_BC}\end{aligned}$$ where $E=\sqrt{m^2+k^2}$. The general solution of these equations can be written by the linear combination of the Green function defined by $$\begin{aligned}
&&
G_{JMls}({\bf x};k)
= G_{JMls}^{\rm NR}({\bf x};k) \bigl|_{\rm R-EX}
\ ,
\label{eq:G_JMls_def1}
\\
&&
G_{JMls}^{\rm NR}({\bf x};k)
=
{\cal Y}_{JM}^{ls}(\nabla) \frac{1}{L^3}
\sum_{ {\bf p}\in \Gamma }
\frac{ 1 }{ p^2 - k^2 }
{\rm e}^{ i {\bf p}\cdot{\bf x} }
\ , \quad
{\cal Y}_{JM}^{ls}({\bf p}) = p^l \cdot Y_{JM}^{ls}(\Omega_p)
\ ,
\label{eq:G_JMls_def2} \end{aligned}$$ where $\Gamma = \{ {\bf p} |
{\bf p}=(2\pi)/L \cdot {\bf n} \ , \ {\bf n}\in\mathbb{Z}^3 \}$ and $\Omega_p$ is the spherical coordinate for ${\bf p}$. This Green function is related to that introduced in Ref. [@fm_LU] $G_{lm}({\bf x};k)$ by $G_{JMls}^{\rm NR}({\bf x};k)
= \sum_{m \mu} G_{lm}({\bf x};k)
\cdot \phi(s,\mu) C(lm;s\mu;JM)$.
Using partial wave expansion of $G_{lm}({\bf x};k)$ given in Ref. [@fm_LU], we obtain $$\begin{aligned}
G_{JMls}({\bf x};k)
=
a_l(k) b_l(k) \cdot {N}_{JMls}({\bf x};k)
+
a_l(k)
\sum_{J'M'l'}
b_{l'}(k) \cdot {J}_{J'M'l's}({\bf x};k)
\cdot M_{J'M'l',JMl}^{(s)}(k)
\ ,
\label{eq:G_JMls_pwexp}\end{aligned}$$ where $a_l(k)=(-1)^l k^{l+1} / (4\pi)$, $b_l(k)$ is the normalization constant appeared in (\[eq:J\_JMls\_NR\_def\_2\]) and $$M_{J'M'l',JMl}^{(s)}(k)
= \sum_{mm' \mu}
M_{l'm',lm}(k) \cdot
C( l' m' ; s \mu ; J' M' )
C( l m ; s \mu ; J M )
\ .
\label{eq:M_JML_def}$$ The function $M_{l'm',lm}(k)$ in (\[eq:M\_JML\_def\]) is defined by (3.34) in Ref. [@fm_LU], which is given by $$\begin{aligned}
&&
M_{l'm',lm}(k)
= \sum_{l''m''} I_{l'm',l''m'',lm} W_{l''m''}(q)
\ , \quad q= k L / (2\pi)
\\
&&
I_{l'm', l'' m'',lm}
=
(-1)^l \cdot i^{l+l'} \cdot (2l''+1) \sqrt{ \frac{ 2l+1 }{ 2l'+1 } }
\cdot
C( l 0 ; l'' 0 ; l' 0 )
C( l m ; l'' m'' ; l' m' )
\ , \\
&&
W_{lm}(q)
= \frac{1}{ \pi^{3/2} q^{l+1} \sqrt{2l+1} }
\ \sum_{ {\bf n} \in \mathbb{Z}^3 }
\frac{1}{ n^2 - q^2} {\cal Y}_{lm}({\bf n})
\ , \quad
{\cal Y}_{lm}({\bf n}) = n^l \cdot Y_{lm}(\Omega_n)
\ .
\label{eq:W_lm_def}\end{aligned}$$
Relation between $\phi^{\infty}$ and $\phi^{L}$
=================================================
In the following we restrict ourselves to the wave function for the irreducible representation of the rotational group on the finite volume (cubic group ${\rm O}$), which is defined by $$\phi^{L}_{\Gamma\alpha} ({\bf x};k)
=
\langle 0 |
\ n( {\bf x}/2)
\ p(-{\bf x}/2)
\ | k ; \Gamma \alpha \rangle
\ ,
\label{eq:WF_finV_IRREP_def}$$ where $|k ; \Gamma \alpha \rangle$ is the energy eigenstate with $\sqrt{s}=2\sqrt{m^2+k^2}$ and belongs to the irreducible representation of ${\rm O}$ labeled by $\Gamma$ and $\alpha$ ($\alpha=1\dots \dim\Gamma$ , $\Gamma = \{ A_1, A_2, E, T_1, T_2\}$). Projection of the irreducible representation of SU(2) ($|JM\rangle$) to that of ${\rm O}$ ($|\Gamma \alpha n J \rangle$) is given by $$|JM \rangle
= \sum_{\Gamma \alpha n }
|\Gamma \alpha nJ \rangle \cdot V(JM;\Gamma\alpha nJ)^{*}
\ , \quad
|\Gamma\alpha nJ \rangle
= \sum_{M}
|JM \rangle \cdot V(JM;\Gamma\alpha nJ)
\ ,
\label{eq:SU2_O_Proj}$$ with known coefficient $V(JM;\Gamma\alpha nJ)$, where $n$ is the multiplicity of the representation $\Gamma$.
In the previous section the wave functions were expanded in terms of functions of the $(JMls)$-base ($J_{JMls}$ and $N_{JMls}$). But it is more convenient for the wave function (\[eq:WF\_finV\_IRREP\_def\]) to expand in terms of functions of $(\Gamma\alpha nJls)$-base defined by $${J}_{\Gamma\alpha nJls}({\bf x};k) =
\sum_{M} J_{JMls}({\bf x};k) \cdot
V(JM;\Gamma\alpha nJ)
\ ,
\label{eq:Trans_to_IRR_O}$$ with the coefficient $V(JM;\Gamma\alpha nJ)$.
In the region $|{\bf x}|>R$, the wave function (\[eq:WF\_finV\_IRREP\_def\]) can be written by the linear combination of the Green function and also the wave function in the infinite volume as $$\begin{aligned}
\phi^{L}_{\Gamma\alpha} ({\bf x};k)
=
\sum_{nJls}
E_{\Gamma\alpha nJls}({\bf k})
\cdot G_{\Gamma\alpha nJls}({\bf x};k)
=
\sum_{nJls}
C_{\Gamma\alpha nJls}({\bf k})
\cdot \phi^\infty_{\Gamma\alpha nJls}({\bf x};k)
\ ,
\label{eq:WF_finV_IRREP_exp}\end{aligned}$$ with some coefficients $E_{\Gamma\alpha nJls}({\bf k})$ and $C_{\Gamma\alpha nJls}({\bf k})$, where $G_{\Gamma\alpha nJls}({\bf x};k)$ and $\phi^{\infty}_{\Gamma\alpha nJls}({\bf x};k)$ are functions of the $(\Gamma\alpha nJls)$-base obtained by the transformation (\[eq:Trans\_to\_IRR\_O\]) from $G_{JMls}({\bf x};k)$ defined by (\[eq:G\_JMls\_def1\]) and $\phi^{\infty}_{JMls}({\bf x};k)$ defind by (\[eq:WF\_infV\_ls2\]). After some calculations we obtain $$\begin{aligned}
\phi^{L}_{\Gamma\alpha}({\bf x};k)
&=&
\sum_{nJls}
E_{\Gamma\alpha nJls}({\bf k})
\left(
b_{l}(k) \cdot {N}_{\Gamma\alpha nJls}({\bf x};k)
+
\sum_{n'J'l'}
b_{l'}(k) \cdot {J}_{\Gamma\alpha n'J'l's}({\bf x};k) \cdot
M_{n'J'l',nJl}^{(s)}(\Gamma;k)
\right)
\cr
&=&
\sum_{nJls}
C_{\Gamma\alpha nJls}({\bf k})
\sum_{l's'}
\left(
{J}_{\Gamma\alpha nJl's'}({\bf x};k) \cdot \alpha_{l's',ls}^{(J)}(\Gamma;k) +
{N}_{\Gamma\alpha nJl's'}({\bf x};k) \cdot \beta_{l's',ls}^{(J)}(\Gamma;k)
\right)
\ , \quad
\label{eq:WF_finV_IRREP_exp_2}\end{aligned}$$ where the constant $a_l(k)$ are removed by redefinition of the constant $E_{\Gamma\alpha nJls}({\bf k})$, and $$\begin{aligned}
&&
\delta_{\Gamma'\Gamma}
\delta_{\alpha'\alpha}\cdot
M_{n'J'l',nJl}^{(s)}(\Gamma;k)
=
\sum_{MM'}
M_{J'M'l',JMl}^{(s)}(k) \cdot
V( J'M' ; \Gamma' \alpha' n' J' )
V( J M ; \Gamma \alpha n J )
\ ,
\label{eq:M_Gamma_def}
\\
&&
\alpha_{l's',ls}^{(J)}(\Gamma;k)
=
\sum_{M}
\alpha_{l's',ls}^{(J)}\cdot
V(JM;\Gamma \alpha nJ )
V(JM;\Gamma \alpha nJ )
\ ,
\label{eq:alpha_Gamma_def}\end{aligned}$$ and $\beta_{l's',ls}^{(J)}(\Gamma;k)$ is similarly defined. The diagonal property of $M(\Gamma;k)$ in (\[eq:M\_Gamma\_def\]) for indices $(\Gamma\alpha)$ is result from the invariance of $M_{J'M'l',JMl}^{(s)}(k)$ under the rotation on the finite volume (see Ref. [@fm_LU]).
From (\[eq:WF\_finV\_IRREP\_exp\_2\]), we know that coefficients of functions ${J}_{\Gamma\alpha nJls}({\bf x};k)$ and ${N}_{\Gamma\alpha nJls}({\bf x};k)$ relate each other. After some calculations, we find that it is given by $$\det\left[
\ {\bf M}(\Gamma;k) \ -
\ {\bf A}(\Gamma;k) / {\bf B}(\Gamma;k) \ \right] = 0
\ ,
\label{eq:finitesize_formula}$$ where we introduce a vector space spanned by indices $(nJls)$ at fixed $(\Gamma\alpha)$ and define linear operators on this vector space by $$\bigl[{\bf M}(\Gamma;k)\bigr]_{n'J'l's',nJls} =
\delta_{s's} \cdot M_{n'J'l',nJl}^{(s)}(\Gamma;k)
\ , \quad
\bigl[{\bf A}(\Gamma;k)\bigr]_{nJ'l's',nJls} =
\delta_{n'n} \delta_{J'J} \cdot
\alpha_{l's',ls}^{(J)}(\Gamma;k)/b_{l'}(k)
\ ,
\label{eq:M_Gamma_AB_Gamma_defV}$$ and ${\bf B}(\Gamma;k)$ is similarly defined. Equation (\[eq:finitesize\_formula\]) is a finite size formula for the elastic $NN$ scattering system, which gives us a relation between the energy eigenvalue on the finite volume and the quantity of the elastic scattering ${\bf A}/{\bf B}$.
Finite size formula for $NN$ system
=====================================
In this section we show the explicit matrix form of the finite size formula for the $NN$ system (\[eq:finitesize\_formula\]). $S$-matrix at fixed $J$ forms a $4\times 4$ matrix. This matrix is reduced to sub-matrices by the eigenvalue of the global symmetry : the parity $P$ and the particle exchange $R$ ($=(-1)^I$ with the iso-spin $I$) as $$S^{(J)}
=
\begin{array}[t]
{c l l c l c l l }
& \Bigl( & \mbox{$2\times 2$\ matrix} & ; \ & P=(-1)^{J-1} & , & R=(-1)^{J-1} & \Bigr) \\
+ & \Bigl( & \mbox{$1\times 1$\ matrix} & ; \ & P=(-1)^{J } & , & R=(-1)^{J } & \Bigr) \\
+ & \Bigl( & \mbox{$1\times 1$\ matrix} & ; \ & P=(-1)^{J } & , & R=(-1)^{J-1} & \Bigr) \\
\end{array}
\ .$$ ${\bf A}(\Gamma;k)$ and ${\bf B}(\Gamma;k)$ in the finite size formula (\[eq:finitesize\_formula\]) also take same form.
We note that the basis of the partial wave expansion ${J}_{\Gamma\alpha nJls}({\bf x};k)$ and ${N}_{\Gamma\alpha nJls}({\bf x};k)$ in (\[eq:WF\_finV\_IRREP\_exp\_2\]) are eigenstates of the parity and the particle exchange with $P=(-1)^l$ and $R=(-1)^l \cdot (-1)^{s-1}$. Thus the wave function for the state with $R=-P$, only functions with $s=0$ appear in the partial wave expansion. For the state with $R=P$, only functions with $s=1$ appear. The mixing between $s=0$ and $s=1$ is forbidden by the symmetry of the parity and the particle exchange (iso-spin). Therefore we can separately obtain the finite size formula for $P=-R$ ($s=0$) and $P=R$ ($s=1$).
In the case of $R=-P$ ($s=0$), the components of the matrix ${\bf M}(\Gamma;k)$ in the finite size formula (\[eq:finitesize\_formula\]) are given by $$M_{n'J'l',nJl}^{(s)}(\Gamma;k)
=
\delta_{J'l'} \delta_{Jl} \cdot
\sum_{MM'}
M_{J'M',JM}(k) \cdot
V( \Gamma\alpha nJ' ; J'M' )
V( \Gamma\alpha nJ ; J M )
\ .
\label{eq:M_s0}$$ This is the same matrix as that appeared in the finite size formula for the two-meson system. Further, $\alpha^{(J)}_{l's,ls}(k)
= \delta_{Jl}\delta_{Jl'} \cdot \alpha_l(k)$ and $\beta^{(J)}_{l's,ls}(k)
= \delta_{Jl}\delta_{Jl'} \cdot \beta_l(k)$ with the diagonal components $\alpha_l(k)$ and $\beta_l(k)$ also take same matrix form as that for the two-meson system. Thus the finite size formula for the $NN$ system with $P=-R$ ($s=0$) is same as that for the two-meson system in Ref. [@fm_LU].
In the case of $R=P$ ($s=1$), the matrix ${\bf M}(\Gamma;k)$ and ${\bf A}(\Gamma;k)/{\bf B}(\Gamma;k)$ have complicated structure. In the following we show explicit matrix form of the finite size formula for some channels as example. We neglect contributions of $J\ge 5$. In this case the multiplicity $n$ is 1 for all irreducible representations $\Gamma$, thus we omit the index $n$ in the formula (\[eq:finitesize\_formula\]) for simplicity, as $$\begin{aligned}
&&
\det\left[
\ {\bf M}(\Gamma;k) \ -
\ {\bf A}(\Gamma;k)/{\bf B}(\Gamma;k) \ \right] = 0
\ ,
\label{eq:finitesize_formula_s1}
\\
&&
\bigl[{\bf M}(\Gamma;k)\bigr]_{J'l',Jl} =
M_{n'J'l',nJl}^{(s)}(\Gamma;k)
\ , \quad
\bigl[{\bf A}(\Gamma;k)\bigr]_{J'l',Jl} =
\delta_{J'J} \cdot \alpha_{l's',ls}^{(J)}(\Gamma;k)/b_{l'}(k)
\ ,
\label{eq:M_Gamma_AB_Gamma_defV_s1}\end{aligned}$$ where $n=n'=1$, $s=s'=1$ and the matrix ${\bf B}(\Gamma;k)$ is similarly defined.
The first example is the deuteron state. We have to consider the $NN$ state with the total angular momentum $J=1$ and the parity $P=+1$, which corresponds to ${}^3S_1$ and ${}^3D_1$ states in the non-relativistic limit. The $J=1$ state belongs to the irreducible representation of the cubic group $\Gamma=T_1$, thus we consider the finite size formula for $\Gamma=T_1$ for the study of the deuteron. The other angular momentum states also belong to $T_1$ as $T_1 = 1 + 3 + 4$ up to $J \ge 5$ and the finite size formula includes contributions from all these states. For each values of $J$, possible values of $l$ are given by $$\begin{array}{lc}
l = 0 , \ 2 & \ \mbox{ for \ $J=1$ } \\
l = 2 , \ 4 & \ \mbox{ for \ $J=3$ } \\
l = 4 & \ \mbox{ for \ $J=4$ }
\label{eq:l_deuteron}
\end{array}
\ ,$$ from the parity conservation and the theory of addition of the angular momentum. Thus matrices ${\bf M}(\Gamma;k)$ and ${\bf A}(\Gamma;k)/{\bf B}(\Gamma;k)$ in the finite size formula (\[eq:finitesize\_formula\_s1\]) take : $$\begin{aligned}
&& {\bf M}=
\left(
\begin{array}{lllll}
M_{10,10} & M_{10,12} & M_{10,32} & M_{10,34} & M_{10,44} \\
M_{12,10} & M_{12,12} & M_{12,32} & M_{12,34} & M_{12,44} \\
M_{32,10} & M_{32,12} & M_{32,32} & M_{32,34} & M_{32,44} \\
M_{34,10} & M_{34,12} & M_{34,32} & M_{34,34} & M_{34,44} \\
M_{44,10} & M_{44,12} & M_{44,32} & M_{44,34} & M_{44,44} \\
\end{array}
\right)
%---------------------
, \quad
{\bf A}/{\bf B}=
\left( \ \begin{array}{ccccc}
\cline{1-2}
\multicolumn{2}{|c|}{\multirow{2}{*}{\ \ \it J=1 \ }} & \ \ 0 & 0 & 0 \\
\multicolumn{2}{|c|}{} & \ \ 0 & 0 & 0 \\
\cline{1-2}
\cline{3-4}
0 & 0 & \multicolumn{2}{|c|}{\multirow{2}{*}{\ \ \it J=3 \ }} & 0 \\
0 & 0 & \multicolumn{2}{|c|}{} & 0 \\
\cline{3-4}
\cline{5-5}
0 & 0 & 0 & 0 & \multicolumn{1}{|c|}{\multirow{1}{*}{\it J=4}} \\
\cline{5-5}
\end{array} \ \right)
\ ,
\label{eq:eq:finitesize_formula_D}\end{aligned}$$ where the boxes in the matrix ${\bf A}/{\bf B}$ which enclose the values of $J$ refer to $2\times 2$ or $1\times 1$ matrices expanded by the possible values of $l$. In (\[eq:eq:finitesize\_formula\_D\]) components of the matrix ${\bf M}$ are denoted by $M_{J'l',Jl} \equiv [{\bf M}(\Gamma;k)]_{J'l',Jl}$ and are given by $$\begin{aligned}
%---------------------
&& \quad
\begin{array}[b]{l}
M_{10,10}=W_{00} \\
M_{12,10}=0 \\
M_{32,10}=0 \\
M_{34,10}=-2 W_{40} \\
M_{44,10}=\frac{6}{7}\sqrt{7} W_{40} \\
\end{array}
%
\quad
\begin{array}[b]{l}
M_{12,12}=W_{00} \\
M_{32,12}=-\frac{6}{7}\sqrt{6 }W_{40} \\
M_{34,12}=-\frac{5}{7}\sqrt{2 }W_{40} \\
M_{44,12}=-\frac{3}{7}\sqrt{14}W_{40} \\
\end{array}
%
\quad
\begin{array}[b]{l}
M_{32,32}=W_{00} + \frac{6}{7}W_{40} \\
M_{34,32}=\frac{30}{77}\sqrt{3 }W_{40} + \frac{50}{33}\sqrt{3 }W_{60} \\
M_{44,32}=\frac{18}{77}\sqrt{21}W_{40} + \frac{10}{11}\sqrt{21}W_{60} \\
\end{array}
\cr\cr
%
&&
\quad
\begin{array}[b]{l}
M_{34,34}=W_{00} + \frac{81}{77}W_{40} + \frac{25}{33}W_{60} \\
M_{44,34}=-\frac{27}{77}\sqrt{7}W_{40} - \frac{15}{11}\sqrt{7}W_{60} \\
\end{array}
%
\quad
\begin{array}[b]{l}
M_{44,44}=
W_{00}
+ \frac{81 }{143}W_{40}
+ \frac{1 }{55 }W_{60}
+ \frac{1792}{715}W_{80}
\end{array}
\ , \end{aligned}$$ where $M_{J'l',Jl}= M_{Jl,J'l'}$ and the function $W_{lm}$ is defined by (\[eq:W\_lm\_def\]).
Finally we consider the state with same $J$ but opposite parity to the deuteron, [*ie.*]{} $J=1$, $P=-1$. We also consider the representation $\Gamma=T_1$. Possible values of $l$ for each $J$ are different from those of the deuteron case as $$\begin{array}{lc}
l = 1 & \ \mbox{ for $J=1$ } \\
l = 3 & \ \mbox{ for $J=3$ } \\
l = 3, \ 5 & \ \mbox{ for $J=4$ }
\label{eq:l_Pdeuteron}
\end{array}
\ .$$ Thus matrices in the finite size formula take different forms : $$\begin{aligned}
&& {\bf M} =
\left(
\begin{array}{lllll}
M_{11,11} & M_{11,33} & M_{11,43} & M_{11,45} \\
M_{33,11} & M_{33,33} & M_{33,43} & M_{33,45} \\
M_{43,11} & M_{43,33} & M_{43,43} & M_{43,45} \\
M_{45,11} & M_{45,33} & M_{45,43} & M_{45,45} \\
\end{array}
\right)
%------------------------------
, \quad
{\bf A}/{\bf B} =
\left( \
\begin{array}{ccccc}
\cline{1-1}
\multicolumn{1}{|c|}{\multirow{1}{*}{\it J=1}} & 0 & 0 & 0 \\
\cline{1-1}
\cline{2-2}
0 & \multicolumn{1}{|c|}{\multirow{1}{*}{\it J=3}} & 0 & 0 \\
\cline{2-2}
\cline{3-4}
0 & 0 & \multicolumn{2}{|c|}{\multirow{2}{*}{\ \ \it J=4 \ }} \\
0 & 0 & \multicolumn{2}{|c|}{} \\
\cline{3-4}
\end{array}\ \right)
\ ,
\\
\cr
%---------------------
&&
\quad
\begin{array}[b]{l}
M_{11,11}=W_{00} \\
M_{33,11}= \frac{3}{7}\sqrt{14 }W_{40} \\
M_{43,11}=-\frac{1}{7}\sqrt{210}W_{40} \\
M_{45,11}= \frac{2}{7}\sqrt{42 }W_{40} \\
\end{array}
%
\quad
\begin{array}[b]{l}
M_{33,33}=W_{00} + \frac{3}{11}W_{40} - \frac{25}{11} W_{60} \\
M_{43,33}=-\frac{3}{11}\sqrt{15}W_{40} - \frac{35}{33}\sqrt{15}W_{60} \\
M_{45,33}= \frac{6}{11}\sqrt{ 3}W_{40} + \frac{70}{33}\sqrt{ 3}W_{60} \\
\end{array}
\cr
&&
\quad
\begin{array}[b]{l}
M_{43,43}=W_{00} + \frac{9}{11}W_{40} - \frac{5}{33}W_{60} \\
M_{45,43}=
\frac{18 }{143}\sqrt{5}W_{40}
- \frac{14 }{165}\sqrt{5}W_{60}
- \frac{896}{715}\sqrt{5}W_{80} \\
\end{array}
\cr
&&
\quad
\begin{array}[b]{l}
M_{45,45}=W_{00}
+ \frac{126}{143}W_{40}
- \frac{32 }{165}W_{60}
- \frac{448}{715}W_{80}
\end{array}
\ . \end{aligned}$$
Summary
=========
The finite size formula for the elastic $NN$ scattering system is derived from the relativistic quantum field theory. The extension to other two-baryon system as the $N\Lambda$ system is trivial. Finally I give a comment for the $N\pi$ system. The formulation of this paper for the $NN$ system is also valid for the system with the general value of the spin $s$. Thus the finite size formula for the $N\pi$ system can be easily obtained from that for the $NN$ system (\[eq:finitesize\_formula\]) by set $s=1/2$. In calculations of matrices ${\bf M}(\Gamma;k)$, ${\bf A}(\Gamma;k)$ and ${\bf B}(\Gamma;k)$ in (\[eq:M\_Gamma\_AB\_Gamma\_defV\]), we change the coefficient $V(JM;\Gamma\alpha nJ)$ in (\[eq:SU2\_O\_Proj\]) by that for the double covered cubic group (${}^2{\rm O}$) to deal with the half integer value of the total angular momentum $J$ as discussed in Ref. [@fm_BLMR]. I confirmed that my results are consistent with those obtained from the non-relativistic effective theory by Bernard [*et al.*]{} [@fm_BLMR].
This work is supported in part by Grants-in-Aid of the Ministry of Education (No.20540248).
[99]{} M. Lüscher, Commun. Math. Phys. [**105**]{} (1986) 153; Nucl. Phys. [**B354**]{} (1991) 531. V. Bernard, M. Lange, U.-G. Meissner, and A. Rusetsky, Eur. Phys. J. A35(2008)281, JHEP0808(2008)024 \[arXiv:0806.4495\]. QCDSF Collaboration, M. Gockeler [*et al.*]{}, arXiv:0810.5337; these proceeding. CP-PACS Collaboration, S. Aoki [*et al.*]{} Phys. Rev. [**D71**]{} (2005) 094504. A. Messiah, Quantum mechanics, Vols. I, II ( North-Holland, Amsterdam, 1965 ). M. Jacob and G.C. Wick, Ann. Phys. [**7**]{}(1959) 404.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
Global disk simulations provide a powerful tool for investigating accretion and the underlying magnetohydrodynamic turbulence driven by magneto-rotational instability (MRI). Using them to predict accurately quantities such as stress, accretion rate, and surface brightness profile requires that purely numerical effects, arising from both resolution and algorithm, be understood and controlled. We use the flux-conservative [*Athena*]{} code to conduct a series of experiments on disks having a variety of magnetic topologies to determine what constitutes adequate resolution. We develop and apply several resolution metrics: ${\langle Q_z \rangle}$ and ${\langle Q_\phi \rangle}$, the ratio of the grid zone size to the characteristic MRI wavelength, $\alpha_{mag}$, the ratio of the Maxwell stress to the magnetic pressure, and ${\langle B_R^2\rangle/\langle B_\phi^2\rangle}$, the ratio of radial to toroidal magnetic field energy. For the initial conditions considered here, adequate resolution is characterized by ${\langle Q_z \rangle}\ge 15$, ${\langle Q_\phi \rangle}\ge 20$, $\alpha_{mag}
\approx 0.45$, and ${\langle B_R^2\rangle/\langle B_\phi^2\rangle}\approx 0.2$. These values are associated with $\ge 35$ zones per scaleheight $H$, a result consistent with shearing box simulations. Numerical algorithm is also important. Use of the HLLE flux solver or second-order interpolation can significantly degrade the effective resolution compared to the HLLD flux solver and third-order interpolation. Resolution at this standard can be achieved only with large numbers of grid zones, arranged in a fashion that matches the symmetries of the problem and the scientific goals of the simulation. Without it, however, quantitative measures important to predictions of observables are subject to large systematic errors.
author:
- 'John F. Hawley, Sherwood A. Richers, Xiaoyue Guan'
- 'Julian H. Krolik'
bibliography:
- 'hawleyreferences.bib'
title: Testing Convergence for Global Accretion Disks
---
introduction
numerics
resolution
Simulation Variations
=====================
Acknowledgements {#acknowledgements .unnumbered}
================
This work was partially supported by NASA grant NNX09AD14G and NSF grant AST-0908869 (JFH) and NSF grant AST-0908336 (JHK). Some of the simulations described here were carried out on the Kraken system at NICS, supported by the NSF. We thank Jake Simon for comments on this work. We thank the referee for suggesting the regrid tests described in §4.5.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The [*F-GAMMA*]{} program is a coordinated effort to investigate the physics of Active Galactic Nuclei (AGNs) via multi-frequency monitoring of [*Fermi*]{} blazars. The current study is concerned with the broad-band radio spectra composed of measurement at ten frequencies between 2.64 and 142GHz. It is shown that any of the 78 sources studied can be classified in terms of their variability characteristics in merely 5 types of variability. The first four types are dominated by spectral evolution and can be reproduced by a simple two-component system made of the quiescent spectrum of a large scale jet populated with a flaring event evolving according to [@Marscher1985ApJ]. The last type is characterized by an achromatic change of the broad-band spectrum which must be attributed to a completely different mechanism. Here are presented, the classification, the assumed physical system and the results of simulations that have been conducted.'
author:
- 'E. Angelakis, L. Fuhrmann, I. Nestoras, C. M. Fromm, R. Schmidt, J. A. Zensus, N. Marchili, T. P. Krichbaum'
- 'M. Perucho-Pla'
- 'H. Ungerechts, A. Sievers, D. Riquelme'
title: 'On the phenomenological classification of continuum radio spectra variability patterns of [*Fermi*]{} blazars'
---
INTRODUCTION
============
Among the most evident characteristics of blazars is the intense variability at all wavelengths. Studies of the variability characteristics, preferably with simultaneous data, can shed light on the physics driving the energy production and dissipation in these systems [e.g. @boettcher2010; @boettcher2010HEAD]. The [*F-GAMMA*]{} program [see @fuhrmann2007AIPC; @angelakis2008MmSAI..79.1042A; @2010arXiv1006.5610A] is a coordinated effort to explore exactly this very possibility by monthly monitoring of [*Fermi*]{} blazars. [*F-GAMMA*]{} is covering mostly the radio cm to sub-mm bands primarily with the Effelsberg 100-m, the IRAM 30-m and the APEX 12-m telescopes (although optical telescopes are participating as well, Fuhrmann et al. in prep.) for roughly 60 prominent blazars.
The cause for the variability itself has been long debated. The “shock-in-Jet” model suggested by [@Marscher1985ApJ], is the most accepted one and attributes the variability to shocks propagating down the jet. The basic assumption is that changes at the onset of the jet, (e.g. changes in the injection rate, the magnetic field, bulk Lorentz factor etc.) cause the formation of shocks, which then suffer first [*Compton*]{}, then [*synchrotron*]{} and finally [*Adiabatic*]{} losses. This is the main model prediction which is used in the following.
Here it is argued that (a) the variability patterns can be classified in only 5 phenomenological types and (b) the different phenomenological classes can be reproduced with a simple system composed of a quiescent spectrum populated by a flaring event evolving according to the “shock-in-Jet” model.
OBSERVATIONS AND DATA REDUCTION
===============================
\
\
\
The observations discussed here have been conducted quasi-simultaneously with the Effelsberg 100-m and the IRAM 30-m telescope (the combined spectra coherency time is a few days) within the [*F-GAMMA*]{} program [@fuhrmann2007AIPC; @angelakis2008MmSAI..79.1042A; @2010arXiv1006.5610A]. The 100-m telescope has been observing between 2.64 and 43.05GHz at 8 frequencies and the 30-m telescope at 86 and 142GHz (details in Fuhrmann et al. in prep., Nestoras et al. in prep., Angelakis et al. in prep.). In the current study only data collected until June 2011, have been used.
The data reduction includes: (a) [*Pointing correction*]{}, (b) [*Elevation dependent gain correction*]{}, (c) [*Atmospheric opacity correction*]{}, (d)[*Absolute calibration (sensitivity correction)*]{}. The overall uncertainties reached for the [*F-GAMMA*]{} program are of the order of 0.5 - 5% for Effelsberg and of the order of $\le10$% for IRAM. More details can be found in [@angelakis2009AnA] as well as in Angelakis et al. (in prep.), Fuhrmann et al. (in prep.) and Nestoras et al. (in prep.).
ANALYSIS
========
Phenomenological Classification of the Variability Patterns {#subsec:classification}
-----------------------------------------------------------
The visual inspection of the examined sources reveals a plurality in spectral features as well as in the variability pattern that different sources exhibit. Despite the apparent complexity it appears that any of the 78 sources studied here, can be classified in one among only five phenomenological classes on the basis of its variability pattern, which are termed numerically from 1 to 5 (more details will be given by Angelakis et al. in prep.). Four of them show also sub-types which however do not deserve a separate type and are named after the main type followed by the letter “b”. The prototype sources are shown in Figures \[fig:t1\]–\[fig:t5b\]. Their phenomenological characteristics, are:\
[**Type 1**]{}: is clearly dominated by spectral evolution. At an instant in time the spectrum appears convex and its peak is drifting within the observing band-pass from high towards lower frequencies, covering a significant area in the $S-\nu$ space. The convex component is smoothly changing towards an ultimate flat or mildly steep power-law which is then followed by consequent events. There is no evidence for a stable steep spectrum. The lowest frequencies in the bandpass are remarkably variable indicating that the activity seizes at frequencies much lower than the lowest in our band-pass. The prototype source is shown in Figure \[fig:t1\].\
[**Type 1b**]{}: As a sub-class of the previous one, type 1b shows similar characteristics except that the lowest frequency does not show as intense variability. The activity seizes around this part of the band-pass (see Figure \[fig:t1b\]).\
[**Type 2**]{}: is also dominated by spectral evolution. The basic characteristic of this case is the fact that the flux density at the lowest frequency during the steepest spectrum phase is higher than that during the inverted spectrum phase. Moreover, the maximum flux density reached by the flaring events is significantly above that at the lowest frequency. This implies that the observed steep spectrum is not a quiescent spectrum but rather the “echo” of an older, yet recent, outburst. The prototype of this type is shown in Figure \[fig:t2\].\
[**Type 3**]{}: Type 3, shown in Figure \[fig:t3\], is also dominated by spectral evolution. The identifying characteristics of this type are: (a) the fact that the lowest frequency practically does not vary and, (b) the maximum flux density level reached by outbursts is comparable to that at the lowest band-pass frequency. This phenomenology leaves hints that the events seize very close to the lowest frequency of the band-pass and hence a quiescent spectrum is becoming barely evident.\
[**Type 3b**]{}: Type 3b, shown in Figure \[fig:t3b\], is very similar to type 3. Here however the quiescent spectrum is seen clearly at least at the 2 lowest frequencies.\
[**Type 4**]{}: Sources of this type spend most of the time as steep spectrum ones which are sometimes showing an outburst of relatively low power propagating towards low frequencies. A representative case is shown in Figure \[fig:t4\].\
[**Type 4b**]{}: This type includes persistently steep spectrum cases as it is shown in Figure \[fig:t4b\].
All previous classes, are clearly dominated by spectral evolution. There exists a class of sources for which the variability happens self-similarly without signs of spectral evolution. Those are grouped in a separate type with two sub-types:\
[**Type 5**]{}: In this case the spectrum is convex and follows an “achromatic” evolution. That is, it shifts its position in the $S-\nu$ space preserving its shape. This is shown clearly in Figure \[fig:t5\].\
[**Type 5b**]{}: This type shows, in principle, characteristics similar to the previous one but there occurs a mild yet noticeable shift of the peak ($S_\mathrm{m},\nu_\mathrm{m}$) towards lower frequencies as the peak flux density increases. A characteristic case is shown in Figure \[fig:t5b\].
This classification is done solely on the basis of the phenomenological characteristics of the variability pattern shown by the radio spectra within a given band-pass. As it is discussed in the next section, it appears that all the phenomenology for types 1–4b can be naturally explained with the same underlying system observed under different circumstances.
A Physical Interpretation of the Variability Types 1 – 4b
---------------------------------------------------------
The phenomenological types 1 – 4b discussed earlier can be reproduced by the same simple two-component system, made of: (a) A power-law quiescent spectrum attributed to the optically thin emission (large-scale jet and recent flaring events) and (b) the convex synchrotron self-absorbed spectrum of a current outburst superimposed on the quiescent part. The assumed configuration is presented in Figure \[fig:principal\] where the shaded areas denote the observing band-pass. The phenomenology shown there captures the system (solid line) at an instant in time and the spectral shape that would be observed depends on: (a) the [***position***]{} of the shaded areas relative to the high and low frequency peak (i.e. the peak of the outburst) and (b) the [***width***]{} of the band-pass relative to the width of the bridge between the optically thick part of the outburst and the steep part of the quiescent spectrum.
![\[fig:principal\]The assumed two-component system. The different variability types can be reproduced with the appropriate modulation of the relative position and relative broadness of the band-pass denoted by the grey shaded areas.](principal.ps){width="30.00000%"}
These two quantities can be modulated by the combination of (a) the [***redshift***]{} and (b) the [***source intrinsic properties***]{}. The [*redshift*]{} changes the relative [*position*]{} of the band-pass allowing a different part of the spectrum to be sampled. The [*source intrinsic properties*]{} imply that different sources show different spectral characteristics (e.g. peak frequency of the outburst, peak flux density excess of the outburst over the quiescent spectrum, broadness of the valley etc.). More importantly, the dynamical evolution of a flaring event, is a function of the [*source intrinsic properties*]{} and introduces a third parameter (c) the [*flare specific properties*]{} which is determining the characteristics of the variability pattern.
In order to examine whether the assumed model can reproduce the observed phenomenologies, some characteristic cases have been evaluated.
Reproducing the Observed Phenomenologies 1 – 4b
-----------------------------------------------
Following the hypothesis that all the observed events are the reflection of the same process, namely shocks evolving in jets seen with different frequency band-passes at different evolutionary stages, the shock-in-jet model [@Marscher1985ApJ; @turler2000AnA...361..850T] has been applied to reproduce their temporal evolution. The followed approach is presented in [@2011AnA...531A..95F] where the flaring event passes different radiative evolutionary stages ([*Compton*]{}, [ *Synchrotron*]{} and [*Adiabatic*]{} stage). Simulations have been made for a large fraction of the parameter space. In Figure \[fig:simulations\] are presented only three cases of sources at $z=1.5$ but of different luminosity. From these plots it is already clear that the assumed scenario can reproduce most of the observed phenomenologies.
\
DISCUSSION
==========
The variability patterns of the studied blazars can be categorized in (a) spectral evolution dominated cases and (b) self-similarly varying convex spectra ones. This implies that there must exist two distinct mechanisms causing variability. This refers to the available baseline (roughly five years) meaning that sources of type 5 and 5b could still show spectral evolution over longer time scales. An additional element that points towards a different variability mechanism is the persistency of the spectral shape, in the case of type 5 and the fashion of change in the case of type 5b which is not seen in cases of clear spectral evolution.
Of the 78 examined sources, 8 show achromatic variability. The interesting characteristics is that the cases that show a mild spectral evolution (type “5b”), the turnover flux and frequency $S_\mathrm{m}$ and $\nu_\mathrm{m}$, are evolving in an anti-correlated fashion (see e.g. Figure \[fig:t5b\]). Apart from the fact that all of them show the clear presence of a large scale jet even at 2cm (as it is shown form the MOJAVE images @Kellermann2004ApJ) no other peculiar property has been identified so far. Possible mechanisms that are examined to be producing this variability include: opacity effects, changes in the magnetic field structure, changes in the Doppler factors and geometrical effects.
None of the studied sources has shown a switch of type over the baseline of the [ *F-GAMMA*]{} program, neither between types of the same underlying mechanism (i.e. 1–4b) nor between types with different underlying mechanism (i.e. types 1–4b and types 5, 5b). This suggests that the mechanisms producing the variability is either a fingerprint of the source or the conditions that determine it change over longer time scales. It would be essential to observe whether a source can switch from achromatic to an evolution dominate type or even further whether it exhibits periods of either variability flavor. In any case, the persistency of the evolution dominated types implies that the power deposited in each event for a certain source is not varying significantly from one event to the other. Further investigations to investigate this statement are underway and will be presented elsewhere.
Concerning the evolution dominated case, it seems that the [@Marscher1985ApJ] model provides a precise reproduction of the observed phenomenology and most importantly, over a range of intrinsic parameters covered by the [*F-GAMMA*]{} sample. Studies to examine whether other variability mechanisms can reproduce the observed phenomenology (shapes, time scales etc.) are needed. In any case, any successful model could be used for reversing the process and be used for calculating physical parameters from the observed spectra.
Based on observations with the 100m telescope of the MPIfR (Max-Planck-Institut für Radioastronomie) and the IRAM 30-m telescope. I. Nestoras and R. Schmidt are members of the International Max Planck Research School (IMPRS) for Astronomy and Astrophysics at the Universities of Bonn and Cologne. The [*F-GAMMA*]{} team sincerely thanks Dr. A. Kraus and the Time Allocation Committee of the 100-m and 30-m telescope for supporting the continuation of the program.
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}
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---
abstract: 'Grothendieck duality theory assigns to essentially-finite-type maps $f$ of noetherian schemes a pseudofunctor $f^\times$ right-adjoint to ${\mathsf R}{{f^{}_{{\mkern-1mu}{\mkern-1mu}*}}}$, and a pseudofunctor $f^!$ agreeing with $f^\times$ when $f$ is proper, but equal to the usual inverse image $f^*$ when $f$ is étale. We define and study a canonical map from the first pseudofunctor to the second. This map behaves well with respect to flat base change, and is taken to an isomorphism by “compactly supported" versions of standard derived functors. Concrete realizations are described, for instance for maps of affine schemes. Applications include proofs of reduction theorems for Hochschild homology and cohomology, and of a remarkable formula for the fundamental class of a flat map of affine schemes.'
address:
- 'Department of Mathematics, University of Nebraska, Lincoln, NE 68588, U.S.A.'
- 'Department of Mathematics, Purdue University, West Lafayette IN 47907, U.S.A.'
- 'Centre for Mathematics and its Applications, Mathematical Sciences Institute Australian National University, Canberra, ACT 0200, Australia.'
author:
- 'Srikanth B. Iyengar'
- Joseph Lipman
- Amnon Neeman
title: Relation between two twisted inverse image pseudofunctors in duality theory
---
[^1]
Introduction {#introduction .unnumbered}
============
The relation in the title is given by a canonical pseudofunctorial map $\psi\colon(-)^\times\to (-)^!$ between “twisted inverse image" pseudofunctors with which Grothendieck duality theory is concerned. These pseudofunctors on the category ${\mathscr{E}}$ of essentially-finite[.5pt]{}-type separated maps of noetherian schemes take values in bounded-below derived categories of complexes with quasi-coherent homology, see \[\^!\] and \[\^times\]. The map $\psi$, derived from the pseudofunctorial “fake unit map" $\operatorname{id}\to(-)^!{{{\mkern1mu}{\mkern1mu}\lift1,{\scriptstyle}{\circ},\,}}{\mathsf R}(-)_*$ of Proposition \[fake unit\], is specified in Corollary \[relation\]. A number of concrete examples appear in §\[section:examples\]. For instance, if $f$ is a map in ${\mathscr{E}}$, then $\psi(f)$ is an isomorphism if $f$ is proper; but if $f$ is, say, an open immersion, so that $f^!$ is the usual inverse image functor $f^*$ whereas $f^\times$ is right-adjoint to ${\mathsf R}{{f^{}_{{\mkern-1mu}{\mkern-1mu}*}}}{\mkern1mu}$, then $\psi(f)$ is usually quite far from being an isomorphism (see e.g., \[affine locimm\], \[varGM\] and \[affine/k\]).
After some preliminaries are covered in §\[prelims\], the definition of the pseudofunctorial map $\psi$ is worked out at the beginning of §\[basic map\]. Its good behavior with respect to flat base change is given by Proposition \[base change\].
The rest of Section \[basic map\] shows that under suitable “compact support" conditions, various operations from duality theory take $\psi$ to an isomorphism. To wit:
Let ${{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}(X)$ be the derived category of ${\mathcal O_{{\mkern-1mu}{\mkern-1mu}X}}$-complexes with quasi-coherent homology, and let ${{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{{{{\mkern-1mu}{\mkern-1mu}X}}}^{{\mkern-1mu}{\mathsf{qc}}}}(-,-)$ be the internal hom in the closed category ${{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}(X)$ (§1.5). Proposition \[Gampsi\] says:
*If $f\colon X\to Y$ is a map in ${\mathscr{E}}$, if $W$ is a union of closed subsets of $X$ to each of which the restriction of $f$ is proper, and if $E\in{{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}(X)$ has support contained in $W{\mkern-1mu},$ then each of the functors ${\mathsf R}{\varGamma}^{}_{\!W}(-),$ $E{\otimes^{\mathsf L}_{{{{\mkern-1mu}{\mkern-1mu}X}}}}(-)$ and ${{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{{{{\mkern-1mu}{\mkern-1mu}X}}}^{{\mkern-1mu}{\mathsf{qc}}}}(E,-)$ takes the map $\psi(f)\colon f^\times\to f^!$ to an isomorphism.*
The proof uses properties of a bijection between subsets of $X$ and “localizing tensor ideals" in ${{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}(X)$, reviewed in Appendix \[Support\]. A consequence is that even for nonproper $f$, $f^!$ still has dualizing properties for complexes having support in such a $W$ (Corollary \[supports\]); and there results, for $d=\sup\{\,\ell\mid H^\ell f^!{\mathcal O_Y}\ne0\,\}$ and $\omega^{}_{\!f}$ a relative dualizing sheaf, a “generalized residue map"=-1 $$\int_{{\mkern1mu}W}{\mkern-1mu}\colon H^d{{\mathsf R}f^{}_{{\mkern-1mu}{\mkern-1mu}*}}{\mathsf R}{\varGamma}^{}_{\!W}(\omega^{}_{\!f})\to{\mathcal O_Y}.$$
Proposition \[pullback1\] says that *for ${\mathscr{E}}$-maps of noetherian schemes such that $fg$ is proper, and any $F\in{{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}(X),$ $G\in{{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}^{\lift.95,\text{\cmt\char'053},}}(Y),$ the functors ${\mathsf L}g^*{{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{{{{\mkern-1mu}{\mkern-1mu}X}}}^{{\mkern-1mu}{\mathsf{qc}}}}(F,-)$ and $g^\times{{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{{{{\mkern-1mu}{\mkern-1mu}X}}}^{{\mkern-1mu}{\mathsf{qc}}}}(F,-)$ both take the map $\psi(f)G\colon f^\times G\to f^!G$ to an isomorphism.*
Section 3 gives some concrete realizations of $\psi$. Besides the examples mentioned above, one has that if $R$ is a noetherian ring, $S$ a flat essentially-finite[.5pt]{}-type $R$-algebra, $f\colon\operatorname{Spec}S\to\operatorname{Spec}R$ the corresponding scheme[.5pt]{}-map, and $M$ an $R$-module, with sheafification ${\mathscr{M}}$, then with ${{S}^{\mathsf e}}{\!:=}S\otimes_R S$, the map $\psi(f)({\mathscr{M}})\colon f^\times{\mathscr{M}}\to f^!{\mathscr{M}}$ is the sheafification of a simple ${\boldsymbol{\mathsf{D}}}(S)$-map $$\label{affmap}
{\mathsf R}{\mkern-1mu}\operatorname{Hom}_R(S,M)\to S{\otimes^{\mathsf L}_{{{S}^{\mathsf e}}}}{\mathsf R}{\mkern-1mu}\operatorname{Hom}_R(S,S\otimes_R M),$$ described in Proposition \[R1.1.3.5\]. So if $S\to T$ is an $R$-algebra map with $T$ module[.5pt]{}-finite over $R$, then, as above, the functors $T{\otimes^{\mathsf L}_{S}}-$ and ${\mathsf R}{\mkern-1mu}\operatorname{Hom}_S(T, -)$ take to an isomorphism.
In the case where $R$ is a field, more information about the map appears in Proposition \[affine/k\][.5pt]{}: the map is represented by a split $S$-module surjection with an enormous kernel.
In §4, there are two applications of the map $\psi$. The first is to a “reduction theorem" for the Hochschild homology of flat ${\mathscr{E}}$-maps that was stated in [@AILN Theorem 4.6] in algebraic terms (see below), with only an indication of proof. The scheme[.5pt]{}-theoretic version appears here in \[global4.6\].
The paper [@AILN] also treats the nonflat algebraic case, where ${{S}^{\mathsf e}}$ becomes a derived tensor product. In fact, we conjecture that the natural home of the reduction theorems is in a more general derived-algebraic-geometry setting.
The special case \[affreldual\] of gives a canonical description of the relative dualizing sheaf $f^!{\mathcal O_Y}$ of a flat ${\mathscr{E}}$-map $f\colon X\to Y$ between affine schemes. The proof is based on the known theory of $f^!$, which is constructed using arbitrary choices, such as a compactification of $f$ or a factorization of $f$ as smooth${{{\mkern1mu}{\mkern1mu}\lift1,{\scriptstyle}{\circ},\,}}$finite; but the choice[.5pt]{}-free formula \[affreldual\] might be a jumping-off point for a choice[.5pt]{}-free redevelopment of the underlying theory.
The second application is to a simple formula for the *fundamental class* of a flat map $f$ of affine schemes. The fundamental class of a flat ${\mathscr{E}}$-map —a globalization of the Grothendieck residue map—goes from the Hochschild complex of $g$ to the relative dualizing complex $g^!{\mathcal O_Y}$. This map is defined in terms of sophisticated abstract notions from duality theory (see ). But for maps $f\colon\operatorname{Spec}S\to \operatorname{Spec}R$ as above, Theorem \[explicit fc\] says that, with $\mu\colon S\to \operatorname{Hom}_R(S,S)$ the ${{S}^{\mathsf e}}$-homomorphism taking $s\in S$ to multiplication by $s$, *the fundamental class is isomorphic to the sheafification of the natural composite map* $$S{\otimes^{\mathsf L}_{{{S}^{\mathsf e}}}}S{\xrightarrow}{{\mkern-1mu}\operatorname{id}{\mkern-1mu}\otimes{\mkern1mu}\mu{\mkern1mu}} S{\otimes^{\mathsf L}_{{{S}^{\mathsf e}}}}\operatorname{Hom}_R(S,S){\longrightarrow}S{\otimes^{\mathsf L}_{{{S}^{\mathsf e}}}}{\mathsf R}{\mkern-1mu}{\mkern-1mu}\operatorname{Hom}_R(S,S).$$
Preliminaries: twisted inverse image functors, essentially finite-type compactification, conjugate maps {#prelims}
=======================================================================================================
\[\^!\] For a scheme $X{\mkern-1mu}$, ${\boldsymbol{\mathsf{D}}}(X)$ is the derived category of ${\mathcal O_{{\mkern-1mu}{\mkern-1mu}X}}$-modules, and ${{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}(X)$ (${{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}^{\lift.95,\text{\cmt\char'053},}}(X)$) is the full subcategory spanned by the complexes with quasi-coherent cohomology modules (vanishing in all but finitely many negative degrees). We will use freely some standard functorial maps, for instance the projection isomorphism associated to a map $f\colon X\to Y$ of noetherian schemes (see, e.g., [@li 3.9.4]): $${\mathsf R}{\mkern1mu}{{f^{}_{{\mkern-1mu}{\mkern-1mu}*}}}E{\otimes^{\mathsf L}_{Y}}{\mkern-1mu}F{{\mkern8mu\longrightarrow \mkern-25.5mu{}^\sim\mkern17mu}}{\mathsf R}{\mkern1mu}{{f^{}_{{\mkern-1mu}{\mkern-1mu}*}}}(E{\otimes^{\mathsf L}_{X}} {\mkern-1mu}{\mathsf L}f^*{\mkern-1mu}{\mkern-1mu}F)
\quad\ \big(E\in{{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}(X), \,F\in{{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}(Y)\big).$$
Denote by ${\mathscr{E}}$ the category of *separated essentially-finite[.5pt]{}-type maps of noetherian schemes.* By [@Nk 5.2 and 5.3], there is a contravariant pseudofunctor $(-)^!$ over ${\mathscr{E}}$, determined up to isomorphism by the properties:2
[(i)]{} The pseudofunctor $(-)^!$ restricts over the subcategory of proper maps in ${\mathscr{E}}$ to a right adjoint of the derived direct-image pseudofunctor.1
[(ii)]{} The pseudofunctor $(-)^!$ restricts over the subcategory of formally étale maps in ${\mathscr{E}}$ to the usual inverse[.5pt]{}-image pseudofunctor $(-)^*$.1
[(iii)]{} For any fiber square in ${\mathscr{E}}\!:$2 $$\CD
\bullet @>v>> \bullet \\
@V g VV @VV f V \\
\bullet@>{\vbox to 0pt{\vss\hbox{$\Xi$}\vskip.21in}} > \lift1.2,u, >\bullet
\endCD$$ with $f,g$ proper and $ u,v$ formally étale, the base[.5pt]{}-change map $\beta_{{\mkern1mu}\Xi}$, defined to be the adjoint of the natural composition $$\label{bcadj}
{\mathsf R}g_*v^*\!f^!{{\mkern8mu\longrightarrow \mkern-25.5mu{}^\sim\mkern17mu}}u^*{{\mathsf R}f^{}_{{\mkern-1mu}{\mkern-1mu}*}}f^! \longrightarrow u^*{\mkern-1mu},$$ is equal to the natural composite isomorphism $$\label{beta}
v^*{\mkern-1mu}{\mkern-1mu}f^!= v^!{\mkern-1mu}{\mkern-1mu}f^!
{{\mkern8mu\longrightarrow \mkern-25.5mu{}^\sim\mkern17mu}}(fv)^!=(ug)^!{{\mkern8mu\longrightarrow \mkern-25.5mu{}^\sim\mkern17mu}}g^!u^! =g^!u^*.$$
There is in fact a family of base-change *isomorphisms* $$\label{bch}
\beta_{{\mkern1mu}\Xi}\colon v^*{\mkern-1mu}{\mkern-1mu}f^!{{\mkern8mu\longrightarrow \mkern-25.5mu{}^\sim\mkern17mu}}g^!u^*,$$ indexed by *all* commutative ${\mathscr{E}}$-squares $$\CD
\bullet @>v>> \bullet \\
@V g VV @VV f V \\
\bullet@>{\vbox to 0pt{\vss\hbox{$\Xi$}\vskip.21in}} > \lift1.2,u, >\bullet
\endCD$$ that are such that in the associated diagram (which exists in ${\mathscr{E}}$, see [@Nk §2.2])=-1 $$\CD
\bullet @>i>>\bullet @>w>> \bullet \\
@.@V h VV @VV f V \\
@. \bullet@>{\vbox to 0pt{\vss\hbox{$\Xi'$}\vskip.21in}} > \lift1.2,u, >\bullet
\endCD$$ it holds that $\Xi'$ is a fiber square, $wi=v$ and $hi=g$, the map $u$ is flat and $i$ is formally étale, a family that is the unique such one that behaves transitively with respect to vertical and horizontal composition of such $\Xi$ (cf. [@li (4.8.2)(3)]), and satisfies:3
[(iv)]{} if $\Xi$ is a fiber square with $f$ proper then the map $\beta_{{\mkern1mu}\Xi}$ is adjoint to the composite map ;2
[(v)]{} if $f$—hence $g$—is formally étale, so that $f^!=f^*$ and $g^!=g^*{\mkern-1mu},$ then $\beta_{{\mkern1mu}\Xi}$ is the natural isomorphism $v^*{\mkern-1mu}{\mkern-1mu}f^*{{\mkern8mu\longrightarrow \mkern-25.5mu{}^\sim\mkern17mu}}g^*u^*$; and2
[(vi)]{} if $u$—hence $v$—is formally étale, so that $u^*=u^!$ and $v^*=v^!{\mkern-1mu},$ then $\beta_{{\mkern1mu}\Xi}$ is the natural isomorphism .3
(For further explanation see [@li Thm.4.8.3] and [@Nk §5.2].)4
*Remark.* With regard to (vi), if $\Xi$ is *any* commutative ${\mathscr{E}}$-diagram with $u$ and $v$ formally étale, then in the associated diagram $i$ is necessarily formally étale ([@EGA4 (17.1.3(iii) and 17.1.4]), so that $\beta_{{\mkern1mu}\Xi}$ exists 1 (and can be identified with the canonical isomorphism $v^!{\mkern-1mu}{\mkern-1mu}f^!{{\mkern8mu\longrightarrow \mkern-25.5mu{}^\sim\mkern17mu}}g^!u^!$).
\[\^times\]
For *any* ${\mathscr{E}}$-map $f\colon X\to Y{\mkern-1mu}$, there exists a functor $f^\times\colon {\boldsymbol{\mathsf{D}}}(Y)\to{{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}(X)$ that is bounded below and right-adjoint to ${{\mathsf R}f^{}_{{\mkern-1mu}{\mkern-1mu}*}}{\mkern1mu}$. There results a ${{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}$[.5pt]{}-valued pseudofunctor $(-)^\times$ on ${\mathscr{E}}$, for which the said adjunction is pseudofunctorial [@li Corollary (4.1.2)]. Obviously, the restriction of $(-)^\times$ to ${{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}^{\lift.95,\text{\cmt\char'053},}}$ over proper maps in ${\mathscr{E}}$ is isomorphic to that of $(-)^!$. Accordingly, we will identify these two restricted pseudofunctors.
\[locimm\] Nayak’s construction of $(-)^!$ is based on his extension [@Nk p.536, Thm.4.1] of Nagata’s compactification theorem, to wit, that any map $f$ in ${\mathscr{E}}$ factors as $pu$ where $p$ is proper and $u$ is a localizing immersion (see below). Such a factorization is called a *compactification of* $f$.2
A *localizing immersion* is an ${\mathscr{E}}$-map $u\colon X\to Y$ for which every has a neighborhood $V=\operatorname{Spec}A$ such that $u^{-1}V=\operatorname{Spec}A_M$ for some multiplicatively closed subset $M\subseteq A$, see [@Nk p.532, 2.8.8]. For example, *finite[.5pt]{}-type* localizing immersions are1 just open immersions [@Nk p.531, 2.8.3].
Any localizing immersion $u$ is formally étale, so that $u^!=u^*{\mkern-1mu}$.1
\[cosa:localizing-immersion\] Any localizing immersion $u\colon X\to Y$ is a flat monomorphism, whence *the natural map $\epsilon^{}_1\colon u^*{\mathsf R}u_*{{\mkern8mu\longrightarrow \mkern-25.5mu{}^\sim\mkern17mu}}\operatorname{id}_{{{\mkern-1mu}{\mkern-1mu}X}}{\mkern1mu}$ is an isomorphism*: associated to the fiber square=-1 $$\CD
X\times_Y X @>p^{}_1>> X\\
@Vp^{}_2VV @VVuV\\
X @>>u > Y
\endCD$$ there is the flat base[.5pt]{}-change isomorphism $u^*{\mathsf R}u_*{{\mkern8mu\longrightarrow \mkern-25.5mu{}^\sim\mkern17mu}}{\mathsf R}p^{}_{2{\mkern1mu}*}p_1^*$, and since $u$ is a monomorphism, $p^{}_1$ and $p^{}_2$ are equal isomorphisms, so that ${\mathsf R}p_{2{\mkern1mu}*}p_1^*=\operatorname{id}_{{{\mkern-1mu}{\mkern-1mu}X}}$.
That $\epsilon^{}_1$ is an isomorphism means that the natural map is an isomorphism $$\operatorname{Hom}_{{\boldsymbol{\mathsf{D}}}(X)}(E,F){{\mkern8mu\longrightarrow \mkern-25.5mu{}^\sim\mkern17mu}}\operatorname{Hom}_{{\boldsymbol{\mathsf{D}}}(Y)}({\mathsf R}u_*E, {\mathsf R}u_*F)\qquad(E,F\in{\boldsymbol{\mathsf{D}}}(X)),$$ which implies that the natural map $\eta^{}_{{\mkern1mu}2}\colon\operatorname{id}_{{{\mkern-1mu}{\mkern-1mu}X}}\to u^\times {\mathsf R}u_*$ is an isomorphism.
Conversely, *any flat monomorphism $f$ in ${\mathscr{E}}$ is a localizing immersion,* which can be seen as follows. Using [@Nk 2.7[.5pt]{}] and [@EGA4 8.11.5.1 and 17.6.1] one reduces to where $f$ is a map of affine schemes, corresponding to a composite ring map $A\to B\to B_M$ with $A\to B$ étale and $M$ a multiplicative submonoid of $B$. The kernel of multiplication $B\otimes_A B\to B$ is generated by an idempotent $e$, and $B_M\otimes_A B_M\to B_M$ is an isomorphism, so $e$ is annihilated by an element of the form $m\otimes m\ (m\in M)$. Consequently, $B[1/m]\otimes B[1/m]\to B[1/m]$ is an isomorphism, and so replacing $B_M$ by $B[1/m]$ reduces the problem further to the case where $A\to B_M$ is a finite[.5pt]{}-type algebra. Finally, localizing $A$ with respect to its submonoid of elements that are sent to units in $B_M$, one may assume further that $f$ is surjective, in which case [@EGA4 17.9.1] gives that $f$ is an isomorphism.
\[RHom\^qc\] For a noetherian scheme $X{\mkern-1mu}$, the functor $\operatorname{id}_{{{{\mkern-1mu}{\mkern-1mu}X}}}^\times$ specified in §\[\^times\] is right-adjoint to the inclusion ${{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}(X)\hookrightarrow{\boldsymbol{\mathsf{D}}}(X)$. It is sometimes called the *derived quasi-coherator.*
For any $C\in{{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}(X)$, the unit map is an isomorphism $C{{\mkern8mu\longrightarrow \mkern-25.5mu{}^\sim\mkern17mu}}\operatorname{id}_X^\times{\mkern-1mu}C$.
For any complexes $A$ and $B$ in ${{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}(X)$, set $$\label{RHqc}
{{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{{{{\mkern-1mu}{\mkern-1mu}X}}}^{{\mkern-1mu}{\mathsf{qc}}}}(A,B){\!:=}\operatorname{id}_X^\times{\mathsf R}{{\mathcal H}om}_{{{\mkern-1mu}{\mkern-1mu}X}}(A,B)\in{{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}(X).$$
Then for $A$ in ${{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}(X)$, the functor ${{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{{{{\mkern-1mu}{\mkern-1mu}X}}}^{{\mkern-1mu}{\mathsf{qc}}}}(A,-)$ is right-adjoint to the endofunctor $-{\otimes^{\mathsf L}_{{{{\mkern-1mu}{\mkern-1mu}X}}}} A$ of ${\mkern1mu}{{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}(X)$. Thus, ${{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}(X)$ is a closed category with multiplication given by ${\otimes^{\mathsf L}_{{{{\mkern-1mu}{\mkern-1mu}X}}}}$ and internal hom given by ${{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{{{{\mkern-1mu}{\mkern-1mu}X}}}^{{\mkern-1mu}{\mathsf{qc}}}}$.
As above, the canonical ${\boldsymbol{\mathsf{D}}}(X)$-map ${{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{{{{\mkern-1mu}{\mkern-1mu}X}}}^{{\mkern-1mu}{\mathsf{qc}}}}(A,B)\to{\mathsf R}{{\mathcal H}om}_{{{\mkern-1mu}{\mkern-1mu}X}}(A,B)$ is an *isomorphism* whenever ${\mathsf R}{{\mathcal H}om}_{{{\mkern-1mu}{\mkern-1mu}X}}(A,B)\in{{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}(X)$—for example, whenever $B\in{{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}^{\lift.95,\text{\cmt\char'053},}}(X)$ and the cohomology sheaves $H^iA$ are coherent for all $i$, vanishing for $i\gg 0$ [@RD p.92, 3.3].
\[conjugate\] For categories $P$ and $Q$, let Fun$(P,Q)$ be the category of functors from $P$ to $Q$, and let Fun$^\textup{L}(P,Q)$ (resp. Fun$^\textup{R}(P,Q))$ be the full subcategory spanned by the objects that have right (resp. left) adjoints. There is a contravariant isomorphism of categories $$\xi\colon\textup{Fun}^\textup{L}(P,Q){{\mkern8mu\longrightarrow \mkern-25.5mu{}^\sim\mkern17mu}}\textup{Fun}^\textup{R}(P,Q)$$ that takes any map of functors to the *right-conjugate* map between the respective right adjoints (see e.g., [@li 3.3.5–3.3.7[.5pt]{}]). The image under $\xi^{-1}$ of a map of functors is its *left-conjugate* map. The functor $\xi$ (resp. $\xi^{-1}$) takes isomorphisms of functors to isomorphisms.
For instance, for any ${\mathscr{E}}$-map $f\colon X\to Z$ there is a bifunctorial *sheafified duality* isomorphism, with $E\in{{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}(X)$ and $ F\in{{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}(Z)$: $$\label{sheaf dual}
{\mathsf R}{\mkern1mu}{{f^{}_{{\mkern-1mu}{\mkern-1mu}*}}}{{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{{{{\mkern-1mu}{\mkern-1mu}X}}}^{{\mkern-1mu}{\mathsf{qc}}}}(E, f^\times{\mkern-1mu}{\mkern-1mu}F{\mkern1mu}){{\mkern8mu\longrightarrow \mkern-25.5mu{}^\sim\mkern17mu}}{{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{Z}^{{\mkern-1mu}{\mathsf{qc}}}}({\mathsf R}{\mkern1mu}{{f^{}_{{\mkern-1mu}{\mkern-1mu}*}}}E,F{\mkern1mu}),$$ right-conjugate, for each fixed $E$, to the projection isomorphism $$\mkern-24mu {\mathsf R}{\mkern1mu}{{f^{}_{{\mkern-1mu}{\mkern-1mu}*}}}({\mathsf L}f^*G{\otimes^{\mathsf L}_{{{{\mkern-1mu}{\mkern-1mu}X}}}} E){{\mkern8mu\longleftarrow \mkern-23.5mu{}^\sim\mkern17mu}}G{\otimes^{\mathsf L}_{{\mkern-1mu}{\mkern-1mu}Z}}{\mathsf R}{\mkern1mu}{{f^{}_{{\mkern-1mu}{\mkern-1mu}*}}}E.$$
Likewise, there is a functorial isomorphism $$\label{^timesHom}
{{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{W}^{{\mkern-1mu}{\mathsf{qc}}}}({\mathsf L}f^*G, {\mkern1mu}f^\times{\mkern-1mu}{\mkern-1mu}H){{\mkern8mu\longrightarrow \mkern-25.5mu{}^\sim\mkern17mu}}f^\times {{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{{{{\mkern-1mu}{\mkern-1mu}X}}}^{{\mkern-1mu}{\mathsf{qc}}}}(G, H)$$ right-conjugate to the projection isomorphism ${\mathsf R}{{f^{}_{{\mkern-1mu}{\mkern-1mu}*}}}(E{\otimes^{\mathsf L}_{{{{\mkern-1mu}{\mkern-1mu}X}}}} {\mkern-1mu}{\mathsf L}f^*G){{\mkern8mu\longleftarrow \mkern-23.5mu{}^\sim\mkern17mu}}{\mathsf R}{\mkern1mu}{{f^{}_{{\mkern-1mu}{\mkern-1mu}*}}}E{\otimes^{\mathsf L}_{{\mkern-1mu}{\mkern-1mu}Z}} G.$
The basic map {#basic map}
=============
In this section we construct a pseudofunctorial map $\psi\colon(-)^\times\to (-)^!$. The construction is based on the following “fake unit" map.
\[fake unit\] Over ${\mathscr{E}}$ there is a unique pseudofunctorial map $$\eta\colon\operatorname{id}\to (-)^!{{{\mkern1mu}{\mkern1mu}\lift1,{\scriptstyle}{\circ},\,}}{\mathsf R}(-)_*$$ whose restriction to the subcategory of proper maps in ${\mathscr{E}}$ is the unit of the adjunction between ${\mathsf R}(-)_*$ and $(-)^!,$ and such that if $u$ is a localizing immersion then $\eta(u)$ is inverse to the isomorphism $u^!{\mkern1mu}{\mathsf R}u_*= u^*{\mathsf R}u_*{{\mkern8mu\longrightarrow \mkern-25.5mu{}^\sim\mkern17mu}}\operatorname{id}$ in \[cosa:localizing-immersion\].=-1
The proof uses the next result—in which the occurrence of $\beta_{{\mkern1mu}\Xi}$ is justified by the remark at the end of §\[\^!\]. As we are dealing only with functors between derived categories, we will reduce clutter by writing $h_*$ for ${\mathsf R}h_*$ ($h$ any map in ${\mathscr{E}}$).
\[compat\] Let $\Xi$ be a commutative square in ${\mathscr{E}}\colon$ $$\CD
\bullet @>v>> \bullet \\
@V g VV @VV f V \\
\bullet@>{\vbox to 0pt{\vss\hbox{$\Xi$}\vskip.21in}} > \lift1.2,u, >\bullet
\endCD$$ with $f,g$ proper and $ u,v$ localizing immersions. Let $\phi^{}_{{\mkern1mu}\Xi}\colon v_*g^!\to f^!u_*$ be the functorial map adjoint to the natural composite map Then the following natural diagram commutes.
$$\def{\mathbf{1}}{$\operatorname{id}$}
\def\2{$v^*{\mkern-1mu}v_*$}
\def\3{$g^!{\mkern-1mu}g_*$}
\def\4{$v^*{\mkern-1mu}v_*g^!{\mkern-1mu}g_*$}
\def\5{$v^*{\mkern-1mu}{\mkern-1mu}f^!{\mkern-1mu}{{f^{}_{{\mkern-1mu}{\mkern-1mu}*}}}v_*$}
\def\6{$v^*{\mkern-1mu}{\mkern-1mu}f^!{\mkern-1mu}u_*g_*$}
\def\7{$g^!u^*{\mkern-1mu}u_*g_*$}
\def\8{$v^!{\mkern-1mu}f^!{\mkern-1mu}{{f^{}_{{\mkern-1mu}{\mkern-1mu}*}}}v_*$}
\def\9{$(fv)^!(fv)_*$}
\def\ten{$(ug)^!(ug)_*$}
\def\lvn{$g^!u^!{\mkern-1mu}u_*g_*$}
{\begin{tikzpicture}}[xscale=5, yscale=1.45]
\node(11) at (1,-1){\2};
\node(12) at (2,-1){{\mathbf{1}}};
\node(13) at (3,-1){\3};
\node(22) at (2,-2){\4};
\node(32) at (2,-3){\6};
\node(41) at (1,-3){\5};
\node(43) at (3,-3){\7};
\node(51) at (1,-4){\8};
\node(52) at (1.66,-4){\9};
\node(53) at (2.33,-4){\ten};
\node(54) at (3,-4){\lvn};
\draw[<-] (11)--(12) node[above=1pt, midway, scale=.75]{$\Iso$};
\draw[->] (12)--(13) node[above=1pt, midway, scale=.75]{\textup{unit}};
\draw[->] (41)--(32) node[above=1pt, midway, scale=.75]{$\Iso$};
\draw[->] (32)--(43) node[above=1pt, midway, scale=.75]{$\Iso$}
node[below=1pt, midway, scale=.75]{$\beta_{{\mkern1mu}\Xi}$};
\draw[->] (51)--(52) node[above, midway, scale=.75]{$\Iso$};
\draw[double distance=2pt] (52)--(53);
\draw[->] (53)--(54) node[above, midway, scale=.75]{$\Iso$};
\draw[->] (11)--(41) node[left=1pt, midway, scale=.75]{$v^*\textup{unit}$};
\draw[->] (22)--(32) node[left=1pt, midway, scale=.75]{$v^*\phi^{}_{{\mkern1mu}\Xi}$};
\draw[double distance=2pt] (41)--(51) ;
\draw[->] (13)--(43) node[right=1pt, midway, scale=.75]{$\simeq$};
\draw[double distance=2pt] (43)--(54) ;
\draw[->] (11)--(22) node[above=-6pt, midway, scale=.75]{\rotatebox{-16.5}{$\mkern43mu v^*v_*\textup{unit}$}};
\draw[<-] (22)--(13) node[above=-1pt, midway, scale=.75] {$\rotatebox{16.5}\Iso$};
\node at (2,-1.5)[scale=.9]{{\textcircled{\scriptsize{1}}}};
\node at (1.5,-2.35)[scale=.9]{{\textcircled{\scriptsize{2}}}};
\node at (2.5,-2.35)[scale=.9]{{\textcircled{\scriptsize{3}}}};
\node at (2,-3.56)[scale=.9]{{\textcircled{\scriptsize{4}}}};
{\end{tikzpicture}}$$
Commutativity of subdiagram 1 is clear.
For commutativity of subdiagram 2, drop $v^*$ and note the obvious commutativity of the following adjoint of the resulting diagram: $$\def{\mathbf{1}}{${{f^{}_{{\mkern-1mu}{\mkern-1mu}*}}}v_*$}
\def\2{${{f^{}_{{\mkern-1mu}{\mkern-1mu}*}}}v_*g^!{\mkern-1mu}g_*$}
\def\3{$u_*g_*g^!g_*$}
\def\4{$u_*g_*$}
{\begin{tikzpicture}}[xscale=5, yscale=1.25]
\node(11) at (1,-1){{\mathbf{1}}};
\node(12) at (2,-1){\2};
\node(22) at (2,-2){\3};
\node(31) at (1,-3){{\mathbf{1}}};
\node(32) at (2,-3){\4};
\draw[->] (11)--(12) ;
\draw[->] (31)--(32) ;
\draw[double distance=2pt] (11)--(31) ;
\draw[->] (12)--(22) node[right=1pt, midway, scale=.75]{$\simeq$};
\draw[->] (22)--(32) ;
{\end{tikzpicture}}$$
Showing commutativity of subdiagram 3 is similar to working out [@li Exercise 3.10.4(b)]. (Details are left to the reader.)
Commutativity of subdiagram 4 is given by (vi) in section \[\^!\].
As before, for any map $h$ in ${\mathscr{E}}$ we abbreviate ${\mathsf R}h_*$ to $h_*$. Let $f$ be a map in ${\mathscr{E}}$, and $f=pu$ a compactification. If $\eta$ exists, then $\eta(f)\colon \operatorname{id}\to f^!{\mkern-1mu}f^{}_{{\mkern-1mu}{\mkern-1mu}*}$ must be given by the natural composition $$\label{pseudounit}
\operatorname{id}{{\mkern8mu\longrightarrow \mkern-25.5mu{}^\sim\mkern17mu}}u^* u_* {\xrightarrow}{\operatorname{{\textup{via}}}\textup{ unit}\,} u^*p^!{\mkern1mu}p_* u_* {{\mkern8mu\longrightarrow \mkern-25.5mu{}^\sim\mkern17mu}}f^!{\mkern-1mu}f^{}_{{\mkern-1mu}{\mkern-1mu}*},$$ so that uniqueness holds.
Let us show now that this composite map does not depend on the choice of compactification.
A morphism $r\colon(f=qv)\to(f=pu)$ from one compactification of $f$ to another is a commutative diagram of scheme[.5pt]{}-maps $$\label{dominate}
\CD
{\begin{tikzpicture}}[xscale=2.5, yscale=1]
\node(12) at (2,-1){$\bullet$};
\node(21) at (1,-2){$\bullet$};
\node(23) at (3,-2){$\bullet$};
\node(32) at (2,-3){$\bullet$};
\draw[->] (21)--(12) node[above=1pt, midway, scale=.75]{$v$};
\draw[->] (12)--(23) node[above=1pt, midway, scale=.75]{$q$};
\draw[->] (21)--(32) node[below=1pt, midway, scale=.75]{$u$};
\draw[->] (32)--(23) node[below=1pt, midway, scale=.75]{$p$};
\draw[->] (12)--(32) node[left=1pt, midway, scale=.75]{$r$};
{\end{tikzpicture}}\endCD$$ If such a map $r$—necessarily proper—exists, we say that the compactification $f=qv$ *dominates* $f=pu$.
*Any two compactifications $X{\xrightarrow}{u_1\,} Z_1{\xrightarrow}{p^{}_1\,} Y,$ $X{\xrightarrow}{u_2\,} Z_2{\xrightarrow}{p^{}_2\,} Y$ of a given $f\colon X\to Y$ are dominated by a third one*. Indeed, let $v\colon X\to Z_1\times_Y Z_2$ be the map corresponding to the pair $(u_1,u_2)$, let $Z\subseteq Z_1\times_Y Z_2$ be the schematic closure of $v$—so that $v\colon X\to Z$ has schematically dense image[.5pt]{}—and let $r_i:Z\to Z_i\ (i=1,2)$ be the maps induced by the two canonical projections. Since $u=r_iv$ is a localizing immersion, therefore, by [@Nk p.533, 3.2], so is $v$. Thus $f=(p_ir_i)v$ is a compactification, not depending on $i$, mapped by $r_i$ to the compactification $f=p_iu_i$.
So to show that gives the same result for any two compactifications of $f$, it suffices to do so when one of the compactifications dominates the other. Thus with reference to the diagram , and keeping in mind that $u^*=u^!$ and $v^*=v^!$, one need only show that the following natural diagram commutes. $$\def{\mathbf{1}}{$\operatorname{id}$}
\def\2{$u^!u_*$}
\def\3{$v^!r^!r_*v_*$}
\def\4{$v^!v_*$}
\def\5{$u^!p^!p_*u_*$}
\def\6{$v^!r^!p^!p_*r_*v_*$}
\def\7{$v^!q^!q_*v_*$}
\def\8{$f^!{\mkern-1mu}{\mkern-1mu}{{f^{}_{{\mkern-1mu}{\mkern-1mu}*}}}$}
{\begin{tikzpicture}}[xscale=3, yscale=1.6]
\node(12) at (2,-1){{\mathbf{1}}};
\node(21) at (1,-2){\2};
\node(22) at (2,-2){\3};
\node(23) at (3,-2){\4};
\node(31) at (1,-3){\5};
\node(32) at (2,-3){\6};
\node(33) at (3,-3){\7};
\node(42) at (2,-4){\8};
\draw[->] (21)--(22) ;
\draw[->] (23)--(22) ;
\draw[->] (31)--(32) ;
\draw[->] (33)--(32) ;
\draw[->] (21)--(31) ;
\draw[->] (22)--(32) ;
\draw[->] (32)--(42) ;
\draw[->] (23)--(33) ;
\draw[->] (12)--(21) node[right=1pt, midway, scale=.75]{$ $};
\draw[->] (12)--(23) ;
\draw[->] (31)--(42) node[right=1pt, midway, scale=.75]{$ $};
\draw[->] (33)--(42) ;
\node at (2,-1.55)[scale=.9]{{\textcircled{\scriptsize{1}}}};
\node at (1.5,-2.5)[scale=.9]{{\textcircled{\scriptsize{2}}}};
\node at (2.5,-2.5)[scale=.9]{{\textcircled{\scriptsize{3}}}};
\node at (1.75,-3.4)[scale=.9]{{\textcircled{\scriptsize{4}}}};
\node at (2.25,-3.4)[scale=.9]{{\textcircled{\scriptsize{5}}}};
{\end{tikzpicture}}$$
Commutativity of subdiagram 1 is given by Lemma \[compat\], with $f{\!:=}r$ and $g{\!:=}\operatorname{id}_{{{\mkern-1mu}{\mkern-1mu}X}}$.
Commutativity of 2 is clear.
Commutativity of 3 holds because over proper maps, $(-)^!$ and $(-)_*$ are *pseudofunctorially* adjoint (see [@li Corollary (4.1.2)].
Commutativity of 4 and 5 results from the pseudofunctoriality of $(-)^!$ and $(-)_*$.1
Thus is indeed independent of choice of compactification, so that $\eta(f)$ is well-defined.4
Finally, it must be shown that $\eta$ is pseudofunctorial, i.e., for any composition $X{\xrightarrow}{\,f\,}Y{\xrightarrow}{\,g\,}Z$ in ${\mathscr{E}}$, the next diagram commutes: $$\CD
\operatorname{id}@>\eta(gf)>>(gf)^!(gf)_* \\
@V\eta(f)VV@VV\simeq V \\
f^!{\mkern-1mu}{\mkern-1mu}{{f^{}_{{\mkern-1mu}{\mkern-1mu}*}}}@>>f^!\eta(g)> f^!g^!g_*{{f^{}_{{\mkern-1mu}{\mkern-1mu}*}}}\endCD$$
Consider therefore a diagram $${\begin{tikzpicture}}[xscale=1.5, yscale=1.36]
\node(31) at (1,-3){$\bullet$};
\node(21) at (1,-2){$\bullet$};
\node(22) at (2,-2){$\bullet$};
\node(11) at (1,-1){$\bullet$};
\node(12) at (2,-1){$\bullet$};
\node(13) at (3,-1){$\bullet$};
\draw[->] (11)--(12) node[above=1pt, midway, scale=.75]{$u$};
\draw[->] (12)--(13) node[above=1pt, midway, scale=.75]{$w$};
\draw[->] (21)--(22) node[above, midway, scale=.75]{$v$};
\draw[->] (11)--(21) node[left=1pt, midway, scale=.75]{$f$};
\draw[->] (21)--(31) node[left=1pt, midway, scale=.75]{$g$};
\draw[->] (12)--(21) node[right=5pt, midway, scale=.75]{$p$};
\draw[->] (13)--(22) node[right=5pt, midway, scale=.75]{$r$};
\draw[->] (22)--(31) node[right=5pt, midway, scale=.75]{$q$};
{\end{tikzpicture}}$$ where $pu$ is a compactification of $f$, $qv$ of $g$, and $rw$ of $vp$—so that $(qr)(wu)$ is a compactification of $gf$. The problem then is to show commutativity of (the border of) the following natural diagram. $$\mkern-1.5mu
\def{\mathbf{1}}{$\operatorname{id}$}
\def\2{$(wu)^*(wu)_*$}
\def\3{$(wu)^*(qr)^!(qr)_*(wu)_*$}
\def\4{$(gf)^!(gf)_*$}
\def\5{$u^*u_*$}
\def\6{$u^*{\mkern-1mu}p^!p_*u_*$}
\def\7{$f^!{\mkern-1mu}{\mkern-1mu}{{f^{}_{{\mkern-1mu}{\mkern-1mu}*}}}$}
\def\8{$f^!v^*v_*{{f^{}_{{\mkern-1mu}{\mkern-1mu}*}}}$}
\def\9{$f^!v^*q^!q_*v_*{{f^{}_{{\mkern-1mu}{\mkern-1mu}*}}}$}
\def\ten{$f^!g^!g_*{{f^{}_{{\mkern-1mu}{\mkern-1mu}*}}}$}
\def\lvn{$u^*w^*w_*u_*$}
\def\twv{$u^*w^*r^!r_*w_*u_*$}
\def\thn{$u^*(rw)^!(rw)_*u_*$}
\def\frn{$u^*p^!v^*v_*p_*u_*$}
\def\ffn{$u^*w^*r^!q^!q_*r_*w_*u_*$}
\def\sxn{$u^*p^!g^!g_*p_*u_*$}
\def\svn{$u^*(vp)^!(vp)_*u_*$}
\def\egn{$u^*p^!v^*{\mkern-1mu}q^!q_*v_*p_*u_*$}
{\begin{tikzpicture}}[xscale=3.75, yscale=1.5]
\node(11) at (1,-1){{\mathbf{1}}};
\node(12) at (2.29,-1){\2};
\node(13) at (3.23,-1){\3};
\node(21) at (1,-2){\5};
\node(22) at (2.29,-2){\lvn};
\node(24) at (3.9,-2){\4};
\node(31) at (1.4,-3){\thn};
\node(32) at (2.29,-3){\twv};
\node(33) at (3.23,-3){\ffn};
\node(41) at (1.4,-4){\svn};
\node(42) at (2.29,-4){\frn};
\node(43) at (3.23,-4){\egn};
\node(51) at (1,-5){\6};
\node(52) at (2.29,-5){\6};
\node(53) at (3.23,-5){\sxn};
\node(54) at (3.9,-5){\ten};
\node(62) at (2.29,-6){\7};
\node(63) at (3.17,-6){\8};
\node(64) at (3.85,-6){\9};
\draw[->] (11)--(12) ;
\draw[->] (12)--(13) ;
\draw[->] (21)--(22) ;
\draw[<-] (31)--(32) ;
\draw[->] (32)--(33) ;
\draw[->] (41)--(42) ;
\draw[->] (42)--(43) ;
\draw[double distance=2pt] (51)--(52) ;
\draw[->] (52)--(53) node[below=1pt, midway, scale=.75]{$\operatorname{{\textup{via}}}\eta(g)$};
\draw[->] (53)--(54) ;
\draw[->] (62)--(63) ;
\draw[->] (63)--(64) ;
\draw[->] (11)--(21) ;
\draw[->] (21)--(51) ;
\draw[double distance=2pt] (31)--(41) ;
\draw[->] (12)--(22) ;
\draw[->] (22)--(32) ;
\draw[->] (32)--(42) ;
\draw[->] (42)--(52) ;
\draw[->] (52)--(62) ;
\draw[->] (13)--(33) ;
\draw[->] (33)--(43) ;
\draw[->] (43)--(53) ;
\draw[->] (24)--(54) ;
\draw[->] (3.9,-5.78)--(3.9, -5.22) ;
\draw[->] (13)--(24) ;
\draw[->] (21)--(31) node[above=-3pt, midway, scale=.75]{$\mkern45mu\eta(vp)$};
\node at (1.65,-1.55)[scale=.9]{{\textcircled{\scriptsize{1}}}};
\node at (1.85,-2.55)[scale=.9]{{\textcircled{\scriptsize{2}}}};
\node at (1.85,-3.55)[scale=.9]{{\textcircled{\scriptsize{3}}}};
\node at (1.65,-4.55)[scale=.9]{{\textcircled{\scriptsize{4}}}};
\node at (2.76,-2)[scale=.9]{{\textcircled{\scriptsize{5}}}};
\node at (2.76,-3.55)[scale=.9]{{\textcircled{\scriptsize{6}}}};
\node at (3.565,-3.55)[scale=.9]{{\textcircled{\scriptsize{7}}}};
\node at (2.76,-4.55)[scale=.9]{{\textcircled{\scriptsize{8}}}};
\node at (3.23,-5.55)[scale=.9]{{\textcircled{\scriptsize{9}}}};
{\end{tikzpicture}}$$
That subdiagram 1 commutes is shown, e.g., in [@li §3.6, up to (3.6.5)]. (In other words, the adjunction between $(-)^*$ and $(-)_*$ is pseudofunctorial, see *ibid*., (3.6.7)(d).)
Commutativity of 2 is the definition of $\eta(vp)$ via the compactification $rw$.
Commutativity of 3 holds by definition of the vertical arrow on its right.
Commutativity of 4 (omitting $u^*$ and $u_*$) is the case $(f,g,u,v){\!:=}(r,p,v,w)$ of Lemma \[compat\].
Commutativity of 5 holds because of pseudofunctoriality of the adjunction between $(-)_*$ and $(-)^!$ over *proper* maps (see §\[\^times\]).
Commutativity of 6 is clear.
Commutativity of 7 results from pseudofunctoriality of $(-)^!$ and $(-)_*$.
Commutativity of 8 is the definition of $\eta(g)$ via the compactification $qv$.
Commutativity of 9 is simple to verify.
This completes the proof of Proposition \[fake unit\].
\[relation\] There is a unique pseudofunctorial map $\psi\colon(-)^\times\to(-)^!$ whose restriction over the subcategory of proper maps in ${\mathscr{E}}$ is the identity, and such that for every localizing immersion $u,$ $\psi(u)\colon u^\times\to u^!$ is the natural composition $$u^\times{{\mkern8mu\longrightarrow \mkern-25.5mu{}^\sim\mkern17mu}}u^*{\mathsf R}u_* u^\times{\longrightarrow}u^*=u^!.$$
Let $f$ be an ${\mathscr{E}}$-map, and $f=pu$ a compactification. If $\psi$ exists, then $\psi(f)\colon f^\times\to f^!$ must be given by the natural composition $$\label{psidef}
f^\times {{\mkern8mu\longrightarrow \mkern-25.5mu{}^\sim\mkern17mu}}u^\times p^\times = u^\times p^! \to u^!p^!{{\mkern8mu\longrightarrow \mkern-25.5mu{}^\sim\mkern17mu}}f^!,\\[-1pt]
\tag{\ref{relation}.1}$$ so that uniqueness holds.
As for existence, using \[fake unit\] we can take $\psi(f)$ to be the natural composition $$f^\times{\xrightarrow}{{\mkern-1mu}{\mkern-1mu}\operatorname{{\textup{via}}}{\mkern1mu}\eta\,} f^!{{\mathsf R}f^{}_{{\mkern-1mu}{\mkern-1mu}*}}f^\times{\longrightarrow}f^!.$$ This is as required when $f$ is proper or a localizing immersion, and it behaves pseudofunctorially, because both $\eta$ and the counit map ${{\mathsf R}f^{}_{{\mkern-1mu}{\mkern-1mu}*}}f^\times\to \operatorname{id}$ do.
\[eta from psi\] Conversely, one can recover $\eta$ from $\psi$: it is simple to show that for any ${\mathscr{E}}$-map $f\colon X\to Y$ and $E\in{{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}(X)$, and with $\eta^{}_{{\mkern1mu}2}\colon \operatorname{id}_X\to f^\times{\mathsf R}{{f^{}_{{\mkern-1mu}{\mkern-1mu}*}}}$ the unit map of the adjunction $f^\times\! \dashv {\mathsf R}{{f^{}_{{\mkern-1mu}{\mkern-1mu}*}}}$, one has $$\label{eta from psi.1}
\eta(E) = \psi(f)({\mathsf R}{{f^{}_{{\mkern-1mu}{\mkern-1mu}*}}}E){{{\mkern1mu}{\mkern1mu}\lift1,{\scriptstyle}{\circ},\,}}\eta^{}_{{\mkern1mu}2}(E).
\tag{\ref{eta from psi}.1}$$ (*Notation*: $\mathsf F\dashv\mathsf G$ signifies that the functor $\mathsf F$ is left-adjoint to the functor $\mathsf G$.)
If $u\colon X\to Y$ is a localizing immersion, then the map $$u^\times{\mathsf R}u_*{\xrightarrow}{\!\psi(u){\mkern1mu}} u^*{\mathsf R}u_*{{\mkern8mu\longrightarrow \mkern-25.5mu{}^\sim\mkern17mu}}\operatorname{id}.$$ is an *isomorphism,* inverse to the isomorphism $\eta^{}_{{\mkern1mu}2}$ in §\[cosa:localizing-immersion\]. (A proof is left to the reader.)
\[L1.1.25\] *The map ${\mathsf R}{\mkern1mu}u_*\psi(u)\colon{\mathsf R}{\mkern1mu}u_*u^\times\to{\mathsf R}{\mkern1mu}u_*u^*$ is equal to the composite* $${\mathsf R}{\mkern1mu}u_*u^\times{\xrightarrow}{\epsilon_2^{}{\mkern1mu}{\mkern1mu}}\operatorname{id}{\xrightarrow}{\eta_1^{}{\mkern1mu}}{\mathsf R}{\mkern1mu}u_*u^*,$$ where $\epsilon_2^{}$ is the counit of the adjunction ${\mathsf R}{\mkern1mu}u_*{\mkern-1mu}\dashv u^\times$ and $\eta_1^{}$ is the unit of the adjunction $u^*{\mkern-1mu}\dashv{\mathsf R}{\mkern1mu}u_*$.
Indeed, by §\[cosa:localizing-immersion\] the counit $\epsilon^{}_1$ of the adjunction $u^*{\mkern-1mu}\dashv{\mathsf R}{\mkern1mu}u_*$ is an isomorphism; and since the composite $${\mathsf R}{\mkern1mu}u_*{\xrightarrow}{{\mkern-1mu}\eta_1^{}{\mathsf R}{\mkern1mu}u_*{\mkern1mu}}{\mathsf R}{\mkern1mu}u_*u^*{\mathsf R}{\mkern1mu}u_*{\xrightarrow}{{\mkern-1mu}{\mathsf R}{\mkern1mu}u_*\epsilon_1^{}{\mkern1mu}{\mkern1mu}}{\mathsf R}{\mkern1mu}u_*$$ is the identity map, therefore ${\mathsf R}{\mkern1mu}u_*\epsilon_1^{-{\mkern-1mu}1}=\eta_1^{}{\mathsf R}{\mkern1mu}u_*$, as both are the (unique) inverse of ${\mathsf R}{\mkern1mu}u_*\epsilon_1^{}$; so the next diagram commutes, giving the assertion: $$\CD
{\mathsf R}{\mkern1mu}u_*u^\times @>{\mathsf R}{\mkern1mu}u_*\epsilon_1^{-{\mkern-1mu}1}u^\times=\:\eta_1^{}{\mathsf R}{\mkern1mu}u_*u^\times\,>> {\mathsf R}{\mkern1mu}u_*u^*{\mathsf R}{\mkern1mu}u_*u^\times \\
@V{\epsilon_2^{}}VV @VV{ {\mathsf R}{\mkern1mu}u_*u^*\epsilon_2^{}}V \\
\operatorname{id}@>>{\qquad\quad\eta_1^{}\qquad\qquad}> {\mathsf R}{\mkern1mu}u_*u^*
\endCD$$
Also, using the isomorphism $\epsilon^{}_1\colon u^*{\mathsf R}u_*{{\mkern8mu\longrightarrow \mkern-25.5mu{}^\sim\mkern17mu}}\operatorname{id}$ (resp., its right conjugate $\eta^{}_{{\mkern1mu}2}\colon \operatorname{id}{{\mkern8mu\longrightarrow \mkern-25.5mu{}^\sim\mkern17mu}}u^\times {\mathsf R}u_*$), one can recover $\psi(u)$ from ${\mathsf R}u_*\psi(u)$ by applying the functor $u^*$ (resp. $u^\times$), thereby obtaining definitions of $\psi(u)$.
The next Proposition asserts compatibility of $\psi$ with the flat base-change maps for $(-)^!$ (see ) and for $(-)^\times{\mkern-1mu}.$
\[base change\] Let $f\colon X\to Z$ and $g\colon Y\to Z$ be maps in ${\mathscr{E}},$ with $g$ flat. Let $p\colon X\times_Z Y\to X$ and $q\colon X\times_Z Y\to Y$ be the projections. Let $\beta\colon p^*{\mkern-1mu}{\mkern-1mu}f^\times\to q^\times{\mkern-1mu}g^*$ be the map adjoint to the natural composite map $${\mathsf R}q_*{\mkern1mu}p^*{\mkern-1mu}{\mkern-1mu}f^\times{{\mkern8mu\longrightarrow \mkern-25.5mu{}^\sim\mkern17mu}}g^*{\mathsf R}{{f^{}_{{\mkern-1mu}{\mkern-1mu}*}}}f^\times\to g^*{\mkern-1mu}.$$ Then the following diagram commutes. $$\CD
p^*{\mkern-1mu}{\mkern-1mu}f^\times @>p^*\psi(f)>> p^*{\mkern-1mu}{\mkern-1mu}f^!\\
@V\beta VV @VV\eqref{bch}V \\
q^\times{\mkern-1mu}g^* @>>\psi(q)> q^!g^*
\endCD$$
Let $f=\bar f u$ be a compactification, so that there is a composite cartesian diagram (with $h$ flat and with $\bar q v$ a compactification of $q$): $$\CD
X\times_ZY @>p>> X\\
@VvVV @VVuV\\
W\times_ZY @>h>> W\\
@V\bar q VV @VV\bar f V\\
Y @>>g> Z
\endCD$$ In view of the pseudofunctoriality of $\psi$, what needs to be shown is commutativity of the following natural diagram. $$\def{\mathbf{1}}{$p^*u^{{\mkern-1mu}\times}\!\bar f^\times$}
\def\2{$p^*u^*{\mkern-1mu}{\mkern-1mu}\bar f^{{\mkern1mu}{\mkern1mu}!}$}
\def\3{$p^*{\mkern-1mu}{\mkern-1mu}f^!$}
\def\4{$v^\times{\mkern-1mu}h^*{\mkern-1mu}{\mkern-1mu}\bar f^{\times}$}
\def\5{$v^* {\mkern-1mu}h^*{\mkern-1mu}{\mkern-1mu}\bar f^{{\mkern1mu}{\mkern1mu}!}$}
\def\6{$v^\times {\mkern-1mu}\bar q^{\times}{\mkern-1mu}{\mkern-1mu}g^*$}
\def\7{$v^* \bar q^{{\mkern1mu}{\mkern1mu}!}{\mkern-1mu}{\mkern-1mu}g^*$}
\def\8{$q^!{\mkern-1mu}g^*$}
\def\9{$p^*{\mkern-1mu}{\mkern-1mu}f^{{\mkern-1mu}\times}$}
\def\0{$q^\times{\mkern-1mu}g^*$}
{\begin{tikzpicture}}[xscale=3, yscale=1.5]
\node(10) at (0,-1){\9} ;
\node(11) at (1,-1){{\mathbf{1}}};
\node(12) at (2,-1){\2};
\node(13) at (3,-1){\3};
\node(21) at (1,-2){\4};
\node(22) at (2,-2){\5};
\node(30) at (0,-3){\0} ;
\node(31) at (1,-3){\6};
\node(32) at (2,-3){\7};
\node(33) at (3,-3){\8};
\draw[->] (10)--(11) node[above=1pt, midway, scale=.75]{$\Iso$};
\draw[->] (11)--(12) ;
\draw[->] (12)--(13) node[above=1pt, midway, scale=.75]{$\Iso$};
\draw[->] (21)--(22) ;
\draw[->] (30)--(31) node[above=1pt, midway, scale=.75]{$\Iso$};
\draw[->] (31)--(32) ;
\draw[->] (32)--(33) node[above=1pt, midway, scale=.75]{$\Iso$};
\draw[->] (10)--(30) ;
\draw[->] (11)--(21) ;
\draw[->] (21)--(31) ;
\draw[->] (12)--(22) ;
\draw[->] (22)--(32) ;
\draw[->] (13)--(33) ;
\node at (1.5,-1.51)[scale=.9]{{\textcircled{\scriptsize{1}}}};
\node at (1.5,-2.51)[scale=.9]{{\textcircled{\scriptsize{2}}}};
{\end{tikzpicture}}$$
Commutativity of each of the unlabeled subdiagrams is an instance of transitivity of the appropriate base[.5pt]{}-change map (see e.g., [@li Thm.(4.8.3)]).
Commutativity of 2 is straightforward to verify.
Subdiagram 1, without $\bar f^{{\mkern1mu}{\scriptscriptstyle}\bullet}$, expands naturally as follows (where we have written $u_*$ (resp. $v_*$) for ${\mathsf R}u_*$ (resp. ${\mathsf R}v_*$)): $$\def{\mathbf{1}}{$p^*{\mkern-1mu}u^{{\mkern-1mu}\times}$}
\def\2{$p^*{\mkern-1mu}u^*u_*u^{{\mkern-1mu}\times}$}
\def\3{$p^*u^*$}
\def\4{$v^*v_* {\mkern1mu}p^*{\mkern-1mu}u^\times$}
\def\5{$v^*{\mkern-1mu}h^*{\mkern-1mu}u_*u^\times$}
\def\6{$v^\times{\mkern-1mu}h^*$}
\def\7{$v^*v_*v^\times {\mkern-1mu}h^*$}
\def\8{$v^*{\mkern-1mu}h^*$}
{\begin{tikzpicture}}[xscale=3.3, yscale=1.5]
\node(10) at (0,-1){{\mathbf{1}}} ;
\node(12) at (2,-1){\2};
\node(13) at (3,-1){\3};
\node(21) at (1,-2){\4};
\node(22) at (2,-2){\5};
\node(30) at (0,-3){\6};
\node(32) at (2,-3){\7};
\node(33) at (3,-3){\8};
\draw[<-] (10)--(12) node[above=1pt, midway, scale=.75]{$$};
\draw[->] (12)--(13) node[above=1pt, midway, scale=.75]{$$};
\draw[<-] (21)--(22) ;
\draw[<-] (30)--(32) node[above=1pt, midway, scale=.75]{$$};
\draw[->] (32)--(33) node[above=1pt, midway, scale=.75]{$$};
\draw[->] (10)--(30) ;
\draw[->] (12)--(22) ;
\draw[->] (13)--(33) ;
\draw[<-] (10)--(21) ;
\draw[->] (21)--(32) ;
\draw[->] (22)--(33) ;
\node at (1.3,-1.51)[scale=.9]{{\textcircled{\scriptsize{3}}}};
\node at (2,-2.51)[scale=.9]{{\textcircled{\scriptsize{4}}}};
{\end{tikzpicture}}$$
Here the unlabeled diagrams clearly commute.
Commutativity of 3 results from the fact that the natural isomorphism $h^*u_*\to v_*p^*$ is adjoint to the natural composition (see [@li 3.7.2(c)]).
Commutativity of 4 results from the fact that the base[.5pt]{}-change map $p^*u^\times\to v^\times h^*$ is adjoint to $v_*p^*{{\mkern8mu\longrightarrow \mkern-25.5mu{}^\sim\mkern17mu}}h^*u_*u^\times\to h^*{\mkern-1mu}$.
Thus 1 commutes; and Proposition \[base change\] is proved.
\[proper-support\] Next we treat the interaction of the map $\psi$ with standard derived functors. Our approach involves the notion of *support,* reviewed in Appendix \[Support\].
\[L1.1\] Let $u\colon X\to Z$ be a localizing immersion, the counit of the adjunction ${\mathsf R}{\mkern1mu}u_*{\mkern-1mu}\dashv u^\times,$ and $\eta^{}_1\colon \operatorname{id}\to{\mathsf R}{\mkern1mu}u_*u^*$ the unit of the adjunction $u^*\dashv{\mathsf R}{\mkern1mu}u_*$. For all $E\in{{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}(X)$ and $F\in{{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}(Z),$ the maps ${\mathsf R}u_*E{\otimes^{\mathsf L}_{{\mkern-1mu}Z}}\eta^{}_1( F)$ and ${{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{Z}^{{\mkern-1mu}{\mathsf{qc}}}}\big({\mathsf R}u_* E,\epsilon^{}_2( F)\big)$ are isomorphisms.
Projection isomorphisms make the map ${\mathsf R}u_*E{\otimes^{\mathsf L}_{{\mkern-1mu}Z}}\eta^{}_1$ isomorphic to=-1 $${\mathsf R}u_*(E{\otimes^{\mathsf L}_{{{{\mkern-1mu}{\mkern-1mu}X}}}}u^*)
{\xrightarrow}{\!\operatorname{{\textup{via}}}{\mkern1mu}u^*\eta^{}_1}{\mathsf R}u_*(E{\otimes^{\mathsf L}_{{{{\mkern-1mu}{\mkern-1mu}X}}}}u^*{\mathsf R}u_*u^*).$$ Since $u^*\eta^{}_1$ is an isomorphism (with inverse the isomorphism $u^*{\mathsf R}u_*u^*{{\mkern8mu\longrightarrow \mkern-25.5mu{}^\sim\mkern17mu}}u^*$ from \[cosa:localizing-immersion\]), therefore so is ${\mathsf R}u_*E{\otimes^{\mathsf L}_{{\mkern-1mu}Z}}\eta^{}_1$.1
Similarly, to show that ${{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{Z}^{{\mkern-1mu}{\mathsf{qc}}}}\big({\mathsf R}u_* E,\epsilon^{}_2\big)$ is an isomorphism, one can use the duality isomorphism to reduce to noting that $u^\times\epsilon^{}_2$ is an isomorphism because it is right-conjugate to the inverse of the isomorphism ${\mathsf R}u_* \eta^{}_1:{\mathsf R}u_*u^*{\mathsf R}u_*{{\mkern8mu\longrightarrow \mkern-25.5mu{}^\sim\mkern17mu}}{\mathsf R}u_*$.
\[Gampsi\] Let $f\colon X\to Y$ be a map in ${\mathscr{E}},$ $W$ a union of closed subsets of $X{\mkern1mu}$ to each of [.5pt]{}which the restriction of $f{\mkern1mu}$ is proper, and a complex with support $\operatorname{supp}(E)$ contained in $W$. Then the functors ${\mathsf R}{\varGamma}^{}_{\!W}(-)$, $E{\otimes^{\mathsf L}_{{{{\mkern-1mu}{\mkern-1mu}X}}}}(-)$ and ${{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{{{{\mkern-1mu}{\mkern-1mu}X}}}^{{\mkern-1mu}{\mathsf{qc}}}}(E,-)$ take $\psi(f)\colon f^\times\to f^!$ to an isomorphism.=-1
By \[Hom and tensor 0\](ii), it is enough to prove that Proposition \[Gampsi\] holds for one $E$ with like1 $E={\mathsf R}{\varGamma}^{}_{\!W}{\mkern1mu}{\mathcal O_{{\mkern-1mu}{\mkern-1mu}X}}$ (see \[suppRg\]). For such an $E$, \[Hom and tensor 0\] shows it enough to prove that ${{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{{{{\mkern-1mu}{\mkern-1mu}X}}}^{{\mkern-1mu}{\mathsf{qc}}}}(E,\psi(f))$ is an isomorphism.1
Let $X{\xrightarrow}{u{\mkern1mu}{\mkern1mu}} Z{\xrightarrow}{p{\mkern1mu}{\mkern1mu}}Y$ be a compactification of $f$ (§\[locimm\]). In view of , we need only treat the case $f=u$. In this case it suffices to show, with $\epsilon^{}_2$ and $\eta^{}_1$ as in Remark \[L1.1.25\], that $$\begin{aligned}
{{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{Z}^{{\mkern-1mu}{\mathsf{qc}}}}\big({\mathsf R}{\mkern1mu}u_*E,\eta^{}_1\epsilon^{}_2\big) &\cong &
{{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{Z}^{{\mkern-1mu}{\mathsf{qc}}}}\big({\mathsf R}{\mkern1mu}u_*E,{\mathsf R}{\mkern1mu}u_*\psi(u)\big)\\
&\cong& {\mathsf R}u_*{{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{{{{\mkern-1mu}{\mkern-1mu}X}}}^{{\mkern-1mu}{\mathsf{qc}}}}\big(u^*{\mathsf R}{\mkern1mu}u_*E,\psi(u)\big)\\
&\cong& {\mathsf R}u_*{{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{{{{\mkern-1mu}{\mkern-1mu}X}}}^{{\mkern-1mu}{\mathsf{qc}}}}\big(E,\psi(u)\big)\end{aligned}$$ is an isomorphism.
Lemma \[L1.1\] gives that ${{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{Z}^{{\mkern-1mu}{\mathsf{qc}}}}({\mathsf R}{\mkern1mu}u_*E,\epsilon^{}_2)$ is an isomorphism. It remains to be shown that ${{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{Z}^{{\mkern-1mu}{\mathsf{qc}}}}({\mathsf R}{\mkern1mu}u_{*}E,\eta^{}_{1})$ is an isomorphism.
The localizing immersion $u$ maps $X$ homeomorphically onto $u(X)$ [@Nk 2.8.2], so we can regard $X$ as a topological subspace of $Z$. Let $i\colon V\hookrightarrow X$ be the inclusion into $X$ of a subscheme such that the restriction $fi=pui$ is proper. Then $ui$ is proper, and so $V$ is a closed subset of $Z{\mkern-1mu}$. Thus $W=\operatorname{supp}_{{{\mkern-1mu}{\mkern-1mu}X}}(E)=\operatorname{supp}_Z({\mathsf R}{\mkern1mu}u_*E)$ (see Remark \[supp u\_\*\]) is a union of subsets of $X$ that are closed in $Z$. So Proposition \[Hom and tensor 0\] can be applied to show that, since, by Lemma \[L1.1\], ${\mathsf R}u_*E{\otimes^{\mathsf L}_{{\mkern-1mu}Z}}\eta^{}_1$ is an isomorphism, therefore ${{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{Z}^{{\mkern-1mu}{\mathsf{qc}}}}({\mathsf R}{\mkern1mu}u_{*}E,\eta^{}_{1})$ is an isomorphism, as required.
Let $W\subseteq X$ be as in Proposition \[Gampsi\]. Let ${{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}{}(X)_{W}\subseteq{{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}(X)$ be the essential image of ${\mathsf R}{\varGamma}^{}_{\!W}(X)$—the full subcategory spanned by the complexes that are exact outside $W{\mkern-1mu}$. By Lemma \[Supp\], any $E\in{{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}(X)_W$ satisfies $\operatorname{supp}E\subseteq W$. Arguing as in [@AJS §2.3] one finds that the two natural maps from ${\mathsf R}{\varGamma}^{}_{\!W}{\mathsf R}{\varGamma}^{}_{\!W}$ to ${\mathsf R}{\varGamma}^{}_{\!W}$ are *equal isomorphisms;* and deduces that the natural map is an isomorphism $$\operatorname{Hom}_{{\boldsymbol{\mathsf{D}}}(X)}(E,{\mathsf R}{\varGamma}^{}_{\!W}F)
{{\mkern8mu\longrightarrow \mkern-25.5mu{}^\sim\mkern17mu}}\operatorname{Hom}_{{\boldsymbol{\mathsf{D}}}(X)}(E,F)
\quad \big(E\in{{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}(X)_{W},\,F\in{{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}(Y)\big),$$ with inverse the natural composition $$\operatorname{Hom}_{{\boldsymbol{\mathsf{D}}}(X)}(E,F)\to\operatorname{Hom}_{{\boldsymbol{\mathsf{D}}}(X)}({\mathsf R}{\varGamma}^{}_{\!W}E,{\mathsf R}{\varGamma}^{}_{\!W}F){{\mkern8mu\longrightarrow \mkern-25.5mu{}^\sim\mkern17mu}}\operatorname{Hom}_{{\boldsymbol{\mathsf{D}}}(X)}(E,{\mathsf R}{\varGamma}^{}_{\!W}F).$$
\[supports\] With the preceding notation, ${{\mathsf R}f^{}_{{\mkern-1mu}{\mkern-1mu}*}}\colon{{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}{}(X)_{W}\to{{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}(Y)$ has as right adjoint the functor ${\mathsf R}{\varGamma}^{}_{\!W}f^\times{\mkern-1mu}$. When restricted to ${{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}^{\lift.95,\text{\cmt\char'053},}}(Y),$ this right adjoint is isomorphic to ${\mathsf R}{\varGamma}^{}_{\!W}f^!{\mkern-1mu}$.
For $E\in{{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}(X)_{W}$ and $G\in{{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}(Y)$, there are natural isomorphisms $$\begin{aligned}
\operatorname{Hom}_{{\boldsymbol{\mathsf{D}}}(Y)}({{\mathsf R}f^{}_{{\mkern-1mu}{\mkern-1mu}*}}E,G)
&\cong
\operatorname{Hom}_{{\boldsymbol{\mathsf{D}}}(X)}(E,f^\times{\mkern-1mu}G)\\
&\cong
\operatorname{Hom}_{{\boldsymbol{\mathsf{D}}}(X)}(E,{\mathsf R}{\varGamma}^{}_{\!W}f^\times{\mkern-1mu}G)
\underset{\!\textup{\ref{Gampsi}}\,}{{\mkern8mu\longrightarrow \mkern-25.5mu{}^\sim\mkern17mu}}\operatorname{Hom}_{{\boldsymbol{\mathsf{D}}}(X)}(E,{\mathsf R}{\varGamma}^{}_{\!W}f^!G). \end{aligned}$$
\[barint\] The preceding Corollary entails the existence of a counit map $$\bar{{\mkern1mu}{\int}_{\!\!\!W}}\colon {{\mathsf R}f^{}_{{\mkern-1mu}{\mkern-1mu}*}}{\mathsf R}{\varGamma}^{}_{\!W} f^!{\mathcal O_Y}\to {\mathcal O_Y}.$$ Factoring $f$ over suitable affine open subsets $U$ as $U{\xrightarrow}{{\mkern-1mu}i_U{\mkern1mu}}Z{\xrightarrow}{{\mkern-1mu}h_U{\mkern1mu}}Y$ where $i_U$ is finite and $h_U$ is essentially smooth, one gets that $i_{U*}f^!{\mathcal O_Y}|_U$ is of the form ${\mathsf R}{\mkern1mu}{{\mathcal H}om}_Z({\mathsf R}{\mkern1mu}i_{U*}{\mathcal O_{{\mkern-1mu}{\mkern-1mu}X}},\Omega_{h_U}^n[n])$ for some $n=n_U$ such that the sheaf $\Omega_{h_U}^n$ of relative $n$-forms is free of rank 1; and hence local depth considerations imply that there is an integer $d$1 such that $H^{-e}f^!{\mathcal O_Y}=0$ for all $e> d$, while $\omega^{}_{\!f}{\!:=}H^{-d}f^!{\mathcal O_Y}\ne 0$. This $\omega_{f}$, determined up to isomorphism by $f{\mkern-1mu}$, is a *relative dualizing sheaf* (or *relative canonical sheaf*[.5pt]{}) of $f$.
There results a natural composite map of ${\mathcal O_Y}$-modules $$\begin{aligned}
\label{int}
\int_{{\mkern1mu}W}{\mkern-1mu}\colon H^d{{\mathsf R}f^{}_{{\mkern-1mu}{\mkern-1mu}*}}{\mathsf R}{\varGamma}^{}_{\!W}(\omega^{}_{\!f})
&=H^0{{\mathsf R}f^{}_{{\mkern-1mu}{\mkern-1mu}*}}{\mathsf R}{\varGamma}^{}_{\!W}(\omega^{}_{\!f}[d])\\[-5pt]
&{\longrightarrow}H^0({{\mathsf R}f^{}_{{\mkern-1mu}{\mkern-1mu}*}}{\mathsf R}{\varGamma}^{}_{\!W} f^!{\mathcal O_Y})
{\xrightarrow}{\operatorname{{\textup{via}}}\!\!\bar{\,\,{\mkern1mu}\int^{}_{{\mkern-1mu}W}}}
H^0{\mathcal O_Y}={\mathcal O_Y},\end{aligned}$$ that generalizes the map denoted “$\textup{res}_Z$" in [@Sa §3.1].
A deeper study of this map involves the realization of $\omega^{}_{\!f}{\mkern1mu}$, for certain $f{\mkern-1mu}$, in terms of regular differential forms, and the resulting relation of $\int^{}_{W}$ with residues of differential forms, cf. [@HK1] and [@HK2]. See also §\[intres\] below.
\[pullback1\] Let $W{\xrightarrow}{\,g\,} X{\xrightarrow}{\,f\,} Y$ be ${\mathscr{E}}$-maps such that $f{\mkern-1mu}g$ is proper.1 For any $F\in{{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}(X)$ and $G\in{{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}^{\lift.95,\text{\cmt\char'053},}}(Y),$ the maps $$\label{via psi}
{\mathsf L}g^*{{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{{{{\mkern-1mu}{\mkern-1mu}X}}}^{{\mkern-1mu}{\mathsf{qc}}}}(F,f^\times{\mkern-1mu}G) {\xrightarrow}{\operatorname{{\textup{via}}}\psi}{\mathsf L}g^*{{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{{{{\mkern-1mu}{\mkern-1mu}X}}}^{{\mkern-1mu}{\mathsf{qc}}}}(F,f^! G)
\tag{\ref{pullback1}.1}$$ $$\label{via psi'}
g^\times{{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{{{{\mkern-1mu}{\mkern-1mu}X}}}^{{\mkern-1mu}{\mathsf{qc}}}}(F,f^\times{\mkern-1mu}G) {\xrightarrow}{\operatorname{{\textup{via}}}\psi}g^\times{{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{{{{\mkern-1mu}{\mkern-1mu}X}}}^{{\mkern-1mu}{\mathsf{qc}}}}(F,f^! G)
\tag{\ref{pullback1}.2}$$ $$\label{via psi''}
g^\times{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{{{{\mkern-1mu}{\mkern-1mu}X}}}(F,f^\times{\mkern-1mu}G) {\xrightarrow}{\operatorname{{\textup{via}}}\psi}g^\times{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{{{{\mkern-1mu}{\mkern-1mu}X}}}(F,f^! G)
\tag*{(\ref{pullback1}.2)$'$}$$ are isomorphisms.
Since $g^\times{\mkern-1mu}{\mkern-1mu}\operatorname{id}_{{{\mkern-1mu}{\mkern-1mu}X}}^\times\cong (\operatorname{id}_{{{\mkern-1mu}{\mkern-1mu}X}}\!{{{\mkern1mu}{\mkern1mu}\lift1,{\scriptstyle}{\circ},\,}}g)^\times=g^\times{\mkern-1mu}$, therefore is an isomorphism if and only if so is \[via psi”\]. (Recall that ${{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{{{{\mkern-1mu}{\mkern-1mu}X}}}^{{\mkern-1mu}{\mathsf{qc}}}}=\operatorname{id}_{{{\mkern-1mu}{\mkern-1mu}X}}^\times{\mathsf R}{{\mathcal H}om}_{{{\mkern-1mu}{\mkern-1mu}X}}$.) 1
As for and , note first that the proper map $g$ induces a surjection $g^{}_2$ of ${\mkern1mu}W$ onto a closed subscheme $V$ of $X{\mkern-1mu}$; so $g=g^{}_1g^{}_2$ with $g^{}_1$ a closed immersion and $g^{}_2$ surjective. 1
Let $X{\xrightarrow}{u{\mkern1mu}{\mkern1mu}}Z{\xrightarrow}{p{\mkern1mu}{\mkern1mu}}Y$ be a compactification of $f$. Since $p{\mkern1mu}ug^{}_1g^{}_2$ is proper, so is $ug^{}_1g^{}_2$, whence $ug^{}_1$ maps $V=g^{}_2(g^{-{\mkern-1mu}1}_2V)$ homeomorphically onto a closed subset of $Z$, and for each $x\in V$ the natural map ${\mathscr{O}}_{{\mkern-1mu}Z{\mkern-1mu}, {\mkern1mu}ug^{}_1{\mkern-1mu}x}\to{\mathscr{O}}_{{\mkern-1mu}V{\mkern-1mu}{\mkern-1mu},{\mkern1mu}{\mkern1mu}x}$ is a surjection (see [@Nk 2.8.2]); thus $ug^{}_1$ is a closed immersion, and therefore $f{\mkern-1mu}g^{}_1=p{\mkern1mu}ug^{}_1$ is of finite type, hence, by [@EGA2 5.4.3], proper (since $fg^{}_1g^{}_2$ is).1
Since ${\mathsf L}g^*={\mathsf L}g_2^*{\mathsf L}g_1^*$ and $g^\times=g_2^\times g_1^\times$, it suffices that the Proposition hold when $g=g_1^{},$ i.e., we may assume that $g\colon W\to X$ *is a closed immersion.* It’s enough then to show that and become isomorphisms after application of the functor $g_*{\mkern1mu}$.
Via projection isomorphisms, the map $$g_*{\mathsf L}g^*{{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{{{{\mkern-1mu}{\mkern-1mu}X}}}^{{\mkern-1mu}{\mathsf{qc}}}}(F,f^\times{\mkern-1mu}G) {\xrightarrow}{{\mkern-1mu}g_{\mkern-.5mu*}{\mkern-1mu}\eqref{via psi}{\mkern1mu}}g_*{\mathsf L}g^*{{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{{{{\mkern-1mu}{\mkern-1mu}X}}}^{{\mkern-1mu}{\mathsf{qc}}}}(F,f^! G)$$ is isomorphic to the map $$\label{map3}
g_*{\mathcal O_W}{\otimes^{\mathsf L}_{{{{\mkern-1mu}{\mkern-1mu}X}}}}{{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{{{{\mkern-1mu}{\mkern-1mu}X}}}^{{\mkern-1mu}{\mathsf{qc}}}}(F,f^\times{\mkern-1mu}G)
{\xrightarrow}{{\mkern-1mu}{\mkern-1mu}\operatorname{{\textup{via}}}\psi{\mkern1mu}{\mkern1mu}} g_*{\mathcal O_W}{\otimes^{\mathsf L}_{{{{\mkern-1mu}{\mkern-1mu}X}}}}{{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{{{{\mkern-1mu}{\mkern-1mu}X}}}^{{\mkern-1mu}{\mathsf{qc}}}}(F,f^!G);
\tag{\ref{pullback1}.3}$$ and making the substitution $$\big(f\colon X\to Z, E, F\big)\mapsto \big(g\colon W\to X, {\mathcal O_W}, {{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{{{{\mkern-1mu}{\mkern-1mu}X}}}^{{\mkern-1mu}{\mathsf{qc}}}}(F,f^\times{\mkern-1mu}G)\big)$$ in the isomorphism leads to an isomorphism between the map $$g_*g^\times{{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{{{{\mkern-1mu}{\mkern-1mu}X}}}^{{\mkern-1mu}{\mathsf{qc}}}}(F,f^\times{\mkern-1mu}G) {\xrightarrow}{{\mkern-1mu}g_{\mkern-.5mu*}{\mkern-1mu}\eqref{via psi'}{\mkern1mu}}g_*g^\times{{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{{{{\mkern-1mu}{\mkern-1mu}X}}}^{{\mkern-1mu}{\mathsf{qc}}}}(F,f^!G)$$ and the map3 $$\label{map4}
{{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{{{{\mkern-1mu}{\mkern-1mu}X}}}^{{\mkern-1mu}{\mathsf{qc}}}}(g_*{\mathcal O_W}, {\mkern1mu}{{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{{{{\mkern-1mu}{\mkern-1mu}X}}}^{{\mkern-1mu}{\mathsf{qc}}}}(F,f^\times{\mkern-1mu}G)) {\xrightarrow}{\operatorname{{\textup{via}}}\psi}
{{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{{{{\mkern-1mu}{\mkern-1mu}X}}}^{{\mkern-1mu}{\mathsf{qc}}}}(g_*{\mathcal O_W}, {\mkern1mu}{{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{{{{\mkern-1mu}{\mkern-1mu}X}}}^{{\mkern-1mu}{\mathsf{qc}}}}(F,f^! G)).
\tag{\ref{pullback1}.4}$$ Via adjunction and projection isomorphisms, is isomorphic to $$\label{map5}
{{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{{{{\mkern-1mu}{\mkern-1mu}X}}}^{{\mkern-1mu}{\mathsf{qc}}}}(g_*{\mathsf L}g^*{\mkern-1mu}{\mkern-1mu}F, f^\times{\mkern-1mu}G) {\xrightarrow}{\operatorname{{\textup{via}}}\psi}
{{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{{{{\mkern-1mu}{\mkern-1mu}X}}}^{{\mkern-1mu}{\mathsf{qc}}}}(g_*{\mathsf L}g^*{\mkern-1mu}{\mkern-1mu}F, f^! G).
\tag{\ref{pullback1}.5}$$ By Lemma \[Supp\], $\operatorname{supp}(g_*{\mathsf L}g^*{\mkern-1mu}{\mkern-1mu}F)\subseteq\operatorname{Supp}(g_*{\mathsf L}g^*{\mkern-1mu}{\mkern-1mu}F)\subseteq W{\mkern-1mu}$, so \[Gampsi\] gives that is an isomorphism, whence so is .
Thus is an isomorphism. Also, so \[Hom and tensor 0\] shows that is an isomorphism, whence so is .
\[for fTd\] (Added in proof.) For an ${\mathscr{E}}$-map $f$, with compactification $f=pu$, set $$(p,u)^!{\mkern1mu}G{\!:=}u^!p^!{\mkern1mu}G\qquad\big(G\in{{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}(Y)\big).$$ It is shown in [@Nm4 Section 4] that $(p,u)^!$ depends only on $f{\mkern-1mu}$, in the sense that up to canonical isomorphism $(p,u)^!$ is independent of the factorization $f=pu$. (When this paper was written it was known only that $(p,u)^!G$ is canonically isomorphic to $f^!G$ when $G\in{{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}^{\lift.95,\text{\cmt\char'053},}}(Y)$.) Likewise, for all $G\in{{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}(Y)$ the functorial map $$\psi(p,u)(G)\colon f^\times G {{\mkern8mu\longrightarrow \mkern-25.5mu{}^\sim\mkern17mu}}u^\times p^\times G= u^\times p^! G{\xrightarrow}{\!\psi(u)p^!} u^!p^!G= (p,u)^!G$$ depends only on $f$ [@Nm4 Section 8]. So one may set $f^!{\!:=}(p,u)^!$ and $\psi(f){\!:=}\psi(p,u)$; and then the preceding proof of Proposition \[pullback1\] works for *all* $G\in{{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}(Y)$.
Examples {#section:examples}
========
Corollaries \[psi via RHom\]–\[varGM\] provide concrete interpretations of the map $\psi(u)$ for certain localizing immersions $u$.
Proposition \[R1.1.3.5\] gives a purely algebraic expression for $\psi(f)$ when $f$ is a *flat* ${\mathscr{E}}$-map between affine schemes. An elaboration for when the target of $f$ is the Spec of a field is given in Proposition \[affine/k\]. The scheme[.5pt]{}-theoretic results \[relation\], \[base change\] and \[pullback1\] tell us some facts about the pseudofunctorial behavior of $\psi(f)$; but how to prove these facts by purely algebraic arguments is left open.
\[RHom\] Let $f\colon X\to Z$ be an ${\mathscr{E}}$-map, and let $F\in{{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}(Z)$. The functorial isomorphism $\zeta(F{\mkern1mu})$ inverse to that gotten by setting $E={\mathcal O_{{\mkern-1mu}{\mkern-1mu}X}}$ in makes the following, otherwise natural, functorial diagram commute${\mkern1mu}:$ $$\def{\mathbf{1}}{${\mathsf R}{\mkern1mu}{{f^{}_{{\mkern-1mu}{\mkern-1mu}*}}}f^\times{\mkern-1mu}{\mkern-1mu}F$}
\def\3{${{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{Z}^{{\mkern-1mu}{\mathsf{qc}}}}({\mathsf R}{\mkern1mu}{{f^{}_{{\mkern-1mu}{\mkern-1mu}*}}}{\mathcal O_{{\mkern-1mu}{\mkern-1mu}X}},F{\mkern1mu})$}
\def\4{${{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{Z}^{{\mkern-1mu}{\mathsf{qc}}}}({\mathcal O_{{\mkern-1mu}Z}},F{\mkern1mu})$}
\def\5{$F$}
{\begin{tikzpicture}}[xscale=6, yscale=1.0]
\node(11) at (1,-1){\3};
\node(12) at (2,-1){{\mathbf{1}}};
\node(31) at (1,-3){\4};
\node(32) at (2,-3){\5};
\draw[->] (11)--(12) node[above=1pt, midway, scale=.75] {$\zeta(F{\mkern1mu})$};
\draw[->] (31)--(32) node[above=1pt, midway, scale=.75] {$\Iso$};
\draw[->] (11)--(31) ;
\draw[->] (12)--(32) ;
{\end{tikzpicture}}$$
Abbreviating ${\mathsf R}{\mkern1mu}{{f^{}_{{\mkern-1mu}{\mkern-1mu}*}}}$ to ${{f^{}_{{\mkern-1mu}{\mkern-1mu}*}}}$ and ${\mathsf L}f^*$ to $f^*{\mkern-1mu}{\mkern-1mu}$, one checks that the diagram in question is right-conjugate to the natural diagram, functorial in $G\in{{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}(Z)$, $$\def{\mathbf{1}}{$G{\otimes^{\mathsf L}_{Z}}{{f^{}_{{\mkern-1mu}{\mkern-1mu}*}}}{\mathcal O_{{\mkern-1mu}{\mkern-1mu}X}}$}
\def\2{${{f^{}_{{\mkern-1mu}{\mkern-1mu}*}}}(f^*G{\otimes^{\mathsf L}_{X}}{\mathcal O_{{\mkern-1mu}{\mkern-1mu}X}})$}
\def\3{${{f^{}_{{\mkern-1mu}{\mkern-1mu}*}}}f^*G$}
\def\4{$G,$}
\def\5{$G{\otimes^{\mathsf L}_{Z}}{\mathcal O_{{\mkern-1mu}Z}}$}
{\begin{tikzpicture}}[xscale=3.7, yscale=2]
\node(11) at (1,-1){{\mathbf{1}}};
\node(12) at (2,-1){\2};
\node(13) at (3,-1){\3};
\node(21) at (1,-2){\5};
\node(23) at (3,-2){\4};
\draw[<-] (11)--(12) node[above=1pt, midway, scale=.75] {$\textup{projection}$};
\draw[<-] (12)--(13) node[above=1pt, midway, scale=.75] {$\Iso$};
\draw[<-] (21)--(23) node[above=1pt, midway, scale=.75] {$\Iso$};
\draw[<-] (11)--(21) ;
\draw[<-] (13)--(23) ;
{\end{tikzpicture}}$$ whose commutativity is given by [@li 3.4.7(ii)].
\[psi via RHom\] For any localizing immersion $u\colon X\to Z$ and $F\in{{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}(Z),$ the map $\psi(u)\big(F\big)$ from is isomorphic to the natural composite map $$u^*{{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{Z}^{{\mkern-1mu}{\mathsf{qc}}}}({\mathsf R}u_*{\mathcal O_{{\mkern-1mu}{\mkern-1mu}X}},F{\mkern1mu}){\longrightarrow}u^*{{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{Z}^{{\mkern-1mu}{\mathsf{qc}}}}({\mathcal O_{{\mkern-1mu}Z}},F{\mkern1mu}){{\mkern8mu\longrightarrow \mkern-25.5mu{}^\sim\mkern17mu}}u^*{\mkern-1mu}{\mkern-1mu}F.$$
This is immediate from \[RHom\] (with $f=u$).
For the next Corollary recall that, when $Z=\operatorname{Spec}R$, the sheafification functor ${}^\sim={}^{\sim_R}$ is an isomorphism from ${\boldsymbol{\mathsf{D}}}(R)$ to the derived category of quasi-coherent ${\mathcal O_{{\mkern-1mu}Z}}$-modules, whose inclusion into ${{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}(Z)$ is an equivalence of categories [@BN p.230, 5.5].
\[affine locimm\] In if $X=\operatorname{Spec}S$ and $Z=\operatorname{Spec}R$ are affine—so that $u$ corresponds to a flat epimorphic ring homomorphism $R\to S$—and $M\in{\boldsymbol{\mathsf{D}}}(R),$ then $\psi(u)(M^\sim)$ is the sheafification of the natural ${\boldsymbol{\mathsf{D}}}(S)$-map $${\mathsf R}{\mkern-1mu}{\mkern-1mu}\operatorname{Hom}_R(S,M)\cong S\otimes_R {\mathsf R}{\mkern-1mu}{\mkern-1mu}\operatorname{Hom}_R(S,M)\to S\otimes_R ({\mathsf R}{\mkern-1mu}{\mkern-1mu}\operatorname{Hom}_R(R,M))=S\otimes_R M.$$
Use the following well-known facts:2
1\. ${{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{Z}^{{\mkern-1mu}{\mathsf{qc}}}}(A^\sim{\mkern-1mu},B^\sim)\cong{\mathsf R}{\mkern-1mu}{\mkern-1mu}\operatorname{Hom}^{}_R(A,B)^\sim\qquad\big(A,B\in{\boldsymbol{\mathsf{D}}}(R)\big)$.2
This results from the sequence of natural isomorphisms, for any $C\in{\boldsymbol{\mathsf{D}}}(R)$: $$\begin{aligned}
\operatorname{Hom}_{{\boldsymbol{\mathsf{D}}}(Z)}(C^\sim{\mkern-1mu},{\mathsf R}{\mkern-1mu}{\mkern-1mu}\operatorname{Hom}_R(A,B)^\sim)&\cong \operatorname{Hom}_{{\boldsymbol{\mathsf{D}}}(R)}(C,{\mathsf R}{\mkern-1mu}{\mkern-1mu}\operatorname{Hom}_R(A,B))\\
&\cong \operatorname{Hom}_{{\boldsymbol{\mathsf{D}}}(R)}(C{\otimes^{\mathsf L}_{R}}\!A,B)\\
&\cong \operatorname{Hom}_{{\boldsymbol{\mathsf{D}}}(Z)}((C{\otimes^{\mathsf L}_{R}}\!A)^\sim{\mkern-1mu},B^\sim)\\
&\cong \operatorname{Hom}_{{\boldsymbol{\mathsf{D}}}(Z)}(C^\sim\!{\otimes^{\mathsf L}_{Z}}\!{\mkern-1mu}A^\sim{\mkern-1mu},B^\sim)\\
&\cong \operatorname{Hom}_{{\boldsymbol{\mathsf{D}}}(Z)}(C^\sim{\mkern-1mu},{\mkern1mu}{{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{Z}^{{\mkern-1mu}{\mathsf{qc}}}}(A^\sim{\mkern-1mu},B^\sim)).\end{aligned}$$
-8pt 2. ${\mathsf R}{\mkern-1mu}{\mkern-1mu}\operatorname{Hom}_R(S,M)^{\sim_R}= u_*{\mathsf R}{\mkern-1mu}{\mkern-1mu}\operatorname{Hom}_R(S,M)^{\sim_S}$.1
3\. $u^*(A^{\sim_R})=(S\otimes_R A)^{\sim_S}\qquad\big(A\in{\boldsymbol{\mathsf{D}}}(R)\big).$1
4\. For any $N\in{\boldsymbol{\mathsf{D}}}(S)$, the natural ${\boldsymbol{\mathsf{D}}}(Z)$-map $u^*{\mathsf R}u_*N^{\sim_S}\to N^{\sim_S}$ is the sheafification of the natural ${\boldsymbol{\mathsf{D}}}(S)$-map $S\otimes_R N\to N$.
\[varGM\] Let $R$ be a noetherian ring that is complete with respect to the topology defined by an ideal $I{\mkern-1mu},$ let $p\colon Z\to \operatorname{Spec}R$ be a proper map, and let $X{\!:=}(Z\setminus p^{-1}\operatorname{Spec}R/I)\overset {\lift .5,u{\mkern1mu}{\mkern1mu}{\mkern1mu},}{\hookrightarrow} Z$ be the inclusion. For any $F\in{{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}(Z)$ whose cohomology modules are all coherent, $u^\times{\mkern-1mu}{\mkern-1mu}F=0$.
Since $u^*{\mathsf R}u_*u^\times F\cong u^\times F$ (§\[cosa:localizing-immersion\]), it suffices that ${\mathsf R}u_*u^\times F=0$, that is, by \[psi via RHom\], that ${{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{Z}^{{\mkern-1mu}{\mathsf{qc}}}}({\mathsf R}u_*{\mathcal O_{{\mkern-1mu}{\mkern-1mu}X}}, F{\mkern1mu})=0$.
Set $W{\!:=}p^{-1}\operatorname{Spec}(R/I)$. There is a natural triangle $${{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{Z}^{{\mkern-1mu}{\mathsf{qc}}}}({\mathsf R}u_*{\mathcal O_{{\mkern-1mu}{\mkern-1mu}X}}, F{\mkern1mu})
\to {{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{Z}^{{\mkern-1mu}{\mathsf{qc}}}}({\mathcal O_{{\mkern-1mu}Z}}, F{\mkern1mu}){\xrightarrow}{\alpha{\mkern1mu}} {{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{Z}^{{\mkern-1mu}{\mathsf{qc}}}}({\mathsf R}{\varGamma}^{}_{\!W}{\mathcal O_{{\mkern-1mu}Z}}, F{\mkern1mu})
{\xrightarrow}{+\,}$$ It is enough therefore to show that $\alpha$ is an isomorphism.
Let $\kappa\colon Z_{{\mkern-1mu}/W}\to Z$ be the formal completion of $Z$ along $W{\mkern-1mu}$. For any ${\mathcal O_{{\mkern-1mu}Z}}$-module $G$ let $G_{{\mkern-1mu}/W}$ be the completion of $G$—an $\mathcal O_{Z_{{\mkern-1mu}/W}}$-module; and let $\Lambda_W$ be the functor given objectwise by $\kappa_*G_{{\mkern-1mu}/W}$. The composition of $\alpha$ with the “Greenlees[.5pt]{}-May" isomorphism $${{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{Z}^{{\mkern-1mu}{\mathsf{qc}}}}({\mathsf R}{\varGamma}^{}_{\!W}{\mathcal O_{{\mkern-1mu}Z}}, F{\mkern1mu}){{\mkern8mu\longrightarrow \mkern-25.5mu{}^\sim\mkern17mu}}\operatorname{id}_Z^\times{\mathsf L}\Lambda_W F,$$ given by [@AJL1 0.3] is, by *loc.cit.*, the map $\operatorname{id}_Z^\times{\mkern-1mu}{\mkern-1mu}\lambda$, where $\lambda\colon F\to{\mathsf L}\Lambda_W F$ is the unique map whose composition with the canonical map ${\mathsf L}\Lambda_W F\to\Lambda_W F$ is the completion map $F\to\Lambda_W F$. So we need $\operatorname{id}_Z^\times{\mkern-1mu}{\mkern-1mu}\lambda$ to be an isomorphism. Hence, the isomorphisms $F=\operatorname{id}_Z^\times{\mkern-1mu}{\mkern-1mu}F{{\mkern8mu\longrightarrow \mkern-25.5mu{}^\sim\mkern17mu}}\operatorname{id}_Z^\times{\mkern-1mu}\kappa_*\kappa^*{\mkern-1mu}{\mkern-1mu}F$ in [@AJL2 3.3.1(2)] (where $\operatorname{id}_Z^\times$ is denoted ${\mathsf R}Q$) and $
\lambda^*_*\colon\kappa_*\kappa^*{\mkern-1mu}{\mkern-1mu}F{{\mkern8mu\longrightarrow \mkern-25.5mu{}^\sim\mkern17mu}}{\mathsf L}\Lambda_W F
$ in [@AJL1 0.4.1] (which requires coherence of the cohomology of $F$) reduce the problem to showing that *the natural composite map*=-1 $$F\to\kappa_*\kappa^*{\mkern-1mu}{\mkern-1mu}F{\xrightarrow}{{\mkern-1mu}\lambda^*_*{\mkern1mu}}{\mathsf L}\Lambda_W F\to\Lambda_W F$$ *is the completion map.*
By the description of $\lambda^*_*$ preceding [@AJL1 0.4.1], this amounts to commutativity of the border of the following natural diagram: $$\def{\mathbf{1}}{$F$}
\def\2{$\kappa_*\kappa^*{\mkern-1mu}{\mkern-1mu}F$}
\def\3{$\kappa_*\kappa^*\Lambda_W F$}
\def\4{$\kappa_*\kappa^*\kappa_*F_{{\mkern-1mu}/W}$}
\def\5{$\kappa_*F_{{\mkern-1mu}/W}$}
\def\6{$\Lambda_W F$}
{\begin{tikzpicture}}[xscale=4, yscale=1.4]
\node(11) at (1,-1){{\mathbf{1}}};
\node(12) at (2,-1){\6};
\node(13) at (3,-1){\5};
\node(21) at (1,-2){\2};
\node(22) at (2,-2){\3};
\node(23) at (3,-2){\4};
\draw[->] (11)--(12) node[above=1pt, midway, scale=.75] {$$};
\draw[double distance = 2pt] (12)--(13);
\draw[->] (21)--(22) node[above=1pt, midway, scale=.75] {$$};
\draw[double distance = 2pt] (22)--(23);
\draw[->] (11)--(21) ;
\draw[->] (12)--(22) ;
\draw[<-] (2.97,-1.8)--(2.97,-1.2) ;
\draw[->] (3.03,-1.8)--(3.03,-1.2) ;
{\end{tikzpicture}}$$ Verification of the commutativity is left to the reader.
\[sigma,sigma\] Next we generalize Corollary \[affine locimm\], replacing $u$ by an arbitrary flat map $f\colon X=\operatorname{Spec}(S)\to\operatorname{Spec}(R)=Z$ in ${\mathscr{E}}$, corresponding to a flat ring-homomorphism $\sigma\colon R\to S$. Lemma \[L1.1.1\] gives an expression for $\psi(f)$ for an *arbitrary* flat ${\mathscr{E}}$-map $f$, that in the foregoing affine case implies, as shown in Lemma \[L1.1.3\], that for $M\in{\boldsymbol{\mathsf{D}}}(R)$, and ${{S}^{\mathsf e}}{\!:=}S\otimes_R S$, $\psi(f)M$ is (naturally isomorphic to) the sheafification of the natural composite ${\boldsymbol{\mathsf{D}}}(S)$-map $$\begin{gathered}
{\mathsf R}{\mkern-1mu}\operatorname{Hom}_R(S,M){{\mkern8mu\longrightarrow \mkern-25.5mu{}^\sim\mkern17mu}}S{\otimes^{\mathsf L}_{{{S}^{\mathsf e}}}}{\mkern-1mu}({{S}^{\mathsf e}}\otimes_S{\mathsf R}{\mkern-1mu}\operatorname{Hom}_R(S,M))\\
{{\mkern8mu\longrightarrow \mkern-25.5mu{}^\sim\mkern17mu}}S{\otimes^{\mathsf L}_{{{S}^{\mathsf e}}}}{\mkern-1mu}(S\otimes_R {\mathsf R}{\mkern-1mu}\operatorname{Hom}_R(S,M))
{\longrightarrow}\,
S{\otimes^{\mathsf L}_{{{S}^{\mathsf e}}}}{\mathsf R}{\mkern-1mu}\operatorname{Hom}_R(S,{\mkern1mu}S\otimes_R M),\end{gathered}$$ or, more simply, (see Proposition \[R1.1.3.5\]), $${\mathsf R}{\mkern-1mu}{\mkern-1mu}\operatorname{Hom}_R(S,M) \to {\mathsf R}{\mkern-1mu}{\mkern-1mu}\operatorname{Hom}_R(S,S\otimes_R M)
\to
S{\otimes^{\mathsf L}_{{{S}^{\mathsf e}}}}{\mathsf R}{\mkern-1mu}{\mkern-1mu}\operatorname{Hom}_R(S,S\otimes_R M).$$ (The expanded notation in \[L1.1.3\] and \[R1.1.3.5\] indicates the $S$-actions involved.)
So let $f:X\to Z$ be a *flat* ${\mathscr{E}}$-map, let $\delta:X\to X\times_ZX$ be the diagonal, and let $\pi_1^{},\pi_2^{}$ be the projections from to $X$. There is a base-change isomorphism $\beta'=\pi_2^*f^!{{\mkern8mu\longrightarrow \mkern-25.5mu{}^\sim\mkern17mu}}\pi_1^! f^*{\mkern-1mu}$, as in \[bch\]. There is also a base-change map $\beta\colon \pi_2^*f^\times\to\pi_1^\times{\mkern-1mu}{\mkern-1mu}f^*{\mkern-1mu}$ as in Proposition \[base change\]1 (with $g=f,\; p=\pi_2,\; q=\pi_1$); this $\beta$ need not be an isomorphism.2
The next Lemma concerns functors from ${{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}^{\lift.95,\text{\cmt\char'053},}}(Z)$ to ${{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}^{\lift.95,\text{\cmt\char'053},}}(X)$.
\[L1.1.1\] With preceding notation, there is an isomorphism of functors $\nu\colon{\mathsf L}\delta^*\pi_1^\times{\mkern-1mu}{\mkern-1mu}f^*{{\mkern8mu\longrightarrow \mkern-25.5mu{}^\sim\mkern17mu}}f^!$ such that the map $\psi(f):f^\times\to f^!$ from is the composite $$f^\times=\operatorname{id}_{{{\mkern-1mu}{\mkern-1mu}X}}^*{\mkern-1mu}{\mkern-1mu}f^\times\cong{\mathsf L}\delta^*\pi_2^*f^\times{\xrightarrow}{\!{\mathsf L}\delta^*\beta{\mkern1mu}}
{\mathsf L}\delta^*\pi_1^\times{\mkern-1mu}{\mkern-1mu}f^*{\xrightarrow}{\,\nu\,} f^!{\mkern-1mu}.$$
Consider the diagram, where $\theta$ and $\theta'$ are the natural isomorphisms, $$\CD
f^\times @>\Iso>\theta> {\mathsf L}\delta^*\pi_2^*f^\times @>{\mathsf L}\delta^*\beta>>
{\mathsf L}\delta^*\pi_1^\times{\mkern-1mu}{\mkern-1mu}f^* \\
@V\psi(f)VV @V{\mathsf L}\delta^*\pi_2^*\psi(f)VV @VV{\mathsf L}\delta^*\psi(\pi_1^{})V \\
f^! @>\Iso>\theta'> {\mathsf L}\delta^*\pi_2^*f^! @>>{{\mathsf L}\delta^*\beta'}>
{\mathsf L}\delta^*\pi_1^!f^*
\endCD$$ The left square obviously commutes, and the right square commutes by Proposition \[base change\]. Since $\pi_1\delta=\operatorname{id}_X$ is proper, Proposition \[pullback1\] guarantees that ${\mathsf L}\delta^*\psi(\pi_1^{})$ is an isomorphism, while ${\mathsf L}\delta^*\beta'$ is an isomorphism since $\beta'$ is.
The Lemma results, with $\nu{\!:=}(\theta')^{-{\mkern-1mu}1}{{{\mkern1mu}{\mkern1mu}\lift1,{\scriptstyle}{\circ},\,}}({\mathsf L}\delta^*\beta')^{-{\mkern-1mu}1}
{{{\mkern1mu}{\mkern1mu}\lift1,{\scriptstyle}{\circ},\,}}{\mathsf L}\delta^*\psi(\pi_1^{}).$
\[R1.1.1.5\] The map $\psi(f)$ in Lemma factors as $$f^\times\! {\xrightarrow}{\,\eta\,} {\mathsf R}{\mkern1mu}\pi_{2{\mkern1mu}*}^{}\pi_2^*f^\times\! {\xrightarrow}{{\mathsf R}{\mkern1mu}\pi_{2{\mkern1mu}*}^{}\beta}
{\mathsf R}{\mkern1mu}\pi_{2{\mkern1mu}*}^{}\pi_1^\times{\mkern-1mu}{\mkern-1mu}f^*{\xrightarrow}{\,\eta\,}
{\mathsf R}{\mkern1mu}\pi_{2{\mkern1mu}*}^{}{\mathsf R}{\mkern1mu}\delta_*{\mathsf L}{\mkern1mu}\delta^*\pi_1^\times{\mkern-1mu}{\mkern-1mu}f^*\!{{\mkern8mu\longrightarrow \mkern-25.5mu{}^\sim\mkern17mu}}\! {\mathsf L}{\mkern1mu}\delta^*\pi_1^\times{\mkern-1mu}{\mkern-1mu}f^*{\xrightarrow}{\,\nu\,} f^!{\mkern-1mu},$$ where the maps labeled $\eta$ are induced by units of adjunction, and the isomorphism obtains because $\pi_2\delta=\operatorname{id}_{{{\mkern-1mu}{\mkern-1mu}X}}$.
By Lemma \[L1.1.1\] it suffices that the following diagram commute. $$\def{\mathbf{1}}{$f^\times$}
\def\2{${\mathsf R}{\mkern1mu}\pi_{2{\mkern1mu}*}^{}\pi_2^*f^\times$}
\def\3{${\mathsf R}{\mkern1mu}\pi_{2{\mkern1mu}*}^{}\pi_1^\times{\mkern-1mu}{\mkern-1mu}f^*$}
\def\4{${\mathsf L}\delta^*\pi_2^*f^\times$}
\def\5{${\mathsf R}{\mkern1mu}\pi_{2{\mkern1mu}*}^{}{\mathsf R}{\mkern1mu}\delta_*{\mathsf L}{\mkern1mu}\delta^*\pi_2^*f^\times$}
\def\6{${\mathsf R}{\mkern1mu}\pi_{2{\mkern1mu}*}^{}{\mathsf R}{\mkern1mu}\delta_*{\mathsf L}{\mkern1mu}\delta^*\pi_1^\times{\mkern-1mu}{\mkern-1mu}f^*$}
\def\7{${\mathsf L}\delta^*\pi_1^\times f^*$}
{\begin{tikzpicture}}[xscale=4.5, yscale=1.2]
\node(11) at (1,-1){{\mathbf{1}}};
\node(12) at (2,-1){\2};
\node(13) at (3,-1){\3};
\node(21) at (1,-2){\4};
\node(22) at (2,-2){\5};
\node(23) at (3,-2){\6};
\node(32) at (2,-3){\7};
\draw[->] (11)--(12) node[above, midway, scale=.75] {$\eta$};
\draw[->] (12)--(13) node[above, midway, scale=.75] {${\mathsf R}\pi_{2{\mkern1mu}*}\beta$};
\draw[->] (21)--(22) node[above=1pt, midway, scale=.75] {$$};
\draw[->] (22)--(23) node[above=1pt, midway, scale=.75] {$\operatorname{{\textup{via}}}\beta$};
\draw[->] (11)--(21) node[left=1pt, midway, scale=.75] {$\simeq$};
\draw[->] (12)--(22) node[left=1pt, midway, scale=.75] {$\eta$};
\draw[->] (13)--(23) node[right=1pt, midway, scale=.75] {$\eta$};
\draw[->] (21)--(32) node[below=-1pt, midway, scale=.75]
{${\mathsf L}\delta^*\beta\mkern40mu$};
\draw[->] (32)--(23) node[below=1pt, midway, scale=.75] {$$};
\node at (1.43,-1.51)[scale=.9]{{\textcircled{\scriptsize{1}}}};
{\end{tikzpicture}}$$ Commutativity of the unlabeled subdiagrams is clear.
Subdiagram 1 (without $f^\times$) expands as $$\def{\mathbf{1}}{$\operatorname{id}$}
\def\2{${\mathsf R}{\mkern1mu}\pi_{2{\mkern1mu}*}^{}\pi_2^*$}
\def\4{${\mathsf L}\delta^*\pi_2^*$}
\def\5{${\mathsf R}{\mkern1mu}\pi_{2{\mkern1mu}*}^{}{\mathsf R}{\mkern1mu}\delta_*{\mathsf L}{\mkern1mu}\delta^*\pi_2^*$}
\def\8{$(\pi^{}_2\delta)_*{\mathsf L}\delta^*\pi_2^*$}
\def\9{$(\pi^{}_2\delta)^*$}
\def\0{$(\pi^{}_2\delta)_*(\pi^{}_2\delta)^*$}
{\begin{tikzpicture}}[xscale=4, yscale=1.2]
\node(11) at (1,-1){{\mathbf{1}}};
\node(13) at (3,-1){\2};
\node(21) at (1,-2){\9};
\node(22) at (2,-2){\0};
\node(31) at (1,-3){\4};
\node(32) at (2,-3){\8};
\node(33) at (3,-3){\5};
\draw[->] (11)--(13) node[above, midway, scale=.75] {$\eta$};
\draw[double distance=2pt] (21)--(22) ;
\draw[double distance=2pt] (31)--(32) ;
\draw[->] (32)--(33) node[above, midway, scale=.75] {$\Iso$};
\draw[double distance=2pt] (11)--(21) ;
\draw[->] (21)--(31) node[left=1pt, midway, scale=.75] {$\simeq$};
\draw[->] (22)--(32) node[right=1pt, midway, scale=.75] {$\simeq$};
\draw[->] (13)--(33) node[right=1pt, midway, scale=.75] {$\eta$};
\draw[->] (11)--(22) node[above=-2pt, midway, scale=.75]
{$\mkern20mu\eta$};
\node at (2.5,-2)[scale=.9]{{\textcircled{\scriptsize{2}}}};
{\end{tikzpicture}}$$ Commutativity of subdiagram 2 is given by [@li (3.6.2)]. Verification of commutativity of the remaining two subdiagrams is left to the reader.
\[rem:affine-case\] We now concretize the preceding results in case $X=\operatorname{Spec}(S)$ and $Z=\operatorname{Spec}(R)$ are affine, so that the flat map $f\colon X\to Z$ corresponds to a flat homomorphism $\sigma\colon R\to S$ of noetherian rings.
First, some notation. For a ring $P$, ${\boldsymbol{\mathsf{M}}}(P)$ will denote the category of $P$-modules. Forgetting for the moment that $\sigma$ is flat, let $\tau\colon R\to T$ be a flat ring-homomorphism. If $$\operatorname{Hom}_{{\mkern1mu}\sigma{\mkern-1mu},\tau}\colon {\boldsymbol{\mathsf{M}}}(S)^{{\mathsf o\mathsf p}}\times {\boldsymbol{\mathsf{M}}}(T)\to {\boldsymbol{\mathsf{M}}}(T\otimes_R S)$$ is the obvious functor such that $$\operatorname{Hom}_{{\mkern1mu}\sigma{\mkern-1mu},\tau}(A,B){\!:=}\operatorname{Hom}_R(A,B),$$ then, since (by flatness of $\tau$) any K-injective $T$-complex is K-injective over $R$, there is a derived functor $${\mathsf R}{\mkern-1mu}{\mkern-1mu}\operatorname{Hom}_{{\mkern1mu}\sigma{\mkern-1mu},\tau}\colon {\boldsymbol{\mathsf{D}}}(S)^{{\mathsf o\mathsf p}}\times {\boldsymbol{\mathsf{D}}}(T)\to {\boldsymbol{\mathsf{D}}}(T\otimes_R S)$$ such that, with $(F\to J_F{\mkern1mu})_{F\in{\boldsymbol{\mathsf{D}}}(T)}$ a family of K-injective $T$-resolutions, and $E\in{\boldsymbol{\mathsf{D}}}(S)$, $${\mathsf R}{\mkern-1mu}{\mkern-1mu}\operatorname{Hom}_{{\mkern1mu}\sigma{\mkern-1mu},\tau}(E,F{\mkern1mu}){\!:=}\operatorname{Hom}_{{\mkern1mu}\sigma{\mkern-1mu},\tau}(E,J_F).$$
Set $\operatorname{Hom}_{\sigma}{\!:=}\operatorname{Hom}_{\sigma{\mkern-1mu},{\mkern1mu}{\mkern1mu}\operatorname{id}_R}$.1
Let $p^{}_1\colon T\to T\otimes_R S$ be the $R$-algebra homomorphism with $p^{}_1(t)=t\otimes 1$. There is a natural functorial isomorphism in ${\boldsymbol{\mathsf{D}}}(T\otimes_R S)$: $$\label{ss to p}
{\mathsf R}{\mkern-1mu}{\mkern-1mu}\operatorname{Hom}_{{\mkern1mu}\sigma{\mkern-1mu},{\mkern1mu}\tau}(E,{\mkern1mu}F{\mkern1mu}){{\mkern8mu\longrightarrow \mkern-25.5mu{}^\sim\mkern17mu}}{\mathsf R}{\mkern-1mu}{\mkern-1mu}\operatorname{Hom}_{{\mkern1mu}p^{}_1}{\mkern-1mu}{\mkern-1mu}(T\otimes_R E, F{\mkern1mu})
\quad(F\in{\boldsymbol{\mathsf{D}}}(T)).$$ (For this, just replace $F$ by a K-injective $T$-resolution.)
Let $p^{}_2\colon S\to T\otimes_R S$ be the $R$-algebra map with $p^{}_2(s)=1\otimes s$. Let $\rho_\tau\colon{\boldsymbol{\mathsf{D}}}(T)\to{\boldsymbol{\mathsf{D}}}(R)$ be the restriction-of-scalars functor induced by $\tau$; and define $\rho_{p^{}_2}$ analogously. Then, in ${\boldsymbol{\mathsf{D}}}(S)$, $${\mathsf R}{\mkern-1mu}{\mkern-1mu}\operatorname{Hom}_{{\mkern1mu}\sigma}(E, \rho_\tau F{\mkern1mu})=\rho_{p^{}_2}{\mathsf R}{\mkern-1mu}{\mkern-1mu}\operatorname{Hom}_{{\mkern1mu}\sigma{\mkern-1mu},{\mkern1mu}\tau}(E, F{\mkern1mu})
\qquad(E\in{\boldsymbol{\mathsf{D}}}(S),\,F\in{\boldsymbol{\mathsf{D}}}(T)).$$ There results a “multiplication" map in ${\boldsymbol{\mathsf{D}}}(T\otimes_RS)$: $$\mu\colon (T\otimes_R S)\otimes_S {\mathsf R}{\mkern-1mu}{\mkern-1mu}\operatorname{Hom}_{{\mkern1mu}\sigma}(E, \rho_\tau F{\mkern1mu})\to\
{\mathsf R}{\mkern-1mu}{\mkern-1mu}\operatorname{Hom}_{{\mkern1mu}\sigma{\mkern-1mu},{\mkern1mu}\tau}(E, F{\mkern1mu}),$$ and hence a natural composition in ${\boldsymbol{\mathsf{D}}}(S)$ $$\label{extend sigma}
\begin{aligned}
{\mathsf R}{\mkern-1mu}{\mkern-1mu}\operatorname{Hom}_{{\mkern1mu}\sigma}(E, \rho_\tau F{\mkern1mu})&{{\mkern8mu\longrightarrow \mkern-25.5mu{}^\sim\mkern17mu}}S{\otimes^{\mathsf L}_{T\otimes_{{\mkern-1mu}R}{\mkern1mu}S}}\!\big((T\otimes_R S)\otimes_S {\mathsf R}{\mkern-1mu}{\mkern-1mu}\operatorname{Hom}_{{\mkern1mu}\sigma}(E, \rho_\tau F{\mkern1mu})\big)\\
&{\xrightarrow}{S{\otimes^{\mathsf L}_{T\otimes_{{\mkern-1mu}R} S}}\:\mu}
S{\otimes^{\mathsf L}_{T\otimes_{{\mkern-1mu}R}{\mkern1mu}S}}{\mathsf R}{\mkern-1mu}{\mkern-1mu}\operatorname{Hom}_{{\mkern1mu}\sigma{\mkern-1mu},{\mkern1mu}\tau}(E,{\mkern1mu}F{\mkern1mu}).
\end{aligned}$$
Now, assuming $\sigma$ to be flat, we derive algebraic expressions for $f^\times$ and $f^!$.
Application of the functor ${\mathsf R}\Gamma(Z,-)={\mathsf R}{\mkern-1mu}{\mkern-1mu}\operatorname{Hom}({\mathcal O_{{\mkern-1mu}Z}},-)$ to item 1 in the proof of Corollary \[affine locimm\], gives ${\mathsf R}{\mkern-1mu}{\mkern-1mu}\operatorname{Hom}_Z(A^\sim{\mkern-1mu},B^\sim)={\mathsf R}{\mkern-1mu}{\mkern-1mu}\operatorname{Hom}_R(A,B)$. Since $(-)^{\sim_S}\colon{\boldsymbol{\mathsf{D}}}(S)\to{{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}(X)$ is an equivalence of categories [@BN p.230, 5.5], it results from the canonical isomorphism (with $E\in{\boldsymbol{\mathsf{D}}}(S)$, $M\in{\boldsymbol{\mathsf{D}}}(R)$ and $\sigma_{{\mkern-1mu}{\mkern-1mu}*}\colon{\boldsymbol{\mathsf{D}}}(S)\to{\boldsymbol{\mathsf{D}}}(R)$ the functor given by restricting scalars) $$\operatorname{Hom}_{{\boldsymbol{\mathsf{D}}}(S)}\!\big(E, {\mathsf R}{\mkern-1mu}{\mkern-1mu}\operatorname{Hom}_\sigma(S,M)\big){{\mkern8mu\longrightarrow \mkern-25.5mu{}^\sim\mkern17mu}}\operatorname{Hom}_{{\boldsymbol{\mathsf{D}}}(R)}(\sigma_*E,M)$$ that there is a functorial isomorphism $$\label{aff times}
\varrho(M)\colon \big({\mathsf R}{\mkern-1mu}{\mkern-1mu}\operatorname{Hom}_\sigma(S,M)\big)^{\sim_S} \cong f^\times{\mkern-1mu}{\mkern-1mu}\big(M^{\sim_R}\big)
\qquad(M\in{\boldsymbol{\mathsf{D}}}(R))$$ such that $f_*\varrho(M)$ is the isomorphism $\zeta(M^{\sim_R})$ in Lemma \[RHom\].1
Next, let $\pi_i\colon X\times_Z X\to X\ (i=1,2)$ be the projection maps, and let be the diagonal map. Set ${{S}^{\mathsf e}}{\!:=}S\otimes_R S$. Note that if $A\to B$ is a homomorphism of rings, corresponding to $g\colon\operatorname{Spec}B\to\operatorname{Spec}A$, and if $N\in{\boldsymbol{\mathsf{D}}}(A)$, then $$\label{aff^*}
{\mathsf L}g^*\big(N^{\sim_A}) =\big(B{\otimes^{\mathsf L}_{A}} N\big)^{\sim_B}.$$ This follows easily from the fact that the functor $(-)^{\sim_A}$ preserves both quasi-isomorphisms and K-flatness of complexes.
\[L1.1.3\] There is a natural functorial isomorphism of the map $$\psi(f)M^{\sim_R}:f^\times{\mkern-1mu}M^{\sim_R}\to f^!M^{\sim_R}
\qquad \big(M\in {\boldsymbol{\mathsf{D}}}(R)\big)$$ with the sheafification of the natural composite ${\boldsymbol{\mathsf{D}}}(S)$-map $$\begin{aligned}
\psi(\sigma)M\colon {\mathsf R}{\mkern-1mu}\operatorname{Hom}_\sigma(S,M)
&{{\mkern8mu\longrightarrow \mkern-25.5mu{}^\sim\mkern17mu}}S{\otimes^{\mathsf L}_{{{S}^{\mathsf e}}}}{\mkern-1mu}({{S}^{\mathsf e}}\otimes_S{\mathsf R}{\mkern-1mu}\operatorname{Hom}_\sigma(S,M))\\
&{{\mkern8mu\longrightarrow \mkern-25.5mu{}^\sim\mkern17mu}}S{\otimes^{\mathsf L}_{{{S}^{\mathsf e}}}}{\mkern-1mu}(S\otimes_R {\mathsf R}{\mkern-1mu}\operatorname{Hom}_R(S,M))\\
&\,{\longrightarrow}\,
S{\otimes^{\mathsf L}_{{{S}^{\mathsf e}}}}{\mathsf R}{\mkern-1mu}\operatorname{Hom}_{{\mkern1mu}\sigma{\mkern-1mu},{\mkern1mu}\sigma}(S,{\mkern1mu}S\otimes_R M).\end{aligned}$$
Using and , and the fact that sheafification is an equivalence of categories from ${\boldsymbol{\mathsf{D}}}(S)$ to ${\boldsymbol{\mathsf{D}}}(\operatorname{Spec}S)$ ([@BN p.230, 5.5]), one translates the definition of the base[.5pt]{}-change map $\beta$ in \[base change\] to the commutative[.5pt]{}-algebra context, and finds that $$\beta(M^{\sim_R}):\pi_2^*f^\times M^{\sim_R}\to\pi_1^\times{\mkern-1mu}{\mkern-1mu}f^*M^{\sim_R}$$ is naturally isomorphic to the sheafification of the natural composite ${\boldsymbol{\mathsf{D}}}({{S}^{\mathsf e}})$-map $$S\otimes_R {\mathsf R}{\mkern-1mu}{\mkern-1mu}\operatorname{Hom}_\sigma(S,M) \to {\mathsf R}{\mkern-1mu}{\mkern-1mu}\operatorname{Hom}_{{\mkern1mu}\sigma{\mkern-1mu},{\mkern1mu}\sigma}(S,S\otimes_R M){{\mkern8mu\longrightarrow \mkern-25.5mu{}^\sim\mkern17mu}}{\mathsf R}{\mkern-1mu}{\mkern-1mu}\operatorname{Hom}_{{\mkern1mu}p^{}_1}{\mkern-1mu}{\mkern-1mu}({{{S}^{\mathsf e}}},S\otimes_R M)$$ where the isomorphism comes from (with $T=S$).
Lemma \[L1.1.1\] gives that $\psi(f)$ is naturally isomorphic to the composite $$f^\times\cong{\mathsf L}\delta^*\pi_2^*f^\times {\xrightarrow}{{{\mathsf L}\delta^*\beta}}
{\mathsf L}\delta^*\pi_1^\times{\mkern-1mu}{\mkern-1mu}f^*{\mkern-1mu},$$ whence the conclusion.
Here is a neater description of $\psi(\sigma)M$—and hence of $\psi(f)M^{\sim_R}$.
\[R1.1.3.5\] The map $\psi(\sigma)M$ in factors as $${\mathsf R}{\mkern-1mu}{\mkern-1mu}\operatorname{Hom}_\sigma(S,M)
{\xrightarrow}{\,\vartheta\,}\varpi
{\mathsf R}{\mkern-1mu}{\mkern-1mu}\operatorname{Hom}_\sigma(S,S\otimes_R M)
{\xrightarrow}{\!\eqref{extend sigma}{\mkern1mu}}
S{\otimes^{\mathsf L}_{{{S}^{\mathsf e}}}}{\mathsf R}{\mkern-1mu}{\mkern-1mu}\operatorname{Hom}_{{\mkern1mu}\sigma{\mkern-1mu},{\mkern1mu}\sigma}(S,S\otimes_R M),$$ where $\vartheta$ is induced by the natural ${\boldsymbol{\mathsf{D}}}(R)$-map $M\to S\otimes_R M{\mkern-1mu}$.
Note that $\vartheta$ is the natural composite ${\boldsymbol{\mathsf{D}}}(S)$-map $${\mathsf R}{\mkern-1mu}{\mkern-1mu}\operatorname{Hom}_\sigma(S,M) \to S\otimes_R^{}{\mathsf R}{\mkern-1mu}{\mkern-1mu}\operatorname{Hom}_\sigma(S,M)
\to
{\mathsf R}{\mkern-1mu}{\mkern-1mu}\operatorname{Hom}_\sigma(S,S\otimes_R M),$$ recall the description in the proof of \[L1.1.3\] of the map $\beta$, refer to the factorization of $\psi(f)M^{\sim_R}$ coming from , and fill in the details.
From \[R1.1.3.5\] and it follows easily that:
For any $N\in {\boldsymbol{\mathsf{D}}}(S)$, the map $\eta(N^{\sim_S})$ from sheafifies the natural composite ${\boldsymbol{\mathsf{D}}}(S)$-map $$\begin{aligned}
N{\xrightarrow}{\vartheta'}\operatorname{Hom}_\sigma(S,S\otimes_R N)
&{\longrightarrow}{\mathsf R}{\mkern-1mu}{\mkern-1mu}\operatorname{Hom}_\sigma(S,S\otimes_R N)\\
&{\xrightarrow}{\!\eqref{extend sigma}{\mkern1mu}}
S{\otimes^{\mathsf L}_{{{S}^{\mathsf e}}}}{\mathsf R}{\mkern-1mu}{\mkern-1mu}\operatorname{Hom}_{{\mkern1mu}\sigma{\mkern-1mu},{\mkern1mu}\sigma}(S,S\otimes_R N), \end{aligned}$$ where $\vartheta'$ takes $n\in N$ to the map $s\mapsto s\otimes n$. $\square$
Using Proposition \[Gampsi\], we now develop more information about the above map $\psi(\sigma)M$ when $\sigma\colon k\to S$ is an essentially-finite[.5pt]{}-type algebra over a field $k$, and $M= k$.
For any ${{\mathfrak p}}\in\operatorname{Spec}S$, let $I({{\mathfrak p}})$ be the injective hull of the residue field $\kappa({{\mathfrak p}}){\!:=}S_{{\mathfrak p}}/{{\mathfrak p}}S_{{\mathfrak p}}{\mkern1mu}$. Let $D^\sigma\in{\boldsymbol{\mathsf{D}}}(S)$ be a *normalized residual complex,* thus a complex of the form $$D^\sigma:=\cdots0\to I^{-n}\to I^{-n+1}\to\cdots\to I{\mkern1mu}^0 \to 0\cdots$$ where for each integer $m$, $I^{-m}$ is the direct sum of the $I({{\mathfrak p}})$ as ${{\mathfrak p}}$ runs through the primes such that $S/{{\mathfrak p}}$ has dimension $m$. The sheafification of $D^\sigma$ is $f^!k,$ where $f{\!:=}\operatorname{Spec}\sigma$ and where we identify $k$ with the structure sheaf of $\operatorname{Spec}k$, see [@RD Chapter VI, §1].
\[affine/k\] Under the preceding circumstances, there exists a split exact sequence of $S$-modules $$0{\longrightarrow}\bigoplus_{\makebox[5pt]{${\scriptstyle}{{\mathfrak p}}\textup{ nonmaximal}$}}J({{\mathfrak p}}) {\longrightarrow}\operatorname{Hom}_\sigma(S,k) {\xrightarrow}{\,\psi^0\,} I{\mkern1mu}^0{\longrightarrow}0,$$ such that for each nonmaximal prime ${{\mathfrak p}},$ $J({{\mathfrak p}})$ is a direct sum of uncountably many copies of $I({{\mathfrak p}}),$ and in ${\boldsymbol{\mathsf{D}}}(S),$ $\psi(\sigma)k$ is the composition $${\mathsf R}{\mkern-1mu}\operatorname{Hom}_\sigma(S,k)=\operatorname{Hom}_\sigma(S,k){\xrightarrow}{\psi^0} I{\mkern1mu}^0
\hookrightarrow{\boldsymbol{\mathsf{D}}}^\sigma.$$
Since $\operatorname{Hom}_{\sigma}(S,k)$ is an injective $S$-module, there is a decomposition $$\operatorname{Hom}_{\sigma}(S,k) \cong \bigoplus_{{{\mathfrak p}}{\mkern1mu}\in{\mkern1mu}\operatorname{Spec}S} I({{\mathfrak p}})^{\mu({{\mathfrak p}})}$$ -2pt where, $\sigma_{{\mathfrak p}}$ being the natural composite map $k{\xrightarrow}{\sigma}S\twoheadrightarrow S/{{\mathfrak p}}$, $\mu({{\mathfrak p}})$ is the dimension of the $\kappa({{\mathfrak p}})$-vector space $$\begin{aligned}
\operatorname{Hom}_{S_{{\mathfrak p}}}\!{\mkern-1mu}\big(\kappa({{\mathfrak p}}),\operatorname{Hom}_{k}(S,k)_{{\mathfrak p}}\big)
&=\operatorname{Hom}_S\!\big(S/{{\mathfrak p}}, \operatorname{Hom}_{\sigma}(S,k)\big)\otimes_S S_{{\mathfrak p}}\\
&\cong \operatorname{Hom}_{\sigma_{{\mathfrak p}}}(S/{{\mathfrak p}},k)\otimes_{S/{{\mathfrak p}}}\kappa({{\mathfrak p}}).\end{aligned}$$
In particular, if ${{\mathfrak p}}$ is maximal (so that $S/{{\mathfrak p}}=\kappa({{\mathfrak p}})$) then $\mu({{\mathfrak p}})=1$. Thus $\operatorname{Hom}_{\sigma}(S,k)$ has a direct summand $J{\mkern1mu}^0$ isomorphic to $I{\mkern1mu}^0$. (This $J{\mkern1mu}^0$ does not depend on the foregoing decomposition: it consists of all $h\in\operatorname{Hom}_{\sigma}(S,k)$ such that the $S$-submodule $Sh$ has finite length.)
Now since $D^\sigma$ is a bounded injective complex, the ${\boldsymbol{\mathsf{D}}}(S)$-map $\psi(\sigma)$ is represented by an ordinary map of $S$-complexes $\operatorname{Hom}_{\sigma}(S,k)\to D^\sigma{\mkern-1mu}$, that is, by a map of $S$-modules $\psi^0\colon \operatorname{Hom}_{\sigma}(S,k)\to I{\mkern1mu}^0$. By \[L1.1.3\], the sheafification of $\psi(\sigma)$ is $\psi(f)k\colon f^\times k\to f^!k$, and hence Proposition \[Gampsi\] implies that $\psi^0$ maps $J{\mkern1mu}^0$ isomorphically onto $I{\mkern1mu}^0$. Thus $\psi^0$ has a right inverse, unique up to automorphisms of $I{\mkern1mu}^0$; and $\operatorname{Hom}_{\sigma}(S,k)$ is the direct sum of $J{\mkern1mu}^0$ and $\ker(\psi^0)$, whence $$\ker(\psi^0)\cong \bigoplus_{\makebox[7pt]{${\scriptstyle}{{\mathfrak p}}\textup{ nonmaximal}$}}I({{\mathfrak p}})^{\mu({{\mathfrak p}})}.$$ Last, in [@Nm3 Theorem 1.11] it is shown that for nonmaximal ${{\mathfrak p}}$, $$\mu_{{\mathfrak p}}=\dim_{\kappa({{\mathfrak p}})}\!\big({\mkern-1mu}\operatorname{Hom}_{\sigma_{{\mathfrak p}}}{\mkern-1mu}(S/{{\mathfrak p}},k)\otimes_{S/{{\mathfrak p}}}\kappa({{\mathfrak p}})\big)\ge|\#k|^{\aleph_0},$$ with equality if $S$ is finitely generated over $k$.
Applications
============
(Reduction Theorems.) At least for flat maps, $\psi\colon(-)^\times\to(-)^!$ can be used to prove one of the main results in [@AILN], namely Theorem 4.6 (for which only a hint of a proof is given there). With notation as in §\[sigma,sigma\], and again, ${{S}^{\mathsf e}}{\!:=}S\otimes_R S$, that Theorem 4.6 asserts the existence of a complex $D^\sigma\in{\boldsymbol{\mathsf{D}}}(S)$, depending only on $\sigma{\mkern-1mu}$, and for all $\sigma$-perfect $M\in{\boldsymbol{\mathsf{D}}}(S)$ (i.e., $M$ is isomorphic in ${\boldsymbol{\mathsf{D}}}(R)$ to a bounded complex of flat $R$-modules, the cohomology modules of $M$ are all finitely generated over $S$, and all but finitely many of them vanish), and all $N\in{\boldsymbol{\mathsf{D}}}(S)$, a functorial ${\boldsymbol{\mathsf{D}}}(S)$-isomorphism4 $$\label{AILN4.6}
\boxed{{\mathsf R}{\mkern-1mu}{\mkern-1mu}\operatorname{Hom}_S(M, D^\sigma){\otimes^{\mathsf L}_{S}} N \cong S{\otimes^{\mathsf L}_{S^e}}{\mathsf R}{\mkern-1mu}{\mkern-1mu}\operatorname{Hom}_{{\mkern1mu}\sigma{\mkern-1mu},{\mkern1mu}\sigma}(M,N).}$$ In particular, $$\label{affreldual}
D^\sigma\cong S{\otimes^{\mathsf L}_{S^e}}{\mathsf R}{\mkern-1mu}{\mkern-1mu}\operatorname{Hom}_{{\mkern1mu}\sigma{\mkern-1mu},{\mkern1mu}\sigma}(S,S).\tag*{(\ref{AILN4.6})$'$}$$ This explicit description is noteworthy in that the sheafification is a *relative dualizing complex* $f^!{\mathcal O_Y}$, where $f{\!:=}\operatorname{Spec}\sigma\colon \operatorname{Spec}S\to \operatorname{Spec}R$ (see [@AIL Example 2.3.2] or Lemma \[L1.1.3\] above); and otherwise-known definitions of $f^!$ involve choices, of which $f^!$ must be proved independent.2
The present proof will be based on the isomorphism in Lemma below,[^2] which is similar to (and more or less implied by) the isomorphism in [@AILN 6.6].2
Let $f\colon X\to Z$ be an arbitrary map in ${\mathscr{E}}$. Let $Y{\!:=}X\times_ZX$, and let $\pi_1$ and $\pi_2$ be the projections from $Y$ to $X$. For $M, N\in{{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}(X)$ there are natural maps2 $$\label{setup}
\begin{aligned}
\pi_1^*{{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{{{{\mkern-1mu}{\mkern-1mu}X}}}^{{\mkern-1mu}{\mathsf{qc}}}}(M,f^!{\mathcal O_{{\mkern-1mu}Z}}){\otimes^{\mathsf L}_{{\mkern-1mu}Y}}\pi_2^*N
&{\longrightarrow}{{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{Y}^{{\mkern-1mu}{\mathsf{qc}}}}(\pi_1^*M,{\mkern1mu}\pi_1^*f^!{\mathcal O_{{\mkern-1mu}Z}}){\otimes^{\mathsf L}_{{\mkern-1mu}Y}}\pi_2^*N\\
&{\longrightarrow}{{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{Y}^{{\mkern-1mu}{\mathsf{qc}}}}(\pi_1^*M,{\mkern1mu}\pi_1^*f^!{\mathcal O_{{\mkern-1mu}Z}}{\otimes^{\mathsf L}_{{\mkern-1mu}Y}}\pi_2^*N).
\end{aligned}$$
The first of these is the unique one making the following otherwise natural diagram (whose top left entry is in ${{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}(Y)$) commute:2 $$\label{setup'}
\CD
\pi_1^*{{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{{{{\mkern-1mu}{\mkern-1mu}X}}}^{{\mkern-1mu}{\mathsf{qc}}}}(M,f^!{\mathcal O_{{\mkern-1mu}Z}}){\otimes^{\mathsf L}_{{\mkern-1mu}Y}}\pi_2^*N
@>>> {{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{Y}^{{\mkern-1mu}{\mathsf{qc}}}}(\pi_1^*M,{\mkern1mu}\pi_1^*f^!{\mathcal O_{{\mkern-1mu}Z}}){\otimes^{\mathsf L}_{{\mkern-1mu}Y}}\pi_2^*N\\
@VVV @VVV\\
\pi_1^*{\mathsf R}{{\mathcal H}om}_{{{\mkern-1mu}{\mkern-1mu}X}}(M,f^!{\mathcal O_{{\mkern-1mu}Z}}){\otimes^{\mathsf L}_{{\mkern-1mu}Y}}\pi_2^*N
@>>> {\mathsf R}{{\mathcal H}om}_Y(\pi_1^*M,{\mkern1mu}\pi_1^*f^!{\mathcal O_{{\mkern-1mu}Z}}){\otimes^{\mathsf L}_{{\mkern-1mu}Y}}\pi_2^*N
\endCD$$
In [@AJL3 §5.7[.5pt]{}] it is shown that for *perfect* ${\mathscr{E}}$-maps $e\colon X\to Z$ (that is, $e$ has finite flat dimension), the functor $e^!\colon{{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}^{\lift.95,\text{\cmt\char'053},}}(Z)\to {{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}^{\lift.95,\text{\cmt\char'053},}}(X)$ extends pseudofunctorially to a functor—still denoted $e^!$—from ${{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}(Z)$ to ${{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}(X)$ such that 1 $$\label{extend!}
e^!F=e^!{\mathcal O_{{\mkern-1mu}Z}}{\otimes^{\mathsf L}_{{{{\mkern-1mu}{\mkern-1mu}X}}}}{\mathsf L}e^*{\mkern-1mu}{\mkern-1mu}F\qquad (F\in{{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}(X).$$
For proper $e$, the extended $e^!$ is still right-adjoint to ${\mathsf R}e_*$ (see [@AJL3 proof of Prop.5.9.3]).
The complex $M\in{\boldsymbol{\mathsf{D}}}(X)$ is *perfect relative to* $f$ (or simply $f$-*perfect*) if $M$ has coherent cohomology and has finite flat dimension over $Z$. In particular, the map $f$ is perfect if and only if ${\mathcal O_{{\mkern-1mu}{\mkern-1mu}X}}$ is $f$-perfect.
\[Gmap\] If the ${\mathscr{E}}$-map $f\colon X\to Z$ is flat and $M\in{\boldsymbol{\mathsf{D}}}(X)$ is $f{\mkern-1mu}$-perfect, then for all the composite map is an isomorphism.
It holds that ${\mathsf R}{{\mathcal H}om}_{{{\mkern-1mu}{\mkern-1mu}X}}(M,f^!{\mathcal O_{{\mkern-1mu}Z}})\in{{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}(X)$ and ${\mathsf R}{{\mathcal H}om}_Y(\pi_1^*M,{\mkern1mu}\pi_1^*f^!{\mathcal O_{{\mkern-1mu}Z}})\in{{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}(Y)$ (see proof of [@AILN 6.6]); and so the vertical arrows in are isomorphisms. So is the bottom arrow in (see e.g., [@li (4.6.6)]). Hence the first map in is an isomorphism.
As for the second, from the flatness of $f$ it follows that $\pi_1^*M$ is $\pi^{}_2$-perfect, and that there is a base[.5pt]{}-change isomorphism (cf. ) $$\label{bch2}
\pi_1^*f^!{\mathcal O_{{\mkern-1mu}Z}}{{\mkern8mu\longrightarrow \mkern-25.5mu{}^\sim\mkern17mu}}\pi_2^!{\mathcal O_{{\mkern-1mu}{\mkern-1mu}X}}.$$ The conclusion follows then from [@AILN 6.6] (with $g{\!:=}\pi^{}_2$, $E'={\mathcal O_{{\mkern-1mu}{\mkern-1mu}X}}$, $F'=N$, and with ${\mathsf R}{{\mathcal H}om}$ replaced throughout by ${{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{}^{{\mkern-1mu}{\mathsf{qc}}}}$), in whose proof we can replace the duality isomorphism (5.9.1) there by in this paper, and use the definition of $e^!$ for any finite[.5pt]{}-flat-dimensional map $e$ in ${\mathscr{E}}$ (for instance $g$, $h$ and $i$ in *loc.cit.*), thereby rendering unnecessary the boundedness condition in *loc.cit.* on the complex $F'$. (In this connection, note that if $e=hi$ with $h$ smooth and $i$ a closed immersion then $i$ is perfect .)
For $f\colon X\to Z$ a flat ${\mathscr{E}}$-map and $M\in{{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}(X)$ set $$M^\vee{\!:=}{{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{X}^{{\mkern-1mu}{\mathsf{qc}}}}(M, f^!{\mathcal O_{{\mkern-1mu}Z}}{\mkern-1mu}),$$ and consider the composite map, with $N\in{{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}^{\lift.95,\text{\cmt\char'053},}}(X),$ $$\label{setup2}
{{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{Y}^{{\mkern-1mu}{\mathsf{qc}}}}(\pi_1^*M,\pi_2^{\times}{\mkern-1mu}N)
{\longrightarrow}{{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{Y}^{{\mkern-1mu}{\mathsf{qc}}}}(\pi_1^*M,\pi_2^!N) {{\mkern8mu\longrightarrow \mkern-25.5mu{}^\sim\mkern17mu}}\pi_1^*M^{\vee} {\otimes^{\mathsf L}_{{\mkern-1mu}Y}}\pi_2^*N$$ where the first map is induced by $\psi({\mkern-1mu}\pi^{}_2{\mkern-1mu})$, and the isomorphism on the right is gotten by inverting the one given by \[Gmap\] and then replacing by the isomorphic object $\pi_2^!N$ (see and ). Remark 6.2 in [@AILN] authorizes replacement in (\[setup2\][.5pt]{}) of $M$ by $M^\vee{\mkern-1mu}$, and recalls that the natural map is an isomorphism $M{{\mkern8mu\longrightarrow \mkern-25.5mu{}^\sim\mkern17mu}}\smash{{M^\vee}}^\vee$; thus one gets the composite map $$\label{setup2v}
{{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{Y}^{{\mkern-1mu}{\mathsf{qc}}}}(\pi_1^*M^\vee{\mkern-1mu},\pi_2^{\times}{\mkern-1mu}N)
{\longrightarrow}{{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{Y}^{{\mkern-1mu}{\mathsf{qc}}}}(\pi_1^*M^\vee{\mkern-1mu},\pi_2^!N) \!{{\mkern8mu\longrightarrow \mkern-25.5mu{}^\sim\mkern17mu}}\! \pi_1^*M {\otimes^{\mathsf L}_{{\mkern-1mu}Y}}\pi_2^*N.
\tag*{(\ref{setup2}{\kern.5pt}){$^\vee$}}$$
\[global4.6\] If $M\in{{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}(X)$ is $f$-perfect, and $N\in{{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}^{\lift.95,\text{\cmt\char'053},}}(X),$ then application of ${\mkern1mu}{\mathsf L}\delta^*\!{\mkern-1mu}$ $($resp. $\delta^{!})$ to the composite $($resp. $^\vee)$1 produces an isomorphism $$\begin{aligned}
{\mathsf L}\delta^*{{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{Y}^{{\mkern-1mu}{\mathsf{qc}}}}(\pi_1^*M,{\mkern1mu}\pi_2^\times {\mkern-1mu}{\mkern-1mu}N) &{{\mkern8mu\longrightarrow \mkern-25.5mu{}^\sim\mkern17mu}}M^\vee{\otimes^{\mathsf L}_{{{{\mkern-1mu}{\mkern-1mu}X}}}}N \\
({\kern.5pt}resp.)\qquad\qquad\delta^!(\pi_1^*M{\otimes^{\mathsf L}_{Y}}\pi_2^*N)&{{\mkern8mu\longleftarrow \mkern-23.5mu{}^\sim\mkern17mu}}{{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{{{{\mkern-1mu}{\mkern-1mu}X}}}^{{\mkern-1mu}{\mathsf{qc}}}}(M^{\vee},N) .\end{aligned}$$
By \[pullback1\], application of ${\mathsf L}\delta^*$ to the first map in (\[setup2\][.5pt]{}) produces an isomorphism. Similarly, in view of , applying $\delta^\times$ ($=\delta^!{\mkern1mu}$) to the first map in (\[setup2\][.5pt]{})$^\vee$ produces an isomorphism.
Using Remark \[for fTd\] for the first map in (\[setup2\][.5pt]{}), *one can extend Theorem \[global4.6\] to all* $N\in{{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}(X)$. This results immediately from the fact, given by [@Nm4 Proposition 7.11], that *if $e=pu$ is a compactification of a perfect ${\mathscr{E}}$-map then the following natural map is an isomorphism*: $$e^!N\!:\underset{\eqref{extend!}}{=\!\!=} e^!{\mathcal O_{{\mkern-1mu}Z}}{\otimes^{\mathsf L}_{{{{\mkern-1mu}{\mkern-1mu}X}}}}{\mathsf L}e^*N
\cong u^*(p^\times{\mathcal O_{{\mkern-1mu}Z}}{\otimes^{\mathsf L}_{{{{\mkern-1mu}{\mkern-1mu}X}}}}{\mathsf L}p^*{\mkern-1mu}{\mkern-1mu}N)\to u^*p^\times{\mkern-1mu}{\mkern-1mu}N\qquad(N\in{{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}(Z)).$$
\[globalization\] The first isomorphism in Theorem \[global4.6\] is a globalization (for flat $f$ and cohomologically bounded-below $N$) of [@AILN Theorem4.6]. let $\sigma\colon R\to S$ be an essentially-finite-type flat homomorphism of noetherian rings, $f=\operatorname{Spec}(\sigma)$, ${{S}^{\mathsf e}}{\!:=}S\otimes_R S$ and $p_i\colon S\to{{S}^{\mathsf e}}\ (i=1,2)$ the canonical maps. Let $M,N,D^\sigma\in{\boldsymbol{\mathsf{D}}}(S)$, where $M$ is and $D^\sigma$ is a relative dualizing complex, sheafifying to [@AIL Example 2.3.2]. Set $X{\!:=}\operatorname{Spec}S$, $Z{\!:=}\operatorname{Spec}R$, $Y{\!:=}X\times_ZX$, and let $\delta\colon X\to Y$ be the diagonal. Then (as the cohomology of $M$ is bounded and finitely generated over $S$) $\delta_*{\mkern-1mu}\big({\widetilde M}^\vee{\otimes^{\mathsf L}_{{{{\mkern-1mu}{\mkern-1mu}X}}}}\widetilde N\big)$ sheafifies ${\mathsf R}{\mkern-1mu}{\mkern-1mu}\operatorname{Hom}_S(M,D^\sigma){\otimes^{\mathsf L}_{S}}N\in{\boldsymbol{\mathsf{D}}}({{S}^{\mathsf e}})$, and, with notation as in §\[sigma,sigma\], $\delta_*{\mathsf L}\delta^*{\mathsf R}{\mkern1mu}{{\mathcal H}om}_Y(\pi_1^*M,{\mkern1mu}\pi_2^\times {\mkern-1mu}{\mkern-1mu}N)$ sheafifies $$\begin{aligned}
S{\otimes^{\mathsf L}_{{{{S}^{\mathsf e}}}}}{\mathsf R}{\mkern-1mu}{\mkern-1mu}\operatorname{Hom}_{{{S}^{\mathsf e}}}(M\otimes_S {{S}^{\mathsf e}}{\mkern-1mu}{\mkern-1mu},\,{\mathsf R}{\mkern-1mu}{\mkern-1mu}\operatorname{Hom}_{{\mkern1mu}p^{}_2}{\mkern-1mu}({{{S}^{\mathsf e}}},N))
&\cong S{\otimes^{\mathsf L}_{{{{S}^{\mathsf e}}}}}{\mathsf R}{\mkern-1mu}{\mkern-1mu}\operatorname{Hom}_{{\mkern1mu}p^{}_2}{\mkern-1mu}(M\otimes_S {{S}^{\mathsf e}},N)\\
&\cong S{\otimes^{\mathsf L}_{{{{S}^{\mathsf e}}}}}{\mathsf R}{\mkern-1mu}{\mkern-1mu}\operatorname{Hom}_{{\mkern1mu}p^{}_2}{\mkern-1mu}(M\otimes_R S,N)\\
&\cong S{\otimes^{\mathsf L}_{{{{S}^{\mathsf e}}}}}{\mathsf R}{\mkern-1mu}{\mkern-1mu}\operatorname{Hom}_{{\mkern1mu}\sigma{\mkern-1mu},{\mkern1mu}\sigma}(M,N).\end{aligned}$$ Thus in this situation, application of $\delta_{*}$ to gives the existence of a functorial isomorphism (that should be closely related to—if not identical with—the one in [@AILN 4.6]).
\[redn-iso\] Let $f$ be as in \[Gmap\], and let $\delta\colon X\to Y{\!:=}X\times_Z X$ be the diagonal map. Keeping in mind the last paragraph of section \[RHom\^qc\] above, one checks that the reduction isomorphism [@AILN Corollary 6.5] $$\label{HomReduction}
\boxed{\delta^!(\pi_1^*M{\otimes^{\mathsf L}_{{{{\mkern-1mu}{\mkern-1mu}X}}}}\pi_2^*N){{\mkern8mu\longrightarrow \mkern-25.5mu{}^\sim\mkern17mu}}{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{{{\mkern-1mu}{\mkern-1mu}X}}(M^\vee{\mkern-1mu},{\mkern1mu}N\big)}\\[-1pt]
\tag{\ref{redn-iso}.1}$$ is inverse to the second isomorphism in \[global4.6\]. (In [@AILN], see the proof of Corollary 6.5, and the last four lines of the proof of Theorem 6.1 with $(X'{\mkern-1mu},Y'{\mkern-1mu},Y,Z)=(Y,X,Z,X)$, $E={\mathcal O_Y}$, and $(g,u,{\mkern-1mu}f,v)=(\pi^{}_2, f, f, \pi^{}_1)$, so that $\nu=\gamma=\operatorname{id}_{{{\mkern-1mu}{\mkern-1mu}X}}$.)
In the affine case, with assumptions on $\sigma$, $M$ and $N$ as above, “desheafification" of (i.e., applying derived global sections to) produces a functorial isomorphism $$\boxed{{\mathsf R}{\mkern-1mu}\operatorname{Hom}_{{{S}^{\mathsf e}}}(S, M{\otimes^{\mathsf L}_{R}} N){{\mkern8mu\longrightarrow \mkern-25.5mu{}^\sim\mkern17mu}}{\mathsf R}{\mkern-1mu}\operatorname{Hom}_S({\mathsf R}{\mkern-1mu}\operatorname{Hom}_S(M,D^\sigma),N)}$$ with the same source and target as the one in [@AILN p.736, Theorem 1]. (We suspect, but don’t know, that the two isomorphisms are the same—at least up to sign.)
\[intres\] In this section we review, from the perspective afforded by results in this paper, some known basic facts about *integrals, residues and fundamental classes.* The description is mostly in abstract terms. What will be new is a direct *concrete* description of the fundamental class of a flat essentially-finite[.5pt]{}-type homomorphism $\sigma\colon R\to S$ of noetherian rings (Theorem \[explicit fc\]).2
Let $I\subset S$ be an ideal such that $S/I$ is a finite $R$-module, and let $\Gamma^{}_{\!{\mkern-1mu}{\mkern-1mu}I}$ be the subfunctor of the identity functor on $S$-modules $M$ given objectwise by $$\Gamma^{}_{\!{\mkern-1mu}{\mkern-1mu}I}(M){\!:=}\{\,m\in M\mid I^nm=0\textup{ for some }n>0\,\}.$$ There is an obvious map from the derived functor ${\mathsf R}{\mkern1mu}\Gamma^{}_{\!{\mkern-1mu}{\mkern-1mu}I}$ to the identity functor on ${\boldsymbol{\mathsf{D}}}(S)$.
In view of the isomorphism , one can apply derived global sections in to get, in the present context, the following diagram, whose rectangle commutes. In this diagram, $\sigma_{{\mkern-1mu}{\mkern-1mu}*}\colon{\boldsymbol{\mathsf{D}}}(S)\to{\boldsymbol{\mathsf{D}}}(R)$ is the functor given by restricting scalars; and $\omega_\sigma$ is a canonical module of $\sigma$ (that is, an $S$-module whose sheafification is a relative dualizing sheaf of $f{\!:=}\operatorname{Spec}\sigma$, as in Remark \[barint\], where the integer $d$ is defined as well); and $D^\sigma$ is, as in \[globalization\], a relative dualizing complex.4 $$\def{\mathbf{1}}{$\sigma_{{\mkern-1mu}{\mkern-1mu}*}{\mathsf R}{\mkern1mu}\Gamma^{}_{\!{\mkern-1mu}{\mkern-1mu}I}{\mkern1mu}{\mathsf R}{\mkern-1mu}{\mkern-1mu}\operatorname{Hom}_\sigma(S,R)$}
\def\2{$\sigma_{{\mkern-1mu}{\mkern-1mu}*}{\mathsf R}{\mkern1mu}\Gamma^{}_{\!{\mkern-1mu}{\mkern-1mu}I}D^\sigma$}
\def\3{$\sigma_{{\mkern-1mu}{\mkern-1mu}*}{\mathsf R}{\mkern1mu}\Gamma^{}_{\!{\mkern-1mu}{\mkern-1mu}I}\,\omega_\sigma[d]$}
\def\4{${\mathsf R}{\mkern-1mu}{\mkern-1mu}\operatorname{Hom}_R(S,R)$}
\def\5{${\mathsf R}{\mkern-1mu}{\mkern-1mu}\operatorname{Hom}_R(R,R)$}
\def\6{$R$}
\def\7{$\sigma_{{\mkern-1mu}{\mkern-1mu}*}{\mathsf R}{\mkern-1mu}{\mkern-1mu}\operatorname{Hom}_\sigma(S,R)$}
{\begin{tikzpicture}}[xscale=4.7, yscale=2.5]
\node(11) at (1,-1) {{\mathbf{1}}};
\node(12) at (2.3,-1){\2};
\node(13) at (2.92,-1){\3};
\node(151) at (1,-1.5){\7};
\node(21) at (1,-2){\4};
\node(215) at (1.75,-2){\5};
\node(22) at (2.3,-2){\6};
\draw[->] (11)--(12) node[above, midway, scale=.7]{$\Iso$}
node[below=1pt, midway, scale=.7]{$\operatorname{{\textup{via}}}\psi$};
\draw[<-] (12)--(13) ;
\draw[->] (21)--(215) node[below=1pt, midway, scale=.7]{$\operatorname{{\textup{via}}}\sigma$};
\draw[double distance=2pt] (215)--(22);
\draw[->] (11)--(151) ;
\draw[double distance=2pt] (151)--(21);
\draw[->] (12)--(22) node[right=2pt, midway, scale=.7]{$\bar{\!\int^{}}_{\!\!I}$} ;
{\end{tikzpicture}}$$ If $\sigma$ is Cohen-Macaulay and equidimensional, the natural map is an isomorphism $\omega_\sigma[d{\mkern1mu}]{{\mkern8mu\longrightarrow \mkern-25.5mu{}^\sim\mkern17mu}}D^\sigma$; and application of $\textup{H}^0$ to the preceding diagram produces a commutative diagram of $R$-modules3 $$\def{\mathbf{1}}{$\Gamma^{}_{\!{\mkern-1mu}{\mkern-1mu}I}{\mkern-1mu}\operatorname{Hom}_\sigma(S,R)$}
\def\2{$\Gamma^{}_{\!{\mkern-1mu}{\mkern-1mu}I}{\mkern1mu}{\mkern1mu}\sigma^!{\mkern-1mu}R$}
\def\3{$\textup{H}^d_I\,\omega_\sigma$}
\def\4{$\operatorname{Hom}_R(S,R)$}
\def\6{$R$}
{\begin{tikzpicture}}[xscale=4.7, yscale=2.5]
\node(11) at (1,-1) {{\mathbf{1}}};
\node(13) at (2.3,-1){\3};
\node(21) at (1,-2){\4};
\node(22) at (2.3,-2){\6};
\draw[->] (11)--(13) node[above, midway, scale=.7]{$\Iso$}
node[below=1pt, midway, scale=.7]{$\operatorname{{\textup{via}}}\psi$};
\draw[->] (21)--(22) node[below=1pt, midway, scale=.7]{$\textup{evaluation at 1}$};
\draw[->] (11)--(21) ;
\draw[<-] (22)--(13) node[midway, right=1pt, scale=.7]{$\int_I$};
{\end{tikzpicture}}$$ This shows that *an explicit description of $(\operatorname{{\textup{via}}}\psi)^{-1}$ 1 is more or less the same as an explicit description of $\int_{{\mkern-1mu}I}$—and so, when $I$ is a maximal ideal, of residues.* “Explicit" includes the realization of the relative canonical module $\omega_\sigma$ in terms of regular differential forms (cf. Remark \[barint\]).
Such a realization comes out of the theory of the *fundamental class* ${{\boldsymbol{\mathsf{c}}}_{f}}$ of a flat ${\mathscr{E}}$-map $f$, as indicated below. This ${{\boldsymbol{\mathsf{c}}}_{f}}$ is a key link between the abstract duality theory of $f$ and its canonical reification via differential forms. It may be viewed as an orientation, compatible with essentially étale base change, in a suitable bivariant theory on the category of flat ${\mathscr{E}}$-maps [@AJL4].2
Given a flat ${\mathscr{E}}$-map $f\colon X\to Z$, with $\pi_1$ and $\pi_2$ the projections from $Y{\!:=}X\times_Z X$ to $X$, and $\delta\colon X\to Y$ the diagonal map, let ${{\boldsymbol{\mathsf{c}}}_{{\mkern-1mu}f}}$ be, as in [@AJL4 Example 2.3], the natural composite ${\boldsymbol{\mathsf{D}}}(X)$-map4 $$\label{fclass}
{\mathsf L}\delta^*\delta_*{\mathcal O_{{\mkern-1mu}{\mkern-1mu}X}}{{\mkern8mu\longrightarrow \mkern-25.5mu{}^\sim\mkern17mu}}{\mathsf L}\delta^*\delta_*\delta^!\pi_1^!{\mathcal O_{{\mkern-1mu}{\mkern-1mu}X}}{\longrightarrow}{\mathsf L}\delta^*\pi_1^!{\mathcal O_{{\mkern-1mu}{\mkern-1mu}X}}\;
\underset{\eqref{bch}}{{\mkern8mu\longrightarrow \mkern-25.5mu{}^\sim\mkern17mu}}{\mathsf L}\delta^*\pi_2^*f^!{\mathcal O_{{\mkern-1mu}Z}}{{\mkern8mu\longrightarrow \mkern-25.5mu{}^\sim\mkern17mu}}f^!{\mathcal O_{{\mkern-1mu}Z}}.$$
Let $\mathcal J$ be the kernel of the natural surjection ${\mathcal O_Y}\twoheadrightarrow \delta_*{\mathcal O_{{\mkern-1mu}{\mkern-1mu}X}}$. Using a flat ${\mathcal O_Y}$-resolution of $\delta_*{\mathcal O_{{\mkern-1mu}{\mkern-1mu}X}}$ one gets a natural isomorphism of ${\mathcal O_{{\mkern-1mu}{\mkern-1mu}X}}$-modules $$\Omega^1_f=
\mathcal J/{\mkern1mu}\mathcal J^{{\mkern1mu}2}\cong \mathcal T\!or_1^{{\mathcal O_Y}}\!(\delta_*{\mathcal O_{{\mkern-1mu}{\mkern-1mu}X}},\delta_*{\mathcal O_{{\mkern-1mu}{\mkern-1mu}X}})=H^{-1}{\mathsf L}\delta^*\delta_*{\mathcal O_{{\mkern-1mu}{\mkern-1mu}X}},$$ whence a map of graded-commutative ${\mathcal O_{{\mkern-1mu}{\mkern-1mu}X}}$-algebras, with $\Omega^i_f{\!:=}\wedge^{\!i}\mkern1.5mu \Omega^1_f{\mkern1mu}$, $$\label{Omega and H}
\oplus_{i\ge 0} \,\Omega^i_f \to\oplus_{i\ge 0}\,\mathcal T\!or_i^{{\mathcal O_Y}}\!(\delta_*{\mathcal O_{{\mkern-1mu}{\mkern-1mu}X}},\delta_*{\mathcal O_{{\mkern-1mu}{\mkern-1mu}X}})
=\oplus_{i\ge 0}\,H^{-i}{\mathsf L}\delta^*\delta_*{\mathcal O_{{\mkern-1mu}{\mkern-1mu}X}}{\mkern1mu}.$$
In particular one has, with $d$ as above, a natural composition $$\gamma^{}_{{\mkern-1mu}f}\colon\Omega^d_f\to H^{-d}{\mathsf L}\delta^*\delta_*{\mathcal O_{{\mkern-1mu}{\mkern-1mu}X}}{\xrightarrow}{{\mkern-1mu}{\mkern-1mu}\operatorname{{\textup{via}}}{\mkern1mu}{{\boldsymbol{\mathsf{c}}}_{{\mkern-1mu}f}}}H^{-d}f^!{\mathcal O_{{\mkern-1mu}Z}}=:\omega_{{\mkern-1mu}f}.$$ (In the literature, the term “fundamental class" often refers to this $\gamma^{}_{{\mkern-1mu}f}$ rather than to ${{\boldsymbol{\mathsf{c}}}_{f}}{\mkern1mu}$.) When $f$ is essentially smooth, this map is an isomorphism, as is $\omega_{{\mkern-1mu}f}\to f^!{\mathcal O_{{\mkern-1mu}Z}}{\mkern1mu}$. (The proof uses the known fact that there exists an isomorphism $\Omega^d_f{{\mkern8mu\longrightarrow \mkern-25.5mu{}^\sim\mkern17mu}}f^!{\mathcal O_{{\mkern-1mu}Z}}{\mkern1mu}$, but does not reveal the relation between that isomorphism and $\gamma^{}_{{\mkern-1mu}f}$, see [@AJL4 2.4.2, 2.4.4].) It follows that if $f$ is just *generically* smooth, then $\gamma^{}_{{\mkern-1mu}f}$ is a generic isomorphism. For example, if $X$ is a reduced algebraic variety over a field $k$, of pure dimension $d$, with structure map $f\colon X\to \operatorname{Spec}k$, then one deduces that $\omega_{{\mkern-1mu}f}$ is canonically represented by a coherent sheaf of meromorphic $d$-forms—the sheaf of regular $d$-forms—containing the sheaf ${\mkern1mu}\Omega^d_{{\mkern-1mu}{\mkern-1mu}f}$ of holomorphic $d$-forms, with equality over the smooth part of $X{\mkern-1mu}$.
From $\gamma^{}_{{\mkern-1mu}f}$ and the above $\int_{{\mkern-1mu}I}$ one deduces a map $$\textup{H}^d_I{\mkern1mu}\Omega^d_\sigma\to R$$ that generalizes the classical residue map.
Theorem \[explicit fc\] below provides a direct concrete definition of the fundamental class of a flat essentially-finite-type homomorphism $\sigma\colon R\to S$ of noetherian rings.
First, some preliminaries. As before, set ${{S}^{\mathsf e}}{\!:=}S\otimes_R S$, let $p^{}_1\colon S\to{{S}^{\mathsf e}}$ be the homomorphism such that for $s\in S$, $p^{}_1(s)=s\otimes 1$, and $p^{}_2\colon S\to{{S}^{\mathsf e}}$ such that $p^{}_2(s)=1\otimes s$.
Let $f\colon\operatorname{Spec}S=:{\mkern-1mu}X\to Z{\!:=}\operatorname{Spec}R$ be the scheme[.5pt]{}-map corresponding to $\sigma$. Let $\pi_1$ and $\pi_2$ be the projections (corresponding to $p^{}_1$ and $p^{}_2$) from $X\times_Z X$ to $X{\mkern-1mu}$.1
Let $\operatorname{Hom}_{{\mkern1mu}\sigma{\mkern-1mu},{\mkern1mu}\sigma}$ and $\operatorname{Hom}_{p^{}_1}$ be as in §\[rem:affine-case\].
For an $S$-complex $F$, considered as an ${{S}^{\mathsf e}}$-complex via the multiplication map ${{S}^{\mathsf e}}\to S$, let $\mu_F\colon F\to \operatorname{Hom}_{{\mkern1mu}\sigma{\mkern-1mu},{\mkern1mu}\sigma}(S,{\mkern1mu}F)$ be the ${{S}^{\mathsf e}}$-homomorphism taking $f\in F$ to the map $s\mapsto sf$.
For an $S$-complex $E$, there is an obvious ${{S}^{\mathsf e}}$-isomorphism $$\operatorname{Hom}_{{\mkern1mu}\sigma{\mkern-1mu},{\mkern1mu}\sigma}(S,{\mkern1mu}E){{\mkern8mu\longrightarrow \mkern-25.5mu{}^\sim\mkern17mu}}\operatorname{Hom}_{p^{}_1}{\mkern-1mu}{\mkern-1mu}({{S}^{\mathsf e}},{\mkern1mu}E).$$ Taking $E$ to be a K-injective resolution of $F$ (over $S$, and hence, since $\sigma$ is flat, also over $R$), one gets the isomorphism in the following statement.
\[L3.2.3\] Let $F\in{\boldsymbol{\mathsf{D}}}(S)$ have sheafification $F^{{\mkern1mu}\sim}\in{\boldsymbol{\mathsf{D}}}(X)$. The sheafification of the natural composite ${\boldsymbol{\mathsf{D}}}({{S}^{\mathsf e}})$-map $$\xi(F)\colon F {\xrightarrow}{\,\mu_{{\mkern-1mu}F}\,}\operatorname{Hom}_{{\mkern1mu}\sigma{\mkern-1mu},{\mkern1mu}\sigma}(S,{\mkern1mu}F)
{\longrightarrow}{\mathsf R}{\mkern-1mu}{\mkern-1mu}\operatorname{Hom}_{{\mkern1mu}\sigma{\mkern-1mu},{\mkern1mu}\sigma}(S,{\mkern1mu}F)
{{\mkern8mu\longrightarrow \mkern-25.5mu{}^\sim\mkern17mu}}{\mathsf R}{\mkern-1mu}{\mkern-1mu}\operatorname{Hom}_{p^{}_1}{\mkern-1mu}({{S}^{\mathsf e}}{\mkern-1mu},{\mkern1mu}F)$$ is the natural composite $({\mkern-1mu}$with $\epsilon^{}_2$ the counit map$)$ $$\label{iso323}
\delta_*F^{{\mkern1mu}\sim}{{\mkern8mu\longrightarrow \mkern-25.5mu{}^\sim\mkern17mu}}\delta_*\delta^\times{\mkern-1mu}\pi_1^\times F^{{\mkern1mu}\sim}
{\xrightarrow}{\ \epsilon^{}_2\ }
\pi_1^\times F^{{\mkern1mu}\sim}.
\tag{\ref{L3.2.3}.1}$$
The sheafification of ${\mathsf R}{\mkern-1mu}{\mkern-1mu}\operatorname{Hom}_{p^{}_1}{\mkern-1mu}{\mkern-1mu}({{S}^{\mathsf e}}{\mkern-1mu},{\mkern1mu}F)$ is $\pi_1^\times{\mkern-1mu}{\mkern-1mu}F^{{\mkern1mu}\sim}$, see . Likewise, with $m\colon {{S}^{\mathsf e}}\to S$ the multiplication map, and $G\in{\boldsymbol{\mathsf{D}}}({{S}^{\mathsf e}})$, one has that $\delta^\times {\mkern-1mu}G^{{\mkern1mu}\sim_{{{S}^{\mathsf e}}}}$ is the sheafification of ${\mathsf R}{\mkern-1mu}{\mkern-1mu}\operatorname{Hom}_m(S, G)$; and Lemma \[RHom\] implies that $\epsilon^{}_2$ is the sheafification of the “evaluation at 1" map$$\textup{ev}\colon {\mathsf R}{\mkern-1mu}{\mkern-1mu}\operatorname{Hom}_m(S,{\mathsf R}{\mkern-1mu}{\mkern-1mu}\operatorname{Hom}_{p^{}_1}{\mkern-1mu}{\mkern-1mu}({{S}^{\mathsf e}}{\mkern-1mu},{\mkern1mu}F{\mkern1mu}))\to
{\mathsf R}{\mkern-1mu}{\mkern-1mu}\operatorname{Hom}_{p^{}_1}{\mkern-1mu}{\mkern-1mu}({{S}^{\mathsf e}},{\mkern1mu}F{\mkern1mu}).$$
Moreover, one checks that the isomorphism $\delta_*F^{{\mkern1mu}\sim}\!{{\mkern8mu\longrightarrow \mkern-25.5mu{}^\sim\mkern17mu}}{\mkern-1mu}{\mkern-1mu}\delta_*\delta^\times{\mkern-1mu}\pi_1^\times F^{{\mkern1mu}\sim}\,$ is the sheafification of the natural isomorphism1 $F\!{{\mkern8mu\longrightarrow \mkern-25.5mu{}^\sim\mkern17mu}}{\mkern-1mu}{\mkern-1mu}{\mathsf R}{\mkern-1mu}{\mkern-1mu}\operatorname{Hom}_m(S,{\mathsf R}{\mkern-1mu}{\mkern-1mu}\operatorname{Hom}_{p^{}_1}\!({{S}^{\mathsf e}}{\mkern-1mu},{\mkern1mu}F{\mkern1mu}))$.
Under the allowable assumption that $F$ is K-injective, one finds then that is the sheafification of the map $\xi'(F)\colon F \to\operatorname{Hom}_{p^{}_1}{\mkern-1mu}({{S}^{\mathsf e}}{\mkern-1mu},{\mkern1mu}F)$ that takes $f\in F$ to the map $[s\otimes s'\mapsto ss'{\mkern-1mu}{\mkern-1mu}f{\mkern1mu}]$. It is simple to check that $\xi'(F)=\xi(F)$.
\[explicit fc\] Let $\sigma\colon R\to S$ be a flat essentially-finite-type map of noetherian rings, and $f\colon\operatorname{Spec}S\to\operatorname{Spec}R$ the corresponding scheme-map. Let $\mu\colon S\to \operatorname{Hom}_{{\mkern1mu}\sigma{\mkern-1mu},{\mkern1mu}\sigma}(S,S)$ be the ${{S}^{\mathsf e}}$-homomorphism taking $s\in S$ to multiplication by $s$. Then the fundamental class ${\boldsymbol{\mathsf{c}}}_{{\mkern-1mu}f}{\mkern-1mu}$ given by is naturally isomorphic to the sheafification of the natural composite map $$S{\otimes^{\mathsf L}_{{{S}^{\mathsf e}}}}S{\xrightarrow}{{\mkern-1mu}\operatorname{id}\otimes\mu{\mkern1mu}} S{\otimes^{\mathsf L}_{{{S}^{\mathsf e}}}}\operatorname{Hom}_{{\mkern1mu}\sigma{\mkern-1mu},{\mkern1mu}\sigma}(S,S){\longrightarrow}S{\otimes^{\mathsf L}_{{{S}^{\mathsf e}}}}{\mathsf R}{\mkern-1mu}{\mkern-1mu}\operatorname{Hom}_{{\mkern1mu}\sigma{\mkern-1mu},{\mkern1mu}\sigma}(S,S).$$
It suffices to show that the map in Theorem \[explicit fc\] sheafifies to a map isomorphic to the canonical composite map $$\label{fc eqn}
{\mathsf L}\delta^*\delta_*{\mathcal O_{{\mkern-1mu}{\mkern-1mu}X}}{{\mkern8mu\longrightarrow \mkern-25.5mu{}^\sim\mkern17mu}}{\mathsf L}\delta^*\delta_*\delta^!\pi_1^!{\mathcal O_{{\mkern-1mu}{\mkern-1mu}X}}{\longrightarrow}{\mathsf L}\delta^*\pi_1^!{\mathcal O_{{\mkern-1mu}{\mkern-1mu}X}}\tag{\ref{explicit fc}.1}$$ (see , in which the last two maps are isomorphisms). 1
Applying pseudofunctoriality of $\psi$ (Corollary \[relation\]) to $\operatorname{id}_X=\pi_1\delta$, one sees that the map in factors as $${\mathsf L}\delta^*\delta_*{\mathcal O_{{\mkern-1mu}{\mkern-1mu}X}}\to {\mathsf L}\delta^*\delta_*\delta^\times\pi_1^\times{\mathcal O_{{\mkern-1mu}{\mkern-1mu}X}}\to
{\mathsf L}\delta^*\pi_1^\times{\mathcal O_{{\mkern-1mu}{\mkern-1mu}X}}{{\mkern8mu\longrightarrow \mkern-25.5mu{}^\sim\mkern17mu}}{\mathsf L}\delta^*\pi_1^!{\mathcal O_{{\mkern-1mu}{\mkern-1mu}X}},$$ where the isomorphism is from Proposition \[pullback1\]. Thus the conclusion follows from Lemma \[L3.2.3\].
Let $T$ be a finite étale $R$-algebra. The (desheafified) fundamental class ${\boldsymbol{\mathsf{c}}}_{R\to T}$ is the ${\boldsymbol{\mathsf{D}}}(T)$-isomorphism from $T{\otimes^{\mathsf L}_{{{T}^{\mathsf e}}}} T =T$ to $T{\otimes^{\mathsf L}_{{{T}^{\mathsf e}}}}\operatorname{Hom}_R(T,T)\cong\operatorname{Hom}_R(T,R)$ taking 1 to the trace map. (Cf. [@AJL4 Example 2.6].)1
If $S$ is an essentially étale $T$-algebra (for instance, a localization of $T$), then there is a canonical identification of ${\boldsymbol{\mathsf{c}}}_{R\to S}$ with $({\boldsymbol{\mathsf{c}}}_{R\to T})\otimes_{{\mkern1mu}T} S$. (This fact results from [@AJL4 2.5 and 3.1], but can be proved more directly.) However, ${\boldsymbol{\mathsf{c}}}_{R\to S}$ depends only on $R\to S$, not on $T$.
Supports {#Support}
========
The goal of this appendix is to establish some basic facts—used repeatedly in §\[proper-support\]—about the relation between subsets of a noetherian scheme $X$ and “localizing tensor ideals" in ${{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}(X)$.2
*Notation*: Let $X$ be a noetherian scheme. For any $x\in X{\mkern-1mu}$,1 let ${\mathscr{O}}_x$ be the stalk ${\mathscr{O}}_{{\mkern-1mu}{\mkern-1mu}X{\mkern-1mu}{\mkern-1mu},{\mkern1mu}x{\mkern1mu}}$, let $\kappa(x)$ be the residue field of ${\mathscr{O}}_x$, let $\widetilde{\kappa(x)}$ be the corresponding sheaf on $X_x{\!:=}\operatorname{Spec}{\mathscr{O}}_x$—a quasi-coherent, *flasque* sheaf, let $\iota_x\colon X_x\to X$ be the canonical (flat) map—a localizing immersion,1 and let $$k(x){\!:=}\iota_{x*}\widetilde{\kappa(x)}={\mathsf R}\iota_{x*}\widetilde{\kappa(x)},$$ a quasi-coherent flasque ${\mathcal O_{{\mkern-1mu}{\mkern-1mu}X}}$-module whose stalk at a point $y$ is $\kappa(x)$ if $y$ is a specialization of $x$, and 0 otherwise.1
For $E\in{\boldsymbol{\mathsf{D}}}(X)$, we consider two notions of the *support of* $E{\mkern1mu}$: $$\begin{aligned}
\operatorname{supp}(E{\mkern1mu})&{\!:=}\{\,x\in X\mid E{\otimes^{\mathsf L}_{{{{\mkern-1mu}{\mkern-1mu}X}}}} k(x)\ne 0\in{\boldsymbol{\mathsf{D}}}(X)\,\},\\
\operatorname{Supp}(E{\mkern1mu})&{\!:=}\{\,x\in X\mid E_x\ne 0\in{\boldsymbol{\mathsf{D}}}({\mathscr{O}}_x)\,\}.\end{aligned}$$
Let ${\boldsymbol{\mathsf{D}}}_{{\mathsf{c}}}(X)$ (${{\boldsymbol{\mathsf{D}}}_{\mathsf{c}}^{\lift.95,\text{\cmt\char'053},}}(X)$) be the full subcategory of ${\boldsymbol{\mathsf{D}}}(X)$ spanned by the complexes with coherent cohomology modules (vanishing in all but finitely many negative degrees). For affine $X$ and $E\in{{\boldsymbol{\mathsf{D}}}_{\mathsf{c}}^{\lift.95,\text{\cmt\char'053},}}(X)$ the next Lemma appears in [@Fx top of page 158].
\[Supp\] For any $E\in{{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}(X),$ $$\operatorname{supp}(E{\mkern1mu})\subseteq\operatorname{Supp}(E{\mkern1mu});$$ and equality holds whenever $E\in{\boldsymbol{\mathsf{D}}}_{{\mathsf{c}}}(X)$.
For $E\in{{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}(X)$, there is a projection isomorphism $$E{\otimes^{\mathsf L}_{{{{\mkern-1mu}{\mkern-1mu}X}}}}k(x)\cong {\mathsf R}\iota_{x*}{\mkern-1mu}\big(\iota_x^*E{\otimes^{\mathsf L}_{X_x}}\widetilde{\kappa(x)}\big).$$ -1ptApplying $\iota_x^*$ to this isomorphism, and recalling from §\[cosa:localizing-immersion\] that $\iota_x^*{\mathsf R}\iota_{x*}$ is isomorphic to the identity, we get $$\iota_x^*\big(E{\otimes^{\mathsf L}_{{{{\mkern-1mu}{\mkern-1mu}X}}}}k(x)\big)\cong \iota_x^*E{\otimes^{\mathsf L}_{{{{\mkern-1mu}{\mkern-1mu}X}}_{{\mkern-1mu}{\mkern-1mu}x}}}\widetilde{\kappa(x)}.$$ These two isomorphisms tell us that $E{\otimes^{\mathsf L}_{{{{\mkern-1mu}{\mkern-1mu}X}}}}k(x)$ vanishes in ${{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}(X)$ if and only if $\iota_x^*E{\otimes^{\mathsf L}_{{{{\mkern-1mu}{\mkern-1mu}X}}_{{\mkern-1mu}{\mkern-1mu}x}}}\widetilde{\kappa(x)}$ vanishes in ${{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}(X_x)$.
Moreover, $\iota_x^*E{\otimes^{\mathsf L}_{{{{\mkern-1mu}{\mkern-1mu}X}}_{{\mkern-1mu}{\mkern-1mu}x}}}\widetilde{\kappa(x)}$ is the sheafification of $E_x{\otimes^{\mathsf L}_{{\mathscr{O}}_x}}\kappa(x)\in{\boldsymbol{\mathsf{D}}}({\mathscr{O}}_x)$, and so its vanishing in ${\boldsymbol{\mathsf{D}}}(X_x)$ (i.e., its being exact) is equivalent to that of $E_x{\otimes^{\mathsf L}_{{\mathscr{O}}_x}}\kappa(x)$ in ${\boldsymbol{\mathsf{D}}}({\mathscr{O}}_x)$. Thus $$x\in\operatorname{supp}(E) \iff E_x{\otimes^{\mathsf L}_{{\mathscr{O}}_x}}\kappa(x)\neq0.$$
It follows that if $x\in\operatorname{supp}(E)$, then $E_x\neq0$, that is to say, $x\in\operatorname{Supp}(E)$. So for all $E\in{{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}(X)$ we have $\operatorname{supp}(E)\subseteq\operatorname{Supp}(E)$. 2
Now suppose $E\in{\boldsymbol{\mathsf{D}}}_{{\mathsf{c}}}(X)$ and $x\not\in\operatorname{supp}(E)$, i.e., $E_{x} {\otimes^{\mathsf L}_{{\mathscr{O}}_{x}}}\kappa(x)=0$.1 Let $K$ be the Koszul complex on a finite set of generators for the maximal ideal of the local ring ${\mathscr{O}}_{x}$. It is easy to check that the full subcategory of ${\boldsymbol{\mathsf{D}}}({\mathscr{O}}_{x})$ consisting of complexes $C$ such that $E_{x}{\otimes^{\mathsf L}_{{\mathscr{O}}_{x}}}C=0$ is a thick subcategory. It contains $\kappa(x)$, and hence also $K$, since the ${\mathscr{O}}_{x}$-module $\oplus_{i\in{\mathbb Z}}{\mkern1mu}{\mkern1mu}\textup{H}^i(K)$ has finite length, see [@DGI 3.5]. Thus1 $E_{x}{\otimes^{\mathsf L}_{{\mathscr{O}}_{x}}} K = 0$ in ${\boldsymbol{\mathsf{D}}}({\mathscr{O}}_{x})$; and since the cohomology of $E_{x}$ is finitely generated in all degrees, [@FI 1.3(2)] gives Thus, $x\not\in\operatorname{Supp}(E)$; and so $\operatorname{supp}(E)\supseteq\operatorname{Supp}(E)$.
A *localizing tensor ideal* ${\mathscr{L}}\subseteq{{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}(X)$ is a full triangulated subcategory of $\,{{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}(X)$, closed under arbitrary direct sums, and such that for all $G\in{\mathscr{L}}$ and $E\in{{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}(X)$, it holds that1 $G{\otimes^{\mathsf L}_{{{{\mkern-1mu}{\mkern-1mu}X}}}}E\in{\mathscr{L}}$.
The next Proposition is proved in [@Nm1 §2] in the affine case; and in [@AJS] (where localizing tensor ideals are called *rigid localizing subcategories*) the proof is extended to noetherian schemes. (Use e.g., *ibid.,* Corollary 4.11 and the bijection in Theorem 4.12, as described at the beginning of its proof.)1
\[ideals\] Let ${\mathscr{L}}\subseteq{{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}(X)$ be a localizing tensor ideal. A complex $E\in{{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}(X)$ is in ${\mathscr{L}}$ if and only if so is $k(x)$ for all $x$ in $\operatorname{supp}(E{\mkern1mu}).$
For closed subsets of affine schemes the next result is part of [@DG Proposition 6.5].
\[Hom and tensor 0\] Let $E\in{{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}(X)$ be such that $W:=\operatorname{supp}(E)$ is a union of closed subsets of $X{\mkern-1mu}$.
- For any $F\in{{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}(X),$ $$\begin{aligned}
E{\otimes^{\mathsf L}_{{{{\mkern-1mu}{\mkern-1mu}X}}}}F=0&\iff{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{{{{\mkern-1mu}{\mkern-1mu}X}}}(E,F)=0\\
&\iff {{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{{{{\mkern-1mu}{\mkern-1mu}X}}}^{{\mkern-1mu}{\mathsf{qc}}}}(E,F)=0\\
&\iff{\mathsf R}{\varGamma}^{}_{\!W}F=0.\end{aligned}$$
- For any morphism $\phi\in{{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}(X),$ $$\begin{aligned}
E{\otimes^{\mathsf L}_{{{{\mkern-1mu}{\mkern-1mu}X}}}}{\mkern-1mu}\phi\textup{ is an isomorphism }
&\iff {\mathsf R}{\mkern1mu}{{\mathcal H}om}_{{{{\mkern-1mu}{\mkern-1mu}X}}}(E,\phi)\textup{ is an isomorphism }\\
&\iff {{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{{{{\mkern-1mu}{\mkern-1mu}X}}}^{{\mkern-1mu}{\mathsf{qc}}}}(E,\phi)\textup{ is an isomorphism }\\
&\iff {\mathsf R}{\varGamma}^{}_{\!W}\phi\textup{ is an isomorphism.}\end{aligned}$$
Let ${\mathscr{L}}\subseteq{{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}(X)$ (resp. ${\mathscr{L}}' \subseteq{{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}(X))$ be the full subcategory spanned by the complexes $C$ such that $C{\otimes^{\mathsf L}_{{{{\mkern-1mu}{\mkern-1mu}X}}}}F=0$ (resp. ${\mathsf R}{\mkern1mu}{{\mathcal H}om}_{{{{\mkern-1mu}{\mkern-1mu}X}}}(C,F)=0$). It is clear that ${\mathscr{L}}$ is a localizing tensor ideal; and using the natural isomorphisms (with $G\in{{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}(X)$), $$\begin{aligned}\label{eqnA3}
{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{{{{\mkern-1mu}{\mkern-1mu}X}}}\big(\!\oplus_{i\in I}C_i{\mkern1mu},{\mkern1mu}F\big)&\cong\prod_{i\in I}\,{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{{{{\mkern-1mu}{\mkern-1mu}X}}}(C_i{\mkern1mu},{\mkern1mu}F),\\
{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{{{{\mkern-1mu}{\mkern-1mu}X}}}(G{\otimes^{\mathsf L}_{{{{\mkern-1mu}{\mkern-1mu}X}}}}C,F)&\cong{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{{{{\mkern-1mu}{\mkern-1mu}X}}}(G,{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{{{{\mkern-1mu}{\mkern-1mu}X}}}(C,F))\\[1pt]
\end{aligned}\tag{\ref{Hom and tensor 0}.1}$$ -2ptone sees that ${\mathscr{L}}'$ is a localizing tensor ideal too.
We claim that when $E$ is in ${\mathscr{L}}$ it is also in ${\mathscr{L}}'$. For this it’s enough, by Proposition \[ideals\], that for any $x\in W{\mkern-1mu}$, $k(x)$ be in ${\mathscr{L}}'$. By [@T Lemma 3.4], there is a perfect ${\mathcal O_{{\mkern-1mu}{\mkern-1mu}X}}$-complex $C$ such that $\operatorname{Supp}(C)$ is the closure $\overline{\{x\}}$. We have=-1 $$\operatorname{supp}(C)=\operatorname{Supp}(C)=\overline{\{x\}}\subseteq W,$$ where the first equality holds by Lemma \[Supp\] and the inclusion holds because $W$ is a union of closed sets. Thus \[ideals\] yields $C\in{\mathscr{L}}$; and the dual complex $C'{\!:=}{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{{{\mkern-1mu}{\mkern-1mu}X}}(C,{\mathcal O_{{\mkern-1mu}{\mkern-1mu}X}})$ is in ${\mathscr{L}}'$, because ${\mathsf R}{\mkern1mu}{{\mathcal H}om}_{{{\mkern-1mu}{\mkern-1mu}X}}(C'{\mkern-1mu},F)\cong C{\otimes^{\mathsf L}_{{{{\mkern-1mu}{\mkern-1mu}X}}}} F=0$. Since $$x\in\operatorname{supp}(C) =\operatorname{Supp}(C)=\operatorname{Supp}(C')=\operatorname{supp}(C'),$$ -1pttherefore \[ideals\] gives that, indeed, $k(x)\in{\mathscr{L}}'{\mkern-1mu}$.
Similarly, if $E\in{\mathscr{L}}'$ then $E\in{\mathscr{L}}$, proving the first part of (i). 2
The same argument holds with ${{\mathsf R}{\mkern1mu}{{\mathcal H}om}_{{{{\mkern-1mu}{\mkern-1mu}X}}}^{{\mkern-1mu}{\mathsf{qc}}}}$ in place of ${\mathsf R}{\mkern1mu}{{\mathcal H}om}_{{{{\mkern-1mu}{\mkern-1mu}X}}}$. (After that replacement, the isomorphisms still hold if ${\mkern1mu}\displaystyle\prod$ is prefixed by $\operatorname{id}_X^\times$: this can be checked by applying the functors $\operatorname{Hom}_{{{{\mkern-1mu}{\mkern-1mu}X}}}(H,-)$ for all $H\in{{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}(X)$.)2
As for the rest, recall that ${\mathsf R}{\varGamma}^{}_{\!W}{\mathcal O_{{\mkern-1mu}{\mkern-1mu}X}}\in{{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}(X)$: when $W$ itself is closed, this results from the standard triangle (with $w\colon X\setminus W\hookrightarrow X$ the inclusion) $${\mathsf R}{\varGamma}^{}_{\!W}{\mathcal O_{{\mkern-1mu}{\mkern-1mu}X}}\to{\mathcal O_{{\mkern-1mu}{\mkern-1mu}X}}\to {\mathsf R}w_*w^*{\mathcal O_{{\mkern-1mu}{\mkern-1mu}X}}{\xrightarrow}{\,+\,}({\mathsf R}{\varGamma}^{}_{\!W}{\mathcal O_{{\mkern-1mu}{\mkern-1mu}X}})[1]$$ (or from the local representation of ${\mathsf R}{\varGamma}^{}_{\!W}{\mathcal O_{{\mkern-1mu}{\mkern-1mu}X}}$ by a $\dirlm{}\!$ of Koszul complexes); and then for the general case, use that ${\varGamma}^{}_W=\dirlm{}\,{\varGamma}^{}_{\!Z}$ where $Z$ runs through all closed subsets of $W{\mkern-1mu}$.
By the following Lemma, $\operatorname{supp}({\mathsf R}{\varGamma}^{}_{\!W}{\mathcal O_{{\mkern-1mu}{\mkern-1mu}X}}) = W{\mkern-1mu}$, so Proposition \[ideals\] implies that $E\in{\mathscr{L}}$ if and only if ${\mathsf R}{\varGamma}^{}_{\!W}{\mathcal O_{{\mkern-1mu}{\mkern-1mu}X}}\in{\mathscr{L}}$, i.e., $E{\otimes^{\mathsf L}_{{{{\mkern-1mu}{\mkern-1mu}X}}}}F=0$ if and only if ${\mathsf R}{\varGamma}^{}_{\!W}{\mathcal O_{{\mkern-1mu}{\mkern-1mu}X}}{\otimes^{\mathsf L}_{{{{\mkern-1mu}{\mkern-1mu}X}}}} F = 0$.
The last part of (i) results then from the standard isomorphism ${\mathsf R}{\varGamma}^{}_{\!W}{\mathcal O_{{\mkern-1mu}{\mkern-1mu}X}}{\otimes^{\mathsf L}_{{{{\mkern-1mu}{\mkern-1mu}X}}}} F \cong {\mathsf R}{\varGamma}^{}_{\!W}F$ (for which see, e.g., [@AJL1 3.1.4(i) or 3.2.5(i)]1 when $W$ itself is closed, then pass to the general case using ${\varGamma}^{}_W=\dirlm{}\,{\varGamma}^{}_{\!Z}{\mkern1mu}).$ And applying (i) to the third vertex of a triangle based on $\phi$ gives (ii).
\[suppRg\] If $W$ is a union of closed subsets of $X{\mkern-1mu},$ then $
\operatorname{supp}({\mathsf R}{\varGamma}^{}_{\!W}{\mathcal O_{{\mkern-1mu}{\mkern-1mu}X}}) = W.{\vspace{-3pt}}
$
As seen a few lines back, $({\mathsf R}{\varGamma}^{}_{\!W}{\mathcal O_{{\mkern-1mu}{\mkern-1mu}X}}){\otimes^{\mathsf L}_{{{{\mkern-1mu}{\mkern-1mu}X}}}}k(x)\cong
{\mathsf R}{\varGamma}^{}_{\!W}k(x)$ for As $k(x)$ is flasque, the canonical map ${\varGamma}^{}_{\!W}k(x)\to{\mathsf R}{\varGamma}^{}_{\!W}k(x)$ is an isomorphism. The assertion is then that ${\varGamma}^{}_{\!W}k(x)\ne0\iff x\in W$ (i.e., $\overline{\{x\}}\subset W)$, which is easily verified since $k(x)$ is constant on $\overline{\{x\}}$ and vanishes elsewhere.
\[L1.-1\] Let $u\colon W\to X$ be a localizing immersion, and $F\in{{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}(X)$. The following conditions are equivalent.
[(i)]{} $\operatorname{supp}(F)\subseteq W{\mkern-1mu}$.
[(ii)]{} The canonical map is an isomorphism $F{{\mkern8mu\longrightarrow \mkern-25.5mu{}^\sim\mkern17mu}}{\mathsf R}u_*u^*{\mkern-1mu}{\mkern-1mu}F$.
[(iii)]{} $F\cong{\mathsf R}u_*G$ for some $G\in{{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}(W)$.
As in Remark \[L1.1.25\] , the canonical map ${\mathsf R}u_*G\to {\mathsf R}u_*u^*{\mathsf R}u_*G$ is an isomorphism, whence (iii)$\Rightarrow$(ii); and the converse implication is trivial.1
Next, if $x\notin W$ then $\overline{\{x\}}\cap W=\phi{\mkern1mu}$: to see this, one reduces easily to the case in which $u$ is the natural map $\operatorname{Spec}A_M\to \operatorname{Spec}A$, where $M$ is a multiplicatively closed subset of the noetherian ring $A$ (see §\[locimm\]). Since $k(x)$ vanishes outside $\overline{\{x\}}$, it follows that $u^*k(x)=0$ whenever $x\notin W$. Using the projection isomorphism $
{\mathsf R}{\mkern1mu}u_*G{\otimes^{\mathsf L}_{{{{\mkern-1mu}{\mkern-1mu}X}}}} k(x)\cong {\mathsf R}{\mkern1mu}u_*(G{\otimes^{\mathsf L}_{W}}u^*k(x)),
$ one sees then that (iii)$\Rightarrow$(i).
The complexes $F\in{{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}(X)$ satisfying (i) span a localizing tensor ideal. So do those $F$ satisfying (iii): the full subcategory spanned by them is triangulated, as one finds by applying ${\mathsf R}u_*u^*$ to a triangle based on a ${{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}(X)$-map ${\mathsf R}u_*G_1\to{\mathsf R}u_*G_2{\mkern1mu}$; ${\boldsymbol{\mathsf{D}}}_3$ is closed under direct sums (since ${\mathsf R}u_*$ respects direct sums, see [@Nm2 Lemma 1.4], whose proof—in view of the equivalence of categories mentioned above just before \[affine locimm\]—applies to ${{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}(X))$; and ${\boldsymbol{\mathsf{D}}}_3$ is a tensor ideal since ${\mathsf R}u_*G{\otimes^{\mathsf L}_{{{{\mkern-1mu}{\mkern-1mu}X}}}} E \cong {\mathsf R}u_*(G{\otimes^{\mathsf L}_{W}}u^*{\mkern-1mu}{\mkern-1mu}E)$ for all $E\in{{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}(X)$. So \[ideals\] shows that for the implication (i)$\Rightarrow$(iii) we need only treat the case $F=k(x)$.
Since $\operatorname{supp}(k(x))=x$ (see, e.g., [@AJS 4.6, 4.7[.5pt]{}]), it suffices now to note that if $x\in W$ then $
{\mathscr{O}}_{W{\mkern-1mu},{\mkern1mu}x}={\mathscr{O}}_{{\mkern-1mu}{\mkern-1mu}X{\mkern-1mu},{\mkern1mu}x},
$ so the canonical map1 in the definition of $k(x)$ (near the beginning of this Appendix) factors as $X_x\to W{\xrightarrow}{\lift.5,u,{\mkern1mu}{\mkern1mu}}X{\mkern-1mu}$, whence $k(x)={\mathsf R}\iota_{x*}\widetilde{\kappa(x)}$ satisfies (iii).
\[supp u\_\*\] With $u$ as in \[L1.-1\], one checks that if $x\in W$ then (with self-explanatory notation) $u^*k(x)_{{{\mkern-1mu}{\mkern-1mu}X}}=k(x)_W$. Also, as above, if $x\notin W$ then $u^*k(x)_{{{\mkern-1mu}{\mkern-1mu}X}}=0$. So for $E\in{{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}(W)$, $${\mathsf R}{\mkern1mu}u_*E{\otimes^{\mathsf L}_{{{{\mkern-1mu}{\mkern-1mu}X}}}}k(x)_{{{\mkern-1mu}{\mkern-1mu}X}}\cong {\mathsf R}{\mkern1mu}u_*\big(E{\otimes^{\mathsf L}_{W}} u^*k(x)_{{{\mkern-1mu}{\mkern-1mu}X}}\big) \cong
\begin{cases}
0 & \text{if $x\notin W,$} \\
{\mathsf R}u_*\big(E{\otimes^{\mathsf L}_{W}} k(x)_W\big) & \text{if $x\in W;$}
\end{cases}$$ and since for $F\in{{\boldsymbol{\mathsf{D}}}_{\mathsf{qc}}}(W)$, $[0=F\cong u^*{\mathsf R}u_*F{\mkern1mu}]\iff [{\mathsf R}u_*F=0{\mkern1mu}]$, therefore $$\operatorname{supp}_{{{\mkern-1mu}{\mkern-1mu}X}}({\mathsf R}{\mkern1mu}u_*E)=\operatorname{supp}_W(E). \\[4pt]$$
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[^1]: This article is based on work supported by the National Science Foundation under Grant No. 0932078000, while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring semester of 2013. The first author was partly supported by NSF grant DMS 1201889 and a Simons Fellowship.
[^2]: After [@AILN] appeared, Leo Alonso and Ana Jeremías informed us that Lemma is an instance of [@SGA5 p.123, (6.4.2)]—whose proof, however, is not given in detail.
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---
abstract: 'The procedure used to “do physics” in the macroscopic world is familiar: You take an object, start it off with a particular position and velocity, subject it to known forces (say gravity or friction, or both), and follow its trajectory. You then measure the dynamical properties (say position or energy) of that object at a later time and compare those measurements with the prediction using Isaac Newton’s laws of motion. Newton’s laws directly predict what those quantities should be at that later time, so the comparison is straightforward. However, the microscopic laws of physics, quantum mechanics, aren’t so simple. The quantum concepts are so alien and counterintuitive that the language used to describe the mathematics and physics is often ambiguous and confusing. Therefore, for you to *really* understand what is going on, some mathematics must be used. This is my attempt to explain some “weird facts” using a minimum of mathematics, yet eliminating as much ambiguity as possible.'
author:
- 'M. A. Reynolds'
title: 'Three weird facts about quantum mechanics: What Bohr, Schrödinger, and Einstein actually said'
---
Niels Bohr 1913
===============
One of the first weird facts about quantum mechanics came out of Niels Bohr’s attempt to understand the atom in 1913. Only two years before, in 1911, Ernest Rutherford (with the help of Hans Geiger and Ernest Marsden) had shown that atoms were composed of a tiny, massive nucleus with positive charge surrounded by electrons of negative charge. Hydrogen, with only one electron, was the simplest of these. Bohr had joined Rutherford’s group in Manchester after receiving his doctorate from the University of Copenhagen in 1911. While there, he tried to understand the spectrum of hydrogen using Rutherford’s “planetary” model as a starting point.
From the classical electrodynamics developed in the late 1800s, it was known that the combination of a single electron orbiting a proton (as the hydrogen nucleus was called) in planetary fashion was not stable. The accelerating electron radiated electromagnetic waves, losing energy, and was predicted to spiral into the proton in about $10^{-11}$ s. Of course, as there is plenty of hydrogen around, this prediction must be wrong. Bohr therefore hypothesized that the electron could only occupy a “stationary state” in which it did *not* spiral into the proton. (What this stationary state was, or how it remained stationary, was left unexplained.) He chose what would later be called “energy eigenstates” in order to correctly predict the spectrum of hydrogen.[^1]
In modern notation, a single particle in a stationary state (in this example, an electron in a hydrogen atom) can be written $$\label{eq:StationaryState}
| \psi \rangle = | \psi_n \rangle .$$ That is, the state of the electron, $| \psi \rangle$, is a stationary state given by the quantum number $n$, $| \psi_n \rangle$, and has energy $$\label{eq:HydrogenSpectrum}
E_n = \frac{-13.6 \textrm{ eV}}{n^2} ,$$ where $n$ can take on the discrete integer values 1, 2, $\ldots$, $\infty$. The “ket” notation $| \psi \rangle$ is used to remind you that we are not really talking about a function, but simply labeling the state that the electron is in.[^2] The stationary states (which will turn out to be solutions to Schrödinger’s equation, described in the next section) can also be written simply as $| \psi \rangle =| n \rangle$.[^3]
The hydrogen spectrum that Bohr was trying to understand can be obtained by assuming that the electron can “jump” from one stationary state to another stationary state (with a lower energy). In this process, energy is conserved and therefore the atom must emit a photon with an energy equal to the energy difference between the two states $$h\nu = \Delta E = E_m - E_n ,$$ where $\nu$ is the frequency of the emitted photon and $h$ is Planck’s constant. This assumes that light comes in discrete units called “quanta” (now called a photon) and each has energy $h\nu$ (an idea that was put forth by Max Planck in 1900 and used by Albert Einstein in 1905). Bohr had therefore combined a previous idea (the quantum of light) with his own hypothesis (stationary states) along with standard physics (energy conservation) and was able to predict the hydrogen spectrum perfectly! However, physicists were still confused about what was *meant* by a state. And how did the electron know when to jump, and to what other state to jump to? Why were these the only states that were stationary? These are questions that we still have today, but we answer them in a probabilistic fashion. As physicist Abraham Pais put it
> At a moment which cannot be predicted an excited atom makes a transition to its ground state by emitting a photon. Where was the photon before that time? It was not anywhere; it was created in the act of transition.... Is there a theoretical framework for describing how particles are made and how they vanish? There is: quantum field theory. It is a language, a technique, for calculating the probabilities of creation, annihilation, scatterings of all sorts of particles: photons, electrons, positrons, protons, mesons, others ...[^4]
#### {#section .unnumbered}
This, then, is our first weird fact. In certain situations electrons (particles, in general) occupy stationary states which have discrete energies (completely contrary to the predictions of classical mechanics in which particles can have any energy) and they emit or absorb photons when they make transitions between these states. They are not allowed to have any other energy.
Erwin Schrödinger 1926
======================
The second weird fact about quantum mechanics arose with Erwin Schrödinger’s equation $$H | \psi_n \rangle = E_n | \psi_n \rangle ,$$ which he obtained by making some assumptions about the mathematical form of the state $|\psi \rangle$ and invoking energy conservation. It looks very complicated (for example, $H$ is something called the Hamiltonian operator), but for our purposes, all we need to know is that it is what mathematicians call an eigenvalue equation. What does this mean? It means that there are solutions only for discrete values of the energy $E_n$, which is called the eigenvalue.[^5] This is just what Bohr had postulated! In fact, when Schrödinger applied his equation to the hydrogen atom, he was able to derive Bohr’s result, and Bohr’s energies — see Equation (\[eq:HydrogenSpectrum\]) — turned out to be just the energy eigenvalues.
Since this is a differential equation, there are many possible solutions, one for each value of $n$ in the set $n = 1$, 2, 3, $\ldots$, $\infty$. As is true for all linear differential equations, the most general solution is a combination of all viable solutions, and this means that the electron in a hydrogen atom doesn’t have be in just *one* state, as assumed in Equation (\[eq:StationaryState\]), but can be in a “superposition state” $$\label{eq:SuperState}
| \psi \rangle = c_1 |\psi_1 \rangle + c_2 |\psi_2 \rangle + c_3 |\psi_3 \rangle + \cdots
= \sum_n c_n | \psi_n \rangle ,$$ that is, many states at once. (This, of course, is counter to our everyday experience in which we never find objects in superposition states, but always find them in a single state.)
#### {#section-1 .unnumbered}
Again, contrary to the predictions of classical physics, and counter to our everyday experience, electrons (particles in general) are allowed to be in superposition states, in which they can take on the characteristics of each of the stationary states (for example, the energy) with a certain probability.
The measurement process {#the-measurement-process .unnumbered}
-----------------------
This weirdness manifests itself when we talk about taking measurements of a system. If an electron is in such a superposition state as depicted by Equation (\[eq:SuperState\]), quantum mechanics tells us that if you measure the energy, you won’t obtain just *any* value for the result of your measurement, but the only possible values will be $E_1$ or $E_2$ or $E_3$, etc. In addition, the probability of measuring a certain energy, say $E_n$, is given by the coefficient squared, $|c_n|^2$. Since the probability of measuring *any* energy must be unity, there must be a restriction $\sum_n |c_n|^2 = 1$. This is called “normalization.” The way it is usually described is as follows. When measuring the energy of one electron, there is no way of knowing which energy you will obtain, but after measuring the energies of many *identically prepared* electrons (this set of systems is called an ensemble), the probability of obtaining the different energies will be as described.
Even though Schrödinger had developed the equation that led to this formalism, he thought that the situation was ridiculous. To show this, he came up with a thought experiment (involving the infamous cat) to highlight how strange this is. In his own words:
> One can even set up quite ridiculous cases. A cat is penned up in a steel chamber, along with the following device (which must be secured against direct interference by the cat): in a Geiger counter there is a tiny bit of radioactive substance, so small, that perhaps in the course of the hour one of the atoms decays, but also, with equal probability, perhaps none; if it happens, the counter tube discharges and through a relay releases a hammer which shatters a small flask of hydrocyanic acid. If one has left this entire system to itself for an hour, one would say that the cat still lives if meanwhile no atom has decayed.[^6]
That is, imagine a cat in an opaque box. Also in this box is a vial of poison gas, and if this vial breaks, the cat will die. Also in this box is a radioactive atom, arranged such that if the atom decays, it will trigger a small hammer to break the vial, killing the cat. Now then, after one hour, the cat is in a superposition state (in this case there are only two states, i.e., $n=1$, 2, or more descriptively, “alive” or “dead”), $$|\psi_{cat} \rangle = c_{alive} |\psi_{alive} \rangle + c_{dead} | \psi_{dead} \rangle ,$$ where we have chosen the radioactive substance so that the values of $c_{dead}$ and $c_{alive}$ are each equal to $\frac{1}{\sqrt{2}}$, which means that the probabilities of the cat being alive or dead are each $$\left| c_{alive} \right|^2 = \left| c_{dead} \right|^2 = \left| \frac{1}{\sqrt{2}} \right|^2 = \frac{1}{2} .$$
Now, even though Schrödinger — and many others — thought this was weird, it is an observational fact that these rules predict accurately the outcomes of subatomic experiments. That is, Weird Fact \#2 (the possibility of particles being in states of superposition) is an accurate description of the world, as long as we interpret these states in the probabilistic fashion just described.
Entanglement
============
Our third weird fact comes into play when we consider two identical particles simultaneously, for example, the two electrons in a helium atom. Following our previous rules, you might think that each electron must occupy a stationary state, or perhaps occupy a superposition state. In general, this is true, but the presence of one electron affects the other electron — specifically, one electron modifies the electric force experienced by the second electron, so the stationary states (and their energies) are modified from the one-electron case.
But more important, there really is only *one* wave function, or *one* state, but it depends on the properties of *both* electrons. That is, we can write the wave function of the two electrons as a “product” state $$\label{eq:product}
| \psi \rangle = | \psi_n^a \rangle | \psi_m^b \rangle,$$ which means that electron $a$ is in state $n$ and electron $b$ is in state $m$. However, in quantum mechanics, identical particles are truly identical. In classical physics, two white billiard balls, while they look the same, can be distinguished by looking closely at any possible scuffs or scratches. But two electrons, for example, are *indistinguishable*, and no one, not even God, can tell them apart. This means that we don’t know whether electron $a$ is in state $n$ and electron $b$ is in state $m$ or vice versa. Therefore, Equation (\[eq:product\]) is not an accurate representation of the state of the system. We must allow for the possibility that the particles switch places, i.e., electron $a$ is in state $m$ and electron $b$ is in state $n$. This means that the wave function must be written as $$\label{eq:entangled}
| \psi \rangle = | \psi_n^a \rangle | \psi_m^b \rangle
+ | \psi_m^a \rangle | \psi_n^b \rangle ,$$ which is called an “entangled” state. The particles are entangled because we don’t know which particle is in which state, although we *do* know that if one of them is in state $a$, then the other *must* be in state $b$.
In reality, the wave function must be normalized, and the state must be $$\label{eq:entangled}
| \psi \rangle = \alpha | \psi_n^a \rangle | \psi_m^b \rangle
+ \beta | \psi_m^a \rangle | \psi_n^b \rangle ,$$ where $|\alpha|^2$ is the probability of finding electron $a$ in state $n$ and electron $b$ in state $m$ and $|\beta|^2$ is the probability of finding electron $a$ in state $m$ and electron $b$ in state $n$. Of course, normalization requires that $|\alpha|^2 + |\beta|^2 = 1$. And since either case is equally probable, $|\alpha|^2 = |\beta|^2 = \frac{1}{2}$. Mathematically, there are two equally correct choices: either $\alpha = \frac{1}{\sqrt{2}}$ and $\beta = \frac{1}{\sqrt{2}}$, in which case $| \psi \rangle$ is called a symmetric wave function, or $\alpha = \frac{1}{\sqrt{2}}$ and $\beta = -\frac{1}{\sqrt{2}}$, in which case $| \psi \rangle$ is called an anti-symmetric wave function.[^7]
#### {#section-2 .unnumbered}
When the system consists of two identical (i.e., indistinguishable) particles, they are in a superposition state together, which is called an entangled state. It is not known which particle is in which state, but it is equally likely to be either.
Einstein 1935 {#einstein-1935 .unnumbered}
-------------
Einstein, among others, thought that this was weird (just like Schrödinger had thought Weird Fact \#2 was weird), and he spent many years arguing with Bohr about the meaning of superposition states and entangled states. Much has been written about these debates, and I can’t do justice to them here. However, I will discuss one aspect of the problem that philosophers of physics must wrestle with, and which Einstein got wrong.[^8]
Einstein was a realist. This means that while he believed that the mathematical prescription of a superposition state correctly predicted the probabilities of certain measurements (as described above), he thought that such a superposition did not completely describe the state of the object. That is, an electron really *does* have a specific energy, but we just don’t know what it is. Our knowledge is incomplete, and a deeper, more fundamental theory than quantum mechanics would tell us exactly what the energy is. Another way of thinking about it is that the standard quantum mechanical description claims that the electron doesn’t have a specific energy until we measure it. This was the position of Bohr, Einstein’s sparring partner in this debate, but Einstein though that “\[n\]o reasonable definition of reality could be expected to permit this.”[^9]
Who was correct, Bohr or Einstein? At the time, it was impossible to determine, since both agreed that the probabilistic interpretation correctly predicted experimental outcomes, which it did. Einstein thought there was something more, but couldn’t prove it (except by appeal to his version of “reality”). Bohr thought there was nothing more, but also couldn’t prove it. It was not until the 1960s that John Bell came up with a way to determine who was right and who was wrong. He derived an inequality that would be violated if standard quantum mechanics was correct (that is, Bohr), and satisfied if Einstein’s reality were correct. It wasn’t until the 1970s (long after Einstein’s death in 1955 and Bohr’s death in 1962) that technology had developed to the point where experiments could be done that were sensitive enough to distinguish between the two cases. All experiments that have been performed up to the present come down solidly on the side of Bohr.[^10] There is no underlying reality to the energy of an electron. It truly is in a superposition state, and the energy is not determined until it is measured.
[^1]: The details of his calculation are not important for our purposes here, only the implications of the stationary state hypothesis.
[^2]: This notation, $| \psi \rangle$, was invented by Paul Dirac in the 1920s.
[^3]: There are really four quantum numbers that completely describe the state of the electron in a hydrogen atom, $n$, $\ell$, $m_\ell$, and $m_s$, but for our purposes, all the weird physics can be understood with only one quantum number.
[^4]: A. Pais, *Inward Bound* (Oxford University Press, New York), pp. 324-5.
[^5]: In this context, $|\psi_n \rangle$ is called the “eigenfunction,” or “wave function.”
[^6]: This quote is from a three-part paper, E. Schrödinger, “Die gegenwärtige Situation in der Quantenmechanik,” *Naturwissenschaften*, **23** 807-812; 823-828; 844-849 (1935). Translated by John D. Trimmer.
[^7]: It turns out that these two choices describe two different kinds of particles. Symmetric wave functions describe “bosons,” and anti-symmetric wave functions describe “fermions.”
[^8]: Modern writers make a big deal whenever they discover that Einstein got something wrong, but in this case he made a perfectly reasonable choice that was only shown to be wrong by experiments performed in the 1970s.
[^9]: A. Einstein, B. Podolsky and N. Rosen, “Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?” *Physical Review* **47**, 777-780, 1935.
[^10]: See, for example, J. Bell, *Speakable and unspeakable in quantum mechanics: Collected papers on quantum philosophy* (Cambridge University Press, New York, 2004).
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- 'Sudeep Katakol, Basem Elbarashy, Luis Herranz, Joost van de Weijer, and Antonio M. López, '
bibliography:
- 'refs.bib'
title: Distributed Learning and Inference with Compressed Images
---
Introduction {#sec:introduction}
============
intelligent devices such as smartphones, autonomous vehicles and robots are equipped with high-quality cameras and powerful deep neural networks that enable advanced on-board visual analysis and understanding. These large models are trained with a large amount of data and require powerful hardware resources (e.g. GPUs). These models also require days or even weeks to train, which is not possible in resource-limited devices. Thus, training is often performed in a centralized server, which also allows using data captured by multiple devices to train better models (e.g. a fleet of autonomous cars). In this case, training and test take place in two physically separated locations, i.e. server and device, respectively. In other cases, such as in mobile cloud computing, the data is captured by the device, while the inference takes place in a server.
One important requirement in these scenarios is that, at some point, the visual data needs to be transmitted from the device to the server. Fig. \[fig:feature\_vis\]a shows an archetypal scenario of autonomous driving, where each vehicle of the fleet captures and encodes data and transmits it to the server. The server decodes the data and uses it for training the analysis models. The trained models are then deployed to autonomous vehicles, where they perform inference. Captured data often requires to be annotated by humans in order to train supervised models, which adds to the reasons to process the data in a server.
The captured data can be stored on-board in a storage device and physically delivered to the server, or directly transmitted through a communication channel. In either case, storage space or channel capacity are constraints that condition the amount of collected samples in practice, and effective collection requires data compression to exploit the limited storage and communication resources efficiently.
The amount of data captured (possibly from multiple cameras) can be enormous, requiring high compression rates with lossy compression. However, this entails a certain degradation in the images, which depends on the bitstream rate (lower the rate, higher the degradation). In this paper, we study the impact of such degradation on the performance of the downstream analysis task. At times, the degradation affects only one of the training and test data. For instance, in Fig \[fig:feature\_vis\]a, training data is degraded while test data (on-board) can be accessed without degradation.
When training data is compressed and test is not (or vice versa), a first effect we observe is *covariate shift* (i.e. train and test data are sampled from different distributions). For instance, the first column of Fig. \[fig:feature\_vis\]b represents the original captured images, while the second represents the compressed images (*i.e.* reconstructed[^1]). A clear difference in terms of lack of details and blurred textures is observed, which causes covariate shift (e.g. original for test, compressed for training in the example of Fig \[fig:feature\_vis\]a). A possible solution to this problem is compressing both training and test data at the same rate. For the autonomous driving scenario of Fig. \[fig:feature\_vis\]a, this would mean deploying an image compressor in the car (including encoder and decoder) and performing inference on the reconstructed images. While this approach alleviates the covariate shift, it is not always effective and also increases the computational cost in the on-board system.
The degradation caused by lossy compression not only induces covariate shift, but can also harm the performance of the downstream task through the means of *semantic information loss*. Here semantic information refers to the information that is relevant to solve a particular downstream task and it can be lost during the process of compression. Semantic information is task-dependent and its loss is typically irreversible. For example, the actual plate number WAF BA 747 in the second column of Fig. \[fig:feature\_vis\]b is lost in the process of compression, and cannot be recovered. However, if the task is car detection, the actual plate number is not necessarily relevant semantic information.
In this paper, we study the effect of compression on downstream analysis tasks (focusing on semantic segmentation) under different configurations, which in turn can be related to real scenarios. We observe that both covariate shift and semantic information loss indeed result in a performance drop (see Fig. \[fig:feature\_vis\]c) compared to training and test with original images (configuration OO). The performance depends on the compression rate and the particular training/test configuration. For instance, in the configuration of the autonomous driving scenario of Fig. \[fig:feature\_vis\]a compressing the test data prior to inference (we refer to this approach as *compression before inference*, and corresponds to the training/test configuration compressed-compressed, or CC for short) degrades the performance more than using the original data (configuration CO), showing that it is preferable to keep the test data more semantically informative than correcting the covariate shift.
The previous result also motivates us to explore whether there exists a solution that improves over the baseline CO configuration. As a result, we propose *dataset restoration*, an effective approach based on image restoration using generative adversarial networks (GANs) [@goodfellow2014generative]. Dataset restoration is applied to the images in the training set without modifying the test images, effectively alleviating the covariate shift while keeping the test data semantically informative. In this case, we show that the configuration restored-original (RO) does improve performance over the baseline (see Fig. \[fig:feature\_vis\]c). An additional advantage is that there is no computational cost penalty nor additional hardware or software requirements in the deployed on-board system (in contrast to compressing the test data). Note also that our approach is generic and independent of the particular compression (deep or conventional) used to compress the images.
Adversarial restoration decreases the covariate shift by hallucinating texture patterns that resemble those lost during compression while removing compression artifacts, both of which contribute to the covariate shift. The distribution of restored images is closer to the distribution of original images and thus the covariate shift is lower. Fig. \[fig:feature\_vis\]b shows an example where the trees have lost their texture and appear essentially as blurred green areas. A segmentator trained with these images will expect trees to have this appearance, but during test they appear with the original texture and details of leaves and branches, which leads to poor performance. The restored image has textures that resemble real trees and contains less compression artifacts, which makes its distribution closer to that of the actual test images, contributing to a significant improvement in downstream performance (see Fig \[fig:feature\_vis\]c). Note that adversarial restoration cannot recover certain semantic information. This example also illustrates the effect on semantic information. The license plate appears completely blurred due to compression. Note that adversarial restoration can recover the texture of digits (or even hallucinate random digits), which can be useful to improve car segmentation, but the original plate number is lost (i.e. semantic information), which makes it impossible to perform license plate recognition at that compression rate.
In summary, our contributions are as follows:
- Systematic analysis of training/test configurations with compression and relation of downstream performance with rate, semantic information loss and covariate shift.
- Dataset restoration, a principled method based on our theoretical analysis, to improve downstream performance in on-board analysis scenarios. This method is task-agnostic and can be used alongside multiple image compression methods. It also does not increase the inference time and memory requirements of the downstream model.
Related Work {#sec:related_work}
============
Lossy compression
-----------------
A fundamental problem in digital communication is the transmission of data as binary streams (*bitstreams*) under limited capacity channels [@shannon1948mathematical; @ct91], a problem addressed by data compression. Often, practical compression ratios are achievable only with lossy compression, *i.e.* a certain loss with respect to the original data is tolerated. Traditional lossy compression algorithms for images typically use a DCT or a wavelet transform to transform the image into a compact representation, which is simplified further to achieve the desired bitrate. Examples of lossy image compression algorithms are JPEG [@wallace1992jpeg], JPEG 2000 [@skodras2001jpeg; @taubman2012jpeg2000], and BPG [@bellard2017bpg]. BPG is the current state-of-the-art and is based on tools from the HEVC video coding standard [@sullivan2012overview].
Recently, deep image compression [@balle2016end; @balle2018variational; @toderici2015variable; @theis2017lossy; @minnen2018joint] has emerged as a powerful alternative to the traditional algorithms. These methods also use a transformation based approach like the traditional methods, but use deep neural networks to parameterize the transformation [@balle2016end]. The parameters of the networks are learned by optimizing for a particular rate-distortion tradeoff on a chosen dataset. Mean Scale Hyperprior (MSH) [@minnen2018joint], a deep image compression method based on variational autoencoders and BPG are used as representative methods of deep learning based and traditional image compression respectively.
Visual degradation and deep learning
------------------------------------
A loss in the quality of images can occur through many factors including blur, noise, downsampling and compression. Researchers have reported a drop in task performance of convolutional neural networks (CNN) models when such degradations are present in the test images [@dodge2016understanding; @hendrycks2019benchmarking; @roy2018effects]. Further, numerous methods have been proposed to make these CNN models robust to degradations [@hendrycks2019benchmarking; @borkar2019deepcorrect; @ghosh2018robustness]. These approaches include forcing adversarial robustness during training [@hendrycks2019benchmarking], modifying and retraining the network [@borkar2019deepcorrect], and using an additional network altogether [@ghosh2018robustness].
While the aforementioned works target robustness across degradations, there have been studies focusing exclusively on compression as well. These include [@torfason2018towards] (on the deep compression method [@theis2017lossy]), [@ehrlich2019deep] (JPEG) and [@lohdefink2019gan] (both deep [@agustsson2019generative] and JPEG). Unlike the previous methods, these works use the compressed images (in some form) for training the deep models and thus obtain a better performance on compressed images. Moreover, [@torfason2018towards] and [@ehrlich2019deep] encode the images using the compressors and the deep networks are trained to predict the task output using the encoded representation directly, resulting in faster inference.
Image restoration
-----------------
Image restoration involves the process of improving the quality of degraded images. Restoration methods can be grouped into denoising [@buades2005review], deblurring [@richardson1972bayesian; @lucy1974iterative], super-resolution [@park2003super], compression artifact removal [@shen1998review], etc. depending on the kind of degradation, although they share many similarities. Lately, deep learning methods have been successful for image restoration tasks. Some of these methods can be applied to any degradation [@chen2016trainable; @zhang2018residual_restoration] while others are specific to the degradation (deblurring [@nah2017deep; @chen2019blind], super-resolution [@shi2016real; @zhang2018residual], denoising [@zhang2017beyond; @zhang2018ffdnet] and compression artifact removal [@dong2015compression]). More recently, image restoration algorithms based on generative adversarial networks (GANs) have become popular thanks to their improved performance (super-resolution [@ledig2017photo; @wang2018esrgan], compression artifact removal [@galteri2019deep; @zhao2019compression] and deblurring [@kupyn2018deblurgan]).
A compressed image can be processed using a restoration method before using it for inference to improve its performance; although our analysis reveals that this is a sub-optimal approach. Galteri *et al.* [@galteri2019deep] propose a GAN-based restoration network to correct JPEG compression artifacts. They also evaluate different restoration algorithms on the basis of the performance of restored images on a trained object detection network. They show that their GAN-based algorithm performs better than other methods compared in the paper. Our analysis provides an explanation for this observation.
Domain adaptation.
------------------
Domain adaptation [@wang2018deep] is a problem motivated by the lack of sufficient annotations. Typically, domain adaptation methods leverage the abundant annotated data available from a different yet related domain (called as the source domain) to improve performance on the domain of interest (target domain), where there is a lack of annotated data. Examples of source and target domains include synthetic images vs real images, images in the wild vs images on a webpage, etc. Domain adaptation methods can be divided into unsupervised [@ganin2014unsupervised; @bousmalis2017unsupervised], semi-supervised [@saito2019semi] and supervised [@chen2012marginalized; @glorot2011domain; @tzeng2017adversarial; @ganin2016domain; @hoffman2017cycada] depending on the quantity of available data (and annotations) in the target domain. Approaches for domain adaptation can be categorized into latent feature alignment using autoencoders [@chen2012marginalized; @glorot2011domain], adversarial latent feature alignment [@tzeng2017adversarial; @ganin2016domain; @murez2018image; @hoffman2017cycada] and pixel-level adversarial alignment [@bousmalis2017unsupervised; @hoffman2017cycada].
The scenario when only compressed images are available at training time, with original images available at test time is related to domain adaptation. Dataset restoration, our proposed method for this scenario, corrects the covariate shift and domain adaptation algorithms account for domain shift in some form. Probably, the closest domain adaptation method to dataset restoration is a unsupervised method that addresses alignment only at pixel-level [@bousmalis2017unsupervised]. However, an important distinction is that domain adaptation tackles problem arising due to lack of annotations for the images in the target domain, while for us the concern lies in the non-availability of images themselves. Thus, we study the effectiveness of dataset restoration, using a varying number of original images for training the restoration model.
Learning and inference with compressed images {#sec:methods}
=============================================
Problem definition
------------------
We are concerned with downstream understanding tasks where we want to infer from an input image $\mathbf{x}\sim p_X\left(\mathbf{x}\right)$ the corresponding semantic information $\mathbf{y}\sim p_{Y\vert X}\left(\mathbf{y}\right)$. In the rest of the paper, we will assume that $\mathbf{y}$ is a semantic segmentation map, but our approach can be applied to other semantic inference tasks, such as image classification or object detection. The objective is to find a parametric mapping $\phi:\mathbf{x} \mapsto \mathbf{y}$ by supervised learning from a training dataset $X_{tr}=\left\{\left(\mathbf{x}^{(1)}_{tr},\mathbf{y}^{(1)}_{tr}\right),\ldots,\left(\mathbf{x}^{(N)}_{tr},\mathbf{y}^{(N)}_{tr}\right)\right\}$, where each image $\mathbf{x}^{(i)}$ has a corresponding ground truth annotation $\mathbf{y}^{(i)}$. The mapping is typically implemented as a deep neural network. The performance of the resulting model is evaluated on a test set $X_{ts}=\left\{\left(\mathbf{x}^{(1)}_{ts},\mathbf{y}^{(1)}_{ts}\right),\ldots,\left(\mathbf{x}^{(M)}_{ts},\mathbf{y}^{(M)}_{ts}\right)\right\}$. Under conventional machine learning assumptions, $X_{tr}\sim p_X\left(\mathbf{x}\right)$ and $X_{ts}\sim p_X\left(\mathbf{x}\right)$, i.e. both training and test sets are sampled from the same underlying distribution $p_X$.
In our setting, we consider that $X_{tr}$ $X_{ts}$ undergo a certain degradation $\psi:\mathbf{x} \mapsto \hat{\mathbf{x}}$. In our case, the degradation is related with the lossy compression process necessary to transmit the image to the remote location where the actual training or inference takes place; and so we have $\hat{\mathbf{x}}=\psi\left(\mathbf{x}\right)=g\left(f\left(\mathbf{x}\right)\right)$, where, $f\left(\mathbf{x}\right)$ is the image encoder, $g\left(\mathbf{z}\right)$ is the image decoder[^2] and $\mathbf{z}$ is the compressed bitstream. The result $\hat{\textbf{x}}$ is the reconstructed image, which follows a new distribution $p_{\hat{X}}$ of degraded images, *i.e.* $\hat{\mathbf{x}}\sim p_{\hat{X}}\left(\mathbf{x}\right)$. Note that parallels can be drawn from the arguments in this section for other image degradations such as blur, downsampling, noise, color and illumination changes, etc.
Lossy compression is characterized by the distortion $D\left(\mathbf{x},\hat{\mathbf{x}}\right)$ of the reconstructed image and the rate $R\left(\mathbf{z}\right)$ of the compressed bitstream. The encoder and decoder are designed to operate around a particular rate-distortion (R-D) tradeoff $\lambda$, either by expert crafting in conventional image compression, or by directly optimizing parameters of a deep neural network.
Covariate shift
---------------
The *covariate shift* problem precisely occurs when the underlying distributions of training and test data differ, *i.e.* $X_{tr}\sim p_{X_{tr}}$ and $X_{ts}\sim p_{X_{ts}}$ with $p_{X_{tr}}\neq p_{X_{ts}}$. This leads to sub-optimal performance because the model is evaluated on a data distribution different from the one it was optimized for. While covariate shift is often found in machine learning (e.g. training with synthetic data and evaluating on real), in our case this problem is a consequence of lossy compression and it increases severely as the rate decreases. The drop in performance is related to the degree of covariate shift, which could be seen as the divergence between distributions $d\left(p_{X_{tr}}, p_{X_{ts}}\right)$. In the conventional machine learning setting without compression there is no covariate shift, since $X_{tr}\sim p_{X}$ and $X_{ts}\sim p_{X}$, nor when both training and test set are compressed with the same method and at the same rate, since $X_{tr}\sim p_{\hat{X}}$ and $X_{ts}\sim p_{\hat{X}}$. However, covariate shift exists in the other two configurations, namely CO and OC (see Table \[tab:nomenclature\]).
The degradation due to lossy compression can be observed clearly in Fig. \[fig:feature\_vis\]b, when comparing the original captured image and the image after compression. This also gives an idea of the difference between the original domain and the domain induced by compression. It has images with lesser details which also suffer from blurring and coding artifacts. More examples are shown in Fig. \[fig:degradationexamples\] for the two compression methods (MSH and BPG), with the images compressed at a similar rate. It can be seen that degradations are consistent yet with some differences (e.g. blocky artifacts for BPG, more blurred in MSH).
Semantic information loss
-------------------------
Covariate shift explains how compression impacts the downstream task when data at training and test time are compressed unequally. Another factor that impacts task performance arises from compression resulting in *semantic information loss*. By the semantic information present in an image we refer *only* to what is relevant to the downstream task. Thus, by definition, semantic information loss is task dependent. Continuing with our example from the introduction (Fig. \[fig:feature\_vis\]b), the letters in the license plate of the car plays little to no role in establishing the presence of a car in the image. Thus, the exact letters are not relevant semantic information for the task of car detection. However, if the task is license plate recognition, the letters are an integral part of semantic information. Compression causes semantic information loss as it makes the compressed image devoid of some semantic attributes present in the original image. The loss of letters on the plate in the compressed image (Fig. \[fig:feature\_vis\]b) is evidence of semantic information loss (when the task is license plate detection).
Further evidence of semantic information loss can be found in Fig. \[fig:degradationexamples\], since the degradation often removes details and textures, blends small objects together via blur and lack of contrast, and introduces confusing artifacts, preventing us from recognizing small objects at all (e.g. individual pedestrians), and making larger objects more difficult to recognize due to the loss of discriminative details and textures (e.g. tree leaves). Only in retrospective, after observing the original undistorted crop, we can infer the small objects in the distorted image. Similarly, a semantic segmentation model will struggle to recognize them, or directly fail when the semantic information has disappeared completely (e.g. license plate number).
Let $Y$ be a random variable that represents the semantic information in the original image, $X$. For instance, if the task is semantic segmentation, $Y$ would take values from the set of semantic maps of images. Mathematically, we formulate semantic information loss, $S$, in the compressed images, $\hat{X}$, using mutual information, $I$, as follows: $S_{Y}(X, \hat{X}) = I(X, Y) - I(\hat{X}, Y)$. Predictably, $S_{Y}(X, \hat{X})$ is non-negative, since $\hat{X}$ is produced from $X$ via the map $\psi$ and thus, we have $I(X, Y) \ge I (\hat{X}, Y)$ as a consequence of the data processing inequality.
Training/test configurations, application scenarios and related work
--------------------------------------------------------------------
Now we focus on several combination of training/test configurations and provide examples of real world scenarios (summarized in Table \[tab:nomenclature\]). A configuration is defined by the pair $\left(X_{tr},X_{ts}\right)$, with $X_i \sim p_X$ represented as O and $X_i \sim p_{\hat{X}}$ represented as C. Thus, the conventional machine learning setting (i.e. without compression) corresponds to OO, and the configuration of Fig. \[fig:feature\_vis\]a is CO, since $X_{tr}\sim p_X$ and $X_{ts}\sim p_{\hat{X}}$. The former does not suffer from semantic information loss nor covariate shift, while the latter does suffer from both. The CO configuration can also be generalized to other scenarios involving on-board analysis[^3] where data capture and inference takes place in the device and the training in a server (e.g. autonomous cars, unmanned aerial vehicles and other robotic devices).
The configuration OC involves training performed in the server with the original images, while the device is resource-limited (e.g. a smartphone) but can send the image to the server, and the result of the analysis (e.g. predicted class, bounding box, segmentation map) is used in the server side or sent back to the device. In this case, the test images are compressed, which implies semantic information loss, and also covariate shift. Fig. \[fig:mcc\] illustrates the paradigmatic scenario of (mobile) cloud computing [@hayes2008cloud; @zhu2011multimedia; @fernando2013mobile]. Another example of OC configuration is distributed automotive perception [@lohdefink2019gan], where the sensor module compresses the captured image and transmits it through the automotive bus system to the perception module where the downstream tasks are performed.
Similarly, the configuration CC appears in the previous scenario when training images are also compressed, and at the same rate as test images. In this case, both training and test images suffer from semantic information loss, but there is no covariate shift since both are sampled from the same $p_{\hat{X}}$.
**Compression before training.** We can remove the covariate shift from a configuration OC by transforming it into CC. This can be achieved by compressing the training data at the same rate and we refer to this adaptation approach as *compression before training* (see Fig. \[fig:adaptation\_methods\]a). However, through this process, semantic information loss is additionally introduced to the training set. Generally, the presence or absence of semantic information loss in the test images is a major factor, while it is not always the case with the training images. While some class information may be lost in a particular training image, its presence in other training images can compensate for it. As such we expect the model with configuration CC to outperform configuration OC.
**Compression before inference.** Similarly, we can also transform a configuration CO into CC by compressing the test images. We refer to this process as *compression before inference* (see Fig. \[fig:adaptation\_methods\]b). While this process allows us to correct the covariate shift due to compression, it also introduces semantic information loss at test time. The introduction of semantic information loss in the test is critical and can cause the performance of configuration CC to be even worse than the configuration CO at times (as shown in Fig. \[fig:feature\_vis\]c). Moreover, compression before inference requires installing a full compression encoder and decoder module on-board prior to the downstream task, resulting in a significant computational penalty in the deployed system.
Dataset restoration {#sec:restoration}
===================
Proposed approach
-----------------
Motivated by the two limitations mentioned above, we propose *dataset restoration* as an alternative approach that alleviates covariate shift without inducing semantic information loss in the test data (in contrast to compression before inference). The key idea is to adapt the training dataset using adversarial image restoration, and use the adapted dataset as actual training data for the downstream task (see Fig. \[fig:adaptation\_methods\]c). In this way, the on-board analysis module can exploit all the information available in the captured image. Another important advantage is that adaptation takes place only in the server, and the resulting model can be readily and seamlessly deployed in the car with the same hardware, therefore without requiring to install any additional hardware nor increasing the inference cost.
We now recall that a great deal of degradation is related to the loss of texture in the decoded image and the appearance of compression artifacts (these two factors are clearly apparent in Figs. \[fig:feature\_vis\]b and \[fig:degradationexamples\]). Our goal was to find an appropriate image restoration technique that could learn from a given set of examples and provide us a way to remove the artifacts and recover texture in the images.
Our restoration module is based on adversarial image restoration, where a generative adversarial network (GAN) [@goodfellow2014generative] conditioned on the degraded image is employed to improve the image quality. A GAN is based on two networks competing in an adversarial setting. The generator takes the input image and outputs the restored image. The discriminator observes real and restored images and it is optimized to classify between real and restored images. The generator, in contrast is optimized to fool the discriminator, and indirectly improves the quality of the restored images. Through the process, the generator learns to remove compression artifacts and replace unrealistic textures by realistic ones that could be used by the discriminator to identify the restored images. The architecture of GAN is based on the one proposed in [@wang2018high] (for image-to-image translation [@isola2017image]), which has a generator and multiple discriminators (see Appendix \[appendix:air\] for details).
During the process of dataset restoration, we use our trained generator to restore individually every image in the training dataset for the downstream task. Examples of some of the restored images can be found in Figs. \[fig:feature\_vis\]b and \[fig:degradationexamples\]. While not being able to restore lost semantic information (e.g. the same individual pedestrians), the restored images look sharper, with less artifacts and blurred regions are enhanced with hallucinated textures that resemble the real images. As such, the shift with respect to the distribution of original images, on which the trained model will be evaluated, is reduced. Table \[tab:nomenclature\] includes two new configurations OR and RO, where R refers to restored images.
Adversarial restoration, covariate shift and perceptual index
-------------------------------------------------------------
There is an interesting relation between the perceptual index described in [@blau2018perception] and the covariate shift, which explains why adversarial image restoration is the appropriate approach (compared to non-adversarial). In the CO configuration, the training images $X_{tr}$ are compressed and therefore follow $p_{\hat{X}}$, while the test images $X_{ts}$ follow $p_X$. Thus, the covariate shift is equal to $d(p_X, p_{\hat{X}})$, where $d$ denotes a probabilistic divergence, like the Kullback-Leibler divergence or the Jenson-Shannon divergence [@ct91]. On the other hand, the perceptual index in [@blau2018perception] is defined as the divergence between real images $p_X$ and compressed (and decoded) images $p_{\hat{X}}$ (note that there is not training/test distinction). Therefore, in the case of CO configuration, the covariate shift corresponds to the perceptual index of the test set (in both cases, the lower the better). An important conclusion from [@blau2018perception] is that perception and distortion are at odds with each other, and that there exists a limit beyond which perception and distortion cannot be reduced simultaneously (see Fig. \[fig:illus\_percep\_dist\]). Thus, the effect of dataset restoration (i.e. moving from CO to RO) is to lower the covariate shift (and perceptual index) at the expense of increasing distortion, provided we are close to the perception-distortion limit.
We are ultimately interested in the implications on the performance of the downstream task, semantic segmentation in particular. Our analysis in the previous section reveals that the task performance is greatly dependent on covariate shift and semantic information loss. Reducing the perception index is therefore more crucial than training with images of low distortion, as a lower perception index corresponds to a lower covariate shift. Hence, we use adversarial restoration and decrease the perceptual index at the cost of increased distortion.
Are all image restoration approaches helpful? We argue that only adversarial image restoration are suitable, since they explicitly minimize the perceptual index through the discriminator and consequently the covariate shift with respect to the captured images. In contrast, non-adversarial image restoration methods do not necessarily reduce the perceptual index. Typically, these methods try to further decrease the distortion and this can be counter-productive as perception and distortion are at odds near the limit.
Experiments {#sec:results}
===========
Experimental settings
---------------------
**Datasets.** We evaluate our methods on three datasets:
*Cityscapes* [@cordts2016cityscapes] is a popular dataset in autonomous driving, and contains 5000 street images (2975/500/1525 for training/validation/test sets) of which training and validation have pixel-level segmentation maps annotated with 19 different concepts, including objects and “stuff”. We use the annotated sets to train (training set) and evaluate semantic segmentation (validation set). It also contains another 20000 images with coarse annotation. We ignore these annotations and use a subset of 2000 images to train the deep image compression model (i.e. MSH [@balle2018variational]) and the image restoration methods.
*INRIA Aerial Images Dataset* [@maggiori2017dataset] contains aerial images of diverse urban settlements with segmentation maps with two classes (building and background). The dataset consists of aerial images from 10 cities with 36 images per city. Annotations are provided for 5 of these cities and the segmentation models were trained on 4 cities and evaluated on 1 from these. The images from the other 5 cities were used for compression and restoration.
*Semantic Drone Dataset* [@mostegel2019semanticdronedataset] contains 400 high resolution images captured with an autonomous drone at an altitude of 5 to 30 meters above ground, and their corresponding annotated segmentation maps (20 classes). The 400 publicly released images were resized from a resolution of 6000x4000 to 3000x2000. The segmentation models were trained on 265 images and evaluated on 70 images while the remaining 65 images were used for the compression and restoration models. Each image was further split into 12 patches each with dimension of a 1200x800. All metrics are calculated on these patched images.
**Compression methods.** We use two state-of-the-art image compression methods. The Better Portable Graphics (BPG) format [@bellard2017bpg] is based on a subset of the video compression standard HEVC/H.265 [@sullivan2012overview] and is the state-of-the-art in non-deep image compression. The Mean Scale Hyperprior (MSH) [@balle2018variational; @minnen2018joint] is a state-of-the-art deep image compression method, based on an autoencoder whose parameters are learned to jointly minimize rate and distortion at a particular tradeoff $\lambda$, i.e. $\min R+\lambda D$. MSH models were pretrained for 600k iterations on the CLIC Professional Dataset[^4] with *MSE* loss. Appendix \[appendix:msh\] contains details of the model architecture.
**Segmentation.** For the downstream task we use the state-of-the-art semantic segmentation method DeepLabv3+ [@chen2018encoder]. The model is trained using the same procedure mentioned in the paper. We use an output stride of 16 and perform single scale evaluation.
**Metrics.** The quality of the inferred semantic segmentation map is evaluated using the mean intersection over union (mIoU, the higher the better). For image compression we measure rate in bits per pixel (bpp, the lower the better) and the distortion in PSNR (in dB, the higher the better).
Cityscapes
----------
**Rate-distortion curves.** We first characterize the rate-distortion performance and the tradeoff for the two compression methods in our experiments, and the impact of the proposed restoration approach on them (see Fig. \[fig:cityscapes\_rd\_curves\]). The curves sweep the whole range, from low to high quality images. As expected, the distortion decreases (PSNR increases) with rate. We observe that MSH performs significantly better than BPG on Cityscapes, *i.e.* it produces images with lower average distortion at similar rate. Interestingly, once the images are restored, images compressed with MSH have marginally higher distortion than those compressed with BPG.
**Segmentation performance.** We evaluate the segmentation performance under six different configurations (i.e. OO, CO, RO, CC, OC, OR). The results are shown in Fig. \[fig:cityscapes-segmentation-performance\].
For the Cityscapes dataset, we observe that the model with configuration CO outperforms the model with configuration CC which shows that correcting covariate shift by compression before inference can potentially result in lowering the performance. Table \[tab:cityscapes-per\_class\_miou\] shows the performance per class. We see that the mIoU of classes representing small objects[^5] is significantly lower in the configuration CC when compared to CO. Since smaller objects are relatively easier to lose by compression, the observation confirms that the introduction of semantic information loss in the test set from the process of compression before inference is responsible for the decrease in performance.
The proposed dataset restoration approach is able to improve 1.4-4.8% on the configuration CO, and we achieve close to optimal performance (77% mIoU) requiring only 0.13 bits per pixel. Thus, lossy compression can result in huge storage savings during data collection when compared to lossless image compression. For the same budget required to collect the 2975 training images with MSH at 0.13 bpp, using lossless compression (PNG in our experiments, resulting in 9 bpp) we would have collected only 42 images, which is clearly insufficient to train the segmentation network. Furthermore, for the same performance, dataset restoration effectively reduces the required budget (for example, configuration CO achieves 75.64% with 0.128 bpp, while RO approximately requires around 0.07 bpp).
**Amount of samples for training the restoration network.** In order to learn the restoration network it is necessary to collect high quality images (uncompressed or compressed losslessly, ideally). Thus, the amount of images to train the restoration network is also an important factor. Fig. \[fig:restoration\_data-required\] shows the result for configuration CO (no restoration) and RO with a restoration network train with different amounts of training samples. It suggests that restoring images compressed with MSH can be learned with fewer images (or at least the performance saturates) than in the case of BPG.
**Adversarial vs non-adversarial restoration.** In order to show the connection between perceptual index and adversarial image restoration, we compute the perceptual index of the original, compressed and restored images using the blind quality assessment method HOSA [@xu2016blind] (as described in [@blau2018perception]). We use Residual Dense Networks [@zhang2018residual_restoration] as a representative method of non-adversarial image restoration. RDN is trained with the objective of increasing the PSNR (RDN - PSNR) or MS-SSIM [@wang2003multiscale] (RDN - MS-SSIM). Refer to Appendix \[appendix:rdn\] for more details. Note that adversarial restoration is able to achieve a perceptual index close to that of the original images while RDN does not affect the perceptual index significantly.
While the previous experiment shows that adversarial restoration does improve the perceptual quality of images, we are ultimately interested in the segmentation task. We use the model trained with the original images to extract features from a shallow layer. Then we extract features of original images, compressed images and restored images. Since we only use one model, all features are aligned in the channel dimension. We want to measure the level of alignment in activation, so we plot the average activation (over the validation set of Cityscapes) value for each channel of original, compressed or restored versus the original one (see Fig. \[fig:images\_feat\_corr\]). Obviously, the points corresponding to original images lie on the identity line, while many channels from compressed images are clearly not aligned. Adversarial restoration manages to bring back the features to the identity line, while non-adversarial restoration has little effect. This shows that adversarially restored images are not only perceptually closer to the real images, but also semantically more correlated.
Further, Table \[tab:rdn\_results\] compares the performance of different restoration methods in the configuration RO. Adversarial restoration results in a far better segmentation performance when compared to RDN.
**Efficiency.** Table \[tab:efficiency\] reports the inference times for different configurations. The time for encode-decode times for MSH and the segmentation time for DeepLabv3+ were measured on a Quadro RTX6000 GPU, while the encode-decode times for BPG was measured on a Intel Xeon(R) E5-1620 v4 CPU. Inference using dataset restoration (configuration RO) is faster than compression before inference (configuration CC) by 26% and 66%, when the compression method used is MSH and BPG respectively.
INRIA Aerial Images Dataset
---------------------------
**Segmentation performance.** Fig. \[fig:aid-segmentation-performance\] depicts the segmentation results obtained on the INRIA Aerial Images Dataset (AID) for the same six configurations mentioned in Section B. We observe that the model with configuration CO performs better than CC when the compression method used is BPG. However, the same cannot be said for MSH. We hypothesize that the smoothing artifacts caused when MSH is used destroy the discriminative features and a segmentation model capable of taking advantage of these features in the original image, cannot be learnt. This is especially critical with the AID since there are only a few features that discriminate a building from the background.
The proposed approach of RO performs consistently better than both these configurations with gains up to 3.9 % mIoU. Fig. \[fig:aidexample\] shows a portion of an image from the dataset along with the segmentation maps predicted by various models.
Interestingly, contrary to the results on the Cityscapes dataset, the performance of configuration OR does not improve over OC. This suggests that the process of restoration causes further damage (over compression) to the discriminative features used by the model trained on the original images for its prediction.
Semantic Drone Dataset
----------------------
**Segmentation performance.** The segmentation results obtained on the Semantic Drones Dataset (SDD) are shown in Fig. \[fig:sdd-segmentation-performance\]. We observe that the models with configuration CC outperform CO consistently for both MSH and BPG. We attribute this result to the lack of semantic information loss in this dataset. A typical object from the SDD covers a significant portion of the image and compression does not significantly affect its recognizability. When we assume little to no semantic information loss, the effects of covariate shift are dominant and as such, the configuration CC performs better. This result shows that the performance of various configurations are dependent on the properties of the dataset.
The proposed approach of RO performs similarly to configuration CC and lies within $\pm 1.5\% $ mIoU of the configuration CC. The lack of significant semantic information loss in SDD affects the effectiveness of dataset restoration.
Conclusions
===========
The rapid development in sensor quality and increasing data collection rate makes lossy compression necessary to reduce transmission and storage costs. By means of dataset restoration, we enable the incorporation of lossy compression for on-board analysis, greatly mitigating the drop in performance. Dataset restoration is a principled approach, based on our analysis of the various scenarios involving learning and inference with compressed images. Our analysis framework involving covariate shift and semantic information loss can be further extended to other degradations like blur, noise, color and illumination changes, etc.
Mean scale hyperprior {#appendix:msh}
=====================
Fig. \[fig:msh\_arch\] describes the architecture of the MSH [@minnen2018joint] used in this paper.
Adversarial image restoration {#appendix:air}
=============================
We use the FineNet from Akbari *et. al* [@akbari2019dsslic] (adapted from [@wang2018high]) with slight changes for our image restoration module.
Following the same notation in [@akbari2019dsslic], the generator architecture is written as $ c_{64}, \; d_{128}, \; d_{256}, \; d_{512}, \; 9 \times r_{512}, \; u_{256}, \; u_{128}, \; u_{64}, \; o_{3} $ where
- $c_k$: Conv: 7 × 7 x k, Instance Normalization, ReLU
- $d_k$: Conv: 3 x 3 x k / $\downarrow$ 2, Instance Normalization, ReLU
- $r_k$: Conv: 3 x 3 x k, Reflection padding, Instance Normalization, ReLU
- $u_k$: Conv: 3 x 3 x k / $\uparrow$ 2, Instance Normalization, ReLU
- $o_3$: Conv: 7 x 7 x 3, Instance Normalization, Tanh
We use two discriminators, as in [@akbari2019dsslic], operating at two different scales. Akbari *et al.* rescale the image to half the resolution while we don’t. The discriminators act on the original resolution, $H \times W $ and $H/4 \times W/4$ resolution. Again following notation in [@akbari2019dsslic], the discriminators have the following architecture, $ C_{64}, \; C_{128}, \; C_{256}, \; C_{512}, \; O_{1} $, where
- $C_k$: Conv: 4 x 4 x k / $\downarrow$ 2, Instance Normalization, LeakyReLU
- $O_1$: Conv: 1 x 1 x 1
Let the captured image be $x$. The restored image, $ \bar{x} $ is obtained by adding the residual computed by the generator to the compressed image, $$\bar{x} = \hat{x} + G(\hat{x}).$$
All images are scaled to $[-1, 1]$.
The loss function used for training are as follows:
- Generator, $G$: $$L_{GAN}^{(G)} + 10 \cdot (2 \cdot L_{1} + L_{VGG} + L_{MS-SSIM} + L_{DIST})$$
- Discriminator, $D_i$: $ L_{GAN}^{(D_i)} $
$L_{GAN}^{(G)}$ is the sum of the standard GAN loss from each of the discriminator, *i.e.* $$L_{GAN}^{(G)} = \sum_{i=1}^{2}{- \log(D_{i}(\hat{x}, \bar{x}))}.$$
$$L_{1} = {\left\lVert\bar{x} - x\right\rVert_1}.$$ $$L_{MS-SSIM} = \textrm{\textit{MS-SSIM}}(\bar{x}, x).$$
Let VGG denote a VGG-Net trained on the ImageNet dataset and $M_{j}$ denote the size of the output of the $j^{th}$ layer of VGG. The output of each of the 5 convolution blocks are considered for the VGG feature distillation loss, which is given by
$$L_{VGG} = \sum_{j=1}^{5} \frac{1}{M_{j}} {\left\lVertVGG^{(j)}(\bar{x}) - VGG^{(j)}(x)\right\rVert_1}.$$
Similarly, the features of the discriminators are also distilled for stable GAN training.
$$L_{DIST} = \sum_{i=1}^{2} \sum_{j=1}^{4} \frac{1}{N_{j}^{(i)}} {\left\lVertD_{i}^{(j)}(\hat{x}, \bar{x}) - D_{i}^{(j)}(\hat{x}, x)\right\rVert_1}.$$
The discriminators are trained using the standard GAN loss.
$$L_{GAN}^{(D_i)} = \log(1 - D_{i}(\hat{x}, \bar{x})) + \log(D_{i}(\hat{x}, x)).$$
We use a batchsize of 1 and train the GAN for around 135k iterations. Adam optimizer with $\beta_1 = 0.1$ and $\beta_2=0.9$ is employed. Initially, the learning rate is set to 0.0002 and is reduced by a factor of 10 after 80k iterations.
Residual dense network {#appendix:rdn}
======================
We use the RDN architecture from [@zhang2018residual_restoration]. We ask the reader to refer to the paper for the architecture. The following hyperparameters are used: Global layers = 16, Local layers = 6, Growth rate = 32.
We train the CAR model with the objective of maximising MS-SSIM or PSNR. The models are trained using 256x256 patches. A mini-batchsize of 1 is used and the model is trained for around 200k iterations. Adam optimizer is used with initial learning rate of 0.001 which is reduced by a factor 10 at 80k and 150k iterations.
Acknowledgments {#acknowledgments .unnumbered}
===============
Acknowledgment {#acknowledgment .unnumbered}
==============
The authors thank Audi Electronics Venture GmbH for supporting this work, the Generalitat de Catalunya CERCA Program and its ACCIO agency. L. Herranz acknowledges the support of the Spanish project RTI2018-102285-A-I00, and the EU’s Horizon 2020 R&I programme under the Marie Skłodowska-Curie grant No.665919. A. M. López acknowledges the support of project TIN2017-88709-R (MINECO/AEI/FEDER, UE) and ICREA under the ICREA Academia programme.
[^1]: When referring to data used in the downstream tasks, *compressed* images will implicitly refer to the reconstructed images after the compression decoder.
[^2]: We only consider lossy compression, since in lossless compression $\hat{\mathbf{x}}=\mathbf{x}$.
[^3]: Often called *on-board perception*, but we prefer *on-board analysis* to avoid confusion later.
[^4]: https://www.compression.cc/2019/challenge/
[^5]: We consider classes *person*, *rider*, *motorcycle*, *bicycle*, *pole*, *traffic light*, *traffic sign* as small objects and the remaining classes as big objects.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Suppose that $X$ is a projective manifold whose tangent bundle $T_X$ contains a locally free strictly nef subsheaf. We prove that $X$ is isomorphic to a projective bundle over a hyperbolic manifold. Moreover, if the fundamental group $\pi_1(X)$ is virtually abelian, then $X$ is isomorphic to a projective space.'
address:
- 'Jie Liu, Morningside Center of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China'
- 'Wenhao Ou, Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China'
- 'Xiaokui Yang, Department of Mathematics and Yau Mathematical Sciences Center, Tsinghua University, Beijing, 100084, China'
author:
- Jie Liu
- Wenhao Ou
- Xiaokui Yang
bibliography:
- 'strcitlynef.bib'
title: Projective manifolds whose tangent bundle contains a strictly nef subsheaf
---
Introduction
============
Since the seminal works of Mori and Siu-Yau on the solutions to Hartshorne conjecture and Frankel conjecture ([@Mori1979], [@SiuYau1980]), it becomes apparent that the positivity of the tangent bundle of a complex projective manifold carries important geometric information. In the past decades, many remarkable generalizations have been established. For instance, Mok classified compact Kähler manifold with semipositive holomorphic bisectional curvature in [@Mok1988]. Initiated by the fundamental works of Campana, Demailly, Peternell and Schneider ([@CampanaPeternell1991], [@DemaillyPeternellSchneider1994], [@Peternell1996]), the structure of projective manifolds with nef tangent bundles is investigated by many mathematicians. The last building block to be understood for such manifolds are Fano manifolds with nef tangent bundles. Campana and Peternell proposed in [@CampanaPeternell1991] the following conjecture, which is still an important open problem: a Fano manifold with nef tangent bundle must be a rational homogeneous space. It is proved for all Fano manifolds of dimension at most five and has also been verified for certain special varieties. We refer to [@CampanaPeternell1991; @CampanaPeternell1993; @Mok2002; @Hwang2006; @Pandharipande2013; @Watanabe2014; @MunozOcchettaSolaCondeWatanabeEtAl2015; @Kanemitsu2017; @Li2017] and the references therein.
Recall that a line bundle ${{\mathscr L}}$ on a projective variety $X$ is said to be *strictly nef* if $c_1({{\mathscr L}})\cdot C>0$ for all complete curves $C$ in $X$, and a vector bundle ${{\mathscr F}}$ is strictly nef if its tautological line bundle ${{\mathscr O}}_{\mathbb{P}({{\mathscr F}})}(1)$ is strictly nef. The definition of strict nefness is quite natural and it is a notion of positivity which is stronger than nefness but weaker than ampleness. The main difficulty to deal with it is that the strictly nefness is not closed under exterior product. Actually, there exist *Hermitian flat* vector bundles which are also strictly nef, and this phenomenen will be studied intensively in this paper. Even though there are significant differences between strict nefness and ampleness, we still expect that the strict nefness could play similar roles as ampleness in many situations. Indeed, together with Li, the second and third authors obtained the following theorem in [@LiOuYang2019] which extends Mori’s result.
[@LiOuYang2019 Theorem 1.4] \[thm:Li-Ou-Yang\] Let $X$ be an $n$-dimensional complex projective manifold such that $T_X$ is strictly nef. Then $X\cong {{\mathbb P}}^n$.
Meanwhile, it is also known that the existence of positive subsheaves of the tangent bundle already impose strong geometric restrictions on the ambient manifold. For example, Andreatta and Wiśniewski achieved the following characterization of projective spaces.
[@AndreattaWisniewski2001 Theorem] \[thm:Andreatta-Wisniewski\] Let $X$ be an $n$-dimensional complex projective manifold. If there exists a rank $r$ ample locally free subsheaf ${{\mathscr F}}$ of $T_X$, then $X\cong {{\mathbb P}}^n$ and either ${{\mathscr F}}\cong T_{{{\mathbb P}}^n}$ or ${{\mathscr F}}\cong {{\mathscr O}}_{{{\mathbb P}}^n}(1)^{\oplus r}$.
When ${{\mathscr F}}$ is a line bundle, this theorem is settled by Wahl in [@Wahl1983] via the theory of algebraic derivations in characteristic zero. In [@CampanaPeternell1998], Campana and Peternell proved the theorem in the cases $r\geqslant n-2$. Later, it is shown that the assumption on the local freeness can be dropped, see [@AproduKebekusPeternell2008; @Liu2019].
In view of Theorem \[thm:Li-Ou-Yang\], it is natural to ask whether Theorem \[thm:Andreatta-Wisniewski\] still holds if the subsheaf ${{\mathscr F}}$ is only assumed to be strictly nef. Unfortunately, Mumford constructed an example (see [@Hartshorne1970 Chapter I, Example 10.6]) which gives a negative answer to this question. Indeed, for any smooth projective curve $C$ of genus $g\geqslant 2$, there exists a rank $2$ Hermitian flat and strictly nef vector bundle ${{\mathscr E}}$ over $C$. Then the relative tangent bundle $T_{{{\mathbb P}}({{\mathscr E}})/C}$ is a strictly nef subbundle of $T_{{{\mathbb P}}({{\mathscr E}})}$ since it is isomorphic to the line bundle ${{\mathscr O}}_{{{\mathbb P}}({{\mathscr E}})}(2)$.
In this paper, we investigate the geometry of projective manifolds whose tangent bundle contains a strictly nef subsheaf, and obtain the following structure theorem, which is an extension of Theorem \[thm:Andreatta-Wisniewski\].
\[thm:main-theorem\] Let $X$ be a complex projective manifold. Assume that the tangent bundle $T_X$ contains a locally free strictly nef subsheaf ${{\mathscr F}}$ of rank $r>0$. Then $X$ admits a ${{\mathbb P}}^d$-bundle structure $\varphi\colon X\rightarrow T$ for some integer $d\geq r$. Furthermore $T$ is a hyperbolic projective manifold.
Here we recall that a single point is also considered to be hyperbolic in the sense that every holomorphic map from ${{\mathbb C}}$ to it is a constant map. Indeed, we obtain in Theorem \[cor:main:part1\] a more concrete description on the structure of the subsheaf ${{\mathscr F}}$, and there are only two possibilities, which correspond to those of Theorem \[thm:Andreatta-Wisniewski\]:
1. ${{\mathscr F}}\cong T_{X/T}$ and $X$ is isomorphic to a flat projective bundle over $T$;
2. ${{\mathscr F}}$ is a numerically projectively flat vector bundle and its restriction on every fiber of $\varphi$ is isomorphic to ${{\mathscr O}}_{{{\mathbb P}}^d}(1)^{\oplus r}$.
When ${{\mathscr F}}$ is a line bundle, Druel obtained in [@Druel2004] that $X$ is isomorphic to either a projective space or a ${{\mathbb P}}^1$-bundle. However, when ${{\mathscr F}}$ has greater rank, there are extra structures as shown in the second case above (see Example \[example:not-subbundles\] for more details). To classify these structures, different methods are needed and transcendental tools are also crucially involved.
As an application of Theorem \[thm:main-theorem\], we obtain a new characterization of projective spaces.
\[thm:simply-connected-Pn\] Let $X$ be an $n$-dimensional complex projective manifold such that $T_X$ contains a locally free strictly nef subsheaf ${{\mathscr F}}$. If $\pi_1(X)$ is virtually abelian, then $X$ is isomorphic to ${{\mathbb P}}^n$, and ${{\mathscr F}}$ is isomorphic to either $T_{{{\mathbb P}}^n}$ or ${{\mathscr O}}_{{{\mathbb P}}^n}(1)^{\oplus r}$.
By using Theorem \[thm:main-theorem\] and [@BrunebarbeKlinglerTotaro2013 Theorem 0.1], we get the existence of non-zero symmetric differentials.
\[cor:existence-symmetric-forms\] Let $X$ be an $n$-dimensional projective manifold whose tangent bundle contains a locally free strictly nef subsheaf. If $X$ is not isomorphic to ${{\mathbb P}}^n$, then $X$ has a non-zero symmetric differential, i.e. $H^0(X,\operatorname*{Sym}^i\Omega_X)\not=0$ for some $i>0$.
One of the important tools for the proof of Theorem \[thm:main-theorem\] is the theory of numerically projectively flat vector bundles. The following criterion is a variant of [@HoeringPeternell2019 Theorem 1.8], which is also the first step to understand the structures of projective bundles induced by strictly nef subsheaves.
\[Num-Projectily-flatness-criterion\] Let $X$ be an $n$-dimensional complex projective manifold and ${{\mathscr F}}$ be a reflexive coherent sheaf of rank $r$. Assume that there exists a ${{\mathbb Q}}$-Cartier divisor class $\delta\in N^1(X)_{{{\mathbb Q}}}$ such that the $\mathbb{Q}$-twisted coherent sheaf ${{\mathscr F}}\hspace{-0.8ex}<\hspace{-0.8ex}\delta\hspace{-0.8ex}>$ is almost nef and that $$(c_1({{\mathscr F}})+r\delta)\cdot A^{n-1}=0$$ for some ample divisor $A$. Then ${{\mathscr F}}$ is locally free and numerically projectively flat.
It is known that numerically projectively flat vector bundles are extensions of *projectively Hermitian flat* vector bundles. In this paper, we show that they are actually isomorphic to projectively flat vector bundles with compatible holomorphic connections, and this structure is crucial in the proof of Theorem \[thm:main-theorem\]. We refer the reader to [@Simpson1992 Section 3] (see also [@Deng2018]) for a similar result on numerically flat vector bundles.
\[thm:num-proj-flat=proj-flat\] Let $X$ be a projective manifold and ${{\mathscr E}}$ be a numerically projectively flat vector bundle on $X$. Then ${{\mathscr E}}$ is isomorphic to a projectively flat holomorphic vector bundle ${{\mathscr F}}$, i.e. there exists a projectively flat connection $\nabla$ on ${{\mathscr F}}$ such that $\nabla^{0,1}=\bar\partial_{{{\mathscr F}}}$ where $\nabla^{0,1}$ is the $(0,1)$-part of $\nabla$.
The paper is organized as follows. After giving some elementary results in Section \[section:pre\], we recall the basics of $\mathbb{Q}$-twisted sheaves in Section \[section:Q-twisted\]. We prove Theorem \[Num-Projectily-flatness-criterion\] and Theorem \[thm:num-proj-flat=proj-flat\] in Section \[section:proj-flat\]. In Section \[Examples\], we study a special case when $X$ is of the form $\mathbb{P}({{\mathscr E}})$ and include some examples. Starting from Section \[section:uniruled\], we focus on the proof of Theorem \[thm:main-theorem\] in the general setting. We first show that the MRC fibration of $X$ provides a $\mathbb{P}^d$-bundle structure on a large open subset of $X$. Then we recall some results about degeneration of ${{\mathbb P}}^d$ in Section \[section:degeneration-P\^d\] and prove in Section \[section:bundle structure\] that the ${{\mathbb P}}^d$-bundle structure holds on entire $X$. Finally, in Section \[section:hyperbolicity\], we show that the base $T$ is hyperbolic, and complete the proof of Theorem \[thm:main-theorem\], Theorem \[thm:simply-connected-Pn\] and Corollary \[cor:existence-symmetric-forms\].\
[**Acknowledgements.**]{} We would like to thank Professors Ya Deng, Stéphane Druel, Baohua Fu, Andreas Höring, Thomas Peternell and Zhiyu Tian for inspiring discussions and useful communications. We would also like to thank Professor Shing-Tung Yau for his valuable help, support and guidance. The first-named author is supported by China Postdoctoral Science Foundation (2019M650873).
2
Preliminaries {#section:pre}
=============
Throughout this paper, we work over $\mathbb{C}$, the field of complex numbers. All manifolds and varieties are supposed to be irreducible. The following statement is a refined version of the negativity lemma (see [@KollarMori1998 Lemma 3.39]).
\[lemma:negativity-lemma-line-bundle\] Let $f\colon X\to Y$ be a projective birational morphism between normal varieties of dimension $n\geqslant 2$. Assume that $D$ is a $f$-exceptional $\mathbb{Q}$- Cartier $\mathbb{Q}$-divisor which has at least one positive coefficient. Then there is a family $\{C_\gamma\}_{\gamma\in \Gamma}$ of complete $f$-exceptional curves such that $C_\gamma\cdot D<0$ for all $\gamma\in \Gamma$.
Furthermore, for any fixed subvariety $W\subseteq X$ of codimension at least $2$, and for any fixed subvariety $V\subseteq X$ of codimension at least $3$, a general member $C$ of $\{C_\gamma\}_{\gamma\in \Gamma}$ is not contained in $W$, and is disjoint from $V$.
We first assume that $n=2$. Let $r\colon X'\to X$ be a desingularization and $D'=r^*D$. Set $h=f\circ r$. Then as in the proof of [@KollarMori1998 Lemma 3.41], there is a component $C'$ of $D'$ with positive coefficient such that $C'\cdot D'<0$. Since $D'$ is relatively numerically trivial over $X'$, we see that $C'$ is not $r$-exceptional. Let $C=r(C')$. Then $C$ is a component of $D$ with positive coefficient such that $C\cdot D<0$.
Next we study the general case. Let $D_1$ be a component of $D$ with positive coefficient and let $d=\dim f(D_1)$. We chose $n-2$ hypersurfaces $H_1,\dots, H_{n-2}$ in $X$ such that $H_{i}$ is the pullback of some general very ample divisor in $Y$ for $i\le d$ and is a general very ample divisor in $X$ for $i> d$. Let $S$ be the surface cut out by these hyperplanes. Then $S$ is a normal surface. Let $T$ be the normalization of $f(S)$ and denote by $g \colon S\to T$ the natural morphism.
Let $A$ be the cycle theoretic intersection of $D$ and $S$. Since $D$ is $\mathbb{Q}$-Cartier, so is $A$. We note that $A$ has at least one positive coefficient. From the first paragraph, we see that $A$ is not $g$-nef, and there is a component $Z$ of $A$ with positive coefficient such that $Z\cdot A <0$. Therefore, $Z\cdot D<0$. By deforming the hypersurfaces $H_i$, we can deform the curve $Z$ into a family $\{C_\gamma\}_{\gamma\in \Gamma}$. The last assertion of the lemma follows from the fact that every curve $C_\gamma$ is contained in the complete intersection of $n-2$ base-point-free big divisors.
\[lemma:numerical-proportional-preseved\] Let $f\colon X\rightarrow Y$ be a morphism between normal projective varieties. Let $C_1$ and $C_2$ be two complete curves in $Y$ and let $C_1'$ and $C_2'$ be two curves in $X$ such that $f(C_i')=C_i$ for each $i$. If $C'_1$ is numerically proportional to $C_2'$ in $X$, then $C_1$ is numerically proportional to $C_2$ in $Y$.
By assumption, there exists a positive rational number $r\in {{\mathbb Q}}$ such that $$\label{equation:numerically-proportional}
C_1'\equiv_{{{\mathbb Q}}} r C_2'.$$ Denote by $d_i$ the degree of the finite morphism $f\vert_{C_i'}\colon C_i'\rightarrow C_i$. Let ${{\mathscr L}}$ be a line bundle on $Y$. By projection formula, we have, for each $i$, $$c_1(f^*{{\mathscr L}})\cdot C_i'=d_i c_1({{\mathscr L}})\cdot C_i.$$ Combining with the first equation derives $$c_1({{\mathscr L}})\cdot C_1=\frac{rd_2}{d_1}c_1({{\mathscr L}})\cdot C_2.$$ As ${{\mathscr L}}$ is arbitrary, we conclude that $C_1$ is numerically proportional to $C_2$.
\[lemma:rational-section-birational-morphism\] Let $f\colon X\rightarrow Y$ be a birational projective morphism between normal quasi-projective varieties. Assume that $Y$ is smooth. Let $C$ be a complete curve in $Y$. Then there is a complete irreducible and reduced curve $C'$ in $X$ such that $f\vert_{C'}\colon C'\rightarrow C$ is a birational morphism.
By [@Hartshorne1977 II, Theorem 7.17], there exists a coherent sheaf of ideals ${{\mathscr I}}$ on $Y$ such that $X$ is isomorphic to the blowing-up of $X$ with respect to ${{\mathscr I}}$. By Hironaka’s resolution theorem, there exists a finite sequence $g\colon W\rightarrow Y$ of blowups with smooth center such that $g^*{{\mathscr I}}$ is invertible. In particular, by the univeral property of blowing-up (see [@Hartshorne1977 II, Proposition 7.14]), $g$ factors through $f\colon X\rightarrow Y$. Thus, by replacing $X$ by $W$, we may assume that $f$ is the composition of a sequence of blowups at smooth center. Using induction, we then reduce it to the case when $f$ is a blowup of smooth center $Z\subseteq Y$.
If $C\not\subset Z$ then we can take $C'$ to be the strict transform of $C$ in $X$. Assume that $C\subseteq Z$. We note that $f$ is a projective bundle over $Z$. Hence if $\overline{C}$ is the normalization of $C$, then $\overline{C}\times_Y X$ is a projective bundle on $\overline{C}$. Such a projective bundle admits a section, whose image is $D$. Let $C'$ be the image of $D$ in $X$. Then $f\vert_{C'}$ is a birational morphism onto $C$.
2
Positivity of ${{\mathbb Q}}$-twisted coherent sheaves {#section:Q-twisted}
======================================================
For readers’ convenience, we collect some basic properties on positivity of ${{\mathbb Q}}$-twisted coherent sheaves.
Nefness of ${{\mathbb Q}}$-twisted coherent sheaves
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Let ${{\mathscr F}}$ be a coherent sheaf on a variety $X$. ${{\mathscr F}}$ is called a vector bundle if it is locally free. The singular locus $\mathrm{Sing}({{\mathscr F}})$ of ${{\mathscr F}}$ is the smallest closed subset of $X$ such that ${{\mathscr F}}$ is locally free over $X\backslash \mathrm{Sing}({{\mathscr F}})$. We denote by ${{\rm rk}}({{\mathscr F}})$ the rank of ${{\mathscr F}}$. The dual sheaf ${\ensuremath{\mathscr{H}om_{\mathscr{O}_X}({{\mathscr F}},{{\mathscr O}}_X)}}$ is denoted by ${{\mathscr F}}^*$ and the reflexive hull of ${{\mathscr F}}$ is the double dual ${{\mathscr F}}^{**}$. Given a morphism $\gamma\colon Y\rightarrow X$, we denote by $\gamma^{[*]}{{\mathscr F}}\coloneqq (\gamma^*{{\mathscr F}})^{**}$ the reflexive pullback. The $m$-th reflexive symmetric power and $q$-th reflexive exterior product of ${{\mathscr F}}$ are
$S^{[m]}{{\mathscr F}}\coloneqq(\operatorname*{Sym}^m{{\mathscr F}})^{**}$ and $\wedge^{[q]}{{\mathscr F}}\coloneqq (\wedge^q{{\mathscr F}})^{**}$.
The determinant $\det({{\mathscr F}})$ of ${{\mathscr F}}$ is defined as $\wedge^{[r]}{{\mathscr F}}$, where $r={{\rm rk}}({{\mathscr F}})$. We denote by $N^1(X)_{{{\mathbb Q}}}$ the finite-dimensional ${{\mathbb Q}}$-vector space of numerical equivalence classes of ${{\mathbb Q}}$-Cartier divisors on $X$. If we assume moreover that a coherent sheaf ${{\mathscr F}}$ is locally free in codimension $1$ and that $\det({{\mathscr F}})$ is ${{\mathbb Q}}$-Cartier, then the first Chern class $c_1({{\mathscr F}})$ of ${{\mathscr F}}$ is defined as $$c_1({{\mathscr F}}){{\vcentcolon=}}c_1(\det({{\mathscr F}}))\in N^1(X)_{{{\mathbb Q}}}$$ and the averaged first Chern class $\mu({{\mathscr F}})$ of ${{\mathscr F}}$ is given by $$\mu({{\mathscr F}}) {{\vcentcolon=}}\frac{1}{r}c_1({{\mathscr F}})\in N^1(X)_{{{\mathbb Q}}}.$$
The projectivization ${{\mathbb P}}({{\mathscr F}})$ is defined by ${{\mathbb P}}({{\mathscr F}})\coloneqq {\ensuremath{\mbox{\rm Proj}(Sym^{\bullet}{{\mathscr F}})}}$. If ${{\mathscr F}}$ is a vector bundle, then $\mathbb{P}({{\mathscr F}})$ is the projective bundle of hyperplanes in ${{\mathscr F}}$. We denote by ${{\mathscr O}}_{{{\mathbb P}}({{\mathscr F}})}(1)$ the tautological line bundle on ${{\mathbb P}}({{\mathscr F}})$ and by $\zeta_{{{\mathscr F}}}$ the tautological class $$c_1({{\mathscr O}}_{{{\mathbb P}}({{\mathscr F}})}(1))\in N^1({{\mathbb P}}({{\mathscr F}})).$$
Now we introduce formally ${{\mathbb Q}}$-twisted coherent sheaves, which extends the notion of ${{\mathbb Q}}$-twisted vector bundles (see [@Lazarsfeld2004a §6.2]).
1. A ${{\mathbb Q}}$-twisted coherent sheaf ${{\mathscr F}}\hspace{-0.8ex}<\hspace{-0.8ex}\delta\hspace{-0.8ex}>$ on a projective variety $X$ is an ordered pair consisting of a coherent sheaf ${{\mathscr F}}$ on $X$, and a numerical equivalence class $\delta\in N^1(X)_{{{\mathbb Q}}}$.
2. A ${{\mathbb Q}}$-twisted coherent sheaf ${{\mathscr F}}\hspace{-0.8ex}<\hspace{-0.8ex}\delta\hspace{-0.8ex}>$ on a projective variety $X$ is called nef (resp. strictly nef, ample) if $$\zeta_{{{\mathscr F}}}+p^*\delta\in N^1({{\mathbb P}}({{\mathscr F}}))_{{{\mathbb Q}}}$$ is a nef (resp. strictly nef, ample) ${{\mathbb Q}}$-Cartier divisor class where $p\colon \mathbb{P}({{\mathscr F}}) \to X$ is the projection.
The following result is a Barton-Kleiman type criterion for ${{\mathbb Q}}$-twisted coherent sheaves (see also [@Lazarsfeld2004a Proposition 6.1.18] and [@LiOuYang2019 Proposition 2.1]).
\[BK criterion\] Let ${{\mathscr F}}\hspace{-0.8ex}<\hspace{-0.8ex}\delta\hspace{-0.8ex}>$ be a ${{\mathbb Q}}$-twisted coherent sheaf over a projective variety $X$. Then ${{\mathscr F}}\hspace{-0.8ex}<\hspace{-0.8ex}\delta\hspace{-0.8ex}>$ is nef if and only if given any finite morphism $\nu\colon C\rightarrow X$ from a smooth complete curve $C$ to $X$, and given any quotient line bundle ${{\mathscr L}}$ of $\nu^*{{\mathscr F}}$, one has $$\det({{\mathscr L}})+\deg(\nu^*\delta)\geqslant 0\label{BK}$$ In particular, ${{\mathscr F}}\hspace{-0.8ex}<\hspace{-0.8ex}\delta\hspace{-0.8ex}>$ is nef if and only if the restriction ${{\mathscr F}}\hspace{-0.8ex}<\hspace{-0.8ex}\delta\hspace{-0.8ex}>\vert_C$ is nef for any complete curve $C\subseteq X$. The same criterion holds for strict nefness if the inequality in (\[BK\]) is replaced by the strict one.
Let $p\colon \mathbb{P}({{\mathscr F}}) \to X$ be the natural projection. Since $\zeta_{{\mathscr F}}+p^*\delta$ is $p$-relatively ample, we only need to consider complete curves in $\mathbb{P}({{\mathscr F}})$ which are not contracted by $p$. Let $B$ be such a curve and $C$ its normalization. We denote by $\nu \colon C\to X$ the morphism induced by the projection from ${{\mathbb P}}({{\mathscr F}})\rightarrow X$. Then $\nu$ is a finite morphism. By [@Grothendieck1961 (4.1.3) and Proposition 4.2.3], quotient line bundles $\nu^*{{\mathscr F}}\rightarrow {{\mathscr L}}$ correspond one-to-one to sections $\varphi\colon C\rightarrow {{\mathbb P}}(\nu^*{{\mathscr F}})$ with the following commutative diagram $$\begin{tikzcd}[column sep=large, row sep=large]
{{\mathbb P}}(\nu^*{{\mathscr F}})\ar[r,"{\pi}"] \ar[d,"{p'}"] & {{\mathbb P}}({{\mathscr F}})\ar[d,"{p}"]\\
C\ar[r,"{\nu}"] \arrow[bend left]{u}{\varphi} & X
\end{tikzcd}$$ such that ${{\mathbb P}}(\nu^*{{\mathscr F}})={{\mathbb P}}({{\mathscr F}})\times_{X} C$, ${{\mathscr O}}_{{{\mathbb P}}(\nu^*{{\mathscr F}})}(1)=\pi^*{{\mathscr O}}_{{{\mathbb P}}({{\mathscr F}})}(1)$ and $${{\mathscr L}}\cong \varphi^*{{\mathscr O}}_{{{\mathbb P}}(\nu^*{{\mathscr F}})}(1)=\varphi^*\pi^*{{\mathscr O}}_{{{\mathbb P}}({{\mathscr F}})}(1).$$ This implies $
\deg({{\mathscr L}})+\deg(\nu^*\delta) = B\cdot (\zeta_{{\mathscr F}}+p^*\delta),
$ and we can conclude the criterion for nefness. The proof for strict nefness is similar.
As an application, one has the following useful criterion.
\[Image-nef\] Let $C$ be an irreducible projective curve and ${{\mathscr F}}\hspace{-0.8ex}<\hspace{-0.8ex}\delta\hspace{-0.8ex}>$ a nef (resp. strictly nef) ${{\mathbb Q}}$-twisted sheaf on $C$. If $\varphi\colon{{\mathscr F}}\rightarrow {{\mathscr Q}}$ is a morphism of coherent sheaves which is generically surjective, then the ${{\mathbb Q}}$-twisted sheaf ${{\mathscr Q}}\hspace{-0.8ex}<\hspace{-0.8ex}\delta\hspace{-0.8ex}>$ is nef (resp. strictly nef).
Let $\nu\colon D\rightarrow C$ be a finite morphism from a smooth irreducible curve $D$ to $C$ and let $\nu^*{{\mathscr Q}}\rightarrow {{\mathscr L}}$ be a quotient line bundle. Since $\varphi$ is generically surjective, the composition $
\nu^*{{\mathscr F}}\rightarrow \nu^*{{\mathscr Q}}\rightarrow {{\mathscr L}}$ is non-zero. Denote its image by ${{\mathscr L}}'$. Since ${{\mathscr L}}$ is torsion free, so is ${{\mathscr L}}'$. Thus ${{\mathscr L}}'$ is a line bundle on the smooth curve $D$. If ${{\mathscr F}}\hspace{-0.8ex}<\hspace{-0.8ex}\delta\hspace{-0.8ex}>$ is nef (resp. strictly nef), then it follows from Proposition \[BK criterion\] that
$\deg({{\mathscr L}}')+\deg(\nu^*\delta)\geqslant 0$ (resp. $>0$).
We note that $\deg({{\mathscr L}})\geqslant \deg({{\mathscr L}}')$. By applying Proposition \[BK criterion\] again, we conclude that ${{\mathscr Q}}\hspace{-0.8ex}<\hspace{-0.8ex}\delta\hspace{-0.8ex}>$ is nef (resp. strictly nef).
The following proposition shows that nef ${{\mathbb Q}}$-twisted coherent sheaves are limits of ample ${{\mathbb Q}}$-twisted coherent sheaves.
[@Lazarsfeld2004a Proposition 6.2.11]\[Limit-ample\] Let ${{\mathscr F}}\hspace{-0.8ex}<\hspace{-0.8ex}\delta\hspace{-0.8ex}>$ be a ${{\mathbb Q}}$-twisted coherent sheaf over a projective variety $X$. Then ${{\mathscr F}}\hspace{-0.8ex}<\hspace{-0.8ex}\delta\hspace{-0.8ex}>$ is nef if and only if ${{\mathscr F}}\hspace{-0.8ex}<\hspace{-0.8ex}\delta+h\hspace{-0.8ex}>$ is ample for any ample class $h\in N^1(X)_{{{\mathbb Q}}}$.
By definition, if ${{\mathscr F}}\hspace{-0.8ex}<\hspace{-0.8ex}\delta+h\hspace{-0.8ex}>$ is ample for every ample class $h$, then $\zeta_{{{\mathscr F}}}+p^*(\delta+h)$ is ample. Hence $\zeta_{{{\mathscr F}}}+p^*\delta$ is nef. Assume conversely that ${{\mathscr F}}\hspace{-0.8ex}<\hspace{-0.8ex}\delta\hspace{-0.8ex}>$ is nef. Since $\zeta_{{{\mathscr F}}}+p^*\delta$ is $p$-ample, by [@Grothendieck1961 Proposition 4.4.10], $\varepsilon(\zeta_{{{\mathscr F}}}+p^*\delta)+p^*h$ is ample for $0<\varepsilon \ll 1$. As $\zeta_F+p^*\delta$ is nef, it follows that $$\zeta_{{{\mathscr F}}}+p^*\delta+p^*h=(1-\varepsilon)(\zeta_{{{\mathscr F}}}+p^*\delta)+\varepsilon(\zeta_{{{\mathscr F}}}+p^*\delta)+p^*h$$ is ample by [@Lazarsfeld2004 Corollary 1.4.10].
Similarly, one has
\[Symmetry-wedge\] Let ${{\mathscr E}}\hspace{-0.8ex}<\hspace{-0.8ex}\delta\hspace{-0.8ex}>$ and ${{\mathscr F}}\hspace{-0.8ex}<\hspace{-0.8ex}\delta\hspace{-0.8ex}>$ be two nef ${{\mathbb Q}}$-twisted sheaves on a projective variety $X$. Then the tensor product $({{\mathscr E}}\otimes {{\mathscr F}})\hspace{-0.8ex}<\hspace{-0.8ex}2\delta\hspace{-0.8ex}>$ is nef. In particular, $S^m{{\mathscr E}}\hspace{-0.8ex}<\hspace{-0.8ex}m\delta\hspace{-0.8ex}>$ and $\wedge^q{{\mathscr E}}\hspace{-0.8ex}<\hspace{-0.8ex}q\delta\hspace{-0.8ex}>$ are nef for all $m\geqslant 1$ and $1\leqslant q\leqslant {{\rm rk}}({{\mathscr E}})$.
Almost nef ${{\mathbb Q}}$-twisted coherent sheaves
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For a strictly nef subsheaf of the tangent bundle, its saturation is *a priori* not nef. We therefore consider a weaker positivity. The following notion of almost nef line bundle was introduced in [@DemaillyPeternellSchneider2001]. We extend this to the setting of $\mathbb{Q}$-twisted sheaves.
Let ${{\mathscr F}}$ be a coherent sheaf on a projective variety $X$, and let $\delta\in N^1(X)_{{{\mathbb Q}}}$ be a ${{\mathbb Q}}$-Cartier divisor class. The ${{\mathbb Q}}$-twisted sheaf ${{\mathscr F}}\hspace{-0.8ex}<\hspace{-0.8ex}\delta\hspace{-0.8ex}>$ is said to be almost nef, if there is a countable family $(Z_i)_{i\in \mathbb{N}}$ of proper subvarieties of $X$ such that ${{\mathscr F}}\hspace{-0.8ex}<\hspace{-0.8ex}\delta\hspace{-0.8ex}>\vert_C$ is nef for all irreducible curves $C\not\subset \cup_{i\in \mathrm{N}} Z_i$.
If ${{\mathscr F}}\hspace{-0.8ex}<\hspace{-0.8ex}\delta\hspace{-0.8ex}>$ is almost nef, its negative locus ${{\mathbb S}}({{\mathscr F}}\hspace{-0.8ex}<\hspace{-0.8ex}\delta\hspace{-0.8ex}>)$ is the smallest countable union of closed subvarieties such that ${{\mathscr F}}\hspace{-0.8ex}<\hspace{-0.8ex}\delta\hspace{-0.8ex}>\vert_C$ is nef for all irreducible curves $C\not\subset {{\mathbb S}}({{\mathscr F}}\hspace{-0.8ex}<\hspace{-0.8ex}\delta\hspace{-0.8ex}>)$.
By [@DemaillyPeternellSchneider2001 Proposition 3.3] and [@BoucksomDemaillyPuaunPeternell2013 Theorem 0.2], a ${{\mathbb Q}}$-Cartier divisor $D$ on a projective manifold $X$ is almost nef if and only if $D$ is pseudo-effective. However, we remark that in general the negative locus ${{\mathbb S}}(D)$ of $D$ is a proper subset of the non-nef locus ${{\mathbb B}}_{-}(D)$ of $D$ (e.g. [@BoucksomDemaillyPuaunPeternell2013 Remark 6.3]).
We collect some basic properties of ${{\mathbb Q}}$-twisted almost nef coherent sheaves.
\[Almost-nef-properties\] Let $\delta\in N^1(X)_{{{\mathbb Q}}}$ be a ${{\mathbb Q}}$-Cartier divisor class on a projective variety $X$, and let ${{\mathscr E}}$, ${{\mathscr F}}$ and ${{\mathscr G}}$ be coherent sheaves on $X$.
1. If ${{\mathscr E}}\hspace{-0.8ex}<\hspace{-0.8ex}\delta\hspace{-0.8ex}>$ is almost nef and if $\sigma\colon {{\mathscr E}}\rightarrow {{\mathscr Q}}$ generically surjective, then ${{\mathscr Q}}\hspace{-0.8ex}<\hspace{-0.8ex}\delta\hspace{-0.8ex}>$ is almost nef. Moreover, ${{\mathbb S}}({{\mathscr Q}}\hspace{-0.8ex}<\hspace{-0.8ex}\delta\hspace{-0.8ex}>)$ is contained in the union of ${{\mathbb S}}({{\mathscr E}}\hspace{-0.8ex}<\hspace{-0.8ex}\delta\hspace{-0.8ex}>)$ and the support of the torsion sheaf ${{\mathscr G}}/\sigma({{\mathscr G}})$.
2. If ${{\mathscr E}}\hspace{-0.8ex}<\hspace{-0.8ex}\delta\hspace{-0.8ex}>$ is almost nef, then $S^{[m]}{{\mathscr E}}\hspace{-0.8ex}<\hspace{-0.8ex}m\delta\hspace{-0.8ex}>$ and $\wedge^{[q]}{{\mathscr E}}\hspace{-0.8ex}<\hspace{-0.8ex}q\delta\hspace{-0.8ex}>$ are almost nef for all $m,q>0$. Their negative loci are contained in the union of $\mathrm{Sing}({{\mathscr E}})$ and ${{\mathbb S}}({{\mathscr E}}\hspace{-0.8ex}<\hspace{-0.8ex}\delta\hspace{-0.8ex}>)$.
3. Let $p \colon \mathbb{P}({{\mathscr E}}) \to X$ be the natural projection. If $\zeta_{{{\mathscr E}}}+p^*\delta$ is almost nef and its negative locus ${{\mathbb S}}(\zeta_{{{\mathscr E}}}+p^*\delta)$ does not dominate $X$, then ${{\mathscr E}}\hspace{-0.8ex}<\hspace{-0.8ex}\delta\hspace{-0.8ex}>$ is almost nef and ${{\mathbb S}}({{\mathscr E}}\hspace{-0.8ex}<\hspace{-0.8ex}\delta\hspace{-0.8ex}>)$ is contained in $p({{\mathbb S}}(\zeta_{{{\mathscr E}}}+p^*\delta))$.
4. Let $0\rightarrow {{\mathscr F}}\rightarrow {{\mathscr E}}\rightarrow {{\mathscr Q}}\rightarrow 0$ be an exact sequence of coherent sheaves. If both ${{\mathscr F}}\hspace{-0.8ex}<\hspace{-0.8ex}\delta\hspace{-0.8ex}>$ and ${{\mathscr Q}}\hspace{-0.8ex}<\hspace{-0.8ex}\delta\hspace{-0.8ex}>$ are almost nef, then ${{\mathscr E}}\hspace{-0.8ex}<\hspace{-0.8ex}\delta\hspace{-0.8ex}>$ is almost nef and ${{\mathbb S}}({{\mathscr E}}\hspace{-0.8ex}<\hspace{-0.8ex}\delta\hspace{-0.8ex}>)$ is contained in ${{\mathbb S}}({{\mathscr F}}\hspace{-0.8ex}<\hspace{-0.8ex}\delta\hspace{-0.8ex}>)\cup{{\mathbb S}}({{\mathscr Q}}\hspace{-0.8ex}<\hspace{-0.8ex}\delta\hspace{-0.8ex}>)$.
For (1), let $C\subseteq X$ be a complete curve such that $C\not\subset {{\mathbb S}}({{\mathscr E}}\hspace{-0.8ex}<\hspace{-0.8ex}\delta\hspace{-0.8ex}>)\cup \operatorname*{Supp}({{\mathscr G}}/\sigma({{\mathscr G}}))$. Then the induced morphism ${{\mathscr E}}\vert_C\rightarrow {{\mathscr G}}\vert_C$ is generically surjective and we conclude by Corollary \[Image-nef\] that ${{\mathscr G}}\hspace{-0.8ex}<\hspace{-0.8ex}\delta\hspace{-0.8ex}>\vert_C$ is nef. For (2), we only prove the statement for $S^{[m]}{{\mathscr E}}\hspace{-0.8ex}<\hspace{-0.8ex}m\delta\hspace{-0.8ex}>$, and the case of exterior power is similar. Let $C\subseteq X$ be a complete curve such that $C\not\subset \mathrm{Sing}({{\mathscr E}})\cup {{\mathbb S}}({{\mathscr E}}\hspace{-0.8ex}<\hspace{-0.8ex}\delta\hspace{-0.8ex}>)$. Then the induced morphism $(S^m{{\mathscr E}})\vert_C\rightarrow (S^{[m]}{{\mathscr E}})\vert_C$ is generically surjective. Since symmetric powers commute with pullbacks (see for instance [@Hartshorne1977 II, Exercise 5.16]), the following morphism $$S^m({{\mathscr E}}\vert_C)=(S^m{{\mathscr E}})\vert_C\rightarrow (S^{[m]}{{\mathscr E}})\vert_C$$ is generically surjective. As ${{\mathscr E}}\hspace{-0.8ex}<\hspace{-0.8ex}\delta\hspace{-0.8ex}>\vert_C$ is nef by assumption, we obtain by Corollary \[Symmetry-wedge\] and Corollary \[Image-nef\] that $(S^{[m]}{{\mathscr E}}\hspace{-0.8ex}<\hspace{-0.8ex}m\delta\hspace{-0.8ex}>)\vert_C$ is nef. For (3), let $C\subseteq X$ be a complete curve not contained in $p({{\mathbb S}}(\zeta_{{{\mathscr E}}}+p^*\delta))$. Then we have the following commutative diagram $$\begin{tikzcd}[column sep=large, row sep=large]
{{\mathbb P}}({{\mathscr E}}|_C)\dar[swap]{p'}\ar[r,"{j}"] & {{\mathbb P}}({{\mathscr E}})\ar[d,"{p}"]\\
C\ar[r,"{i}"] & X
\end{tikzcd}$$ such that $$\zeta_{{{\mathscr E}}|_C}+p'^*(\delta\vert_C)=j^*(\zeta_{{{\mathscr E}}}+p^*\delta).$$ This implies that $\zeta_{{{\mathscr E}}|_C}+p'^*(\delta\vert_C)$ is nef, that is, ${{\mathscr E}}\hspace{-0.8ex}<\hspace{-0.8ex}\delta\hspace{-0.8ex}>\vert_C$ is nef. For (4), let $\nu \colon C\to X$ be finite morphism from a smooth complete curve $C$ to $X$ such that $\nu(C)$ is not contained in $ {{\mathbb S}}({{\mathscr F}}\hspace{-0.8ex}<\hspace{-0.8ex}\delta\hspace{-0.8ex}>)\cup {{\mathbb S}}({{\mathscr G}}\hspace{-0.8ex}<\hspace{-0.8ex}\delta\hspace{-0.8ex}>)$. Then we have an exact sequence $$\nu^*{{\mathscr F}}\rightarrow \nu^*{{\mathscr E}}\rightarrow \nu^*{{\mathscr G}}\rightarrow 0.$$ Let $\nu^*{{\mathscr E}}\rightarrow {{\mathscr L}}$ be a quotient line bundle. Then either the composition $\nu^*{{\mathscr F}}\rightarrow \nu^*{{\mathscr E}}\rightarrow {{\mathscr L}}$ is not zero, or there exists a factorization $\nu^*{{\mathscr E}}\rightarrow \nu^*{{\mathscr G}}\rightarrow {{\mathscr L}}.$ Since both ${{\mathscr F}}\hspace{-0.8ex}<\hspace{-0.8ex}\delta\hspace{-0.8ex}>\vert_C$ and ${{\mathscr G}}\hspace{-0.8ex}<\hspace{-0.8ex}\delta\hspace{-0.8ex}>\vert_C$ are nef, by Corollary \[Image-nef\], we obtain that ${{\mathscr L}}\hspace{-0.8ex}<\hspace{-0.8ex}\delta\hspace{-0.8ex}>\vert_C$ is nef. Hence, ${{\mathscr E}}\hspace{-0.8ex}<\hspace{-0.8ex}\delta\hspace{-0.8ex}>\vert_C$ is nef.
2
Projectively flat vector bundles {#section:proj-flat}
================================
The notion of numerically flat vector bundle was firstly introduced in [@DemaillyPeternellSchneider1994 Definition 1.7]. We can extend this to $\mathbb{Q}$-twisted vector bundles as follows: a $\mathbb{Q}$-twisted vector bundle ${{\mathscr E}}\hspace{-0.8ex}<\hspace{-0.8ex}\delta\hspace{-0.8ex}>$ is numerically flat if it is nef and it has trivial first Chern class. We also introduce the following definition.
Let ${{\mathscr E}}$ be a vector bundle on a projective manifold $X$ with projectivization $p\colon {{\mathbb P}}({{\mathscr E}})\rightarrow X$. The normalized tautological class $\Lambda_{{{\mathscr E}}}$ is defined as $\zeta_{{{\mathscr E}}}-p^*\mu({{\mathscr E}})\in N^1({{\mathbb P}}({{\mathscr E}}))_{{{\mathbb Q}}}$. ${{\mathscr E}}$ is called *numerically projectively flat* if $\Lambda_{{{\mathscr E}}}$ is nef.
Equivalently, a vector bundle ${{\mathscr E}}$ is numerically projectively flat if and only if the $\mathbb{Q}$-twisted vector bundle ${{\mathscr E}}\hspace{-0.8ex}<\hspace{-0.8ex}-\mu({{\mathscr E}})\hspace{-0.8ex}>$ is numerically flat.
Characterization of numerically projectively flat vector bundles
----------------------------------------------------------------
Recall that a $C^\infty$ complex vector bundle ${{\mathscr E}}$ is called *projectively flat* if there exists an affine connection $\nabla$ such that its curvature $\nabla^2=\alpha\cdot {{\rm id}}_{{{\mathscr E}}}$ for some complex $2$-form $\alpha$. A holomorphic vector bundle ${{\mathscr E}}$ is *projectively Hermitian flat* if it admits a smooth Hermitian metric $h$ such that its Chern curvature tensor $R=\nabla^2$ can be written as $R=\alpha \cdot{{\rm id}}_{{{\mathscr E}}}$ for some $2$-form $\alpha$. In particular, the associated projectivized bundle ${{\mathbb P}}({{\mathscr E}})$ is given by a representation $\pi_1(X)\rightarrow PU(r)$ (see [@Nakayama2004 Corollary 4.3]). The following theorem is derived from the study of stable vector bundles and Einstein-Hermitian metrics by Narasimhan-Seshadri [@NarasimhanSeshadri1965], Mehta-Ramanathan [@MehtaRamanathan1981/82; @MehtaRamanathan1984], Donaldson [@Donaldson1985], Uhlenbeck-Yau [@UhlenbeckYau1986], and Bando-Siu [@BandoSiu1994]. One can find a complete proof in [@Nakayama2004 IV, Theorem 4.1].
\[thm:num-proj-flat-criterion-Nakayama\] Let ${{\mathscr E}}$ be a reflexive sheaf of rank $r$ on a projective manifold $X$ of dimension $n$. Then the following assertions are equivalent.
1. ${{\mathscr E}}$ is a numerically projectively flat vector bundle;
2. ${{\mathscr E}}$ is semistable with respect to some ample divisor $A$ and the following equality holds: $$\left(c_2({{\mathscr E}})-\frac{r-1}{2r}c_1^2({{\mathscr E}})\right)\cdot A^{n-2}=0;$$
3. ${{\mathscr E}}$ is a vector bundle and there exists a filtration of subbundles $$\{0\}={{\mathscr E}}_0\subsetneq {{\mathscr E}}_1\subsetneq \dots\subsetneq {{\mathscr E}}_{p-1}\subsetneq {{\mathscr E}}_p={{\mathscr E}}$$ such that ${{\mathscr E}}_i/{{\mathscr E}}_{i+1}$ are projectively Hermitian flat and that the averaged first Chern classes $\mu({{\mathscr E}}_i/{{\mathscr E}}_{i-1})$ are all equal to $\mu({{\mathscr E}})$.
One can easily derive the following lemma from definition and Theorem \[thm:num-proj-flat-criterion-Nakayama\].
\[lemma:prop-num-proj-flat\] Let ${{\mathscr E}}$ be a numerically projectively flat vector bundle on a projective manifold $Y$.
1. If $\det({{\mathscr E}})$ is nef, then ${{\mathscr E}}$ is nef.
2. If ${{\mathscr L}}$ is a line bundle on $Y$, then ${{\mathscr E}}\otimes {{\mathscr L}}$ is numerically projectively flat.
3. If $f\colon X\rightarrow Y$ is a morphism from a projective manifold $X$ to $Y$, then $f^*{{\mathscr E}}$ is numerically projectively flat.
4. ${{\mathscr E}}^*$ is numerically projectively flat.
In [@HoeringPeternell2019], Höring and Peternell characterized numerically flat vector bundles by using almost nefness, instead of nefness in the original definition of [@DemaillyPeternellSchneider1994]. Here we quote their theorem in a special case and refer the readers to [@HoeringPeternell2019] for the complete statement.
[@HoeringPeternell2019 Theorem 1.8]\[HP-Num-flatness\] Let $X$ be an $n$-dimensional projective manifold. Let ${{\mathscr F}}$ be an almost nef reflexive coherent sheaf on $X$ such that $c_1({{\mathscr F}})\cdot A^{n-1}=0$ for some ample divisor $A$ on $X$. Then ${{\mathscr F}}$ is locally free and numerically flat.
According to [@HoeringPeternell2019 Theorem 1.8], there exists a finite cover $\gamma\colon \widetilde{X}\rightarrow X$, étale in codimension one, such that the reflexive pullback $\gamma^{[*]}{{\mathscr F}}$ is locally free and numerically flat. Since $X$ is smooth, $\gamma$ is actually étale and $\gamma^*{{\mathscr F}}=\gamma^{[*]}{{\mathscr F}}$. Since $\gamma$ is étale, this implies that ${{\mathscr F}}$ itself is locally free and numerically flat.
Before giving the proof of Theorem \[Num-Projectily-flatness-criterion\], we need the following elementary lemma.
\[Chern-class-symmetric-power\] Let $X$ be a projective manifold of dimension $n\geqslant 2$, and let ${{\mathscr F}}$ be a reflexive sheaf of rank $r\geqslant 2$ on $X$. For any positive integer $m\geqslant 2$, we have $$\label{Chern class}
c_2(S^{[m]}{{\mathscr F}})=Ac_1^2({{\mathscr F}})+Bc_2({{\mathscr F}}),$$ where $A$ and $B$ are non-zero rational numbers depending only on $m$ and $r$, and satisfy $$\label{AB}
A+\frac{r-1}{2r}B-\frac{(R-1)Rm^2}{2r^2}=0,$$ where $R=\binom{r+m-1}{r}$ is the rank of $S^{[m]}{{\mathscr F}}$.
The existence of the expression is clear and the splitting principle asserts that $A$ and $B$ depend only on $m$ and $r$. To prove , it suffices to prove it for some special ${{\mathscr F}}$ by the universal property of $A$ and $B$.
Firstly we choose an ample line bundle ${{\mathscr L}}$ on $X$ and let ${{\mathscr F}}={{\mathscr L}}^{\oplus r}$. Then $S^m{{\mathscr F}}\otimes{{\mathscr L}}^{*\otimes m}$ is a trivial vector bundle. In particular, we have $$c_2(S^m{{\mathscr F}}\otimes {{\mathscr L}}^{*\otimes m})=0.$$ By the formula of second Chern class of tensor products, we obtain that $$\begin{aligned}
c_2(S^m{{\mathscr F}})+(R-1)c_1(S^m{{\mathscr F}})\cdot (-m)c_1({{\mathscr L}})+\binom{R}{2}m^2c_1^2({{\mathscr L}})=0.
\end{aligned}$$ On the other hand, since ${{\mathscr F}}$ is numerically projectively flat, we have $$\left(c_2({{\mathscr F}})-\frac{r-1}{2r}c_1^2({{\mathscr F}})\right)\cdot A^{n-2}=0$$ for any ample divisor $A$. For $c_1({{\mathscr F}})=rc_1({{\mathscr L}})$ and $c_1(S^m{{\mathscr F}})=\frac{Rm}{r}c_1({{\mathscr F}})$, one has $$\left(A+\frac{r-1}{2r}B+\frac{(R-1)Rm^2}{2r^2}\right)c_1^2({{\mathscr F}})\cdot A^{n-2}=0.$$ Since $c_1({{\mathscr F}})$ is ample, we must have $c_1^2({{\mathscr F}})\cdot A^{n-2}\not=0$. This shows that holds. To see that $A$ and $B$ are non-zero, we may consider the vector bundles
${{\mathscr F}}'\cong {{\mathscr O}}_X^{\oplus (r-1)}\oplus {{\mathscr L}}$ and ${{\mathscr F}}''\cong
{{\mathscr O}}_{X}^{\oplus (r-2)}\otimes {{\mathscr L}}^*\oplus{{\mathscr L}}$.
Then $c_1({{\mathscr F}}')\not=0$, $c_2({{\mathscr F}}')\not=0$. A straightforward computation shows that $c_2(S^m({{\mathscr F}}'))\not=0$ for $m\geqslant 2$. In particular, $A$ is non-zero. Similarly, one can show $B$ is non-zero.
Suppose $C$ is a curve cut out by general elements in $\vert kA\vert$ for $k\gg 0$. Then $C$ is disjoint from $\mathrm{Sing}({{\mathscr F}})$. In particular, ${{\mathscr F}}$ is locally free along $C$ and ${{\mathscr F}}\hspace{-0.8ex}<\hspace{-0.8ex}\delta\hspace{-0.8ex}>\vert_C$ is a ${{\mathbb Q}}$-twisted nef vector bundle. Moreover, we have $c_1({{\mathscr F}}\hspace{-0.8ex}<\hspace{-0.8ex}\delta\hspace{-0.8ex}>\vert_C)=0$ as $(c_1({{\mathscr F}})+r\delta)\cdot C=0$. This implies that ${{\mathscr F}}\vert_C$ is semistable. By Mehta-Ramanathan theorem, ${{\mathscr F}}$ is $A$-semistable.
Let $m$ be a positive integer such that $m\delta$ is Cartier. Let ${{\mathscr L}}$ be a line bundle such that $c_1({{\mathscr L}})=m\delta$. Since ${{\mathscr F}}\hspace{-0.8ex}<\hspace{-0.8ex}\delta\hspace{-0.8ex}>$ is almost nef, so is ${{\mathscr E}}\coloneqq S^{[m]}{{\mathscr F}}\otimes {{\mathscr L}}$ by Proposition \[Almost-nef-properties\]. On the other hand, we know $$c_1({{\mathscr E}})\cdot A^{n-1}=\left(\frac{Rm}{r}c_1({{\mathscr F}})+Rc_1({{\mathscr L}})\right)\cdot A^{n-1}=\frac{Rm}{r}(c_1({{\mathscr F}})+r\delta)\cdot A^{n-1}=0,$$ where $R$ is the rank of $S^{[m]}{{\mathscr F}}$. Then we have $c_1({{\mathscr E}})=0$ since $\det({{\mathscr E}})$ is almost nef (see [@Peternell1994 Lemma 6.5]). In particular, $\delta=-\mu({{\mathscr F}})$.
Moreover, by Theorem \[HP-Num-flatness\], ${{\mathscr E}}$ is locally free and numerically flat. In particular, we have $c_2({{\mathscr E}})\cdot H^{n-2}=0$. As $c_1({{\mathscr F}})=-r\delta$, we have $$\begin{aligned}
c_2({{\mathscr E}}) & =c_2(S^{[m]}{{\mathscr F}})+(R-1)c_1(S^{[m]}{{\mathscr F}})\cdot c_1({{\mathscr L}})+\frac{R(R-1)}{2}c_1^2({{\mathscr L}})\\
& = c_2(S^{[m]}{{\mathscr F}})-\frac{(R-1)Rm^2}{r^2}c^2_1({{\mathscr F}})+\frac{(R-1)Rm^2}{2r^2}c_1^2({{\mathscr F}})\\
& = c_2(S^{[m]}{{\mathscr F}})- \frac{(R-1)Rm^2}{2r^2}c_1^2({{\mathscr F}}).
\end{aligned}$$ Then Lemma \[Chern-class-symmetric-power\] yields $$\left(c_2({{\mathscr F}})-\frac{r-1}{2r}c_1^2({{\mathscr F}})\right)\cdot A^{n-2}=0.$$ Thanks to Theorem \[thm:num-proj-flat-criterion-Nakayama\], we conclude that ${{\mathscr F}}$ is a numerically projectively flat vector bundle.
Projectively flat connections
-----------------------------
In this subsection, we prove Theorem \[thm:num-proj-flat=proj-flat\] and it can be deduced from the following theorem.
\[thm:Kahler:proj-flat-connection\] Let $(X,\omega)$ be a compact Kähler manifold and ${{\mathscr E}}$ a holomorphic vector bundle on $X$. Assume that there exists a filtration of subbundles $$\{0\}={{\mathscr E}}_0\subsetneq {{\mathscr E}}_1\subsetneq \dots\subsetneq {{\mathscr E}}_{p-1}\subsetneq {{\mathscr E}}_p={{\mathscr E}}$$ such that ${{\mathscr E}}_i/{{\mathscr E}}_{i+1}$ are projectively Hermitian flat and that the averaged first Chern classes $\mu({{\mathscr E}}_i/{{\mathscr E}}_{i-1})$ are all equal to $\mu({{\mathscr E}})$. Then ${{\mathscr E}}$ is isomorphic to a projectively flat holomorphic vector bundle ${{\mathscr F}}$, i.e. there exists a projectively flat connection $\nabla$ on ${{\mathscr F}}$ such that $\nabla^{0,1}=\bar\partial_{{{\mathscr F}}}$ where $\nabla^{0,1}$ is the $(0,1)$-part of $\nabla$.
It is well-known that an affine connection $\nabla$ on a $C^\infty$ complex vector bundle ${{\mathscr Q}}$ defines a holomorphic vector bundle structure on ${{\mathscr Q}}$ if $(\nabla^{0,1})^2=0$ where $\nabla^{0,1}$ is the $(0,1)$ part of $\nabla$ (see e.g. [@Kobayashi2014 Proposition 1.3.7]). If in addition $({{\mathscr Q}}, \nabla)$ is projectively flat, then the projective bundle $\mathbb{P}({{\mathscr Q}})$ is induced by a representation of the fundamental group $\pi_1(X)$ in $\mathrm{PGL}_{r}(\mathbb{C})$, where $r$ is the rank of ${{\mathscr G}}$. The strategy of the proof of Theorem \[thm:Kahler:proj-flat-connection\] is to construct a new holomorphic structure on ${{\mathscr Q}}$, where ${{\mathscr Q}}$ is the underlying $C^\infty$ complex vector bundle of the holomorphic vector bundle ${{\mathscr E}}$, and this new holomorphic structure is isomorphic to ${{\mathscr E}}$. We recall some elementary results.
\[lemma:unique-proj-flat-connection\] Let $(X,\omega)$ a compact Kähler manifold and let ${{\mathscr E}}$ be a $[\omega]$-stable vector bundle of rank $r$ on $X$. If ${{\mathscr E}}$ satisfies $$\label{BG}
\int_X \left((r-1) c_1^2({{\mathscr E}})-2r c_2({{\mathscr E}})\right)
\omega^{n-1}=0,$$ then there exists a smooth Hermitian metric $h^{{\mathscr E}}$ on ${{\mathscr E}}$ with Chern connection $\nabla^{{\mathscr E}}$ such that the Chern curvature satisfies $$R^{{\mathscr E}}= (\nabla^{{\mathscr E}})^2 = \gamma \cdot h^{{\mathscr E}},$$ where $\gamma$ is the unique harmonic representative of the average first Chern class $\frac{1}{r}c_1({{\mathscr E}})\in H_{{{\bar\partial}}}^{1,1}(X,\mathbb{C})$ with respect to the Kähler metric $\omega$, i.e. ${{\bar\partial}}\gamma={{\bar\partial}}^*\gamma=0$.
Since ${{\mathscr E}}$ is $[\omega]$-stable, by the Donaldson-Uhlenbeck-Yau theorem, there exists a smooth Hermitian-Einstein metric $h^{{\mathscr E}}$ on ${{\mathscr E}}$. That means $\Lambda_\omega
R^{{\mathscr E}}=c\cdot h^{{\mathscr E}}$ for some constant $c$. By [@Kobayashi2014 Theorem 4.4.7], the equality (\[BG\]) implies that $({{\mathscr E}},h)$ is projectively flat, that is $R^{{\mathscr E}}=\frac{1}{r} \eta \cdot h^{{\mathscr E}}$ for some smooth closed $(1,1)$-form $\eta$. By taking the trace of $R^{{\mathscr E}}$ with respect to $h^{{\mathscr E}}$, we have $$R^{\det {{\mathscr E}}}=-\sqrt{-1} \partial {{\bar\partial}}\log(\det
h^{{\mathscr E}})=\mathrm{tr}_{h^{{\mathscr E}}} R^{{\mathscr E}}=\eta.$$ This shows that class of $\eta$ is equal to the first Chern class of ${{\mathscr E}}$. Finally, since $h^{{\mathscr E}}$ is Hermitian-Einstein, we also have $\Lambda_\omega \eta=rc$. Hence ${{\bar\partial}}^*\eta=-\sqrt{-1}[\Lambda_\omega,\partial]\eta=0$. This implies that $\eta$ is harmonic. The lemma then follows by setting $\gamma=\frac{1}{r}\eta.$
\[DUY\] Let $(X,\omega)$ a compact Kähler manifold. Suppose ${{\mathscr E}}$ and ${{\mathscr F}}$ are $[\omega]$-stable vector bundles with $$\label{average}
\frac{c_1({{\mathscr E}})}{\mathrm{rank}({{\mathscr E}})}=\frac{c_1({{\mathscr F}})}{\mathrm{rank}({{\mathscr F}})}\in
H_{{{\bar\partial}}}^{1,1}(X,\mathbb{C})\cap H^2(X,\mathbb{Q}).$$ If both ${{\mathscr E}}$ and ${{\mathscr F}}$ satisfy the equality (\[BG\]), then there exist Hermitian-Einstein metrics $h^{{\mathscr E}}$ and $h^{{\mathscr F}}$ on ${{\mathscr E}}$ and ${{\mathscr F}}$ respectively, such that $$R^{{\mathscr E}}=\gamma \cdot h^{{\mathscr E}},\ \ \ R^{{\mathscr F}}=\gamma \cdot h^{{\mathscr F}}$$ where $\gamma$ is the unique harmonic representative of the class of (\[average\]). In particular, the metric on ${{\mathscr E}}^*\otimes {{\mathscr F}}$ induced by $h^{{\mathscr E}}$ and $h^{{\mathscr F}}$ is Hermitian flat.
The following lemma reveals the relationship between isomorphism classes of holomorphic structures and Dolbeault cohomology groups.
\[lemma:coh-class-iso-holomorphic-class\] Let ${{\mathscr F}}$ and ${{\mathscr G}}$ be two holomorphic vector bundles on a complex manifold $X$. We denote by ${{\mathscr Q}}$ the underlying $C^\infty$ vector bundle of ${{\mathscr G}}\oplus {{\mathscr F}}.$ Then there is a bijection between the cohomology group $H^{0,1}_{{{\bar\partial}}}(X, {{\mathscr F}}^*\otimes {{\mathscr G}})$ and the set of isomorphism classes of holomorphic vector bundle structure ${{\mathscr E}}$ on ${{\mathscr Q}}$ which realizes ${{\mathscr E}}$ as an extension $0\to {{\mathscr G}}\to {{\mathscr E}}\to {{\mathscr F}}\to 0$ of holomorphic vector bundles.
To define such a holomorphic structure on ${{\mathscr Q}}$, it is equivalent to give a $(0,1)$-connection $\nabla^{0,1}$ on ${{\mathscr Q}}$ of the form $$\nabla^{0,1}=\left[\begin{array}{cc} {{\bar\partial}}_{{\mathscr G}}&\eta\\
0&{{\bar\partial}}_{{\mathscr F}}\end{array}\right]$$ such that $(\nabla^{0,1})^2=0$, where $\eta \in A^{0,1}(X,{{\mathscr F}}^*\otimes {{\mathscr G}})$ is a smooth $(0,1)$-form.
Now we assume that $\eta$ and $\eta'$ are two elements in $A^{0,1}(X,{{\mathscr F}}^*\otimes {{\mathscr G}}).$ Then they induce isomorphic extension structures, ${{\mathscr E}}$ and ${{\mathscr E}}'$, on ${{\mathscr Q}}$ if and only if there is some smooth form $\alpha \in A^0(X,{{\mathscr F}}^*\otimes {{\mathscr G}})$ such that $\eta'=\eta+{{\bar\partial}}\alpha$. Moreover, the corresponding isomorphism from ${{\mathscr E}}$ to ${{\mathscr E}}'$, as a smooth automorphism of ${{\mathscr Q}}$, is expressed as $$\left[\begin{array}{cc} 1_{{\mathscr G}}& \alpha\\
0&1_{{\mathscr F}}\end{array}\right].$$ This finishes the proof.
In the next lemma, we prove Theorem \[thm:Kahler:proj-flat-connection\] in a special case when ${{\mathscr E}}$ is an extension of two projectively Hermitian flat vector bundles.
\[bb\] Let $(X,\omega)$ be a compact Kähler manifold and let $$0\to {{\mathscr G}}_1\to {{\mathscr E}}\to {{\mathscr G}}_2\to 0$$ be an exact sequence of holomorphic vector bundles on $X$. Suppose that ${{\mathscr E}}$ is $[\omega]$-semistable and satisfies the equality (\[BG\]), and that ${{\mathscr G}}_1$ and ${{\mathscr G}}_2$ are $[\omega]$-stable with the same average first Chern class. Let ${{\mathscr Q}}$ be the underlying $C^\infty$ vector bundle of ${{\mathscr G}}_1\oplus {{\mathscr G}}_2$. Then there exists a projectively flat connection $\nabla^{{\mathscr F}}$ on ${{\mathscr Q}}$ such that it defines a holomorphic structure ${{\mathscr F}}$ isomorphic to ${{\mathscr E}}$.
It is easy to see that if ${{\mathscr E}}$ is $[\omega]$-semistable and satisfies (\[BG\]), then both ${{\mathscr G}}_1$ and ${{\mathscr G}}_2$ satisfy (\[BG\]). By Corollary \[DUY\], there exist Hermitian-Einstein metrics $h^{{{\mathscr G}}_1}$ and $h^{{{\mathscr G}}_2}$ on ${{\mathscr G}}_1$ and ${{\mathscr G}}_2$ respectively, such that their Chern curvatures are given by $$\label{equal}
R^{{{\mathscr G}}_1}=\gamma \cdot h^{{{\mathscr G}}_1},\ \ \
R^{{{\mathscr G}}_2}=\gamma \cdot h^{{{\mathscr G}}_2},$$ where $\gamma$ is the unique harmonic representative of the average first Chern class of ${{\mathscr G}}_1$. Let $\nabla^{{{\mathscr G}}_1}$ and $\nabla^{{{\mathscr G}}_2}$ be the Chern connections on $({{\mathscr G}}_1,h^{{{\mathscr G}}_1})$ and $({{\mathscr G}}_2,h^{{{\mathscr G}}_2})$ respectively. Then the Chern curvature of $({{\mathscr G}}_2^*\otimes {{\mathscr G}}_1, h^{{{\mathscr G}}_2^*}\otimes h^{{\mathscr G}}_1, \nabla^{{{\mathscr G}}_2^*}\otimes \nabla^{{\mathscr G}}_1)$ satisfies $R^{{{\mathscr G}}_2^*\otimes {{\mathscr G}}_1}=0$. The standard Bochner-Kodaira identity on ${{\mathscr G}}_2^*\otimes {{\mathscr G}}_1$ shows the following equalities on Laplacian operators on $A^{\bullet,\bullet}(X,{{\mathscr G}}_2^*\otimes {{\mathscr G}}_1)$, $$\Delta''=\Delta'+\sqrt{-1}[R^{{{\mathscr G}}_2^*\otimes {{\mathscr G}}_1},\Lambda_\omega]=\Delta'.$$
By partition of unity, there is a connection $\nabla^{{{\mathscr E}}}$ on ${{\mathscr Q}}$ with $(\nabla^{{\mathscr E}})^{0,1}={{\bar\partial}}_{{{\mathscr E}}}$ of the form $$\nabla^{{\mathscr E}}:=\left[\begin{array}{cc} \nabla^{{{\mathscr G}}_1}&\beta\\
0&\nabla^{{{\mathscr G}}_2}
\end{array}\right].$$ By using the metrics $\omega$, $h^{{{\mathscr G}}_2^*}$ and $h^{{{\mathscr G}}_1}$, one has an isomorphism $ H^1(X,{{\mathscr G}}_2^*\otimes {{\mathscr G}}_1))\cong H_{{{\bar\partial}}}^{0,1}(X,{{\mathscr G}}_2^*\otimes {{\mathscr G}}_1).$ Let $\eta \in A^{0,1}(X,{{\mathscr G}}_2^*\otimes {{\mathscr G}}_1)$ be the unique harmonic representative of the class $[\beta]\in H_{{{\bar\partial}}}^{0,1}(X,{{\mathscr G}}_2^*\otimes
{{\mathscr G}}_1)$. Then $\Delta'' \eta =0$. Since $\Delta'=\Delta''$, we deduce that $$\label{harmonic}
\nabla^{{{\mathscr G}}_2^*\otimes
{{\mathscr G}}_1}\eta=0.$$ We define a connection $\nabla^{{\mathscr F}}$ on ${{\mathscr Q}}$ in the form $$\nabla^{{\mathscr F}}:=\left[\begin{array}{cc} \nabla^{{{\mathscr G}}_1}&\eta\\
0&\nabla^{{{\mathscr G}}_2}
\end{array}\right].$$ Then by Lemma \[lemma:coh-class-iso-holomorphic-class\], the connection $\nabla^{{\mathscr F}}$ defines a holomorphic structure ${{\mathscr F}}$ on ${{\mathscr Q}}$ isomorphic to ${{\mathscr E}}$. Moreover, we have $$(\nabla^{{\mathscr F}})^2=\gamma\cdot (h^{{{\mathscr G}}_1}\oplus h^{{{\mathscr G}}_2}).$$ \[flat\] Indeed, this is equivalent to $ \nabla^{{{\mathscr G}}_1}\circ \eta+\eta\circ \nabla^{{{\mathscr G}}_2}=\nabla^{{{\mathscr G}}_2^*\otimes
{{\mathscr G}}_1}\eta=0$ and it follows by the choice of $\eta$. This completes the proof of the lemma.
The next statement addresses a generalization of the construction in Lemma \[bb\].
\[lemma:connection-construction\] Let $\{({{\mathscr G}}_i,h^{{{\mathscr G}}_i}, \nabla_i)\}_{i=1}^p (p\geq 2)$ be a collection of projectively Hermitian flat vector bundles on a compact Kähler manifold $(X,\omega)$. Let ${{\mathscr Q}}$ be the underlying $C^\infty$ vector bundle of $\bigoplus_{i=1}^p {{\mathscr G}}_i$. Assume that there is a connection on ${{\mathscr Q}}$ of the form $$\nabla^{{{\mathscr Q}}}_1 =
\begin{bmatrix}
\nabla_1 & \theta_{1,2} & \cdots & \cdots & \theta_{1,p-1} & \beta_{1,p} \\
& \nabla_2 & \cdots & \cdots& \theta_{2,p-1} & \beta_{2,p} \\
& &\ddots & \cdots & \vdots & \vdots \\
& & & \ddots & \vdots & \vdots \\
& & & & \nabla_{p-1} & \beta_{p-1,p} \\
0 & & & & & \nabla_p
\end{bmatrix}$$ such that for each $i<j$, the $(i,j)$ entry belongs to $A^{0,1}(X, {{\mathscr G}}_j^*\otimes {{\mathscr G}}_i)$. Furthermore, we assume that $(\nabla^{{{\mathscr Q}}}_1)^{0,1})^2=0$ and that $$\begin{bmatrix}
\nabla_1 & \theta_{1,2} & \cdots & \cdots & \theta_{1,p-1} \\
&\nabla_2 & \cdots & \cdots& \theta_{2,p-1} \\
& & \ddots & \cdots & \vdots \\
& & & \ddots & \vdots \\
0 & & & & \nabla_{p-1}
\end{bmatrix}^2
=
\begin{bmatrix}
\gamma h^{{{\mathscr G}}_1} & 0 & \cdots & \cdots & 0 \\
& \gamma h^{{{\mathscr G}}_2} & \cdots & \cdots& 0 \\
& &\ddots & \cdots & \vdots \\
& & & \ddots & \vdots \\
0 & & & & \gamma h^{{{\mathscr G}}_{p-1}}
\end{bmatrix},$$ where $r_i$ is the rank of ${{\mathscr G}}_i$ for $i=1,...,p.$
Then there is a connection on ${{\mathscr Q}}$ of the form $$\nabla^{{{\mathscr Q}}}_2 =
\begin{bmatrix}
\nabla_1 & \theta_{1,2} & \cdots & \cdots & \theta_{1,p-1} & \theta_{1,p} \\
& \nabla_2 & \cdots & \cdots& \theta_{2,p-1} & \theta_{2,p} \\
& &\ddots & \cdots & \vdots & \vdots \\
& & & \ddots & \vdots & \vdots \\
& & & & \nabla_{p-1} & \theta_{p-1,p} \\
0 & & & & & \nabla_p
\end{bmatrix}$$ with $\theta_{i,p}\in A^{0,1}(X, {{\mathscr G}}_p^*\otimes {{\mathscr G}}_i)$ for $i<p$ such that its curvature is of the form $$(\nabla^{{{\mathscr Q}}}_2)^2 = \gamma \cdot (h^{{{\mathscr G}}_1}\oplus\cdots\oplus h^{{{\mathscr G}}_p}).$$ Moreover, the holomorphic structures induced by $\nabla^{{\mathscr Q}}_1$ and $\nabla^{{\mathscr Q}}_2$ are isomorphic.
We prove it by induction on $p$. For $p=2$, it follows from Lemma \[bb\]. Assume that $p>2$ and that the lemma is true for smaller integers. By induction hypothesis, we can find $\theta_{2,p},...,\theta_{p-1,p}$ such that $$\begin{bmatrix}
\nabla_2 & \theta_{2,3} & \cdots & \cdots & \theta_{2,p} \\
&\nabla_3 & \cdots & \cdots& \theta_{3,p} \\
& & \ddots & \cdots & \vdots \\
& & & \ddots & \vdots \\
0 & & & & \nabla_{p}
\end{bmatrix}^2
=
\begin{bmatrix}
\gamma h^{{{\mathscr G}}_2} & 0 & \cdots & \cdots & 0 \\
& \gamma h^{{{\mathscr G}}_3} & \cdots & \cdots& 0 \\
& &\ddots & \cdots & \vdots \\
& & & \ddots & \vdots \\
0 & & & & \gamma h^{{{\mathscr G}}_{p}}
\end{bmatrix}.$$ Moreover, by Lemma \[lemma:coh-class-iso-holomorphic-class\], the column $$\begin{bmatrix}
\theta_{2,p}-\beta_{2,p}\\
\vdots\\
\theta_{p-1,p}-\beta_{p-1,p}
\end{bmatrix}$$ represents a $(0,1)$-form ${{\bar\partial}}_{{{\mathscr G}}_p^*\otimes {{\mathscr T}}} \alpha$, where $\alpha \in A^0(X,{{\mathscr G}}_p^*\otimes \bigoplus_{i=2}^{p-1} {{\mathscr G}}_i)$ and ${{\mathscr T}}$ is the holomorphic structure on $\bigoplus_{i=2}^{p-1} {{\mathscr G}}_i$ induced by $\nabla^{{\mathscr Q}}_1$. Note that $\alpha$ can also be view as an element in $A^0(X, {{\mathscr G}}_p^*\otimes ( \bigoplus_{i=1}^{p-1} {{\mathscr G}}_i))$. Then there is some $\delta \in A^{0,1}(X,{{\mathscr G}}_p^*\otimes {{\mathscr G}}_1)$ such that the column $$\begin{bmatrix}
\delta - \beta_{1,p}\\
\theta_{2,p}-\beta_{2,p}\\
\vdots\\
\theta_{p-1,p}-\beta_{p-1,p}
\end{bmatrix}$$ represents ${{\bar\partial}}_{{{\mathscr G}}_p^*\otimes {{\mathscr H}}} \alpha$, where ${{\mathscr H}}$ is the holomorphic vector bundle structure on $\bigoplus_{i=1}^{p-1} {{\mathscr G}}_i$ induced by $\nabla^{{\mathscr Q}}_1$.
We define the following connection $$\nabla^{{{\mathscr Q}}}_3 =
\begin{bmatrix}
\nabla_1 & \theta_{1,2} & \cdots & \cdots & \theta_{1,p-1} & \delta \\
& \nabla_2 & \cdots & \cdots& \theta_{2,p-1} & \theta_{2,p} \\
& &\ddots & \cdots & \vdots & \vdots \\
& & & \ddots & \vdots & \vdots \\
& & & & \nabla_{p-1} & \theta_{p-1,p} \\
0 & & & & & \nabla_p
\end{bmatrix}.$$ Then $(\nabla^{{{\mathscr Q}}}_3)^{(0,1)})^2=0$ and it defines a holomorphic structure on ${{\mathscr Q}}$ which is isomorphic to the one defined by $\nabla^{{\mathscr Q}}_1$ by Lemma \[lemma:coh-class-iso-holomorphic-class\]. Moreover, we have $$(\nabla^{{{\mathscr Q}}}_3)^2 =
\begin{bmatrix}
\gamma h^{{{\mathscr G}}_1} & 0 & \cdots & \cdots & 0 & \xi \\
& \gamma h^{{{\mathscr G}}_2} & \cdots & \cdots& 0 & 0 \\
& &\ddots & \cdots & \vdots & \vdots \\
& & & \ddots & \vdots & \vdots \\
& & & & \gamma h^{{{\mathscr G}}_{p-1}} & 0 \\
0 & & & & & \gamma h^{{{\mathscr G}}_p}
\end{bmatrix}$$ where $$\xi= \nabla^{{{\mathscr G}}_p^*\otimes {{\mathscr G}}_1}(\delta) + \sum_{i=1}^{p-1}\theta_{1,i}\circ \theta_{i,p} \in A^{2}(X,{{\mathscr G}}_p^*\otimes {{\mathscr G}}_1).$$
The vanishing of the $(1,i)$-entry of $(\nabla^{{{\mathscr Q}}}_3)^2$ for $1<i<p$ implies $$\label{eq-vansh-1}
\nabla_1 \circ \theta_{1,i} = - (\theta_{1,i} \circ \nabla_i + \sum_{j=2}^{i-1} \theta_{1,j}\circ \theta_{j,i}),$$ and the vanishing of the $(i,p)$-entry for $1<i<p$ implies $$\label{eq-vansh-2}
\theta_{i,p} \circ \nabla_p = -(\nabla_i \circ \theta_{i,p} + \sum_{k=i+1}^{p-1} \theta_{i,k} \circ \theta_{k,p}).$$ Moreover, the condition that $((\nabla^{{{\mathscr Q}}}_3)^{(0,1)})^2=0$ shows that $$\label{eq-vansh-3}
{{\bar\partial}}_{{{\mathscr G}}_p^*\otimes {{\mathscr G}}_1}(\delta) + \sum_{i=1}^{p-1}\theta_{1,i}\circ \theta_{i,p} =0 \in A^{2}(X,{{\mathscr G}}_p^*\otimes {{\mathscr G}}_1),$$ and we deduce $$\label{eq-vansh-4}
\xi = (\nabla^{{{\mathscr G}}_p^*\otimes {{\mathscr G}}_1})^{1,0}(\delta).$$
Now we compute that $$\begin{aligned}
{{\bar\partial}}_{{{\mathscr G}}_p^*\otimes {{\mathscr G}}_1}((\nabla^{{{\mathscr G}}_p^*\otimes {{\mathscr G}}_1})^{1,0}(\delta))
&=& {{\bar\partial}}_{{{\mathscr G}}_p^*\otimes {{\mathscr G}}_1} (\nabla^{{{\mathscr G}}_p^*\otimes {{\mathscr G}}_1}(\delta))\\
&=&- \nabla^{{{\mathscr G}}_p^*\otimes {{\mathscr G}}_1}({{\bar\partial}}_{{{\mathscr G}}_p^*\otimes {{\mathscr G}}_1}(\delta))\\
&\stackrel{(\ref{eq-vansh-3})}{=}&\nabla^{{{\mathscr G}}_p^*\otimes {{\mathscr G}}_1}(\sum_{i=1}^{p-1}\theta_{1,i}\circ \theta_{i,p})\\
&=& \sum_{i=1}^{p-1}\nabla_1\circ \theta_{1,i} \circ \theta_{i,p}- \sum_{i=1}^{p-1} \theta_{1,i}\circ\theta_{i,p}\circ
\nabla_p\\
&\stackrel{(\ref{eq-vansh-1}),\ (\ref{eq-vansh-2})}{=}& \sum_{i=1}^{p-1} \left(- (\theta_{1,i} \circ \nabla_i + \sum_{j=2}^{i-1} \theta_{1,j}\circ \theta_{j,i}) \circ \theta_{i,p}\right) \\
&& -\sum_{i=1}^{p-1} \left(-\theta_{1,i}\circ (\nabla_i \circ \theta_{i,p} + \sum_{k=i+1}^{p-1} \theta_{i,k} \circ \theta_{k,p})\right)\\
&=& -\sum_{i=1}^{p-1} \sum_{j=2}^{i-1} \theta_{1,j}\circ \theta_{j,i} \circ \theta_{i,p} \\
&& +\sum_{i=1}^{p-1}\sum_{k=i+1}^{p-1}\theta_{1,i}\circ \theta_{i,k} \circ \theta_{k,p}\\
&=&0.
\end{aligned}$$ We note that ${{\mathscr G}}_p^*\otimes {{\mathscr G}}_1$ is Hermitian flat by Corollary \[DUY\]. Hence by $\partial{{\bar\partial}}$-Lemma, there exists $\mu \in A^0(X,{{\mathscr G}}_p^*\otimes {{\mathscr G}}_1)$ such that $$(\nabla^{{{\mathscr G}}_p^*\otimes {{\mathscr G}}_1})^{1,0}(\delta) =(\nabla^{{{\mathscr G}}_p^*\otimes {{\mathscr G}}_1})^{1,0} ( {{\bar\partial}}_{{{\mathscr G}}_p^*\otimes
{{\mathscr G}}_1}(\mu))= \nabla^{{{\mathscr G}}_p^*\otimes {{\mathscr G}}_1} ( {{\bar\partial}}_{{{\mathscr G}}_p^*\otimes
{{\mathscr G}}_1}(\mu)).$$ Thanks to equation (\[eq-vansh-4\]), this implies that $$\xi = \nabla^{{{\mathscr G}}_p^*\otimes {{\mathscr G}}_1} ( {{\bar\partial}}_{{{\mathscr G}}_p^*\otimes {{\mathscr G}}_1}(\mu)).$$ Let $\theta_{1,p}=\delta- {{\bar\partial}}_{{{\mathscr G}}_p^*\otimes {{\mathscr G}}_1}(\mu)$. Then we check that $$\nabla^{{{\mathscr G}}_p^*\otimes {{\mathscr G}}_1}(\theta_{1,p}) + \sum_{i=1}^{p-1}\theta_{1,i}\circ \theta_{i,p}
= \xi - \nabla^{{{\mathscr G}}_p^*\otimes {{\mathscr G}}_1}({{\bar\partial}}_{{{\mathscr G}}_p^*\otimes {{\mathscr G}}_1}(\mu)) = 0.$$ This completes the proof of Lemma \[lemma:connection-construction\].
Now we are ready to prove Theorem \[thm:Kahler:proj-flat-connection\] and Theorem \[thm:num-proj-flat=proj-flat\].
Let $r_i$ be the rank of ${{\mathscr G}}_i = {{\mathscr E}}_i/{{\mathscr E}}_{i+1} $. As in Lemma \[lemma:unique-proj-flat-connection\], we denote the projectively Hermitian flat metric on ${{\mathscr G}}_i$ by $h^{{{\mathscr G}}_i}$ and the Chern connection by $\nabla_i$. Then there is a $(1,1)$-form $\gamma$ such that $\nabla_i^2=\gamma h^{{{\mathscr G}}_i}$ for all $i=1,..,p$. We denote by ${{\mathscr Q}}$ the underlying smooth vector bundle of ${{\mathscr E}}$. To prove the theorem, we will construct a connection $\nabla^{{{\mathscr F}}}$ on ${{\mathscr Q}}$ in the following upper triangular form $$\nabla^{{{\mathscr F}}} =
\begin{bmatrix}
\nabla_1 & \theta_{1,2} & \cdots & \cdots & \theta_{1,p-1} & \theta_{1,p} \\
& \nabla_2 & \cdots & \cdots& \theta_{2,p-1} & \theta_{2,p} \\
& &\ddots & \cdots & \vdots & \vdots \\
& & & \ddots & \vdots & \vdots \\
& & & & \nabla_{p-1} & \theta_{p-1,p} \\
0 & & & & & \nabla_p
\end{bmatrix}$$ such that
1. $\theta_{i,j} \in A^{0,1}(X,{{\mathscr G}}_j^* \otimes {{\mathscr G}}_i)$ are global smooth $(0,1)$-forms;
2. $\nabla^{{\mathscr F}}$ defines a holomorphic structure ${{\mathscr F}}$ on ${{\mathscr Q}}$ which is isomorphic to ${{\mathscr E}}$;
3. $\nabla^{{\mathscr F}}$ is projectively flat.
By partition of unity, we can construct a connection $\nabla^{{{\mathscr Q}}}_1$ on ${{\mathscr Q}}$ of the form $$\nabla^{{{\mathscr Q}}}_1 =
\begin{bmatrix}
\nabla_1 & \beta_{1,2} & \cdots & \cdots & \beta_{1,p-1} & \beta_{1,p} \\
& \nabla_2 & \cdots & \cdots& \beta_{2,p-1} & \beta_{2,p} \\
& &\ddots & \cdots & \vdots & \vdots \\
& & & \ddots & \vdots & \vdots \\
& & & & \nabla_{p-1} & \beta_{p-1,p} \\
0 & & & & & \nabla_p
\end{bmatrix}$$ such that $\beta_{i,j} \in A^{1}(X,{{\mathscr G}}_j^* \otimes {{\mathscr G}}_i)$ and that $(\nabla^{{{\mathscr Q}}}_1)^{0,1}={{\bar\partial}}_{{{\mathscr E}}}$. By replacing $\beta_{i,j}$ by its $(0,1)$ part, we may assume further that $\beta_{i,j} \in A^{0,1}(X,{{\mathscr G}}_j^* \otimes {{\mathscr G}}_i)$.
We prove it by induction on $p$. If $p=2$, then it follows from Lemma \[bb\]. Assume $p>2$ and the assertion holds for smaller integers. Then by induction hypothesis, we can find $\theta_{i,j} \in A^{0,1}(X,{{\mathscr G}}_j^* \otimes {{\mathscr G}}_i)$ for $1\leqslant i<j\leqslant p-1$ such that $$\begin{bmatrix}
\nabla_1 & \theta_{1,2} & \cdots & \cdots & \theta_{1,p-1} \\
&\nabla_2 & \cdots & \cdots& \theta_{2,p-1} \\
& & \ddots & \cdots & \vdots \\
& & & \ddots & \vdots \\
0 & & & & \nabla_{p-1}
\end{bmatrix}^2
=
\begin{bmatrix}
\gamma h^{{{\mathscr G}}_1} & 0 & \cdots & \cdots & 0 \\
& \gamma h^{{{\mathscr G}}_2} & \cdots & \cdots& 0 \\
& &\ddots & \cdots & \vdots \\
& & & \ddots & \vdots \\
0 & & & & \gamma h^{{{\mathscr G}}_{p-1}}
\end{bmatrix}.$$
Moreover, the following connection on ${{\mathscr Q}}$ $$\nabla^{{{\mathscr Q}}}_2 =
\begin{bmatrix}
\nabla_1 & \theta_{1,2} & \cdots & \cdots & \theta_{1,p-1} & \beta_{1,p} \\
& \nabla_2 & \cdots & \cdots& \theta_{2,p-1} & \beta_{2,p} \\
& &\ddots & \cdots & \vdots & \vdots \\
& & & \ddots & \vdots & \vdots \\
& & & & \nabla_{p-1} & \beta_{p-1,p} \\
0 & & & & & \nabla_p
\end{bmatrix}$$ defines a holomorphic structure isomorphic to ${{\mathscr E}}$. The existence of $\nabla^{{\mathscr F}}$ then follows from Lemma \[lemma:connection-construction\]. This completes the proof of Theorem \[thm:Kahler:proj-flat-connection\].
It follows from Theorem \[thm:num-proj-flat-criterion-Nakayama\] and Theorem \[thm:Kahler:proj-flat-connection\].
In the sequel of this paper, a holomorphic vector bundle ${{\mathscr E}}$ over a quasi-projective manifold $X$ is called a projectively flat vector bundle if it admits a projectively flat connection $\nabla$ such that $\nabla^{0,1}=\bar{\partial}_{{{\mathscr E}}}$.
Properties of projectively flat vector bundles
----------------------------------------------
We underline that all vector bundles in this subsection are holomorphic and an isomorphism between vector bundles is always an isomorphism of holomorphic vector bundles.
\[lemma:extension-proj-flat-vec-bundle\] Let $X$ be a smooth quasi-projective variety and let ${{\mathscr E}}$ be a vector bundle on $X$. Assume that there is an open subset $X^\circ \subseteq X$ with complement of codimension at least $2$ such that ${{\mathscr E}}|_{X^\circ}$ is isomorphic to a projectively flat vector bundle ${{\mathscr F}}^\circ$. Then this isomorphism extends to an isomorphism of vector bundles ${{\mathscr E}}\to {{\mathscr F}}$ such that ${{\mathscr F}}$ is projectively flat.
Let $Z^\circ = \mathbb{P}({{\mathscr F}}^\circ)$. Then $Z^\circ$ is given by a representation $\rho^\circ$ of the fundamental group $\pi_1(X^\circ)$ in $\mathrm{PGL}_{r}(\mathbb{C})$, where $r$ is the rank of ${{\mathscr E}}$. Since the complement of $X^\circ$ has codimension at least 2 in $X$, the fundamental groups $\pi_1(X^\circ)$ and $\pi_1(X)$ are canonically isomorphic. Hence $\rho$ descends to a representation $\rho\colon \pi_{1}(X) \to \mathrm{PGL}_r(\mathbb{C})$. Such a representation induces a projective bundle $p\colon Z \to X$ which extends $Z^\circ \to X^\circ$. Let ${{\mathscr L}}^\circ = {{\mathscr O}}_{\mathbb{P}({{\mathscr F}}^\circ)}(1)$. Then it extends to a line bundle ${{\mathscr L}}$ on $Z$ whose restriction on a general fiber of $p$ is isomorphic to ${{\mathscr O}}_{{{\mathbb P}}^{r-1}}(1)$. Let ${{\mathscr F}}= p_*{{\mathscr L}}$. Then ${{\mathscr F}}$ is a projectively flat vector bundle such that ${{\mathscr F}}|_{X^\circ}={{\mathscr F}}^\circ$. Since the complement of $X^\circ$ has codimension at least 2 in $X$, the isomorphism ${{\mathscr E}}|_{X^\circ} \to {{\mathscr F}}^\circ$ extends to an isomorphism ${{\mathscr E}}\to {{\mathscr F}}$.
\[lemma:proj-flat-vec-bundle-property2\] Let $f:X\to Y$ be a projective bundle over a quasi-projective manifold $Y$. Assume that ${{\mathscr E}}$ is a vector bundle on $Y$ such that $f^*{{\mathscr E}}$ is isomorphic to a projectively flat vector bundle on $X$. Then ${{\mathscr E}}$ is isomorphic to a projectively flat vector bundle on $Y$.
Up to isomorphism, we may assume that ${{\mathscr F}}= f^*{{\mathscr E}}$ is projectively flat. Let $Q={{\mathbb P}}({{\mathscr E}})$ and $W = {{\mathbb P}}({{\mathscr F}})$. Then $W$ is given by a representation $\rho$ of $\pi_1(X)$ in $\mathrm{PGL}_{r}(\mathbb{C})$, where $r$ is the rank of ${{\mathscr E}}$. We remark that $f$ induces an isomorphism between the fundamental groups $\pi_1(X)$ and $\pi_1(Y)$. Thus such a representation induces a representation $\eta\colon \pi_1(Y) \to \mathrm{PGL}_r(\mathbb{C})$. Let $\pi\colon P\to Y$ be the projective bundle given by $\eta$. Then by pulling back $P$, we obtain a projective bundle $q\colon Z\to X$. By construction, we have an isomorphism $Z\to W$ of projective bundles on $X$.
We denote by $\varphi\colon Z\to Q$ the composition of $Z\to W \to Q.$ Then a complete curve $C$ in $Z$ is contracted by $\varphi$ if and only if the image of $C$ in $P$ is a point. Hence by rigidity lemma, we see that $\varphi$ descends to a surjective morphism $\psi\colon P \to Q$ on $Y$. Since both $P$ and $Q$ are projective bundles on $Y$, such a surjective morphism must be an isomorphism. Note that the tensor product of a projectively flat vector bundle and a line bundle is again projectively flat, ${{\mathscr E}}$ is isomorphic to a projectively flat vector bundle.
The following lemma is a consequence of [@GrebKebekusPeternell2016b Theorem 1.5].
\[lemma:trivial-bundle-finite-change\] Let $Y$ be a projective manifold and $p \colon Y \to Y'$ a surjective morphism onto a normal projective variety with klt singularities. Let $P\rightarrow Y$ be a flat projective bundle given by a representation $\pi_1(Y)\rightarrow \operatorname*{PGL}_{d+1}({{\mathbb C}})$. Assume that the fibers of $p$ are simply connected over some open subset $V\subseteq Y'$ such that ${{\rm codim}}Y'\backslash V \geqslant 2$. Then for every $p$-exceptional curve $C$, there is a smooth curve $D$ with finite surjective morphism $\mu \colon D \to C$ such that $P$ induces a trivial $\mathbb{P}^d$-bundle structure on $D \times_C P$ over $D$. In particular, if $P\cong {{\mathbb P}}({{\mathscr E}})$ for some vector bundle ${{\mathscr E}}$ on $Y$, then $\mu^*{{\mathscr E}}$ is isomorphic to the direct sum of copies of a line bundle on $D$.
By [@GrebKebekusPeternell2016b Theorem 1.5], there is a quasi-étale cover $Z'\to Y'$ such that the algebraic fundamental groups $\hat{\pi}_1(Z'_{\mathrm{reg}})$ and $\hat{\pi}_1(Z')$ are canonically isomorphic. Let $Z$ be a desingularization of the main component of the fiber product $Y\times_{Y'} Z'$.
$$\begin{tikzcd}[column sep=large, row sep=large]
Z \rar{h}\dar[swap]{q} & Y\dar{p}\\
Z' \rar & Y'
\end{tikzcd}$$
We note that $P$ is given by a representation of the fundamental group $\pi_1(Y)$ in $\mathrm{PGL}_{d+1}(\mathbb{C})$. Since the fibers of $p$ are simply connected over $V\subseteq Y'$, such a representation induces a representation of $\pi_1(Y_{\mathrm{reg}})$ in $\mathrm{PGL}_{d+1}(\mathbb{C})$. In another word, $P$ induces a flat $\mathbb{P}^d$-bundle $P'$ on the smooth locus of $Y'$. By pulling back we obtain a flat $\mathbb{P}^d$-bundle $Q'_1$ on $Z'_{\mathrm{reg}}$, given by a representation of $\pi_1(Z'_{\mathrm{reg}})$ in $\mathrm{PGL}_{d+1}(\mathbb{C})$. As in [@GrebKebekusPeternell2016b Proof of Theorem 1.14 on page 1990], this representation of $\pi_1(Z'_{\mathrm{reg}})$ can descend to a representation of $\pi_1(Z') $ in $\mathrm{PGL}_{d+1}(\mathbb{C})$. Hence there is a flat $\mathbb{P}^d$-bundle $Q'$ on $Z'$ which extends $Q'_1$. We denote by $Q$ the $\mathbb{P}^d$-bundle on $Z$ obtained by pulling back $Q'$. We note that $Q$ is a trivial $\mathbb{P}^d$-bundle on every fiber of $q$.
We also remark that the pullback of $P$ by $h$ is isomorphic to $Q$. Indeed, by construction, they are isomorphic over some open dense subset $U$ of $Z$. Moreover, the natural morphism $\pi_1(U) \to \pi_1(Z)$ is surjective and both of these two $\mathbb{P}^d$-bundles are defined by representation of $\pi_1(Z)$ in $\mathrm{PGL}_{d+1}(\mathbb{C})$.
Let $C$ be a $p$-exceptional curve. Then there is a curve $D_1\subseteq Z$ contained in some fiber of $q$ such that $h(D_1)=C$. Let $D$ be the normalization of $D_1$ and $\mu\colon D\to C$ the induced morphism. Then the pullback of $D\times_C P \cong D\times_{D_1} Q$ is a trivial $\mathbb{P}^d$-bundle on $D$.
2
Projectivized bundles of vector bundles {#Examples}
=======================================
Let $X$ be a complex manifold. A projective bundle $P$ on $X$ is a smooth fibration $\varphi\colon P\rightarrow X$ with fibers isomorphic to ${{\mathbb P}}^d$ for some $d>0$. A projective bundle is called *insignificant* if it is isomorphic to the projectivized bundle ${{\mathbb P}}({{\mathscr E}})$ of some vector bundle ${{\mathscr E}}$. This is equivalent to the existence of a line bundle ${{\mathscr L}}$ on $P$ whose restriction on every fiber of $\varphi$ is isomorphic to ${{\mathscr O}}_{{{\mathbb P}}^d}(1)$. If we denote by ${\rm Proj}(X)$ the set of isomorphism classes of projective bundles on $X$, then there is a map $\delta\colon {\rm Proj}(X)\rightarrow H^2(X,{{\mathscr O}}_X^{\times})$ such that
$P\in {\rm Proj}(X)$ is insignificant $\Leftrightarrow$ $\delta(P)=1\in H^2(X,{{\mathscr O}}_X^{\times}).$
In another word, $\delta(P)$ is the obstruction for $P$ to be insignificant. The image of $\delta$ is called the *Brauer group* of $X$ (see for instance [@Elencwajg1985] for more details). We recall the following theorem.
[@Elencwajg1985 Theorem 1]\[thm:Brauer-group\] Let $P\rightarrow X$ be a projective bundle on a complex manifold $X$. Then the fiber product $P'=P\times_XP$ is an insignificant projective bundle on $P$.
In this section, we study insignificant projective bundles $X=\mathbb{P}({{\mathscr E}})$ whose tangent bundle $T_X$ contains a strictly nef subsheaf ${{\mathscr F}}$ and prove a structure theorem of such couples $(\mathbb{P}({{\mathscr E}}), {{\mathscr F}})$ in the first subsection and then provide some examples in the second subsection.
Strictly nef subsheaves of $T_{\mathbb{P}({{\mathscr E}})}$
-----------------------------------------------------------
We first prove the following theorem, which is an analogue of [@CampanaPeternell1998 Lemma 1.2]. It classifies all almost nef locally free subsheaves of the tangent bundle of an insignificant projective bundle provided stronger positivity of the restrictions to fibers.
\[Proj-bundle\] Let $Y$ be a projective manifold of dimension $m>0$ and let ${{\mathscr E}}$ be a vector bundle of rank $d+1$ on $Y$. Denote by $X$ the projective bundle ${{\mathbb P}}({{\mathscr E}})$ and by $p\colon X\rightarrow Y$ the natural projection. Assume that there is an almost nef locally free subsheaf ${{\mathscr F}}\subseteq T_{X}$ of rank $r$ such that its negative locus ${{\mathbb S}}({{\mathscr F}})$ does not dominate $Y$.
1. If ${{\mathscr F}}\vert_F=T_F$ for a general fiber $F$ of $p$, then ${{\mathscr E}}$ is numerically projectively flat and ${{\mathscr F}}=T_{X/Y}$.
2. If ${{\mathscr F}}\vert_F\cong {{\mathscr O}}_{{{\mathbb P}}^n}(1)^{\oplus r}$ for each fiber $F$ of $p$, then there exists a numerically projectively flat subbundle ${{\mathscr M}}$ of ${{\mathscr E}}^*$ such that ${{\mathscr F}}\cong p^*{{\mathscr M}}\otimes{{\mathscr O}}_{{{\mathbb P}}({{\mathscr E}})}(1)$.
By our assumption, in both cases, the restriction of ${{\mathscr F}}$ to a general fiber $F$ of $p$ is ample. Therefore ${{\mathscr F}}\vert_F$ is contained in $T_F$ for general $F$. Consequently, ${{\mathscr F}}$ is contained in the relative tangent bundle $T_{X/Y}$.
*Proof of (1).* Let $S$ be the support of the torsion sheaf $T_{X/Y}/{{\mathscr F}}$. Then $S$ is a closed subvariety of $X$ which does not dominate $Y$. Consider the relative Euler sequence $$\label{eq:relative-euler-sequence-proof-PF}
0\rightarrow {{\mathscr O}}_X\rightarrow p^*{{\mathscr E}}^*\otimes{{\mathscr O}}_{{{\mathbb P}}({{\mathscr E}})}(1)\rightarrow T_{X/Y}\rightarrow 0.$$ Since the inclusion ${{\mathscr F}}\hookrightarrow T_{X/Y}$ is generically surjective and since ${{\mathscr F}}$ is almost nef, by Proposition \[Almost-nef-properties\] (1), $T_{X/Y}$ is almost nef and ${{\mathbb S}}(T_{X/Y})\subseteq S\cup{{\mathbb S}}({{\mathscr F}})$. Then it follows from Proposition \[Almost-nef-properties\] (4) that the vector bundle $p^*{{\mathscr E}}^*\otimes{{\mathscr O}}_{{{\mathbb P}}({{\mathscr E}})}(1)$ is almost nef with negative locus contained in $S\cup {{\mathbb S}}({{\mathscr F}})$. In particular, from Proposition \[Almost-nef-properties\] (2), its determinant $$\det(p^*{{\mathscr E}}^*\otimes{{\mathscr O}}_{{{\mathbb P}}({{\mathscr E}})}(1))={{\mathscr O}}_{{{\mathbb P}}({{\mathscr E}})}(d+1)\otimes p^*\det({{\mathscr E}}^*)$$ is almost nef with negative locus contained in $S\cup {{\mathbb S}}({{\mathscr F}})$. This implies that the normalized tautological class $\Lambda_{{{\mathscr E}}}$ is almost nef with ${{\mathbb S}}(\Lambda_{{{\mathscr E}}})\subseteq S\cup{{\mathbb S}}({{\mathscr F}})$. Since $S\cup {{\mathbb S}}({{\mathscr F}})$ does not dominate $Y$, the ${{\mathbb Q}}$-twisted vector bundle ${{\mathscr E}}\hspace{-0.8ex}<\hspace{-0.8ex}-\mu({{\mathscr E}}) \hspace{-0.8ex}>$ is almost nef by Proposition \[Almost-nef-properties\] (3), where $\mu({{\mathscr E}})$ is the average first Chern class of ${{\mathscr E}}$. Hence ${{\mathscr E}}$ is numerically projectively flat by Theorem \[Num-Projectily-flatness-criterion\].
Next we show that ${{\mathscr F}}=T_{X/Y}$. The induced morphism $\det({{\mathscr F}})\rightarrow \det(T_{X/Y})$ implies that there exists an effective divisor $D$ such that $\det(T_{X/Y})\cong \det({{\mathscr F}})\otimes{{\mathscr O}}_X(D)$. We note that $$c_1(T_{X/Y})-c_1({{\mathscr F}}) \equiv_p 0,$$ since the relative Picard number of $X$ over $Y$ is one. Hence there exists a line bundle ${{\mathscr L}}$ on $Y$ such that $p^*{{\mathscr L}}={{\mathscr O}}_X(D)$. By the relative Euler sequence , we have $$(d+1)\Lambda_{{{\mathscr E}}}-c_1(p^*{{\mathscr L}})=c_1(T_{X/Y})-c_1({{\mathscr O}}_X(D))=c_1({{\mathscr F}}),$$ which is almost nef since ${{\mathscr F}}$ is. Moreover, since its negative locus is contained in ${{\mathbb S}}({{\mathscr F}})$, the ${{\mathbb Q}}$-twisted vector bundle $({{\mathscr E}}\otimes {{\mathscr L}}^*)\hspace{-0.8ex}<\hspace{-0.8ex} -\mu({{\mathscr E}})\hspace{-0.8ex}>$ is almost nef. Then taking the determinant shows that ${{\mathscr L}}^*$ is pseudoeffective. This implies that $D=0$ and $\det({{\mathscr F}})=\det(T_{X/Y})$. Since ${{\mathscr F}}$ and $T_{X/Y}$ are vector bundles of the same rank, by [@DemaillyPeternellSchneider1994 Lemma 1.20], we have ${{\mathscr F}}=T_{X/Y}$.\
*Proof of (2).* The existence of ${{\mathscr M}}\hookrightarrow {{\mathscr E}}^*$ follows from Lemma \[lemma:factorization\] below. Since ${{\mathscr F}}$ is almost nef, by Proposition \[Almost-nef-properties\] (2), the determinant $$\det({{\mathscr F}})=p^*\det({{\mathscr M}})\otimes{{\mathscr O}}_{{{\mathbb P}}({{\mathscr E}})}(r)$$ is almost nef on ${{\mathbb P}}({{\mathscr E}})$ with negative locus contained in ${{\mathbb S}}({{\mathscr F}})$. So the ${{\mathbb Q}}$-twisted vector bundle ${{\mathscr E}}\hspace{-0.8ex}<\hspace{-0.8ex} \mu({{\mathscr M}}) \hspace{-0.8ex}>$ is almost nef (see Proposition \[Almost-nef-properties\] (3)). Since the natural morphism ${{\mathscr E}}\rightarrow {{\mathscr M}}^*$ is generically surjective, by Proposition \[Almost-nef-properties\] (1), the ${{\mathbb Q}}$-twisted vector bundle ${{\mathscr M}}^*\hspace{-0.8ex}<\hspace{-0.8ex} \mu({{\mathscr M}}) \hspace{-0.8ex}>$ is almost nef as well. Hence, it follows from Theorem \[Num-Projectily-flatness-criterion\] that ${{\mathscr M}}^*$ is numerically projectively flat, and so is ${{\mathscr M}}$ by Lemma \[lemma:prop-num-proj-flat\].
Next we show that ${{\mathscr M}}$ is saturated in ${{\mathscr E}}^*$. Fix an ample divisor $A$ on $X$ and let ${{\mathscr G}}$ be the maximal destabilizing subsheaf of ${{\mathscr E}}^*$ with respect to $A$. Since ${{\mathscr M}}$ is numerically projectively flat, both ${{\mathscr M}}^*$ and ${{\mathscr M}}$ are $A$-semistable by Theorem \[thm:num-proj-flat-criterion-Nakayama\]. In particular, we have $$\mu_{A}^{\min}({{\mathscr M}})=\mu_A({{\mathscr M}}).$$ On the other hand, since ${{\mathscr M}}$ is a subsheaf of ${{\mathscr E}}^*$, we have $$\mu_A({{\mathscr M}})\leqslant \mu_A^{\max}({{\mathscr E}}^*)=\mu_A({{\mathscr G}}).$$ Recall that the ${{\mathbb Q}}$-twisted vector bundle ${{\mathscr E}}\hspace{-0.8ex}<\hspace{-0.8ex} \mu({{\mathscr M}}) \hspace{-0.8ex}>$ is almost nef. Since the natural morphism ${{\mathscr E}}\rightarrow {{\mathscr G}}^*$ is generically surjective, by Proposition \[Almost-nef-properties\] (1), the ${{\mathbb Q}}$-twisted sheaf ${{\mathscr G}}^*\hspace{-0.8ex}<\hspace{-0.8ex} \mu({{\mathscr M}}) \hspace{-0.8ex}>$ is almost nef. Thus the ${{\mathbb Q}}$-Cartier divisor class $$c_1({{\mathscr G}}^*)+{{\rm rk}}({{\mathscr G}}^*)\mu({{\mathscr M}})$$ is pseudoeffective on $Y$. It yields $$(c_1({{\mathscr G}}^*)+{{\rm rk}}({{\mathscr G}}^*)\mu({{\mathscr M}}))\cdot A^{n-1} \geqslant 0.$$ This implies $$\mu_A({{\mathscr M}})=\mu({{\mathscr M}})\cdot A^{n-1}\geqslant -\frac{1}{{{\rm rk}}({{\mathscr G}}^*)}c_1({{\mathscr G}}^*)\cdot A^{n-1}= -\mu_A({{\mathscr G}}^*)=\mu_A({{\mathscr G}}).$$ Hence, we have $\mu_A({{\mathscr M}})=\mu_A({{\mathscr G}})$. By the definition of ${{\mathscr G}}$, we see that ${{\mathscr M}}$ is contained in ${{\mathscr G}}$. Let $\overline{{{\mathscr M}}}$ be the saturation of ${{\mathscr M}}$ in ${{\mathscr E}}^*$. Then $\overline{{{\mathscr M}}}$ is contained in ${{\mathscr G}}$. Moreover, as $\mu_A({{\mathscr M}})\leqslant \mu_A(\overline{{{\mathscr M}}})\leqslant \mu_A({{\mathscr G}})$, we deduce that $$\mu_A({{\mathscr M}})=\mu_A(\overline{{{\mathscr M}}}).$$ This implies that the inclusion $\det({{\mathscr M}})\rightarrow \det(\overline{{{\mathscr M}}})$ is an isomorphism. Then applying [@DemaillyPeternellSchneider1994 Lemma 1.20] shows that ${{\mathscr M}}$ is saturated in ${{\mathscr E}}^*$.
Finally, we choose an arbitrary point $x\in Y$. Let $C\subseteq Y$ be a general complete smooth curve passing through $x$. Then ${{\mathscr M}}\vert_C$ is again a subsheaf of ${{\mathscr E}}^*\vert_C$. Now replacing $Y$ by $C$ in the argument above, we can conclude that ${{\mathscr M}}\vert_C$ is saturated in ${{\mathscr E}}^*\vert_C$. Since $C$ is a curve, ${{\mathscr M}}\vert_C$ is actually a subbundle of ${{\mathscr E}}^*\vert_C$. This shows that ${{\mathscr M}}$ is a subbundle of ${{\mathscr E}}^*$.
\[lemma:factorization\] Let $Y$ be a smooth quasi-projective variety and ${{\mathscr E}}$ a vector bundle on $Y$. Let $X=\mathbb{P}({{\mathscr E}})$ and let $p\colon X\to Y$ be the natural fibration. Assume that there is a locally free subsheaf ${{\mathscr F}}\hookrightarrow T_{X/Y}$ such that ${{\mathscr F}}|_F\cong {{\mathscr O}}_{F}(1)^{\oplus r}$ for any fiber $F$ of $p$. Then there is a vector bundle ${{\mathscr M}}$ on $Y$ such that ${{\mathscr F}}= p^*{{\mathscr M}}\otimes {{\mathscr O}}_{\mathbb{P}({{\mathscr E}})}(1)$. Moreover, there is an injective morphism ${{\mathscr M}}\hookrightarrow {{\mathscr E}}^*$ which induces the inclusion ${{\mathscr F}}\hookrightarrow T_{X/Y}$ via the relative Euler sequence.
Set ${{\mathscr M}}=p_*({{\mathscr F}}\otimes {{\mathscr O}}_{\mathbb{P}({{\mathscr E}})}(-1))$. Then ${{\mathscr F}}= p^*{{\mathscr M}}\otimes {{\mathscr O}}_{{{\mathbb P}}({{\mathscr E}})}(1).$ Tensoring the relative Euler sequence of ${{\mathbb P}}({{\mathscr E}})$ with ${{\mathscr F}}^*$, we derive the following short exact sequence of vector bundles $$0\rightarrow {{\mathscr F}}^*\rightarrow p^*{{\mathscr E}}^*\otimes{{\mathscr O}}_{{{\mathbb P}}({{\mathscr E}})}(1)\otimes{{\mathscr F}}^*\rightarrow T_{X/Y}\otimes{{\mathscr F}}^*\rightarrow 0.$$ For any $i\geqslant 0$, we have $$R^i p_*{{\mathscr F}}^*={{\mathscr M}}^*\otimes R^i p_*{{\mathscr O}}_{{{\mathbb P}}({{\mathscr E}})}(-1)=0.$$ Thus Leray Spectral sequence implies that $H^1(X,{{\mathscr F}}^*)=0.$ As a consequence, the inclusion ${{\mathscr F}}\hookrightarrow T_{X/Y}$ can be lifted to an inclusion ${{\mathscr F}}\hookrightarrow p^*{{\mathscr E}}^*\otimes{{\mathscr O}}_{{{\mathbb P}}({{\mathscr E}})}(1)$. We then obtain an injective morphism $p^*{{\mathscr M}}\hookrightarrow p^*{{\mathscr E}}^*$. By taking the direct image, we obtain an inclusion ${{\mathscr M}}\hookrightarrow {{\mathscr E}}^*$.
\[Pn\]
1. Let ${{\mathscr F}}$ be a strictly nef vector bundle of rank $r$ on ${{\mathbb P}}^n$. Assume that there exists an injective morphism ${{\mathscr F}}\hookrightarrow T_{{{\mathbb P}}^n}$. Then ${{\mathscr F}}$ is isomorphic to ${{\mathscr O}}_{{{\mathbb P}}^n}(1)^{\oplus r}$ or $T_{{{\mathbb P}}^n}$.
2. Let $p\colon X \rightarrow Y$ be a projective bundle. Let ${{\mathscr F}}$ be a vector bundle on $X$ which is strictly nef on every fiber of $p$. Assume further that ${{\mathscr F}}|_F \cong {{\mathscr O}}_{F}(1)^{\oplus r}$ for a fiber $F$ of $p$. Then for every fiber $F'$ of $p$, we have ${{\mathscr F}}|_{F'} \cong {{\mathscr O}}_{F'}(1)^{\oplus r}$.
For (1), thanks to [@AproduKebekusPeternell2008 Theorem 4.2], it suffices to show that the restriction ${{\mathscr F}}\vert_{\ell}$ on an arbitrary line $\ell \subseteq {{\mathbb P}}^n$ is ample. Since $\ell$ is a rational curve, the strictly nef bundle ${{\mathscr F}}\vert_{\ell}$ must be ample.
For (2), let $F'$ be an arbitrary fiber of $p$ and $C$ be a line in $F'$. Then ${{\mathscr F}}|_C$ is a strictly nef vector bundle of rank $r$. Moreover, by assumption the degree of ${{\mathscr F}}|_C$ is equal to $r$. Hence we obtain ${{\mathscr F}}|_C \cong {{\mathscr O}}_{C}(1)^{\oplus r}$. As $C$ is arbitrary, this implies that ${{\mathscr F}}|_{F'} \cong {{\mathscr O}}_{F'}(1)^{\oplus r}$ by [@AndreattaWisniewski2001 Proposition 1.2].
\[remark:strictly-nef\] By the lemma above, if we assume that ${{\mathscr F}}$ is strictly nef in Theorem \[Proj-bundle\], then either ${{\mathscr F}}\vert_F=T_F$ for a general fiber $F$ of $p$, or ${{\mathscr F}}\vert_F\cong {{\mathscr O}}_{{{\mathbb P}}^n}(1)^{\oplus r}$ for any fiber $F$ of $p$.
As an immediate corollary of Theorem \[Proj-bundle\], we obtain the following result.
\[cor:proj-vector-bundle-p1-base\] Let ${{\mathscr E}}$ be a vector bundle on $\mathbb{P}^1$. Then the tangent bundle $T_{\mathbb{P}({{\mathscr E}})}$ does not contain any strictly nef locally free subsheaves.
We write $Y=\mathbb{P}^1$ and $X=\mathbb{P}({{\mathscr E}})$. Assume by contradiction that there is a strictly nef locally free subsheaf ${{\mathscr F}}$ of $T_{X}$. Then ${{\mathscr F}}$ is contained in $T_{X/Y}$. As pointed out in Remark \[remark:strictly-nef\], by Lemma \[Pn\], we are in one of the situations of Theorem \[Proj-bundle\].
We note that every numerically projectively flat vector bundle on $Y$ is the direct sum of copies of a line bundle. Hence, if we are in the first case of the theorem, then $X\cong \mathbb{P}^n\times Y$. One readily check that $T_{X/Y}$ is not strictly nef. Now we assume the second case of Theorem \[Proj-bundle\]. We write ${{\mathscr M}}={{\mathscr L}}^{\oplus r}$. Then we have a surjective morphism $${{\mathscr E}}\to {{\mathscr M}}^* \to {{\mathscr L}}^*,$$ where ${{\mathscr M}}^* \to {{\mathscr L}}^*$ one of the canonical projections. The composition of the morphisms above induce a section $\sigma\colon Y\to {\mathbb{P}({{\mathscr E}})}$ such that $$\sigma^*{{\mathscr O}}_{\mathbb{P}({{\mathscr E}})}(1) \cong {{\mathscr L}}^*.$$ As a consequence $\sigma^*{{\mathscr F}}\cong {{\mathscr O}}_Y^{\oplus r}$. Hence ${{\mathscr F}}$ is not strictly nef.
We conclude this subsection with the following proposition.
\[prop:smothness-image-subbundle\] Let $T$ be a complex manifold and let $X\rightarrow T$ be a ${{\mathbb P}}^d$-bundle. Let $Z$ be the fiber product $X\times_T X\cong {{\mathbb P}}({{\mathscr E}})$, where ${{\mathscr E}}$ is a vector bundle over $X$ (see Theorem \[thm:Brauer-group\]). $$\begin{tikzcd}[column sep=large, row sep=large]
X\dar[swap]{\varphi} & Z \cong {{\mathbb P}}({{\mathscr E}}) \lar{q}\dar[swap]{\pi} \\
T &
X\lar{p=\varphi}
\end{tikzcd}$$ Assume that there exists a vector bundle ${{\mathscr F}}$ over $X$ and a subbunlde ${{\mathscr M}}$ of ${{\mathscr E}}^*$ such that $$q^*{{\mathscr F}}\cong \pi^*{{\mathscr M}}\otimes {{\mathscr O}}_{{{\mathbb P}}({{\mathscr E}})}(1).$$ Let $M\subseteq Z$ be the projective subbundle ${{\mathbb P}}({{\mathscr M}}^*) \subseteq {{\mathbb P}}({{\mathscr E}})$. Then $q(M)$ is a projective bundle over $T$.
The problem is local on $T$, hence we can suppose that $T\subseteq {{\mathbb C}}^m$ is a small polydisc so that $X\cong {{\mathbb P}}({{\mathscr V}})$ for some vector bundle ${{\mathscr V}}$ on $T$. Then there exists a line bundle ${{\mathscr L}}$ on $X$ such that $p^*{{\mathscr V}}\cong {{\mathscr E}}\otimes {{\mathscr L}}$. By replacing ${{\mathscr E}}$ with ${{\mathscr E}}\otimes {{\mathscr L}}$ and ${{\mathscr M}}$ with ${{\mathscr M}}\otimes {{\mathscr L}}^*$, we may assume that ${{\mathscr L}}$ is trivial. That is $p^*{{\mathscr V}}\cong {{\mathscr E}}$.
Since the restrictions of $q^*{{\mathscr F}}$ to the fibers of $\pi$ are isomorphic to direct sum of copies of ${{\mathscr O}}_{{{\mathbb P}}^d}(1)$, the same holds for the restrictions of ${{\mathscr F}}$ to the fibers of $\varphi$. We define the following the vector bundle on $T$ $${{\mathscr Q}}=\varphi_*({{\mathscr F}}\otimes {{\mathscr O}}_{{{\mathbb P}}({{\mathscr V}})}(-1)).$$ Then by assumption, we have $$q^*(\varphi^*{{\mathscr Q}}\otimes {{\mathscr O}}_{{{\mathbb P}}({{\mathscr V}})}(1))\cong q^*{{\mathscr F}}\cong \pi^*{{\mathscr M}}\otimes {{\mathscr O}}_{{{\mathbb P}}({{\mathscr E}})}(1).$$ As $q^*{{\mathscr O}}_{{{\mathbb P}}({{\mathscr V}})}(1)\cong {{\mathscr O}}_{{{\mathbb P}}({{\mathscr E}})}(1)$, we deduce that $\pi^*p^*{{\mathscr Q}}\cong \pi^*{{\mathscr M}}$ and hence $$p^*{{\mathscr Q}}\cong {{\mathscr M}}.$$ Since $p^*{{\mathscr V}}^*\cong {{\mathscr E}}^*$, the subbundle structure ${{\mathscr M}}\hookrightarrow {{\mathscr E}}^*$ induces a subbundle structure ${{\mathscr Q}}\hookrightarrow {{\mathscr V}}^*$. Denote by $Q$ the projective subbundle ${{\mathbb P}}({{\mathscr Q}}^*) \subseteq X \cong {{\mathbb P}}({{\mathscr V}})$. Then $M$ is isomorphic to $Q\times_T X$ and consequently $q(M)=Q$. This finishes the proof.
Examples {#Subsection:examples}
--------
In this subsection, our goal is to extend Mumford’s example to higher dimension. A key ingredient is the following theorem due to Subramanian.
[@Subramanian1989 Lemma 3.2 and Theorem 6.1]\[Subramanian\] Let $C$ be a smooth curve of genus $g\geqslant 2$. Then for any $r\geqslant 2$, there exists a Hermitian flat vector bundle ${{\mathscr E}}$ of rank $r$ such that the tautological line bundle ${{\mathscr O}}_{{{\mathbb P}}({{\mathscr E}})}(1)$ is ample when restricted to a proper subvariety of ${{\mathbb P}}({{\mathscr E}})$. In particular, ${{\mathscr E}}$ is strictly nef.
By using this theorem, we construct two examples. Fix a smooth curve $C$ of genus $g \geqslant 2$. Let $r\geqslant 2$ and ${{\mathscr E}}$ a vector bundle of rank $r$ provided in Theorem \[Subramanian\].
\[example:subbundles\] Let $X={{\mathbb P}}({{\mathscr E}})$. Then we have the following relative Euler exact sequence $$0\rightarrow {{\mathscr O}}_X\rightarrow p^*{{\mathscr E}}^*\otimes{{\mathscr O}}_{{{\mathbb P}}({{\mathscr E}})}(1)\rightarrow T_{X/C}\rightarrow 0,$$ where $p\colon X={{\mathbb P}}({{\mathscr E}})\rightarrow C$ is the natural projection. We claim that $${{\mathscr E}}'\coloneqq p^*{{\mathscr E}}^*\otimes {{\mathscr O}}_{{{\mathbb P}}({{\mathscr E}})}(1)$$ is strictly nef. Indeed, let $\nu\colon C'\rightarrow X$ be a finite morphism from a smooth complete curve $C'$ to $X$ and let $\nu^*{{\mathscr E}}'\rightarrow {{\mathscr L}}$ be a quotient line bundle. If $\nu(C')$ is contained in the fibers of $p$, then $\nu^*{{\mathscr E}}'$ is ample and ${{\mathscr L}}$ is still ample. Assume that $\nu(C')$ is not contained in the fibers of $p$. Then the composition $\nu'\colon C'\rightarrow X\rightarrow C$ is a finite morphism. Moreover, we have $$\nu^*{{\mathscr E}}'\cong \nu'^*{{\mathscr E}}^*\otimes \nu^*{{\mathscr O}}_{{{\mathbb P}}({{\mathscr E}})}(1).$$ This implies that ${{\mathscr L}}\otimes \nu^*{{\mathscr O}}_{{{\mathbb P}}({{\mathscr E}})}(-1)$ is a quotient line bundle of $\nu'^*{{\mathscr E}}^*$. Note that ${{\mathscr E}}^*$ is numerically flat and therefore nef. Thus we have $$\deg({{\mathscr L}})+\deg(\nu^*{{\mathscr O}}_{{{\mathbb P}}({{\mathscr E}})}(-1))\geqslant 0.$$ However, since ${{\mathscr O}}_{{{\mathbb P}}({{\mathscr E}})}(1)$ is strictly nef, we deduce that $\deg({{\mathscr L}})>0$. Therefore ${{\mathscr E}}'$ is strictly nef by [@LiOuYang2019 Proposition 2.1]. Hence, ${{\mathscr F}}= T_{X/C}$ is a strictly nef subbundle of $T_X$ (see [@LiOuYang2019 Propostion 2.2]).
\[example:not-subbundles\] We consider the following extension of vector bundles $$0\rightarrow {{\mathscr Q}}\rightarrow {{\mathscr G}}\rightarrow {{\mathscr E}}^*\rightarrow 0,$$ where ${{\mathscr Q}}$ is a nef vector bundle of positive rank. Since ${{\mathscr E}}^*$ is Hermitian flat, it is numerically flat. In particular, ${{\mathscr E}}^*$ is nef and so is ${{\mathscr G}}$ (see [@DemaillyPeternellSchneider1994 Proposition 1.15]). Let $X={{\mathbb P}}({{\mathscr G}})$ and $p\colon X= {{\mathbb P}}({{\mathscr G}})\rightarrow C$ the natural projection. Then we have the following relative Euler sequence $$0\rightarrow {{\mathscr O}}_X\rightarrow p^*{{\mathscr G}}^*\otimes{{\mathscr O}}_{{{\mathbb P}}({{\mathscr G}})}(1)\rightarrow T_{X/C}\rightarrow 0.$$ Since ${{\mathscr E}}$ is a subbundle of ${{\mathscr G}}^*$, it follows that $${{\mathscr F}}\coloneqq p^*{{\mathscr E}}\otimes {{\mathscr O}}_{{{\mathbb P}}({{\mathscr G}})}(1)$$ is a subbundle of $p^*{{\mathscr G}}^*\otimes {{\mathscr O}}_{{{\mathbb P}}({{\mathscr G}})}(1)$. As in Example \[example:subbundles\] above, one can show that ${{\mathscr F}}$ is strictly nef. Moreover, note that the composition $${{\mathscr F}}\rightarrow p^*{{\mathscr G}}^*\otimes {{\mathscr O}}_{{{\mathbb P}}({{\mathscr G}})}(1)\rightarrow T_{X/C}$$ is injective, it follows that ${{\mathscr F}}$ is a strictly nef locally free subsheaf of $T_X$. On the other hand, since the restriction of ${{\mathscr F}}$ to fibers of $p$ is isomorphic to ${{\mathscr O}}_{{{\mathbb P}}^d}(1)^{\oplus r}$, ${{\mathscr F}}$ is not a subbundle of $T_{X/C}$.
2
Structures of MRC fibrations {#section:uniruled}
============================
In this section we prove the following proposition.
\[MRC-fibration\] Let $X$ be a projective manifold. Assume that $T_X$ contains a locally free strictly nef subsheaf ${{\mathscr F}}$. Then $X$ is uniruled.
Moreover, denote by $\varphi\colon X\dashrightarrow T$ its MRC fibration. Then there exists an open subset $X^\circ\subseteq X$ with ${{\rm codim}}(X\setminus X^\circ)\geqslant 2$ such that the restriction $$\varphi^\circ\coloneqq\varphi\vert_{X^\circ}\colon X^\circ\rightarrow T^\circ\subseteq T$$ is a ${{\mathbb P}}^d$-bundle. In particular, the restriction ${{\mathscr F}}\vert_{X^\circ}$ is contained in $T_{X^\circ/T^\circ}$.
We first show that such a projective manifold $X$ must be uniruled (see Corollary \[Uniruledness\]). We remark that *a priori* this is not straightforward since a strictly nef vector bundle may also be numerically flat, and Miyaoka’s criterion cannot be applied in this case. The uniruledness in Proposition \[MRC-fibration\] is a consequence of the following theorem, which itself may be of independent interest.
\[Almost-nef-non-uniruled\] Let $X$ be a non-uniruled projective manifold. Assume that there is a non-zero map $\sigma\colon {{\mathscr F}}\rightarrow T_X$, where ${{\mathscr F}}$ is an almost nef coherent sheaf over $X$. Let $\widehat{{{\mathscr Q}}}$ be the reflexive hull of the image ${{\mathscr Q}}$ of $\sigma$. Then $\widehat{{{\mathscr Q}}}$ is an involutive subbundle of $T_X$, and is numerically flat with torsion determinant bundle. Furthermore, there exist a finite étale cover $\gamma\colon \widetilde{X}\rightarrow X$ and an almost holomorphic map $g\colon \widetilde{X}\dashrightarrow Y$ whose general fibers are abelian varieties such that the restriction of $\gamma^*\widehat{{{\mathscr Q}}}$ to a general fiber of $g$ is a linear foliation.
Let $\overline{{{\mathscr Q}}}$ be the saturation of ${{\mathscr Q}}$ in $T_X$. By the definition of reflexive hull, we have natural inclusions ${{\mathscr Q}}\hookrightarrow\widehat{{{\mathscr Q}}}\hookrightarrow\overline{{{\mathscr Q}}}$, which induce an injection $$\label{Eq-determinants}
\det(\widehat{{{\mathscr Q}}})\hookrightarrow\det(\overline{{{\mathscr Q}}}).$$ Since ${{\mathscr F}}$ is almost nef and the composition ${{\mathscr F}}\rightarrow \widehat{{{\mathscr Q}}}\rightarrow \overline{{{\mathscr Q}}}$ is generically surjective, Proposition \[Almost-nef-properties\] shows that both $\widehat{{{\mathscr Q}}}$ and $\overline{{{\mathscr Q}}}$ are almost nef, so are $\det(\widehat{{{\mathscr Q}}})$ and $\det(\overline{{{\mathscr Q}}})$. On the other hand, as $X$ is not uniruled, by [@BoucksomDemaillyPuaunPeternell2013 Theorem 2.6], both $\det(\widehat{{{\mathscr Q}}})^*$ and $\det(\overline{{{\mathscr Q}}})^*$ are pseudoeffective. Therefore, by [@Peternell1994 Lemma 6.5], we have $$c_1(\widehat{{{\mathscr Q}}})= c_1(\overline{{{\mathscr Q}}})=0.$$ As a consequence, the natural morphism $\det(\widehat{{{\mathscr Q}}})\hookrightarrow\det(\overline{{{\mathscr Q}}})$ is an isomorphism. We can now apply [@LorayPereiraTouzet2018 Theorem 5.2] to conclude that $\overline{{{\mathscr Q}}}$ is a regular foliation with torsion canonical bundle. Then, from [@DemaillyPeternellSchneider1994 Lemma 1.20], we obtain $\widehat{{{\mathscr Q}}} = \overline{{{\mathscr Q}}}$ .
Finally, since $\widehat{{{\mathscr Q}}}$ is almost nef and $c_1(\widehat{{{\mathscr Q}}})=0$, [@HoeringPeternell2019 Theorem 1.8] says that $\widehat{{{\mathscr Q}}}$ is a numerically flat vector bundle. In particular, $c_2(\widehat{{{\mathscr Q}}})=0$. The remaining part of the theorem then follows from [@PereiraTouzet2013 Theorem C].
As an application, we obtain the following criterion for uniruledness.
\[Uniruledness\] Let ${{\mathscr F}}$ be a strictly nef coherent sheaf on a projective manifold $X$. Assume that there exists a non-zero map $\sigma\colon{{\mathscr F}}\rightarrow T_X$ with image ${{\mathscr Q}}$. Then $X$ is uniruled. Moreover, there exists an open subset $X^\circ\subseteq X$ and a $\mathbb{P}^d$-bundle $\varphi^\circ\colon X^\circ\rightarrow T^\circ$ over a smooth base $T^\circ$ such that ${{\mathscr Q}}\vert_{X^\circ}\subseteq T_{X^\circ/T^\circ}$.
We first assume by contradiction that $X$ is not uniruled. Let $\widehat{{{\mathscr Q}}}$ be the reflexive hull of ${{\mathscr Q}}$. After Theorem \[Almost-nef-non-uniruled\], replacing $X$ by some finite étale cover if necessary, we may assume that there is an almost holomorphic map $g\colon X \dashrightarrow Y$ whose general fibers are abelian varieties. Moreover, the restriction of $ \widehat{{{\mathscr Q}}}$ on a general fiber $F$ of $g$ is a linear foliation.
Denote by $S$ the support of $\widehat{{{\mathscr Q}}}/{{\mathscr Q}}$. Then $S$ is contained in the singular locus $\mathrm{Sing}({{\mathscr Q}})$ of ${{\mathscr Q}}$. In particular, $S$ has codimension at least $2$ as ${{\mathscr Q}}$ is torsion free. As a consequence, the intersection $F\cap S$ also has codimension at least $2$ in $F$. Therefore, if $C$ is a complete intersection curve of general very ample divisors in $F$, we have $\widehat{{{\mathscr Q}}}\vert_C = {{\mathscr Q}}|_C$. The latter is strictly nef by Proposition \[Image-nef\]. This contradicts to the fact that $\widehat{{{\mathscr Q}}}\vert_A$ is a linear foliation.
Finally, since $X$ is uniruled, it carries a covering family ${{\mathcal V}}$ of minimal rational curves. As ${{\mathscr Q}}$ is locally free in codimension one, by [@Kollar1996 II, Proposition 3.7], ${{\mathscr Q}}$ is locally free along a general member $D$ of ${{\mathcal V}}$. Note that ${{\mathscr F}}|_D$ is strictly nef by Proposition \[BK criterion\], so is ${{\mathscr Q}}\vert_D$ by Proposition \[Image-nef\]. This in turn implies that ${{\mathscr Q}}|_D$ is an ample vector bundle as $D$ is a rational curve. The remaining part of the corollary then follows from [@AraujoDruelKovacs2008 Proposition 2.7].
\[rmk:rc-quotient\] In view of the proof of [@AraujoDruelKovacs2008] (see also [@Araujo2006 Theorem 3.4] or [@Liu2019 Theorem 2.1]), the almost holomorphic map on $X$ induced by $\varphi^\circ$ is nothing but the ${{\mathcal V}}$-rationally connected quotient of $X$. Furthermore, every rational curve in ${{\mathcal V}}$ meeting $X^\circ$ is a line in a fiber of $\varphi^\circ$.
Now we can conclude Proposition \[MRC-fibration\].
By Corollary \[Uniruledness\], $X$ is uniruled and it carries a covering family ${{\mathcal V}}$ of minimal rational curves. As pointed out in Remark \[rmk:rc-quotient\], the ${{\mathcal V}}$-rationally connected quotient $\varphi^\circ\colon X^\circ\rightarrow T^\circ$ is a $\mathbb{P}^d$-bundle and the rational curves parameterized by ${{\mathcal V}}$ meeting $X^\circ$ are lines in fibers of $\varphi^\circ$. Moreover, since ${{\mathcal V}}$ is unsplit by Lemma \[lemma:unsplit\] below, thanks to [@Araujo2006 Theorem 3.4] (see also [@Liu2019 Theorem 2.1]), $\varphi^\circ$ can be extended in codimension one; that is, we can choose $X^\circ\subseteq X$ such that ${{\rm codim}}(X\setminus X^\circ)\geqslant 2$.
Now we assume to the contrary that $\varphi$ is not the MRC fibration. Let $g:X\dashrightarrow Y$ be the MRC fibration. Then there is a natural factorization $$X\dashrightarrow T \dashrightarrow Y.$$ Set $Z=X\backslash X^\circ$. Then $Z$ has codimension at least $2$. Thus, for a general fiber $G$ of $g$, $Z\cap G$ also has codimension at least $2$ in $G$. Note that $G$ is smooth and rationally connected. By [@Kollar1996 II, Proposition 3.7], there is a very free rational curve $C$ in $G$ which is contained in $G\backslash Z$. Hence $C$ is also a curve contained in $X^\circ$. Denote by $r\colon {{\mathbb P}}^1\rightarrow \varphi^\circ(C')$ the normalization, and by $W$ the fiber product $X^\circ\times_{{{\mathbb P}}^1} T^\circ$. Then the natural morphism $\pi\colon W\rightarrow {{\mathbb P}}^1$ is a $\mathbb{P}^d$-bundle. This implies that there exists a vector bundle ${{\mathscr E}}$ on ${{\mathbb P}}^1$ such that $W={{\mathbb P}}({{\mathscr E}})$ as the Brauer group of ${{\mathbb P}}^1$ is trivial. Moreover, since ${{\mathscr F}}|_{X^\circ}$ is contained in $T_{X^\circ/T^\circ}$, the pullback ${{\mathscr F}}'$ of ${{\mathscr F}}$ on $W$ is a strictly nef locally free subsheaf of $T_{W/{{\mathbb P}}^1}$. This contradicts with Corollary \[cor:proj-vector-bundle-p1-base\].
\[lemma:unsplit\] Let $X$ be an $n$-dimensional projective manifold, and let ${{\mathcal V}}$ be a covering family of minimal rational curves on $X$. If $T_X$ contains a strictly nef locally free subsheaf ${{\mathscr F}}$, then ${{\mathcal V}}$ is unsplit.
If ${{\mathscr F}}$ is a line bundle, this is a consequence of [@Druel2004 Corollaire]. Therefore, we may assume $r={{\rm rk}}{{\mathscr F}}\geqslant 2$. Let $[C]\in {{\mathcal V}}$ be a general member and let $f\colon {{\mathbb P}}^1\rightarrow X$ be the morphism induced by the normalization of $C$. Then $f^*{{\mathscr F}}$ is a subsheaf of $f^*T_X$. Since ${{\mathcal V}}$ is a minimal family, there is some $d\geqslant r-1$ such that $$f^*T_X \cong {{\mathscr O}}_{{{\mathbb P}}^1}(2)\oplus{{\mathscr O}}_{{{\mathbb P}}^1}(1)^{\oplus d}\oplus{{\mathscr O}}_{{{\mathbb P}}^1}^{\oplus (n-d-1)}.$$ In particular, we have $c_1({{\mathscr F}})\cdot C\leqslant r+1$.
Now let $B=\sum a_i B_i$ be a $1$-cycle obtained as the limit of cycles in ${{\mathcal V}}$ with $B_i$ irreducible and reduced. Then each $B_i$ is a rational curve. Since ${{\mathscr F}}$ is strictly nef, it follows that $c_1({{\mathscr F}})\cdot B_i\geqslant r$ for all $i$. If $B$ is not reduced or not irreducible, then we have $$c_1({{\mathscr F}})\cdot B\geqslant 2r >r+1 \geqslant c_1({{\mathscr F}})\cdot C.$$ This is a contradiction and hence ${{\mathcal V}}$ is unsplit.
2
Degeneration of ${{\mathbb P}}^d$ {#section:degeneration-P^d}
=================================
In order to prove Theorem \[thm:main-theorem\], we need to extend the projective bundle structure $\varphi^\circ \colon X^\circ \to T^\circ$ obtained in Proposition \[MRC-fibration\] from the MRC-fibration to the whole manifold $X$. To this end, we study degenerations of projective spaces. Such problems have been investigated in literatures, and we refer to [@Fujita1987; @HoeringNovelli2013; @AraujoDruel2014] and the references therein. Building on the work of Cho-Miyaoka-Shepherd-Barron [@ChoMiyaokaShepherd-Barron2002] (see also [@Kebekus2002]) and Kollár [@Kollar2011], we have the following result which is essentially proved in [@HoeringNovelli2013] and [@AraujoDruel2014].
\[PROP:HN-degeneration-of-projective-spaces\] Let $\varphi\colon X\rightarrow T$ be an equidimensional fibration between quasi-projective varieties whose general fibers are isomorphic to ${{\mathbb P}}^d$. Assume that $T$ is normal and there exists a line bundle ${{\mathscr L}}$ on $X$, whose restrictions on general fibers of $\varphi$ are isomorphic to ${{\mathscr O}}_{\mathbb{P}^d}(e)$, such that
$c_1({{\mathscr L}})\cdot C > \frac{1}{2} e$ for any $\varphi$-exceptional rational curve $C\subseteq X$.
Then all the fibers of $\varphi$ are irreducible and generically reduced, and the normalization of any fiber is isomorphic to ${{\mathbb P}}^d$. If we assume in addition that $X$ is normal, then $\varphi\colon X\rightarrow T$ is a ${{\mathbb P}}^d$-bundle.
Let ${{\mathcal H}}\subseteq {\ensuremath{\mbox{\rm RatCurves}^n(X/T)}}$ be the unique irreducible component such that a general point corresponds to a line contained in the general fibers of $\varphi$. Let $[l]$ be a line contained in ${{\mathcal H}}$. By assumption, we have $$c_1({{\mathscr L}})\cdot C> \frac{1}{2}e = \frac{1}{2} c_1({{\mathscr L}})\cdot l$$ for any rational curve $C$ contracted by $\varphi$. Thus ${{\mathcal H}}$ is actually proper over $T$. In particular, we can apply the same argument as in [@HoeringNovelli2013 p.222, Proof of Propositon 3.1] to show that the normalization of any irreducible component of any fiber of $\varphi$ is isomorphic to ${{\mathbb P}}^d$.
Since $T$ is normal, by [@Kollar1996 I, Definition 3.10 and Theorem 3.17], $\varphi\colon X\rightarrow T$ is a well-defined family of $d$-dimensional proper algebraic cycles over $T$. In particular, the degree of the fibers with respect to ${{\mathscr L}}$ is constant (see [@Kollar1996 I, Lemma 3.17.1]). Then the computation in [@HoeringNovelli2013 p.223, Proof of Proposition 3.1] shows that the fibers of $\varphi$ are reduced and irreducible. Moreover, the pullback of ${{\mathscr L}}$ on the normalization of any fiber $F$ of $\varphi$ is isomorphic to ${{\mathscr O}}_{{{\mathbb P}}^d}(e)$. Thus the Hilbert polynomials of the normalizations of the fibers are the same. Then we apply [@Kollar2011 Theorem 12] and obtain that $\varphi\colon X\rightarrow T$ admits a simultaneous normalization, which is a finite birational morphism $\eta\colon \widetilde{X}\rightarrow X$ such that the morphism $\varphi\circ\eta\colon \widetilde{X}\rightarrow T$ is flat with normal fibers.
If we assume further that $X$ is normal, then Zariski’s main theorem [@Hartshorne1977 V, Theorem 5.2] implies that $\eta$ is an isomorphism. As a consequence $\varphi\colon X\rightarrow T$ is a smooth morphism.
We have the following corollary.
\[cor-equidim-implies-bundle\] Let $\varphi \colon X\to T$ be an equidimensional Mori fibration from a smooth projective variety $X$ to a normal variety $T$. Assume that general fibers of $\varphi$ are isomorphic to ${{\mathbb P}}^d$ for some $d>0$. Assume further that there is a vector bundle ${{\mathscr F}}$ of rank $r\geqslant 2$ whose restriction on every fiber is strictly nef. Then $\varphi$ is a $\mathbb{P}^d$-bundle between smooth varieties.
Let ${{\mathscr L}}\coloneqq \det({{\mathscr F}})$. Then this line bundle satisfies the hypothesis of Proposition \[PROP:HN-degeneration-of-projective-spaces\]. Therefore $\varphi\colon X\rightarrow T$ is a ${{\mathbb P}}^d$-bundle. Since both $X$ and $\varphi$ are smooth, it follows that $T$ is smooth.
In order to apply previous results, we need to ensure the equidimensionality of fibrations. Therefore, in the remainder of this section, we provide some criteria for equidimensionality.
\[prop:equidimensionality-criterion-I\] Assume that there is a commutative diagram of fibrations of normal projective varieties $$\begin{tikzcd}[column sep=large, row sep=large]
\overline{X}\dar[swap]{\overline{\varphi}} \rar{\rho} & X\dar{{\varphi}} \\
\overline{T} \rar{\gamma}& T
\end{tikzcd}$$ such that
1. $T$ has only klt singularities,
2. general fibers of $\varphi$ are isomorphic to ${{\mathbb P}}^d$,
3. $\overline T$ is smooth and the fibers of $\gamma$ over an open subset $T^\circ$ of $T$ with ${{\rm codim}}(T\setminus T^\circ)\geq 2$ are simply connected,
4. $\overline{\varphi}\colon \overline{X}\rightarrow \overline{T}$ is a flat $\mathbb{P}^d$-bundle given by a representation $\pi_1(\overline{T})\rightarrow \operatorname*{PGL}_{d+1}({{\mathbb C}})$,
5. there is a strictly nef vector bundle ${{\mathscr F}}$ on $X$ with a surjective morphism $\rho^*{{\mathscr F}}\rightarrow T_{\overline{X}/\overline{T}}$.
Then $\varphi\colon X\rightarrow T$ is equidimensional.
Let $C\subseteq \overline{T}$ be a complete curve contracted by $\gamma$. Thanks to Lemma \[lemma:trivial-bundle-finite-change\], there exists a smooth curve $C'$ with a finite surjective morphism $n\colon C'\rightarrow C$ such that the fiber product $\overline{X}_{C'}\coloneqq \overline{X}\times_{\overline{T}} C'$ is isomorphic to $C'\times {{\mathbb P}}^d$ as ${{\mathbb P}}^d$-bundles over $C'$. We denote by $$p_1\colon \overline{X}_{C'} \rightarrow C' \mbox{ and } \nu\colon \overline{X}_{C'} \to \overline{X}$$ the natural morphisms and by $$p_2\colon \overline{X}_{C'} \rightarrow {{\mathbb P}}^d$$ the morphism induced by the natural projection from $C'\times {{\mathbb P}}^d$ to ${{\mathbb P}}^d$.
Let $T_{\overline{X}_{C'}/C'}$ be the relative tangent bundle of $p_1$. Then its restriction on every fiber of $p_2$ is isomorphic to a trivial vector bundle. On the other hand, the surjective morphism $\rho^*{{\mathscr F}}\rightarrow T_{\overline{X}/\overline{T}}$ induces a surjective morphism $\nu^*\rho^*{{\mathscr F}}\rightarrow T_{\overline{X}_{C'}/C'}$. As a consequence, the restriction of $\nu^*\rho^*{{\mathscr F}}$ on any fiber of $p_2$ is not strictly nef. This implies that the fibers of $p_2$ are contracted by the composition $$p\colon \overline{X}_{C'}\xrightarrow{\nu} \overline{X} \xrightarrow{\rho} X.$$ By rigidity lemma, the morphism $p\colon \overline{X}_{C'}\rightarrow X$ factors through $p_2\colon \overline{X}_{C'}\rightarrow {{\mathbb P}}^d$. In particular, the images of all fibers of $\overline{\varphi}$ over $C$ in $X$ under $\rho$ coincide.
Let $t\in T$ be an arbitrary point. Since the fiber $\overline{T}_t:= \gamma^{-1}(t)$ is connected, the previous paragraph implies that the images of all fibers of $\overline{\varphi}$ over $\overline{T}_t$ in $X$ under $\rho$ coincide. It follows that $\dim X_t\leqslant d$, where $X_t$ is the fiber of $X$ over $t$. By semicontinuity, we obtain that $\varphi$ is equidimensional.
In the situation of Proposition \[MRC-fibration\], we can use the criterion in Proposition \[prop:equidimensionality-criterion-I\] to deal with the case where the restriction ${{\mathscr F}}\vert_F$ is isomorphic to $T_F$ for $F$ being a general fiber of the MRC fibration. When ${{\mathscr F}}|_F$ is isomorphic to the direct sum of copies of ${{\mathscr O}}_F(1)$, we need another more detailed treatment.
\[prop:equidimensionality-criterion II\] Let $\varphi\colon X\rightarrow T$ be a surjective morphism from a smooth projective variety $X$ to a normal projective variety $T$, and let $P\subseteq X$ be a subvariety such that the induced morphism $\varphi_P\coloneqq \varphi\vert_P\colon P\rightarrow T$ is surjective. If at a point $x\in P$, the fiber $P_{\varphi(x)}$ is smooth of dimension $\dim P-\dim T$, then the fiber $X_{\varphi(x)}$ is smooth and of dimension $\dim X-\dim T$ at $x$.
Set $\dim X = n$, $\dim T=m$ and $\dim P = r$. By assumption, we see that the relative differential sheaf $\Omega_{P/T}$ on $P$ is locally free of rank $r-m$ around $x$. Consider the following commutative diagram, with exact rows and columns, $$\begin{tikzcd}[column sep=large, row sep=large]
\varphi^*\Omega_T\vert_{P}\rar \ar[d,equal] & \Omega_X\vert_{P}\dar\rar & \Omega_{X/T}\vert_{P}\rar\dar & 0 \\
\varphi_P^*\Omega_T\rar& \Omega_{P}\rar\dar & \Omega_{P/T}\ar[r]\ar[d] & 0\\
& 0 & 0 &
\end{tikzcd}$$ Let ${{\mathbb K}}(x)$ be the residue field of $P$ at $x$. Tensoring the diagram above with ${{\mathbb K}}(x)$, we have the following diagram, with exact rows and columns, $$\begin{tikzcd}[column sep=large, row sep=large]
\varphi^*\Omega_T\vert_{P}\otimes{{\mathbb K}}(x)\rar{h}\ar[d,equal] & \Omega_X\vert_{P}\otimes{{\mathbb K}}(x)\dar\rar & \Omega_{X/T}\vert_{P}\otimes{{\mathbb K}}(x)\rar \dar & 0 \\
\varphi_P^*\Omega_T\otimes{{\mathbb K}}(x)\rar{\overline{h}}& \Omega_{P}\otimes{{\mathbb K}}(x)\rar\dar & \Omega_{P/T}\otimes{{\mathbb K}}(x)\rar \dar & 0\\
& 0 & 0 &
\end{tikzcd}$$ The second row of the last diagram shows that $$\begin{aligned}
\dim_{{{\mathbb K}}(x)}({{\rm im}}(\overline{h})) & = \dim_{{{\mathbb K}}(x)}(\Omega_P\otimes{{\mathbb K}}(x))-\left(r-m\right)\\
& \geqslant r-\left(r-m\right)\\
& = m.
\end{aligned}$$ Since $\dim_{{{\mathbb K}}(x)}({{\rm im}}(h))\geqslant \dim_{{{\mathbb K}}(x)}({{\rm im}}(\overline{h}))$ and $X$ is smooth, the first row of the last diagram implies that $$\label{equation:dimension-fibers-cotangents}
\dim_{{{\mathbb K}}(x)}\left(\Omega_{X/T}\vert_{P}\otimes{{\mathbb K}}(x)\right)= n-\dim({{\rm im}}(h))\leqslant n-m.$$ We note that $\Omega_{X/T}\vert_{P}\otimes{{\mathbb K}}(x) \cong \Omega_{X_{\varphi(x)}}\otimes{{\mathbb K}}(x)$, where $X_{\varphi(x)}$ is the fiber of $\varphi$ over $\varphi(x)$. Then we have $$n-m\leqslant \dim_x X_{\varphi(x)} \leqslant \dim_{{{\mathbb K}}(x)} \Omega_{X_{\varphi(x)}}\otimes{{\mathbb K}}(x) \leqslant n-m$$ This shows that $\Omega_{X/T}$ has rank $n-m$ around $x$. In particular, $\Omega_{X/T}$ is locally free around $x$ by Nakayama’s lemma. Hence, $X_{\varphi(x)}$ is smooth at $x$ and has dimension $n-m$ at $x$.
As an application, we obtain the following criterion for equidimensionality.
\[cor:equi-dim\] Let $\varphi\colon X\rightarrow T$ be a surjective morphism from a smooth projective variety $X$ to a normal projective variety $T$. Assume that there exists a (reduced and irreducible) subvariety $P\subseteq X$ such that the induced morphism $\varphi_P\coloneqq \varphi\vert_P\colon P\rightarrow T$ is surjective and equidimensional with irreducible and generically reduced fibers. Let $t\in T$ be a point. Assume in addition that every component of the fiber $X_t$ contains the fiber $P_t$. Then $\varphi$ is equidimensional around $X_t$.
Since $\varphi_P$ is equidimensional with irreducible and generically reduced fibers, there is a point $x\in P$ lying over $t$ such that the fiber $P_{t}$ is smooth at $x$. By Proposition \[prop:equidimensionality-criterion II\], $\varphi$ is equidimensional around $x$. Let $F$ be an irreducible component of $X_t$. Since $F$ contains $P_t$, we obtain that $x\in F$ and consequently $\dim F = \dim X -\dim T$.
2
Proof of the projective bundle structure {#section:bundle structure}
========================================
In this section, we prove the projective bundle structure in Theorem \[thm:main-theorem\]. Actually, we will prove the following refined statement (compare it with Theorem \[Proj-bundle\]).
\[cor:main:part1\] Let $X$ be a complex projective manifold. Assume that the tangent bundle $T_X$ contains a locally free strictly nef subsheaf ${{\mathscr F}}$ of rank $r>0$. Then $X$ admits a ${{\mathbb P}}^d$-bundle structure $\varphi\colon X\rightarrow T$ over a projective manifold $T$ for some $d\geq r$. Moreover, if $\dim T>0$, then exactly one of the following assertions holds.
1. Either $d=r\geq 1$, ${{\mathscr F}}\cong T_{X/T}$ and $X$ is isomorphic to a flat projective bundle over $T$,
2. or $r\geq 2$, ${{\mathscr F}}$ is a numerically projectively flat vector bundle such that its restriction on every fiber of $\varphi$ is isomorphic to ${{\mathscr O}}_{{{\mathbb P}}^d}(1)^{\oplus r}$, and there exists a flat ${{\mathbb P}}^{r-1}$-subbundle $Q\rightarrow T$ of $X$ with a surjection ${{\mathscr F}}\vert_Q\rightarrow T_{Q/T}$. In particular, the relative tangent bundle $T_{Q/T}$ is strictly nef.
Setup {#section:setup}
-----
For the proof of Theorem \[cor:main:part1\], we discuss two different cases, and each case consists of several steps. For simplicity, we first establish some common setup in this subsection.
Let $X$ be a projective manifold of dimension $n$, and ${{\mathscr F}}\subseteq T_X$ a strictly nef locally free subsheaf of rank $r$. Then Proposition \[MRC-fibration\] shows that there is an open subset $X^\circ$ of $X$ whose complement has codimension at least two such that there is a $\mathbb{P}^d$-bundle structure $$\varphi^\circ \colon X^\circ \to T^\circ$$ and ${{\mathscr F}}|_{X^\circ} \subseteq T_{X^\circ/T^\circ}$. We recall that, by Lemma \[Pn\], for a general fiber $F$ of $\varphi^\circ$, the restriction ${{\mathscr F}}|_F$ is either $T_F$ or isomorphic to ${{\mathscr O}}_{F}(1)^{\oplus r}$. These two cases will be studied separately in Section \[subsection:F=T\_F\] and Section \[subsection:F=O(1)r\].
The crucial part for Theorem \[cor:main:part1\] is to prove the following result.
\[lemma:Core-Technical-Theorem\] Let $X$ be a projective manifold such that $T_X$ contains a locally free strictly nef subsheaf. Then there exists an equidimensional Mori contraction $\varphi\colon X\rightarrow T$ which is also the MRC fibration of $X$.
Indeed, by using Theorem \[lemma:Core-Technical-Theorem\] and Corollary \[cor-equidim-implies-bundle\], one can deduce that $\varphi\colon X\to T$ is a projective bundle, which gives the first part of Theorem \[cor:main:part1\]. Finally we can finish the proof of Theorem \[cor:main:part1\] by applying Theorem \[Proj-bundle\]. To show Theorem \[lemma:Core-Technical-Theorem\], we analyze as follows. Let $T'$ be the normalization of the closure of $T^\circ$ in ${\ensuremath{\mbox{\rm Chow}(X)}}$, and let $X'$ be the normalization of the universal family over $T'$. We claim the following statement.
\[lemma:bundle\] The induced morphism $\varphi'\colon X'\rightarrow T'$ is a ${{\mathbb P}}^d$-bundle.
We first assume that ${{\mathscr F}}$ has rank $r\geqslant 2$. Denote by $e\colon X'\rightarrow X$ the evaluation morphism. Then the restrictions of $e$ on fibers of $\varphi'$ are finite morphisms. In particular, the pullback ${{\mathscr F}}'=e^*{{\mathscr F}}$ is strictly nef when restricted on each fiber of $\varphi'$. Thus for a general fiber $F'$ of $\varphi'$, $\det({{\mathscr F}}')\vert_{F'}$ is isomorphic to either ${{\mathscr O}}_{{{\mathbb P}}^d}(r)$ or ${{\mathscr O}}_{{{\mathbb P}}^d}(r+1)$ after Lemma \[Pn\]. If $C$ is a rational curve contained in a fiber of $\varphi'$, then ${{\mathscr F}}'\vert_C$ is strictly nef, and therefore ample. Hence $c_1(\det({{\mathscr F}}))\cdot C\geqslant r$. By assumption, $r\geqslant \frac{1}{2}(r+1)$ and we can apply Proposition \[PROP:HN-degeneration-of-projective-spaces\] to conclude it.
Now we assume that ${{\mathscr F}}$ is a line bundle. We may assume further that $X$ is not isomorphic to a projective space. Then by [@Druel2004 Corollaire], there is a ${{\mathbb P}}^1$-bundle structure $\beta \colon X\to W$ with $\dim W > 0$ such that ${{\mathscr F}}\cong T_{X/W}$. It then follows that $d=1$ and that $\beta$ is just the MRC fibration of $X$. Thus $X'\cong X$ and $T'\cong W$. This completes the proof of the lemma.
Let $\overline{T}\rightarrow T'$ be a desingularization, and let $\overline{X}$ be the fiber product $X'\times_{T'}\overline{T}$. Then $\overline{\varphi}\colon \overline{X}\rightarrow \overline{T}$ is a ${{\mathbb P}}^d$-bundle by Lemma \[lemma:bundle\]. Let $Z$ be the fiber product $\overline{X}\times_{\overline{T}}\overline{X}$. By Theorem \[thm:Brauer-group\], the induced morphism $Z\rightarrow \overline{X}$ is an insignificant projective bundle. In another word, there exists a vector bundle ${{\mathscr E}}$ of rank $d+1$ on $\overline{X}$ such that $Z={{\mathbb P}}({{\mathscr E}})$. Then we have the following commutative diagram, which will be frequently used throughout this section.
$$\label{Key-diagram}
\begin{tikzcd}[column sep=large, row sep=large]
X^\circ \ar[d, "{\varphi^\circ}"]\ar[r] &
X & X' \ar[l,"e"]\ar[d, "{\varphi'}"]
& \overline{X} \ar[l,"{h}"] \arrow[bend right]{ll}[black,swap]{\rho} \ar[d,"{\overline{\varphi}}"] &
Z ={{\mathbb P}}({{\mathscr E}}) \ar[l,"{q}"]\ar[d,"{\pi}"]
\\
T^\circ & &
T' &
\overline{T} \ar[l] &
\overline{X}\ar[l,"{p=\overline{\varphi}}"]
\end{tikzcd}$$
The first step towards the proof of Theorem \[lemma:Core-Technical-Theorem\] is to show that the inclusion ${{\mathscr F}}\hookrightarrow T_{X}$ induces an inclusion $\rho^*{{\mathscr F}}\to T_{\overline{X}/\overline{T}}$. To achieve this, we will proceed as follows. Let $E_i$’s be the $\rho$-exceptional prime divisors. By shrinking $X^\circ$ if necessary, we may identify $X^\circ$ with an open subset $\overline{X}^\circ$ of $\overline{X}$. Then there are smallest integers $m_i$ such that the morphism ${{\mathscr F}}|_{X^\circ} \to T_{X^\circ/T^\circ}$ extends to a morphism $$\rho^*{{\mathscr F}}\to T_{\overline{X}/\overline{T}}\otimes {{\mathscr O}}_{\overline{X}}(\sum m_iE_i).$$ Our goal is then to prove that $m_i \leqslant 0$ for all $i$.
We also have the following simple observation. By construction, $T^\circ$ can be identified with an open subset of $\overline{T}$ such that $\overline{\varphi}(E_i)$ is contained in $\overline{T}\setminus T^\circ$ for all $i$. As a consequence, there are prime divisors $Q_i$ in $\overline{T}$ such that $E_i=\overline{\varphi}^*Q_i$. In particular, we have $$q^*E_i=\pi^*E_i.$$ Let $E_i'= q^*E_i=\pi^*E_i$ and let ${{\mathscr G}}=q^*\rho^*{{\mathscr F}}$. We denote by $T_{\pi}$ the relative tangent bundle of $\pi\colon Z\rightarrow \overline{X}$. Then the $m_i$ are also the smallest integers such that there is a morphism $$\label{equation:lifing-morphism}
\Phi\colon {{\mathscr G}}\to T_{\pi}\otimes {{\mathscr O}}_{Z}(\sum m_iE_i'),$$ which extends the following morphism on $Z^\circ := \pi^{-1}(\overline{X}^\circ)$ $$q^*\rho^*({{\mathscr F}}|_{X^\circ}) \hookrightarrow q^*\rho^*(T_{X^\circ/T^\circ})\cong T_{Z^\circ/\overline{X}^\circ}.$$ Moreover, $\Phi$ does not vanish in codimension one by the minimality of $m_i$.
The case when ${{\mathscr F}}|_F=T_F$ {#subsection:F=T_F}
-------------------------------------
In this subsection, we will study the case when ${{\mathscr F}}\vert_F=T_F$. Note that we have $r=d$ in this case. Since the proof is a bit involved, we subdivide it into four steps, given in Sections \[section:step-1-case-1\]–\[section:step-4-case-1\] below. We shall follow the notations in Section \[section:setup\], especially those in the commutative diagram .
### Lifting the inclusion ${{\mathscr F}}\hookrightarrow T_X$ to an isomorphism $\rho^*{{\mathscr F}}\rightarrow T_{\overline{X}/\overline{T}}$ {#section:step-1-case-1}
\[claim:pullback-sheaf=relative-tangent\] The injection ${{\mathscr F}}|_{X^\circ} \hookrightarrow T_{X^\circ/T^\circ}$ is an isomorphism.
Let $t\in T^\circ$ be an arbitrary point and let $x$ be a point in $\varphi^{\circ-1}(t)$. Since the complement of $X^\circ$ in $X$ has codimension at least 2, we may choose a general complete curve $C$ in $X$ passing through $x$ such that $B\coloneqq\varphi^\circ(C)$ is a complete curve in $T^\circ$. Let $n\colon B'\rightarrow B$ be the normalization of $B$ and let $Y$ be the fiber product $X^\circ\times_{B} B'$. Then $p\colon Y\rightarrow B'$ is a ${{\mathbb P}}^d$-bundle. Since the Brauer group of $B'$ is trivial, there exists a vector bundle ${{\mathscr V}}$ over $B'$ such that $Y\cong {{\mathbb P}}({{\mathscr V}})$. $$\begin{tikzcd}[column sep=large, row sep=large]
Y={{\mathbb P}}({{\mathscr V}}) \dar[swap]{p} \arrow{rr}[swap]{\nu} & & X^\circ\dar{\varphi^\circ} \arrow[r,hookrightarrow] & X\\
B' \arrow[r,hookrightarrow] & B \rar & T^\circ &
\end{tikzcd}$$ Since $C$ is in general position, the induced morphism $\nu^*{{\mathscr F}}\rightarrow T_{Y/B'}$ is still injective such that $(\nu^*{{\mathscr F}})\vert_G=T_G$ for general fibers $G$ of $p$. As $\nu$ is finite, $\nu^*{{\mathscr F}}$ is strictly nef. Applying Theorem \[Proj-bundle\] to $p\colon Y\rightarrow B'$ shows that $\nu^*{{\mathscr F}}=T_{Y/B'}$. As $t$ is arbitrary, by pushing-forward, we obtain that ${{\mathscr F}}\vert_{X^\circ}\rightarrow T_{X^\circ/T^\circ}$ is an isomorphism.
\[claim:pullback-sheaf-in-tangent-1\] The injection ${{\mathscr F}}\hookrightarrow T_X$ induces an isomorphism $\rho^*{{\mathscr F}}\to T_{\overline{X}/\overline{T}}$.
Let ${{\mathscr G}}=q^*\rho^*{{\mathscr F}}$. As explained at the end of Section \[section:setup\], we have the following commutative diagram
$$\begin{tikzcd}[column sep=large, row sep=large]
X^\circ \dar[swap]{\varphi^\circ}\rar &
X & \overline{X} \lar[swap]{\rho} \dar[swap]{\overline{\varphi}} &
Z={{\mathbb P}}({{\mathscr E}})\lar[swap]{q}\dar{\pi} \\
T^\circ & &
\overline{T} &
\overline{X}\lar{p=\overline{\varphi}}
\end{tikzcd}$$ and an induced morphism (see for details) $$\Phi\colon {{\mathscr G}}\to T_{\pi}\otimes {{\mathscr O}}_{Z}(\sum m_iE_i'),$$ which does not vanish in codimension one and extends the natural isomorphism $$(q^*\rho^*{{\mathscr F}})\vert_{X^\circ}\rightarrow q^*\rho^*(T_{X^\circ/T^\circ})\cong T_{Z^\circ/\overline{X}^\circ}.$$
*1st Step. ${{\mathscr E}}$ is numerically projectively flat.* We consider the line bundle ${{\mathscr L}}= \mathrm{det}\, {{\mathscr G}}$. Its restriction on any fiber of $\pi$ is isomorphic to ${{\mathscr O}}_{\mathbb{P}^d}(d+1)$. Thus there is a line bundle ${{\mathscr H}}$ on $\overline{X}$ such that $${{\mathscr L}}\cong {{\mathscr O}}_{\mathbb{P}({{\mathscr E}})}(d+1) \otimes \pi^*\mathrm{det}\, {{\mathscr E}}^* \otimes \pi^*{{\mathscr H}}.$$ We note that ${{\mathscr G}}|_{Z^\circ} \cong T_{Z^\circ/\overline{X}^\circ}$ by Claim \[claim:pullback-sheaf=relative-tangent\]. This implies that ${{\mathscr H}}|_{\overline{X}^\circ}\cong {{\mathscr O}}_{\overline{X}^\circ}$. Therefore, there is a $\mathbb{Q}$-divisor class $\delta$, supported in the $\rho$-exceptional locus, such that $$c_1({{\mathscr H}}) = (d+1) \delta.$$ Since ${{\mathscr L}}$ is nef, by definition the $\mathbb{Q}$-twisted vector bundle ${{\mathscr E}}\hspace{-0.8ex}<\hspace{-0.8ex} \delta - \mu({{\mathscr E}}) \hspace{-0.8ex}> $ is nef, where $\mu({{\mathscr E}})$ is the average first Chern class of ${{\mathscr E}}$. By taking the first Chern class, we see that $\delta$ is nef. Since $\delta$ is supported in the $\rho$-exceptional locus, by the negativity lemma, we obtain that $\delta=0$. Thus $${{\mathscr O}}_{{{\mathbb P}}({{\mathscr E}})}(d+1)\otimes \pi^*\det{{\mathscr E}}^*$$ is nef and consequently ${{\mathscr E}}$ is numerically projectively flat.\
*2nd Step. $m_i\leqslant 0$ for all $i$.* Assume by contradiction that it is not the case. By Lemma \[lemma:negativity-lemma-line-bundle\], there is a family $\{C_\gamma\}_{\gamma\in \Gamma}$ of complete $\rho$-exceptional curves such that $C_\gamma\cdot \sum m_iE_i<0$ for all $\gamma\in \Gamma$. Let $C'$ be a general member of these curves.
Since ${{\mathscr E}}$ is numerically projectively flat, ${{\mathscr E}}$ is isomorphic to a projectively flat vector bundle by Theorem \[thm:num-proj-flat=proj-flat\]. In particular, $\pi\colon Z\rightarrow X$ is isomorphic to a flat projective bundle over $\overline{X}$. Applying Lemma \[lemma:trivial-bundle-finite-change\] to $\rho\colon \overline{X}\rightarrow X$ and $\pi\colon Z\rightarrow \overline{X}$, we deduce that there is a smooth complete curve $C$, which is finite over $C'$, such that $Z\times_{\overline{X}} C$ is isomorphic to $\mathbb{P}^d\times C$ as ${{\mathbb P}}^d$-bundles over $C$. $$\begin{tikzcd}[column sep=large, row sep=large]
D \arrow[bend left]{rrr}[swap]{\nu}\arrow[r,hookrightarrow] & \mathbb{P}^d\times C \cong Z\times_{\overline{X}} C \arrow[rr] \dar & & Z \dar{\pi}\\
& C \rar & C' \arrow[r,hookrightarrow] & \overline{X}
\end{tikzcd}$$ Let $D$ be a general fiber of the natural projection $\mathbb{P}^d\times C\to \mathbb{P}^d$. We denote by $\nu\colon D \to Z$ the natural morphism. Since the morphism $\Phi\colon{{\mathscr G}}\rightarrow T_{\pi}\otimes {{\mathscr O}}_{Z}(\sum m_i E_i')$ does not vanish in codimension one, by general choices of $C'$ and $D$, we may assume that the morphism $${{\mathscr G}}|_{\nu(D)} \to (T_{\pi} \otimes {{\mathscr O}}_{Z}(\sum m_iE_i'))|_{\nu(D)}$$ is not zero. Since $\nu^*T_{\pi} \cong T_{(\mathbb{P}^d\times C)/C}|_D$ is trivial and ${{\mathscr G}}$ is nef, we obtain that $\nu^* {{\mathscr O}}_{Z}(\sum m_iE_i')$ is pseudoeffective. This contradicts to the fact that $$\nu(D)\cdot \sum m_i E_i'=\deg(\pi\vert_{\nu(D)}) C'\cdot \sum m_iE_i <0.$$ Hence, we have $m_i\leq 0$ for all $i$ and there is an induced injective morphism $$\rho^*{{\mathscr F}}\rightarrow T_{\overline{X}/\overline{T}}.$$
*3rd Step. $\rho^*{{\mathscr F}}\rightarrow T_{\overline{X}/\overline{T}}$ is an isomorphism.* By Theorem \[Proj-bundle\], the induced morphism $q^*\rho^*{{\mathscr F}}\rightarrow T_{\pi}$ is an isomorphism. Hence $\rho^*{{\mathscr F}}\rightarrow T_{\overline{X}/\overline{T}}$ is an isomorphism by taking pushforward.
### Regularity of the MRC fibration
The existence of $\varphi\colon X\rightarrow T$ is a direct consequence of the following claim, which asserts that $X$ admits only one elementary contraction.
\[claim:extremal-ray-F=T\_F\] Let $C$ be a rational curve in $X$. Then $C$ is numerically proportional to a line $l$ contained in a fiber of $\varphi^\circ$.
By Lemma \[lemma:rational-section-birational-morphism\], there exists a curve $C'$ contained in $X'$ such that $e\vert_{C'}\colon C'\rightarrow C$ is birational. In particular, $C'$ is a rational curve. $$\begin{tikzcd}[column sep=large, row sep=large]
X^\circ \dar{\varphi^\circ}\rar &
X & X' \lar{e} \dar[swap]{\varphi'}
& \overline{X} \lar{h} \arrow[bend right]{ll}[swap]{\rho} \dar{\overline{\varphi}}
\\
T^\circ & &
T' &
\overline{T} \lar
\end{tikzcd}$$ Note that $\varphi'\colon X'\rightarrow T'$ is a ${{\mathbb P}}^d$-bundle by Lemma \[lemma:bundle\], thus any complete curve contained in a fiber of $\varphi'$ is numerically proportional to a line contained in a fiber of $\varphi'$. In particular, if $C'$ is contained in a fiber of $\varphi'$, by Lemma \[lemma:numerical-proportional-preseved\], $C'$ is numerically proportional to a line contained in a fiber of $\varphi^\circ$.
Now we assume that $\varphi'(C')=B$ is a curve. Let ${{\mathbb P}}^1\rightarrow B$ be the normalization. Denote by $X'_B$ the fiber product $X'\times_{T'}{{\mathbb P}}^1$ with induced morphism $\nu\colon X'_B\rightarrow X'$. Since $p_1\colon X'_B\rightarrow {{\mathbb P}}^1$ is a ${{\mathbb P}}^d$-bundle and the Brauer group of ${{\mathbb P}}^1$ is trivial, there exists a vector bundle ${{\mathscr V}}$ on ${{\mathbb P}}^1$ such that $X'_B= {{\mathbb P}}({{\mathscr V}})$. Moreover, since $\rho^*{{\mathscr F}}\cong T_{\overline{X}/\overline{T}}$ by Claim \[claim:pullback-sheaf-in-tangent-1\] and since $T_{\overline{X}/\overline{T}}\cong h^*T_{X'/T'}$, we get $$\rho^*{{\mathscr F}}= h^*e^*{{\mathscr F}}\cong T_{\overline{X}/\overline{T}} \cong h^*T_{X'/T'}.$$ Hence $e^*{{\mathscr F}}\cong T_{X'/T'}$. In particular, it yields $$\nu^*e^*{{\mathscr F}}\cong T_{\overline{X}_B/{{\mathbb P}}^1}.$$ Then Theorem \[Proj-bundle\] shows that ${{\mathscr V}}$ is numerically projectively flat. As a consequence, we obtain that $X'_{B}\cong {{\mathbb P}}^1\times {{\mathbb P}}^d$. Let $p_2\colon X'_{B} \to {{\mathbb P}}^d$ be the morphism induced by the projection ${{\mathbb P}}^1\times {{\mathbb P}}^d\rightarrow {{\mathbb P}}^d$ and $f\colon X'_{B} \to X$ the composition of $$X'_B\xrightarrow{\nu} X'\xrightarrow{e} X.$$ Since $\nu^*e^*{{\mathscr F}}\cong T_{\overline{X}_B/{{\mathbb P}}^1}$ is trivial on the fibers of $p_2$ and since ${{\mathscr F}}$ is strictly nef, the fibers of $p_2$ are all contracted by $f$. Hence, by rigidity lemma, the morphism $f\colon {X}'_B\rightarrow X$ factors through $p_2$. As a consequence, every point in $B\subseteq T'$ corresponds to the same cycle in $X$. This contradicts to the definition of ${\ensuremath{\mbox{\rm Chow}(X)}}$. Hence $C'$ is always contracted by $\varphi'$, and we complete the proof of the claim.
### Proof of Theorem \[lemma:Core-Technical-Theorem\] in the case when ${{\mathscr F}}\vert_F=T_F$ {#section:step-3-case-1}
By Claim \[claim:extremal-ray-F=T\_F\], there exists a Mori contraction $\varphi\colon X\rightarrow T$ extending the fibration $\varphi^\circ\colon X^\circ\rightarrow T^\circ$. By replacing $\overline{T}$ with a common resolution of $T'$ and $T$, we have the following commutative diagram: $$\begin{tikzcd}[column sep=large, row sep=large]
X^\circ \dar[swap]{\varphi^\circ}\rar &
X\dar[swap]{\varphi} & X' \lar{e}\dar[swap]{\varphi'}
& \overline{X} \lar{h} \arrow[bend right]{ll}[swap]{\rho} \dar{\overline{\varphi}} &
Z = {{\mathbb P}}({{\mathscr E}})\lar[swap]{q}\dar{\pi}
\\
T^\circ & T &
T' &
\overline{T} \lar \arrow[bend left]{ll}{\gamma} &
\overline{X}\lar{p=\overline{\varphi}}
\end{tikzcd}$$ By Claim \[claim:pullback-sheaf-in-tangent-1\], ${{\mathscr F}}\hookrightarrow T_X$ induces an isomorphism ${{\mathscr G}}= q^*\rho^*{{\mathscr F}}\to T_{Z/\overline{X}}$. By using Theorem \[Proj-bundle\] and Theorem \[thm:num-proj-flat=proj-flat\], we deduce that ${{\mathscr E}}$ is isomorphic to a projectively flat vector bundle. Hence $\pi$ is isomorphic to a ${{\mathbb P}}^d$-bundle structure given by a representation $\pi_1(\overline{X})\rightarrow \operatorname*{PGL}_{d+1}({{\mathbb C}})$. Since $p\colon \overline{X}\rightarrow \overline{T}$ has simply connected fibers, we deduce that $\overline{\varphi}\colon \overline{X}\rightarrow \overline{T}$ is also isomorphic to a flat ${{\mathbb P}}^d$-bundle over $\overline{T}$ given by a representation $\pi_1(\overline{T})\rightarrow \operatorname*{PGL}_{d+1}({{\mathbb C}})$. We also note that $T$ has only klt singularities. Now one can derive the equidimensionality of $\varphi$ from Proposition \[prop:equidimensionality-criterion-I\].
### Proof of Theorem \[cor:main:part1\] in the case when ${{\mathscr F}}\vert_F=T_F$ {#section:step-4-case-1}
We maintain the notations of Section \[section:setup\]. We first show the projective bundle structure on $X$. If ${{\mathscr F}}$ is a line bundle, then we have a ${{\mathbb P}}^1$-bundle structure $\varphi\colon X\to T$ with ${{\mathscr F}}\cong T_{X/T}$ by [@Druel2004 Corollaire]. Thus we may assume that $r\geq 2$. By the corrsponding case in Theorem \[lemma:Core-Technical-Theorem\], there exists an equidimensional Mori contraction $\varphi\colon X\rightarrow T$ which is also the MRC fibration of $X$. Since ${{\mathscr F}}$ is strictly nef of rank at least two, by Corollary \[cor-equidim-implies-bundle\], $\varphi$ is again a projective bundle.
Next we assume that $\dim T>0$. Since $\varphi\colon X\rightarrow T$ is a projective bundle structure between projective manifolds, we may identify $T$ with $\overline{T}$ and $X$ with $\overline{X}$. Then the fiber product $X\times_T X$ can be identified with $Z={{\mathbb P}}({{\mathscr E}})$. In particular, we have the following commutative diagram: $$\begin{tikzcd}[column sep=large, row sep=large]
X\dar[swap]{\varphi} & Z \lar{q}\dar[swap]{\pi} \\
T &
X\lar{p=\varphi}
\end{tikzcd}$$ We have seen ${{\mathscr F}}\cong T_{X/T}$ by Claim \[claim:pullback-sheaf-in-tangent-1\]. Moreover, from the first step of Claim \[claim:pullback-sheaf-in-tangent-1\], ${{\mathscr E}}$ is isomorphic to a projectively flat vector bundle. Therefore, $Z$ is isomorphic to a flat projective bundle over $X$, given by a representation of $\pi_1(X)$ in $\operatorname*{PGL}_{d+1}({{\mathbb C}})$. Since $X$ is a projective bundle over $T$, the fundamental group of $X$ is canonically isomorphic to that of $T$. Hence we get an induced flat projective bundle $Q$ over $T$. Let $f\colon Z\rightarrow Q$ be the natural projection. Then an irreducible $C\subset Z$ is contracted by $q$ if and only if $C$ is contracted by $f$. Thus, by rigidity lemma, one can easily derive that $X$ is isomorphic to $Q$ as projective bundles over $T$.
The case when ${{\mathscr F}}|_F={{\mathscr O}}_{{{\mathbb P}}^d}(1)^{\oplus r}$ {#subsection:F=O(1)r}
--------------------------------------------------------------------------------
In this subsection, we study the case when ${{\mathscr F}}\vert_F\cong {{\mathscr O}}_{{{\mathbb P}}^d}(1)^{\oplus r}$. As in the previous subsection, the proof is subdivided into four parts, given in Sections \[section:step-1-case-2\]–\[section:step-4-case-2\]. We still follow the notations in Section \[section:setup\].
### Lifting the inclusion ${{\mathscr F}}\hookrightarrow T_X$ to an inclusion $\rho^*{{\mathscr F}}\hookrightarrow T_{\overline{X}/\overline{T}}$ {#section:step-1-case-2}
\[claim:pullback-saturated-subsheaf\] Use notations as in Section \[section:setup\] and identify $X^\circ$ with $\overline{X}^\circ$. Let ${{\mathscr E}}^\circ$ be the restriction ${{\mathscr E}}\vert_{X^\circ}$. Then there is a vector bundle ${{\mathscr M}}^\circ$ on $X^\circ$ such that $$p_1^*{{\mathscr F}}\cong p_2^*{{\mathscr M}}^\circ \otimes {{\mathscr O}}_{\mathbb{P}({{\mathscr E}}^\circ)}(1),$$ where $p_1$ and $p_2$ are the natural projection $\pi\vert_{Z^\circ}$ and $q\vert_{Z^\circ}$ respectively. Furthermore, the inclusion ${{\mathscr F}}|_{X^\circ} \hookrightarrow T_{X^\circ/T^\circ}$ induces a subbundle structure ${{\mathscr M}}^\circ \hookrightarrow ({{\mathscr E}}^\circ)^*$.
By construction, we have $Z^\circ={{\mathbb P}}({{\mathscr E}}^\circ)$. Since relative tangent bundles commute with base change, there is an induced inclusion $$p_1^*{{\mathscr F}}\hookrightarrow T_{p_2},$$ where $T_{p_2}$ is the relative tangent bundle of $p_2$. Moreover, for every fiber $G$ of $p_2$, we have $p_1^*{{\mathscr F}}|_{G} \cong {{\mathscr O}}_{{{\mathbb P}}^d}(1)^{\oplus r}$. Thus there is a vector bundle ${{\mathscr M}}^\circ$ on $X^\circ$ such that $$p_1^*{{\mathscr F}}\cong p_2^*{{\mathscr M}}^\circ \otimes {{\mathscr O}}_{\mathbb{P}({{\mathscr E}}^\circ)}(1).$$ Furthermore, by Lemma \[lemma:factorization\], the inclusion ${{\mathscr F}}\hookrightarrow T_{X^\circ/T^\circ}$ induces an inclusion ${{\mathscr M}}^\circ \hookrightarrow ({{\mathscr E}}^\circ)^*$.
Next we show that ${{\mathscr M}}^\circ$ is a subbundle of $({{\mathscr E}}^\circ)^*$. Let $x$ be an arbitrary point in $X^\circ$ and $C$ a general complete intersection curve in $X$ passing through $x$. We may choose $C$ so that $C\subseteq X^\circ$. Let $Y = {{\mathbb P}}({{\mathscr E}}^\circ\vert_C)$. Then we have the following commutative diagram. $$\begin{tikzcd}[column sep=large, row sep=large]
Y={{\mathbb P}}({{\mathscr E}}^\circ\vert_C) \dar{p} \rar & Z^\circ={{\mathbb P}}({{\mathscr E}}^\circ)\rar{p_1}\dar[swap]{p_2} & X^\circ\dar{\varphi^\circ} \arrow[r,hookrightarrow] & X\\
C \arrow[r,hookrightarrow] & X^\circ \rar & T^\circ &
\end{tikzcd}$$ Denote by $f \colon Y\to X$ the composition of the first row. Since $C$ is in general position, we still have an injective morphism $f^*{{\mathscr F}}\to T_{Y/C}$. By applying Lemma \[lemma:factorization\] again, we see that the induced morphism ${{\mathscr M}}^\circ|_C \to ({{\mathscr E}}^\circ)^*\vert_C$ is still injective. Then Theorem \[Proj-bundle\] implies that ${{\mathscr M}}^\circ|_C$ is a subbundle of $({{\mathscr E}}^\circ)^*\vert_C$. Since $x$ is arbitrary, ${{\mathscr M}}^\circ$ is a subbundle of $({{\mathscr E}}^\circ)^*$
\[claim:pullback-sheaf-in-tangent\] The injection ${{\mathscr F}}\hookrightarrow T_X$ induces an injection $\rho^*{{\mathscr F}}\hookrightarrow T_{\overline{X}/\overline{T}}$. Moreover, its restriction on each fiber of $\overline{\varphi}$ is still injective.
Let ${{\mathscr G}}=q^*\rho^*{{\mathscr F}}$. As explained at the end of Section \[section:setup\], we have the following commutative diagram $$\begin{tikzcd}[column sep=large, row sep=large]
X^\circ \dar{\varphi^\circ}\rar &
X & \overline{X} \lar[swap]{\rho}\dar{\overline{\varphi}} &
Z={{\mathbb P}}({{\mathscr E}})\lar[swap]{q}\dar{\pi} \\
T^\circ & &
\overline{T} &
\overline{X}\lar{p=\overline{\varphi}}
\end{tikzcd}$$ and an induced morphism (see ) $$\Phi\colon {{\mathscr G}}\to T_{\pi}\otimes {{\mathscr O}}_{Z}(\sum m_iE_i')$$ which does not vanish in codimension one. Alternatively, we have a morphism $${{\mathscr G}}\otimes \pi^*{{\mathscr O}}_{\overline{X}}(\sum -m_iE_i') \to T_{\pi}$$ on $Z$, which is nonzero in codimension one. We note that the restriction of ${{\mathscr G}}$ on every fiber of $\pi$ is isomorphic to ${{\mathscr O}}_{\mathbb{P}^d}(1)^{\oplus r}$ (see Lemma \[Pn\]). Hence $${{\mathscr G}}\cong \pi^*{{\mathscr M}}\otimes {{\mathscr O}}_{\mathbb{P}({{\mathscr E}})}(1)$$ for some vector bundle ${{\mathscr M}}$ on $ \overline{X}$. Thanks to Lemma \[lemma:factorization\], we obtain an injective morphism $${{\mathscr M}}\otimes {{\mathscr O}}_{\overline{X}}(\sum -m_iE_i) \to {{\mathscr E}}^*$$ on $\overline{X}$ which is nonzero in codimension one. We denote by ${{\mathscr Q}}$ the saturation of ${{\mathscr M}}\otimes {{\mathscr O}}_{\overline{X}}(\sum -m_i E_i)$ in ${{\mathscr E}}^*$.\
*1st Step. ${{\mathscr Q}}$ is a numerically projectively flat vector bundle.* By Claim \[claim:pullback-saturated-subsheaf\], the injection ${{\mathscr F}}\hookrightarrow T_X$ induces a subbundle structure ${{\mathscr M}}|_{\overline{X}^\circ} \hookrightarrow {{\mathscr E}}^*|_{\overline{X}^\circ}$. Thus ${{\mathscr Q}}\vert_{\overline{X}^\circ} = {{\mathscr M}}\vert_{\overline{X}^\circ}$ and there exists a ${{\mathbb Q}}$-divisor $\delta$ supported in the $\rho$-exceptional locus such that $$\mu({{\mathscr M}})-\mu({{\mathscr Q}}) = \delta.$$ Since ${{\mathscr G}}\cong \pi^*{{\mathscr M}}\otimes{{\mathscr O}}_{{{\mathbb P}}({{\mathscr E}})}(1)$ is nef, so is its determinant $$\det {{\mathscr G}}\cong \pi^*\det {{\mathscr M}}\otimes {{\mathscr O}}_{{{\mathbb P}}({{\mathscr E}})}(r).$$ In particular, ${{\mathscr E}}\hspace{-0.8ex}<\hspace{-0.8ex}\mu({{\mathscr M}})\hspace{-0.8ex}>$ is nef. Therefore, from the generically surjective morphism ${{\mathscr E}}\to {{\mathscr Q}}^*$, we know that the $\mathbb{Q}$-twisted sheaf $${{\mathscr Q}}^*\hspace{-0.8ex}<\hspace{-0.8ex}\mu({{\mathscr M}})\hspace{-0.8ex}>$$ is almost nef. By taking the first Chern class, we obtain that $\delta$ is pseudoeffective. As $\delta$ is $\rho$-exceptional, it follows that $\delta$ is effective.
Next we show that $\delta=0$. Assume the opposite. By Lemma \[lemma:negativity-lemma-line-bundle\], there is a family $\{C_\gamma\}_{\gamma\in \Gamma}$ of complete $\rho$-exceptional curves such that $C_\gamma\cdot \delta<0$ for all $\gamma\in \Gamma$. Since ${{\mathscr Q}}$ is locally free in codimension two and it is a subbundle of ${{\mathscr E}}^*$ in codimension one, we may choose a general element $C'$ in $\{C_\gamma\}_{\gamma\in \Gamma}$ such that ${{\mathscr Q}}$ is locally free along $C'$ and the induced morphism ${{\mathscr E}}\vert_{C'}\rightarrow {{\mathscr Q}}^*\vert_{C'}$ is generically surjective. This implies that ${{\mathscr Q}}^*\hspace{-0.8ex}<\hspace{-0.8ex}\mu({{\mathscr M}})\hspace{-0.8ex}>\vert_{C'}$ is an almost nef vector bundle. By taking the determinant, we deduce that $$C'\cdot \delta=C'\cdot (\mu({{\mathscr M}}) - \mu({{\mathscr Q}}))\geqslant 0,$$ which yields a contradiction. Hence $\delta=0$ and we have $\mu({{\mathscr M}})=\mu({{\mathscr Q}})$. In particular, as ${{\mathscr Q}}^*\hspace{-0.8ex}<\hspace{-0.8ex}\mu({{\mathscr M}})\hspace{-0.8ex}>$ is almost nef, by Theorem \[Num-Projectily-flatness-criterion\], ${{\mathscr Q}}^*$ is actually a numerically projectively flat vector bundle. So is ${{\mathscr Q}}$ by Lemma \[lemma:prop-num-proj-flat\].\
*2nd Step. ${{\mathscr M}}$ is isomorphic to a projectively flat vector bundle.* For simplicity, we will argue up to isomorphisms. We may assume that ${{\mathscr Q}}$ is projectively flat by Theorem \[thm:num-proj-flat=proj-flat\]. Since $${{\mathscr G}}|_{Z^\circ} \cong (\pi^*{{\mathscr Q}}\otimes {{\mathscr O}}_{{{\mathbb P}}({{\mathscr E}})}(1))|_{Z^\circ},$$ we see that ${{\mathscr G}}|_{Z^\circ}=(q^*\rho^*{{\mathscr F}})|_{Z^\circ}$ is projectively flat. Then $\rho^*{{\mathscr F}}|_{\overline{X}^\circ}$ is projectively flat by Lemma \[lemma:proj-flat-vec-bundle-property2\]. Therefore, ${{\mathscr F}}|_{X^\circ}$ is projectively flat. By Lemma \[lemma:extension-proj-flat-vec-bundle\], we obtain that ${{\mathscr F}}$ itself is projectively flat. Hence ${{\mathscr G}}$ is projective flat and so is ${{\mathscr M}}$ by Lemma \[lemma:proj-flat-vec-bundle-property2\].\
*3rd Step. $m_i \leqslant 0$ for all $i$.* Assume the opposite. By Lemma \[lemma:negativity-lemma-line-bundle\], there is a family $\{C_\gamma\}_{\gamma\in \Gamma}$ of complete $\rho$-exceptional curves such that $C_\gamma\cdot \sum m_iE_i<0$ for all $\gamma\in \Gamma$. Let $C'$ be a general member of these curves. Since ${{\mathscr M}}$ is isomorphic to a projectively flat vector bundle, ${{\mathbb P}}({{\mathscr M}})$ is isomorphic to a flat projective bundle. Now applying Lemma \[lemma:trivial-bundle-finite-change\] to $p\colon \overline{X}\rightarrow \overline{T}$ and ${{\mathbb P}}({{\mathscr M}})\rightarrow \overline{X}$ shows that there is a smooth curve $C$, finite over $C'$, such that $\eta^*{{\mathscr M}}\cong {{\mathscr L}}^{\oplus r}$ for some line bundle ${{\mathscr L}}$ on $C$, where $\eta\colon C\to \overline{X}$ is the natural morphism. Since ${{\mathscr E}}\rightarrow {{\mathscr M}}^*\otimes {{\mathscr O}}_{\overline{X}}(\sum m_i E_i)$ does not vanish in codimension one, by general choice of $C'$, we may assume that the morphism $$\eta^*{{\mathscr E}}\to \eta^*({{\mathscr M}}^* \otimes {{\mathscr O}}_{\overline{X}}(\sum m_i E_i))$$ is not identically zero. Hence we obtain a generically surjective morphism $$\eta^*{{\mathscr E}}\to {{\mathscr L}}^*\otimes \eta^*{{\mathscr O}}_{\overline{X}}(\sum m_i E_i).$$ Since the $\mathbb{Q}$-twisted sheaf ${{\mathscr E}}\hspace{-0.8ex}<\hspace{-0.8ex}\mu({{\mathscr M}})\hspace{-0.8ex}>$ is nef, it follows that $\eta^*{{\mathscr E}}\otimes {{\mathscr L}}$ is nef. Thus $\eta^*{{\mathscr O}}_{\overline{X}}(\sum m_i E_i)$ is pseudoeffective on $C$. This contradicts to the fact that $C'\cdot \sum m_iE_i<0$. As a consequence, we have induced morphisms $\rho^*{{\mathscr F}}\to T_{\overline{X}/\overline{T}}$ and ${{\mathscr M}}\hookrightarrow {{\mathscr E}}^*$.\
*4th Step. The restriction of $\rho^*{{\mathscr F}}\to T_{\overline{X}/\overline{T}}$ on any fiber of $\overline{\varphi}$ is still injective.* By Theorem \[Proj-bundle\], we see that ${{\mathscr M}}\hookrightarrow {{\mathscr E}}^*$ is a subbundle. The statement then follows.
### Regularity of the MRC fibration
Similarly to the case when ${{\mathscr F}}\vert_F\cong T_F$, we prove the following claim.
\[claim:extremal-ray-F=O1\] Let $C\subseteq X$ be a rational curve. Then $C$ is numerically proportional to a line contained in a fiber of $\varphi^\circ$.
By Lemma \[lemma:rational-section-birational-morphism\], there exists a complete rational curve $C'\subseteq \overline{X}$ which is birational to $C$. As in the proof of Claim \[claim:extremal-ray-F=T\_F\], it is enough to consider the case when $C'$ is not contracted by $\overline{\varphi}$.
Assume that $B=\overline{\varphi}(C')$ is a curve. Then $B$ is a rational curve. Consider the normalization ${{\mathbb P}}^1 \to B$. Let $\overline{X}_B = \overline{X}\times_{\overline{T}} {{\mathbb P}}^1$ and $p_1\colon \overline{X}_B\rightarrow {{\mathbb P}}^1$ the natural projection. $$\begin{tikzcd}[column sep=large, row sep=large]
X^\circ \dar[swap]{\varphi^\circ}\rar &
X & \overline{X} \lar{\rho}\dar[swap]{\overline{\varphi}} & \overline{X}_B \cong{{\mathbb P}}({{\mathscr V}})\dar{p_1}\lar{\nu}\arrow[bend right]{ll}[swap]{f} \rar{g} & Y \\
T^\circ & &
\overline{T} & {{\mathbb P}}^1\lar &
\end{tikzcd}$$ Since $p_1\colon \overline{X}_B\rightarrow {{\mathbb P}}^1$ is a ${{\mathbb P}}^d$-bundle and the Brauer group of ${{\mathbb P}}^1$ is trivial, there exists a vector bundle ${{\mathscr V}}$ such that $\overline{X}_B \cong {{\mathbb P}}({{\mathscr V}})$. Let $\overline{{{\mathscr F}}}=\rho^*{{\mathscr F}}$. Then there is an induced injective morphism $\nu^*\overline{{{\mathscr F}}}\rightarrow T_{p_1}$ by Claim \[claim:pullback-sheaf-in-tangent\]. By Theorem \[Proj-bundle\], there exists a numerically projectively flat subbundle ${{\mathscr N}}\hookrightarrow {{\mathscr V}}^*$ such that $$\nu^*\overline{{{\mathscr F}}}\cong p_1^*{{\mathscr N}}\otimes{{\mathscr O}}_{{{\mathbb P}}({{\mathscr V}})}(1).$$ We remark that ${{\mathscr N}}\cong {{\mathscr L}}^{\oplus r}$ for some line bundle ${{\mathscr L}}$ on ${{\mathbb P}}^1$. Hence, by replacing ${{\mathscr V}}$ with ${{\mathscr V}}\otimes {{\mathscr L}}^*$, we may assume that ${{\mathscr N}}\cong {{\mathscr O}}_{{{\mathbb P}}^1}^{\oplus r}$. The nefness of $\nu^*\overline{{{\mathscr F}}}$ then implies that ${{\mathscr V}}$ is nef. In particular, there exist integers $a_i\geqslant 0$ with $i=1,...,d+1-r$ such that $${{\mathscr V}}\cong {{\mathscr N}}^*\oplus \bigoplus_{i=1}^{d+1-r}{{\mathscr O}}_{{{\mathbb P}}^1}(a_i) \cong {{\mathscr O}}_{{{\mathbb P}}^1}^{\oplus r} \oplus\bigoplus_{i=1}^{d+1-r}{{\mathscr O}}_{{{\mathbb P}}^1}(a_i).$$ Note that ${{\mathscr O}}_{{{\mathbb P}}({{\mathscr V}})}(1)$ is globally generated. Let $g\colon {{\mathbb P}}({{\mathscr V}})\rightarrow Y$ be the Iitaka fibration induced by ${{\mathscr O}}_{{{\mathbb P}}({{\mathscr V}})}(1)$. Then $g$ does not contract any curves contained in the fibers of $p_1$. Consider a complete curve $B'' \subseteq {{\mathbb P}}({{\mathscr V}})$ contracted by $g\colon {{\mathbb P}}({{\mathscr V}})\rightarrow Y$. Then ${{\mathscr O}}_{{{\mathbb P}}({{\mathscr V}})}(1)\vert_{B''}$ is trivial. As $f^*{{\mathscr F}}\cong {{\mathscr O}}_{{{\mathbb P}}({{\mathscr V}})}(1)^{\oplus r}$, it follows that $(f^*{{\mathscr F}})\vert_{B''}$ is trivial. In particular, since ${{\mathscr F}}$ is strictly nef, it follows that $B''$ is contracted by the composition $$f\colon {{\mathbb P}}({{\mathscr V}}) \cong \overline{X}_{B}\xrightarrow{\nu}\overline{X}\xrightarrow{\rho} X.$$ By rigidity lemma, the morphism $f\colon {{\mathbb P}}({{\mathscr V}})\rightarrow X$ factors through $g\colon {{\mathbb P}}({{\mathscr V}})\rightarrow Y$.
Let $C'' \subseteq {{\mathbb P}}({{\mathscr V}})$ be the curve corresponding to $\nu^{-1}(C') \subseteq \overline{X}_B$. Then $C''$ is not contracted by $g$ as $f(C'')=C$ is again a curve. Let $l$ be a line contained in a general fiber of $p_1$, $\widetilde{C} = g(C'')$ and $\widetilde{l} = g(l)$. Note that $Y$ has Picard number $1$, thus $\widetilde{C}$ and $\widetilde{l}$ are numerically proportional in $Y$. Since $f$ factorizes through $g$, by Lemma \[lemma:numerical-proportional-preseved\], we conclude that $f(l)$ is numerically proportional to $f(C'')=C$ in $X$.
### Proof of Theorem \[lemma:Core-Technical-Theorem\] in the case when ${{\mathscr F}}\vert_F={{\mathscr O}}_{{{\mathbb P}}^d}(1)^{\oplus r}$ {#section:step-3-case-2}
If ${{\mathscr F}}$ is a line bundle, by [@Druel2004 Corollaire], $X$ is isomorphic to ${{\mathbb P}}^n$. Thus we may assume that $r\geqslant 2$.
By Claim \[claim:extremal-ray-F=O1\], there exists a Mori contraction $\varphi\colon X\rightarrow T$ extending the fibration $\varphi^\circ\colon X^\circ\rightarrow T^\circ$. By replacing $\overline{T}$ with a common resolution of $T'$ and $T$, we have the following commutative diagram: $$\begin{tikzcd}[column sep=large, row sep=large]
X^\circ \dar[swap]{\varphi^\circ}\rar &
X\dar[swap]{\varphi} & X' \lar{e}\dar[swap]{\varphi'}
& \overline{X} \lar{h} \arrow[bend right]{ll}[swap]{\rho} \dar{\overline{\varphi}} &
Z\lar[swap]{q}\dar{\pi}
\\
T^\circ & T &
T' &
\overline{T} \lar \arrow[bend left]{ll}{\gamma} &
\overline{X}\lar{p=\overline{\varphi}}
\end{tikzcd}$$ By Claim \[claim:pullback-sheaf-in-tangent\], there is an injective morphism $\rho^*{{\mathscr F}}\rightarrow T_{\overline{X}/\overline{T}}$. This induces an injective morphism ${{\mathscr G}}= q^*\rho^*{{\mathscr F}}\rightarrow T_{\pi}$.
By Theorem \[Proj-bundle\], there exists a numerically projectively flat subbundle ${{\mathscr M}}$ of ${{\mathscr E}}^*$ such that ${{\mathscr G}}\cong \pi^*{{\mathscr M}}\otimes{{\mathscr O}}_{{{\mathbb P}}({{\mathscr E}})}(1)$. Denote by $\pi_M\colon M\rightarrow \overline{X}$ the ${{\mathbb P}}^{r-1}$-bundle ${{\mathbb P}}({{\mathscr M}}^*)$. Then, by Theorem \[thm:num-proj-flat=proj-flat\], $M\rightarrow \overline{X}$ is isomorphic to a flat projective bundle given by a representation $\pi_1(\overline{X})\rightarrow \operatorname*{PGL}_{r}({{\mathbb C}})$. Moreover, note that ${{\mathscr G}}\vert_{M}$ is isomorphic to $\pi_M^*{{\mathscr M}}\otimes {{\mathscr O}}_{{{\mathbb P}}({{\mathscr M}}^*)}(1)$ and the relative Euler sequence of ${{\mathbb P}}({{\mathscr M}}^*)$ induces a surjection ${{\mathscr G}}\vert_{M}\rightarrow T_{M/\overline{X}}$. Let $P = (\rho\circ q)(M) \subseteq X$. According to Proposition \[prop:smothness-image-subbundle\], over $T^\circ$, $P$ is a ${{\mathbb P}}^{r-1}$-bundle. In particular, the general fiber of $P\rightarrow T$ is isomorphic to ${{\mathbb P}}^{r-1}$. Let $n\colon P'\rightarrow P$ be the normalization. Then we have a commutative diagram $$\begin{tikzcd}[column sep=large, row sep=large]
M\dar[swap]{\pi_M} \rar{\rho'} & P'\dar{} \\
\overline{X} \rar& T.
\end{tikzcd}$$ Note that $T$ has only ${{\mathbb Q}}$-factorial klt singularities and $n^*{{\mathscr F}}$ is strictly nef with a surjection $$\rho'^*n^*{{\mathscr F}}\cong q^*\rho^*{{\mathscr F}}\vert_M\rightarrow T_{M/\overline{X}},$$ thus we can apply Proposition \[prop:equidimensionality-criterion-I\] to conclude that $P'\rightarrow T$ is equidimensional. Since $P'\rightarrow P$ is finite, $P\rightarrow T$ is again equidimensional. In particular, by Proposition \[PROP:HN-degeneration-of-projective-spaces\], all the fibers of $P\rightarrow T$ are irreducible and generically reduced.
Let $t\in T$ and $F_t$ an irreducible component of the fiber of $\varphi\colon X\rightarrow T$ over $t$. Then there exists an irreducible component $B$ of $(\gamma\circ p)^{-1}(t)$ such that the induced morphism $$Z\times_{\overline{X}} B\rightarrow X$$ is onto $F_t$. In particular, $F_t$ contains the fiber of $P\rightarrow T$ over $t$. Then Corollary \[cor:equi-dim\] shows that $\varphi\colon X\rightarrow T$ is equidimensional.
### Proof of Theorem \[cor:main:part1\] in the case when ${{\mathscr F}}\vert_F={{\mathscr O}}_{{{\mathbb P}}^d}(1)^{\oplus r}$ {#section:step-4-case-2}
We still use the notations of Section \[section:setup\]. If ${{\mathscr F}}$ is a line bundle, by [@Druel2004 Corollaire], $X$ is isomorphic to ${{\mathbb P}}^n$ and we are done. Thus we may assume that $r\geq 2$. Then Corollary \[cor-equidim-implies-bundle\] shows that $\varphi\colon X\rightarrow T$ is a projective bundle between projective manifolds.
Next we assume that $\dim T>0$. Then we must have $r\geq 2$. Moreover, We may identify $T$ with $\overline{T}$ and $X$ with $\overline{X}$. In particular, $Z={{\mathbb P}}({{\mathscr E}})$ is isomorphic to the fiber product $X\times_T X$ and satisfies the following commutative diagram. $$\begin{tikzcd}[column sep=large, row sep=large]
X\dar[swap]{\varphi} & Z \lar{q}\dar[swap]{\pi} \\
T &
X\lar{p=\varphi}.
\end{tikzcd}$$
Let ${{\mathscr G}}= q^*{{\mathscr F}}$. As explained in Section \[section:step-3-case-2\], there exists a numerically projectively flat subbundle ${{\mathscr M}}$ of ${{\mathscr E}}^*$ such that ${{\mathscr G}}\cong \pi^*{{\mathscr M}}\otimes {{\mathscr O}}_{{{\mathbb P}}({{\mathscr E}})}(1)$. In particular, ${{\mathscr G}}$ is numerically projectively flat and so is ${{\mathscr F}}$. Moreover, set $M={{\mathbb P}}({{\mathscr M}}^*)$ and let $\pi_M\colon M\rightarrow X$ be the natural projection. Then we have a surjection ${{\mathscr G}}\vert_M\rightarrow T_{M/X}$ and $M\rightarrow X$ is isomorphic to a flat projective bundle given by a representation of $\pi_1(X)$ in $\operatorname*{PGL}_{r}({{\mathbb C}})$. As $X\rightarrow T$ is a ${{\mathbb P}}^d$-bundle, $\pi_1(X)$ is isomorphic to $\pi_1(T)$. Such a representation of $\pi_1(X)$ induces a flat ${{\mathbb P}}^{r-1}$-bundle $Q\rightarrow T$ with the following commutative diagram $$\begin{tikzcd}[column sep=large, row sep=large]
Q\dar[swap]{} & M \lar{f}\dar[swap]{} \\
T &
X\lar{p=\varphi}.
\end{tikzcd}$$
Let $P=q(M)$. By Proposition \[prop:smothness-image-subbundle\], we see that $P\rightarrow T$ is a ${{\mathbb P}}^{r-1}$-bundle. On the other hand, it is easy to see that an irreducible curve $C\subset M$ is contracted by $q$ if and only if it is contracted by $f$. Therefore, by rigidity lemma, $P$ is isomorphic to $Q$ as projective bundles over $T$. In particular, the pushforward of the surjection ${{\mathscr G}}\vert_{M}\rightarrow T_{M/X}$ induces a surjection ${{\mathscr F}}\vert_Q\rightarrow T_{Q/T}$.
2
Proof of the hyperbolicity {#section:hyperbolicity}
==========================
In this section, we finish the proofs of Theorem \[thm:main-theorem\], Theorem \[thm:simply-connected-Pn\] and Corollary \[cor:existence-symmetric-forms\]. A projective manifold $Y$ is called *Brody hyperbolic* if every holomorphic map $f\colon {{\mathbb C}}\rightarrow Y$ is constant. Since $Y$ is assumed to be compact, the Brody hyperbolicity is equivalent to the Kobayashi hyperbolicity. The following lemma is an application of [@Yamanoi2010 Theorem 1.1], which reveals the relationships between fundamental groups and degeneracy of entire curves.
\[lemma:representation-degenerate-entire-curve\] Let $Z$ be a projective variety. If there exists a subgroup $G\subseteq \pi_1(Z)$ of finite index such that it admits a linear representation whose image is not virtually abelian, then every holomorphic map $f\colon {{\mathbb C}}\to Z$ is degenerate, i.e. $f({{\mathbb C}})$ is not Zariski dense in $Z$.
By taking a finite étale cover, it is enough to prove the case when $G=\pi_1(Z)$. Assume that there exists a holomorphic map $f\colon {{\mathbb C}}\to Z$ which is non-degenerate. Let $\overline{Z} \to Z$ be a desingularization. Then $f$ lifts to a holomorphic map $\overline{f}\colon {{\mathbb C}}\to \overline{Z}$. Since there is a surjective morphism $\pi_1(\overline{Z}) \to \pi_1(Z)$, we concluded that $\pi_1(\overline{Z})$ also admits a linear representation whose image is not virtually abelian. By [@Yamanoi2010 Theorem 1.1], $\overline{f}$ is degenerate and so is $f$. This is a contradiction.
In order to deduce the hyperbolicity in Theorem \[thm:main-theorem\] from Lemma \[lemma:representation-degenerate-entire-curve\], we need some preparatory results.
\[lemma:representation-vir-abelian-not-strictly-nef\] Let $Z$ be a positive dimensional projective variety. Assume that ${{\mathscr G}}$ is a flat vector bundle given by a linear representation $\rho\colon \pi_1(Z) \to \mathrm{GL}_r({{\mathbb C}})$. If the image of $\rho$ is virtually abelian, then ${{\mathscr G}}$ is not strictly nef.
Up to finite étale cover, we may assume that the image of $\rho$ is abelian. Then the image $\rho(\pi_1(Z))$ can be simultaneously triangulated. Hence there is a quotient morphism of flat vector bundles ${{\mathscr G}}\to {{\mathscr L}}$ such that ${{\mathscr L}}$ is a line bundle. Such a quotient induces a section $\sigma\colon Z \to {{\mathbb P}}({{\mathscr G}})$. Moreover, $\sigma^*{{\mathscr O}}_{{{\mathbb P}}({{\mathscr G}})}(1) \cong {{\mathscr L}}$. Since a flat line bundle is never strictly nef, this contradicts to the strict nefness of ${{\mathscr G}}$.
The following result is a consequence of the Borel fixed-point theorem.
\[prop:fixed-point\] Let $G$ be a finitely generated virtually abelian subgroup of $\operatorname*{PGL}_{d+1}({{\mathbb C}})$. Then there exists a subgroup $G'\subseteq G$ of finite index such that the natural action of $G'$ on ${{\mathbb P}}^d$ has a fixed point.
Since $G$ is virtually abelian, there exists a finite index subgroup $G'$ of $G$ such that $G'$ is abelian. Let $\overline{G'}$ be the Zariski closure of $G'$ in $\operatorname*{PGL}_{d+1}({{\mathbb C}})$. Since $G'$ is finitely generated and $\operatorname*{PGL}_{d+1}({{\mathbb C}})$ is a linear algebraic group, $\overline{G'}$ is abelian. By replacing $G'$ with some subgroup of finite index if necessary, we can assume further that $\overline{G'}$ is connected. Therefore, by Borel fixed-point theorem, the action of $\overline{G'} $ on ${{\mathbb P}}^d$ has a fixed point.
We obtain the following result from Lemma \[lemma:representation-vir-abelian-not-strictly-nef\] and Proposition \[prop:fixed-point\].
\[prop:representation-non-abelian\] Let $Z$ be a positive dimensional projective variety and $P\rightarrow Z$ a flat ${{\mathbb P}}^d$-bundle given by a representation $\rho\colon \pi_1(Z) \to \mathrm{PGL}_{d+1}({{\mathbb C}})$. Assume that the relative tangent bundle $T_{P/Z}$ is strictly nef. Then the image $\rho(\pi_1(Z))$ is infinite. Moreover, there exists a subgroup $G\subseteq \pi_1(Z)$ of finite index such that it admits a linear representation whose image is not virtually abelian.
Arguing by contraction, we assume that the image $\rho(\pi_1(Z))$ is finite. Then, after replacing $Z$ by a finite étale cover, the flat ${{\mathbb P}}^d$-bundle $P\rightarrow Z$ is isomorphic to $Z\times {{\mathbb P}}^d$. This contradicts to the strict nefness of $T_{P/Z}$.
Next, if the image of $\rho(\pi_1(Z))$ is not virtually abelian, then we are done. Otherwise, if $\rho(\pi_1(Z))$ is virtually abelian, we shall construct some $G\subseteq \pi_1(Z)$ as required. Indeed, by Proposition \[prop:fixed-point\], there is a subgroup $G'$ of $G$ of finite index such that the natural action of $G'$ on ${{\mathbb P}}^d$ has a fixed point. Let $\Gamma \subseteq \pi_1(Z)$ be the preimage $\rho^{-1}(G')$. Then, by replacing $Z$ with the finite étale cover induced by $\Gamma \subseteq \pi_1(Z)$, we may assume that the natural action of $G$ on ${{\mathbb P}}^d$ has a fixed point $p$.
We remark that $P=({{\mathbb P}}^d\times \widetilde{Z})/\pi_1(Z)$ where $\widetilde{Z}$ is the universal cover of $Z$, and the action of $\pi_1(Z)$ is defined as $$g. (x,z) = (\rho(g)(x), g\cdot z)$$ for every $g\in \pi_1(Z)$. Let $W=(\{p\}\times \widetilde{Z})/\pi_1(Z)$. Then $W\cong Z$ and there is a closed embedding $W\hookrightarrow P$. We note that, the restriction $T_{P/Z}|_W$ is just equal to the bundle $(T_{{{\mathbb P}}^d,p}\times \widetilde{Z})/\pi_1(Z)$, where the action of $\pi_1(Z)$ on $T_{{{\mathbb P}}^d,p}$ is the differentiation of the action of $\pi_1(Z)$ on ${{\mathbb P}}^d$ at $p$. Consequently, $T_{P/Z}|_W$ is a flat vector bundle on $W$. It is strictly nef as well. This implies that the representation of $\pi_1(Z)$ corresponding to $T_{P/Z}|_W$ is not virtually abelian by Lemma \[lemma:representation-vir-abelian-not-strictly-nef\], and we are done.
As an application, one can derive the following corollary.
\[cor:fundamental-group-subvariety\] Under the assumption of Theorem \[thm:main-theorem\], let $\varphi\colon X\rightarrow T$ be the ${{\mathbb P}}^d$-bundle structure provided in Theorem \[cor:main:part1\]. If $Z$ is a subvariety of $T$ with $\dim Z>0$, then there exists a finite index subgroup of $\pi_1(Z)$ admitting a linear representation whose image is not virtually abelian. In particular, every holomorphic map $f:\mathbb{C}\rightarrow Z$ is degenerate.
By Theorem \[cor:main:part1\], there is always a flat projective bundle $Q$ over $T$ such that the relative tangent bundle $T_{Q/T}$ is strictly nef. Hence, $P=Q\times_T Z$ is a flat projective bundle over $Z$. Furthermore, the relative tangent bundle $T_{P/Z}$ is strictly nef as well. Therefore, by Proposition \[prop:representation-non-abelian\], some subgroup of $\pi_1(Z)$ of finite index admits a linear representation whose image is not virtually abelian. By Lemma \[lemma:representation-degenerate-entire-curve\], every holomorphic map $f:{{\mathbb C}}\rightarrow Z$ is degenerate.
1
Thanks to Theorem \[cor:main:part1\], we only need to prove the hyperbolicity of $T$. Let $f\colon {{\mathbb C}}\rightarrow T$ be a holomorphic map. Assume by contradiction that $f$ is not a constant map. Let $Z$ be the Zariski closure of $f({{\mathbb C}})$. Then $\dim Z >0$ and the induced map ${{\mathbb C}}\to Z$ is non degenerate. This contradicts to Corollary \[cor:fundamental-group-subvariety\].
By Theorem \[thm:main-theorem\], there is a ${{\mathbb P}}^d$-bundle structure $\varphi\colon X\to T$. In particular, $\pi_1(T)$ is virtually abelian. By Corollary \[cor:fundamental-group-subvariety\], we deduce that $\dim T=0$ and consequently $X$ is isomorphic to ${{\mathbb P}}^n$.
Since $X$ is not isomorphic to ${{\mathbb P}}^n$, by Theorem \[thm:main-theorem\] and Theorem \[cor:main:part1\], there is a flat projective bundle $Q\rightarrow T$ such that $\dim T>0$ and $T_{Q/T}$ is strictly nef. Hence by Proposition \[prop:representation-non-abelian\], there is a linear representation of $\pi_1(T)$ with infinite image. Since $\pi_1(X) \cong \pi_1(T)$, we obtain the existence of nonzero symmetric differentials on $X$ by using [@BrunebarbeKlinglerTotaro2013 Theorem 0.1].
1
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{
"pile_set_name": "ArXiv"
}
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---
abstract: 'We investigate the effect of electron-electron interactions in ABC stacked graphene trilayers. In the gapless regime, we show that the self-energy corrections lead to the renormalization of the of dynamical exponent $z=3+\alpha_{1}/N$, with $\alpha_{1}\approx0.52$ and $N$ is the number of fermionic species. Although the quasiparticle residue is suppressed near the neutrality point, the lifetime has a sublinear scaling with the energy and the quasiparticles are well defined even at zero energy. We calculate the renormalization of a variety of physical observables, which can be directly measured in experiments.'
author:
- Xu Dou
- Akbar Jaefari
- Yafis Barlas
- Bruno Uchoa
title: Quasiparticle renormalization in ABC graphene trilayers
---
*Introduction.* In graphene single layers, the honeycomb arrangement of the carbon atoms leads to a linear electronic dispersion and to quasiparticles that behave as massless Dirac fermions, akin to massless neutrinos in quantum electrodynamics (QED) [@grapheneRMP; @Kotov]. In graphene multilayers, the electronic spectrum varies depending on the stacking sequence. In the single particle picture, rombohedral ABC-stacked trilayer graphene reveals a gapless band structure of chiral quasiparticles with Berry phase 3$\pi$ and *cubic* low energy excitation spectrum [@Guinea; @Zhang]. Because of the scaling of the kinetic energy, Coulomb interactions are relevant operators in the renormalization group (RG) sense, and can strongly renormalize different physical quantities. Different spontaneous broken symmetry ground states have been already proposed for trilayer graphene [@Cvetovic; @Gorbar; @Olsen]. Very recently, transport experiments revealed a robust many-body gap of $\sim$40 meV at temperatures below $T_{c}\sim34$K [@Lee].
In this letter we study the effect of Coulomb interactions and polarization effects on the behavior of the quasiparticles at small but finite temperature, when the many-body gap is zero. We investigate the analytical structure of the polarization bubble and the leading self-energy corrections due to *dynamically* screened Coulomb interactions. In the gapless regime, we show that the dynamical critical exponent is renormalized to $$z=3+\alpha_{1}/N+O(N^{-2}),$$ where $\alpha_{1}\approx0.52$ and $N=4$ is the number of fermionic flavors. Although the quasiparticle residue is suppressed by interactions, the scattering rate has a sublinear scaling with energy and the quasiparticles remain well defined. We predict the renormalization of several physical observables in the metallic phase, such as the electronic compressibility, the specific heat, the density of states (DOS) and the spectral function, which can be measured with angle resolved photoemission (ARPES) experiments.
*Low energy Hamiltonian.* We start with a simplified two-band model where the high energy bands are separated in energy by interlayer hopping processes, which set the ultraviolet cut-off for the excitations in the low-energy bands, $t_{\perp}\sim0.4$eV. We will assume a temperature regime above the ordering temperature $T\gtrsim T_{c}\sim4$ meV, where the band structure is gapless. The infrared cut-off of the model is the trigonal warping energy $\sim10$ meV, below which the bands disperse quadratically [@Zhang].
The low energy physics of the non-interacting ABC-trilayer in the gapless regime is described by the $2\times2$ Hamiltonian $\mathcal{H}_{0}=\sum_{\mathbf{p}}\Psi_{\mathbf{p}}^{\dagger}\hat{\mathcal{H}}_{0}(\mathbf{p})\Psi_{\mathbf{p}}$, where $\Psi_{\mathbf{k}}=(a_{,\mathbf{k}},\bar{b}_{,\mathbf{k}})$ is a two component spinor defined in terms of one annihilation operator in sublattice $A$ of the top layer ($a_{\mathbf{p}})$ and another in sublattice $B$ for the bottom layer (**$\bar{b}_{\mathbf{p}}$**). The total degeneracy is $N=4$, including spin and valley degrees of freedom. The Hamiltonian density operator is [@Guinea; @Zhang] $$\hat{\mathcal{H}}_{0}=\frac{(\hbar v)^{3}}{t_{\perp}^{2}}\left(\begin{array}{cc}
0 & (\pi)^{3}\\
(\pi^{\dagger})^{3} & 0\end{array}\right),\label{eq:Ho}$$ where $\hbar v\approx6$ eV$\mbox{\AA}$ is the Fermi velocity, and $\pi=p_{x}-ip_{y}$ is defined by the $x$ and $y$ components of the in-plane momentum of the quasiparticles measured away from the neutrality point. In a more compact notation, $\hat{\mathcal{H}}_{0}(\mathbf{k})=\gamma|\mathbf{k}|^{3}\hat{h}_{0}(\mathbf{k})$ with $$\hat{h}_{0}(\mathbf{k})=\mathrm{cos}(3\theta_{\mathbf{k}})\sigma^{1}+\mathrm{sin}(3\theta_{\mathbf{k}})\sigma^{2},\label{eq:h}$$ where $\sigma^{i}$ ($i=1,2$) are $x,\, y$ Pauli spin matrices, and $\mathrm{tan}\theta_{\mathbf{k}}=k_{y}/k_{x}$. The constant $\gamma\equiv(\hbar v)^{3}/t_{\perp}^{2}$, is proportional to the velocity of the quasiparticles $\mathbf{v}_{0}=\partial_{\mathbf{k}}E_{\mathbf{k}}$, which have the energy spectrum $\pm E_{\mathbf{k}}=\pm\gamma|\mathbf{k}|^{3}$.
In ABC trilayers, Coulomb interactions are relevant in the RG flow at the tree level, and hence standard perturbation theory is not possible. We organize the expansion of the self-energy corrections in powers of the dynamically screened Coulomb interaction, which can be rigorously justified in the large $N$ limit. At long wavelengths, $k\ll1/d$, where $d\sim2.4\mbox{\AA}$ is the interlayer distance, the bare Coulomb interaction is $$\mathcal{H}_{I}=\frac{1}{2}\sum_{\mathbf{q}}\, V(q)\hat{n}(\mathbf{q})\hat{n}(-\mathbf{q}),\label{eq:Hi}$$ with $\hat{n}(\mathbf{q})$ a density operator and $V(q)\approx2\pi e^{2}/q$, as in a 2D system. In the long wavelength regime where this approximation is valid, the DOS scales as $\rho(\mathbf{q})=(6\pi\gamma)^{-1}/q$ and the screened Coulomb interaction is $\tilde{V}(q,\omega)=V(q)/\left[1-V(q)\Pi(\mathbf{q},\omega)\right],$ where $\Pi(\mathbf{q},\omega)$ is the dynamical polarization function. In trilayers, the large $N$ approximation becomes asymptotically exact at small momentum, where the DOS diverges and screening becomes strong.
![[Left: Polarization bubble in one loop calculated numerically from Eq. (\[eq:f2-1\]). The real part (black curve) has a logarithmic singularity at the edge of the particle-hole continuum, at $\omega=\gamma q^{3}/4,$ shown in detail in the inset. Red curve: imaginary part. Right panel: Polarization in imaginary frequencies, which is a purely real function. For $\omega/\gamma q^{3}\gg1$, $\Pi^{(0)}(q,i\omega)\to-3Nq^{2}/(16\omega)$ (see text).]{}](fig1 "fig:") \[fig:polarizationfunctions\]
*Polarization bubble.* In order to address the screening effects, we consider the bare polarization function, which is defined as $\Pi^{(0)}(\mathbf{q},\omega)=\frac{1}{\beta}\mbox{tr}\sum_{i\nu}\sum_{\mathbf{p}}\hat{G}_{0}(\mathbf{p},i\nu)\hat{G}_{0}(\mathbf{p}+\mathbf{q},i\omega+i\nu),$ where $$\begin{aligned}
\hat{G}_{0}(\mathbf{q},i\omega) & =\frac{1}{2}\sum_{s=\pm}\frac{1+s\hat{h}_{0}(\mathbf{q})}{i\omega-s\gamma q^{3}}\end{aligned}$$ is the fermionic Greens function, described by a 2$\times2$ matrix. After performing the sums over the Matsubara frequencies, the polarization function is given by $$\begin{aligned}
\Pi^{(0)}(\mathbf{q},\omega)=-\frac{N}{2}\int\frac{d^{2}p}{(2\pi)^{2}}\sum_{s=\pm}\frac{1-\cos(3\theta_{{\bf p}{\bf q}})}{E_{{\bf p}+{\bf q}}+E_{{\bf p}}-s\omega}\label{eqn:polarization1}\end{aligned}$$ where $\theta_{{\bf p}{\bf q}}=\theta_{{\bf p}+{\bf q}}-\theta_{{\bf p}}$ is the angle between vectors ${\bf p}+{\bf q}$ and ${\bf p}$. By sending the ultraviolet cut-off to infinity, a simple dimensional analysis reveals the functional form of the polarization function to be $\gamma q\Pi^{(0)}({\bf q},i\omega)=-Nf(i\omega/(\gamma q^{3})).$ After some algebra, the scaling function $f(z)$ can be written in the form $$\begin{aligned}
f(iz)= & \,\frac{1}{2}\int_{0}^{2\pi}\!\!\mathrm{d}\theta\!\int_{0}^{\infty}\!\!\frac{\mbox{d}x\, x}{(2\pi)^{2}}\sum_{s=\pm}\frac{s}{iz+s\left[x^{3}+h^{3}(x,\theta)\right]}\quad\nonumber \\
& \,\times\left[1-4\left(\frac{1+x\mathrm{cos}\theta}{h(x,\theta)}\right)^{3}+3\left(\frac{1+x\mathrm{cos}\theta}{h(x,\theta)}\right)\right],\label{eq:f2}\end{aligned}$$ where $z=\omega/(\gamma q^{3})$ and $h(x,\theta)\equiv\sqrt{1+x^{2}+2x\mathrm{cos}\theta}$. $f(z)$ is a well-defined function in imaginary frequency but has branch cuts related to the edge of the particle-hole continuum on the real axis. Due to the cubic dispersion, it is difficult to come up with a closed form solution for the polarization function. However the analytical structure of $f(z)$ near the particle-hole threshold $z=1/4$ can be extracted in the collinear scattering approximation, which dominates the processes near that region [@Gangadharaiah]. We consider the singular contribution of the integrand around the momenta $\mathbf{p+q}\approx-\mathbf{p}$. Within this window it is safe to assume $1-\cos(3\theta_{{\bf p}{\bf q}})\approx2$. After expanding $\cos\theta$ around $\theta=\pi$ to the second order, we arrive at the following integral representation for $f(z)$, $$\begin{aligned}
f(z) & \cong\int\frac{xdx}{(2\pi)^{2}}\int\frac{d\theta}{x^{3}+(1-x)^{3}+\frac{3}{2}x(1-x)\theta^{2}-z}.\label{eq:f2-1}\end{aligned}$$ Considering the rapid fall of the integrand with respect to $\theta$ around $\pi$, one can conveniently extend the upper limit of the angular integral to infinity, $\theta\in[0,\infty[$. After performing the integrals, we arrive at the most dominant part of $f(z)$ near $z\sim1/4$, $$f(z)=-\frac{1}{6\sqrt{2}\pi}\ln\left(1-4|z|\right)+\text{regular terms},\label{eq:f3}$$ which describes a logarithmic divergence near the edge of the particle hole continuum. Exploring the two asymptotic regimes, in the $z\to0$ regime, $f(0)=c_{0}\approx0.12$ is a constant [@Min; @Gelderen] and in the $z\gg1$ limit, $f(z)\to-ic_{\infty}/z$ is purely imaginary, with $c_{\infty}=3/16$.
In Fig. 1, we show the behavior of the real and imaginary parts of $f(z)$ calculated numerically from Eq. (\[eq:f2-1\]). The scaling function has only one singularity near $z\sim1/4$. For $z<1/4$, $f(z)$ is purely real and diverges logarithmically at $z=1/4$, in agreement with the analytical expression (\[eq:f3\]), as shown in the inset of Fig. 1. For $z>1/4$, $f(z)$ has also an imaginary part, which decays with $1/z$. The right panel of Fig. 1 shows $f(iz)$ in imaginary frequency, which is a real and well behaved monotonic function.
In the optical regime, for $z\gg1$, where $\Pi^{(0)}(q,\omega)\to iNc_{\infty}q^{2}/\omega$, the optical conductivity can be calculated directly from the charge polarization, $$\sigma(\omega)=\frac{e^{2}}{\hbar}\lim_{q\to0}\frac{i\omega}{q^{2}}\frac{\Pi^{(0)}(\mathbf{q},\omega)}{1-V(q)\Pi^{(0)}(\mathbf{q},\omega)}=\frac{3}{4}\frac{e^{2}}{\hbar},\label{eq:sigma}$$ which is proportional to the Berry phase $3\pi$. In the general case, $\sigma(\omega)=\nu e^{2}/(2h)$, with $\nu=\pi$ for graphene single layer and $\nu=2\pi$ for bilayers.
*Self-energy.* The leading self energy correction due to the screened Coulomb interaction is diagrammatically shown in Fig. 2. In imaginary time, the self-energy is given by $$\hat{\Sigma}^{(1)}(\mathbf{q},i\omega)=-\frac{1}{\beta}\sum_{\nu}\int\!\!\frac{\mbox{d}^{2}p}{(2\pi)^{2}}\tilde{V}(\mathbf{p},i\nu)\hat{G}^{(0)}(\mathbf{q}-\mathbf{p},i\omega-i\nu).\label{eq:Sigma}$$ Through power counting, the leading divergences appear at long wavelengths, where the large $N$ limit is a good approximation. At large $N$, the dynamically screened potential is approximated by $\tilde{V}({\bf q},i\omega)\approx\gamma q/[Nf(i\omega/\gamma q^{3})]+O(N^{-2})$ [@Son; @Foster; @Nmass]. Since $f(iz)$ is a well behaved function, with no singularities or branch cuts, the self energy in one loop can be calculated directly in the zero temperature limit. The leading contribution is logarithmically divergent, $$\Sigma^{(1)}(\mathbf{q},i\omega)=\frac{1}{2\pi^{2}N}\left[\alpha_{d}i\omega+\alpha_{o}\gamma q^{3}\hat{h}(\mathbf{q})\right]\mathrm{ln}\left(\frac{\Lambda}{q}\right),\label{eq:Self}$$ where $t_{\perp}=\gamma\Lambda^{3}$ defines the ultraviolet cut-off in momentum, namely $\Lambda=t_{\perp}/(\hbar v)$. The coefficients $$\alpha_{o}=\int_{0}^{\infty}dz\frac{1}{f(iz)}\frac{z^{2}(10-16z^{2}+z^{4})}{(1+z^{2})^{4}},\label{eq:ctilde}$$ and $$\alpha_{d}=\int_{0}^{\infty}dz\frac{1}{f(iz)}\frac{1-z^{2}}{(1+z^{2})^{2}},\label{eq:alpha_d}$$ can be found though numerical integration using the exact $f(iz)$ from Eq. (\[eq:f2-1\]). Although $\alpha_{o}$ and $\alpha_{d}$ both diverge logarithmically with the upper limit of integration at large $z$, we will postpone their regularization for the moment, since these divergences cancel exactly in the renormalization of $\gamma$ and hence have no consequence to the renormalization of the spectrum.
The self-energy can be separated in two terms, $\hat{\Sigma}(\mathbf{q},i\omega)=i\omega\Sigma_{d}\sigma_{0}+\Sigma_{0}q^{3}\hat{h}_{0}(\mathbf{q}),$ where $\Sigma_{d}$ is the diagonal term, and $\Sigma_{o}$ describes the off-diagonal matrix elements. The diagonal part of the self-energy has frequency dependence and defines the quasiparticle residue renormalization, $$Z_{\psi}^{-1}=1-\partial\hat{\Sigma}/\partial(i\omega)=1-\Sigma_{d}.\label{Z}$$ The renormalized Green’s function is $\hat{G}(\mathbf{q},i\omega)=Z_{\psi}[i\omega-\gamma\hat{h}_{0}(\mathbf{q})Z_{\psi}(1+\Sigma_{o})]^{-1}$. In one loop, the renormalized energy spectrum is $$\frac{\gamma(q)}{\gamma}=\frac{1+\Sigma_{o}}{1-\Sigma_{d}}\approx1-\frac{\alpha_{1}}{N}\ln\!\left(\frac{\Lambda}{q}\right)+O(1/N^{2}),\label{eq:gamma ren}$$ where $$\alpha_{1}=\frac{\alpha_{0}+\alpha_{d}}{2\pi^{2}}=\int_{0}^{\infty}\frac{\mbox{d}z}{2\pi^{2}}\frac{1}{f(iz)}\frac{17z^{4}-11z^{2}-1}{(1+z^{2})^{4}}\approx0.52\label{alpha1}$$ is a finite well defined quantity.
![[One-loop correction to the self-energy with the dressed Coulomb interaction.]{}](fig2 "fig:") \[fig:sunshine\]
The logarithmic renormalization of the quasiparticle velocity in one loop dictates the RG equation of $\gamma$, $$\beta_{\gamma}\equiv\frac{\mathrm{d}\gamma}{\mathrm{d}l}=-\gamma\frac{\alpha_{1}}{N},$$ where $l=\ln(\Lambda/\Lambda^{\prime})$, with $\Lambda^{\prime}<\Lambda$ the renormalized cut-off, whose solution is $$\gamma(q)=\gamma\times[(\hbar v/t_{\perp})q]{}^{\alpha_{1}/N}.\label{eq:gamma3}$$ The energy spectrum acquires an anomalous dimension $\eta=\alpha_{1}/N$, which leads to the renormalization of the dynamical exponent, $\omega\propto q^{z}$, with $z=3+\alpha_{1}/N+O(N^{-2})$. This result can be related with the graphene bilayer case, where $\eta=0.078/N$ [@Lemonik] and with the large $N$ limit of the single layer case, where $\eta=-4/(\pi^{2}N)$ [@Son; @Foster].
This analysis can be explicitly verified by checking the two loop correction in the self energy. The RG equation describes a resummation of leading logs to all orders in $1/N$. The $N^{-2}\log^{2}$ terms cancel exactly in the vertex correction diagram at two loop, and hence vertex corrections do not renormalize in the RG flow [@Lemonik]. The leading logarithmic terms appear in the remaining diagrams of the same order, and lead to a second order correction to Eq. (\[eq:gamma ren\]), $\gamma^{(2)}(q)/q=\frac{1}{2}\alpha_{1}^{2}/N^{2}\ln^{2}(\Lambda/q)$, in agreement with the result of the RG equation up to $1/N^{2}$ order.
*Quasiparticle residue*. To calculate the quasiparticle residue renormalization $Z_{\psi}$ through Eq. (\[Z\]), one needs to regularize integral (\[eq:alpha\_d\]). That can be done introducing an upper cut-off $z_{c}$ which accounts for the condition where the large $N$ limit breaks down, namely $-V(p)\Pi^{(0)}(p,i\nu)=2\pi Ne^{2}\Lambda^{2}/(\hbar vp^{2})f(iz_{c})\sim1.$ At large $z$, where $f(iz)\to3/(16z)$, the leading contribution is $\alpha_{d}\sim-16\ln(\Lambda/p)$. Replacing $\ln(\Lambda/q)\to\int_{q}^{\Lambda}\mbox{d}p/p$ in Eq. (\[eq:Self\]) and carrying out the momentum integration, the quasiparticle residue $Z_{\psi}$ is given by $$Z_{\psi}^{-1}\to1+\frac{4}{\pi^{2}N}\ln^{2}(\Lambda/q)+O(1/N^{2}),\label{Z-1}$$ in one loop, and is suppressed logarithmically in the infrared.
In the RG spirit, we now reestablish the bare value of the quasiparticle residue $Z_{0}$ in the bare Green’s function $\hat{G}_{0}\propto Z_{0}$ [@Nandkishore], and set $Z_{0}\to1$ at the end. Since $\delta\hat{G}=\hat{G}_{0}\hat{\Sigma}\hat{G}_{0}\propto\delta Z_{\psi}$ in lowest order in the Dyson equation, then $\delta Z_{\psi}=Z_{0}^{2}\hat{\Sigma}_{d}\propto Z_{0}$ in large $N$. Eq. (\[Z-1\]) then becomes $\delta Z_{\psi}=-4Z_{0}/(\pi^{2}N)\delta\ln^{2}(\Lambda/q)$, which corresponds to the RG equation $$\beta_{\psi}=\frac{\mbox{d}Z_{\psi}}{\mbox{d}l}=-\frac{8}{\pi^{2}N}lZ_{\psi},\label{eq:betapsi}$$ with $l=\ln(\Lambda/\Lambda^{\prime})$, whose solution is $$Z_{\psi}(q)=\mbox{exp}\left[-4/(\pi^{2}N)\ln^{2}(\Lambda/q)\right],\label{RG sol}$$ in agreement with Eq. (\[Z-1\]) up to $1/N$ order.
![[a) On-shell scattering rate $\tau(\omega)$ vs energy in units of $t_{\perp}\sim0.4$eV. b) Density plot of the spectral function as a function of energy $(\omega/t_{\perp})$ and momentum ($q/\Lambda$). Solid black line: bare energy spectrum. White line: renormalized one. Light regions indicate higher intensity. ]{}](fig3 "fig:") \[fig:sunshine-1\]
*Quasiparticle lifetime.* In real frequency, the polarization function has a logarithmic branch cut. To calculate the quasiparticle scattering rate $\tau=Z_{\psi}\mbox{Im}\hat{\Sigma}$, we use the method in ref. [\[]{} to separate the self-energy into the line part and the residue part, $\hat{\Sigma}=\hat{\Sigma}_{\text{line}}+\hat{\Sigma}_{\text{res}}.$ The line part is obtained by performing Wick rotation $i\omega\to\omega+i0_{+}$ in the self-energy (\[eq:Sigma\]), and is purely real. The residue part follows from the residue calculated around the pole of the Green’s function, $$\begin{aligned}
& \Sigma_{\text{res}}^{(1)}({\bf q},\omega)=-\frac{1}{2}\sum_{s=\pm}\int\frac{\mathrm{d}^{2}p}{(2\pi)^{2}}\tilde{V}(|{\bf q}|,\omega)[1+s\hat{h}(\mathbf{q}-\mathbf{p})]\nonumber \\
& \quad\qquad\times[\theta(\omega-s\gamma|{\bf q}-{\bf p}|^{3})-\theta(-s\gamma|{\bf q}-{\bf p}|^{3})],\end{aligned}$$ with $\theta$ a step function. The scattering rate is given by $\tau(\mathbf{q},\omega)=Z_{\psi}\mbox{Im}\Sigma_{\mathrm{res}}(\mathbf{q},\omega)$. In the on-shell region, near $\omega\sim\gamma q^{3}$, $\tau(\omega)=\omega Z_{\psi}g(\omega/t_{\perp}),$ where $$g(y)=\frac{1}{2N}\,\mbox{Im}\!\int_{|\mathbf{x}|<1}\frac{\mbox{d}^{2}x}{(2\pi)^{2}}\frac{|\hat{q}-\mathbf{x}|}{\bar{\alpha}y^{2/3}|\hat{q}-\mathbf{x}|^{2}+f\!\left(\frac{1-x^{3}}{|\hat{q}-\mathbf{x}^{3}|}\right)}\label{eq:g}$$ is a scaling function in one loop, with $y=\omega/t_{\perp}$, $\bar{\alpha}=\hbar v/(2\pi Ne^{2})$ is a dimensionless constant and $\hat{q}=\mathbf{q}/q$. The function $g(y)$ has a very slow variation, as shown in Fig. 3a, and as a consequence $\tau(\omega)\sim\omega Z_{\psi}[(\omega/\gamma)^{1/3}]$ has a sublinear scaling with energy within logarithmic accuracy. In the large $N$ limit ($\bar{\alpha}\to0$), which is valid at low energy, $g(y)\approx0.043$ is a constant. Since the ratio $\tau(\omega)/\omega\ll1$, the quasiparticles are well defined even in the $\omega\to0$ limit.
The spectral function is given by $A(\mathbf{q},\omega)=-2\mbox{tr}\,\mbox{Im}G^{R}(\mathbf{q},\omega)$, where $$\hat{G}^{R}(\mathbf{q},\omega)=\frac{1}{2}\sum_{s=\pm}\frac{Z_{\psi}(q)[1+s\hat{h}_{0}(\mathbf{q})]}{\omega-s\gamma(q)q^{3}-i\tau(\omega)+i0^{+}}\label{G}$$ is the retarded part of the renormalized Green’s function. The spectral function is shown in Fig. 3b. The solid black line describes the bare energy spectrum, while the light region describes the renormalized one, which corresponds to the pole of the renormalized Green’s function. There is a clear deviation of the two curves, which could be observed with ARPES experiments.
*Other physical observables.* The renormalization of the quasiparticle velocity encoded in the RG flow of $\gamma$ leads to the renormalization of many physical observables. For instance, the specific heat for non-interacting particles with cubic dispersion in 2D scales with $C\sim(T/\gamma)^{2/3}$, where $T$ is the temperature. From Eq. (\[eq:gamma3\]), the scaling of $\gamma$ with energy is $\gamma\sim\omega^{\alpha_{1}/(3N)}$. At $\omega\sim T$, the temperature scaling of the specific heat is renormalized to $$C\sim T^{2(1-\alpha_{1}/(3N)]/3}\approx T^{2/3-0.1/N},\label{C}$$ neglecting slower logarithmic corrections due to the scaling of $Z_{\psi}$, with $T\gtrsim T_{0}$, where $T_{0}$ is defined by the infrared energy cut-off of 10meV due to trigonal warping effects [@Zhang]. In the same way, the renormalized DOS is $\rho(q)=[6\pi\gamma(q)]{}^{-1}/q\sim q^{-(1+\alpha_{1}/N)}$, which can be measured directly on surfaces with scanning tunneling spectroscopy experiments [@Li; @Yankowitz].
In 2D systems, the electronic compressibility can be characterized with single electron transistor measurements [@Martin]. By dimensional analysis, the free electronic compressibility scales with temperature as $\kappa\sim\gamma^{-2/3}T^{-1/3}$ [@Mahan]. In the same spirit, interactions renormalize the scaling of the inverse compressibility, $$\kappa^{-1}\sim T^{[1+2\alpha_{1}/(3N)]/3}\approx T^{1/3+0.1/N},\label{eq:kappa}$$ which strongly deviates from the non-interacting result.
In summary, we derived the effect of electron-electron interactions in the renormalization of a variety of different physical observables in the metallic phase of ABC graphene trilayers.
We thank F. Guinea, V. N. Kotov and K. Mullen for valuable discussions. B. U. acknowledges University of Oklahoma and NSF grant DMR-1352604 for partial support.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'In this paper, we derive new estimates for the remainder term of the midpoint, trapezoid, and Simpson formulae for functions whose derivatives in absolute value at certain power are $h-$convex and we point out the results for some special classes of functions. Some applications to special means of real numbers are also given.'
address: |
Department of Mathematics, Faculty of Arts and Sciences,\
Giresun University, 28100, Giresun, Turkey.
author:
- İmdat İşcan
date: 'June 10, 2012'
title: 'On new general integral inequalities for $h-$ convex functions'
---
Introduction
============
Let $f:I\subseteq \mathbb{R\rightarrow R}$ be a convex function defined on the interval $I$ of real numbers and $a,b\in I$ with $a<b$. The following inequality$$f\left( \frac{a+b}{2}\right) \leq \frac{1}{b-a}\dint\limits_{a}^{b}f(x)dx%
\leq \frac{f(a)+f(b)}{2}\text{.} \label{1-1}$$
holds. This double inequality is known in the literature as Hermite-Hadamard integral inequality for convex functions. See \[2\]-\[7\],\[11\]-\[14\],\[16\],\[22\], the results of the generalization, improvement and extention of the famous integral inequality (\[1-1\]).
In the paper [@V07] a large class of non-negative functions, the so-called $h-$convex functions is considered. This class contains several well-known classes of functions such as non-negative convex functions, $s-$convex in the second sense, Godunova Levin functions and $P-$functions. Let us recall definitions of these special classes of functions.
$f:I\rightarrow R$ is a Godunova–Levin function or that f belongs to the class $Q(I)$ if f is non-negative and for all $x,y\in I$ and $\alpha \in
(0,1)$ we have$$f\left( \alpha x+(1-\alpha )y\right) \leq \frac{f(x)}{\alpha }+\frac{f(y)}{%
1-\alpha }$$
The class $Q(I)$ was firstly described in [@GL85] by Godunova and Levin. Some further properties of it are given in [@DPP95; @MP90; @MPF93]. Among others, it is noted that non-negative monotone and non-negative convex functions belong to this class of functions.
In 1978, Breckner introduced s-convex functions as a generalization of convex functions as follows [@B78]:
Let $s\in (0,1]$ be a fixed real number. A function $f:[0,\infty
)\rightarrow \lbrack 0,\infty )$ is said to be $s-$convex (in the second sense),or that $f$ belongs to the class $K_{s}^{2}$, if $$f\left( \alpha x+(1-\alpha )y\right) \leq \alpha ^{s}f(x)+(1-\alpha )^{s}f(y)$$
for all $x,y\in \lbrack 0,\infty )$ and $\alpha \in \lbrack 0,1]$.
Of course, s-convexity means just convexity when s = 1. In [@DPP95] Dragomir et al. defined the concept of $P-$function as the following:
We say that $f:I\rightarrow
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$ is a $P-$function, or that f belongs to the class $P(I)$, if $f$ is a non-negative function and for all $x,y\in I$, $\alpha \in \lbrack 0,1]$, we have$$f\left( \alpha x+(1-\alpha )y\right) \leq f(x)+f(y).$$
Let $I$ and $J$ be intervals in $%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$, $(0,1)\subseteq J$ and $h$ and $f$ be real non-negative functions defined on $J$ and $I$, respectively. In [@V07], Varošanec defined the concept of $h-$convexity as follows:
Let $h:J\rightarrow
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$ be a non-negative function, $h\neq 0.$ We say that $f:I\rightarrow
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$ is an $h-$convex function or that $f$ belongsto the class $SX(h,I)$, if $f$ is non-negative function and for all $x,y\in I$ and $\alpha \in (0,1)$ we have$$f\left( \alpha x+(1-\alpha )y\right) \leq h(\alpha )f(x)+h(1-\alpha )f(y).
\label{1-a}$$
If inequality (\[1-a\]) is reversed, then $f$ is said to be $h-$concave, i.e.$f\in SV(h,I)$. The notion of $\ h-$convexity unifies and generalizes the known classes of functions, $s-$convex functions,Gudunova-Levin functions and $P-$functions, which are obtained by putting in (\[1-a\]), $%
h(t)=t$, $h(t)=t^{s}$, $h(t)=\frac{1}{t}$, and $h(t)=1$, respectively.
In [@DF99], Dragomir and Fitzpatrick proved a variant of Hadamard’s inequality which holds for $s-$convex functions in the second sense.
Suppose that $f:[0,\infty )\rightarrow \lbrack 0,\infty )$ is an $s-$convex function in the second sense, where $s\in (0,1)$, and let $a,b\in \lbrack
0,\infty )$, $a<b.$If $f\in L[a,b]$ then the following inequalities hold$$2^{s-1}f\left( \frac{a+b}{2}\right) \leq \frac{1}{b-a}\dint%
\limits_{a}^{b}f(x)dx\leq \frac{f(a)+f(b)}{s+1}. \label{1-2a}$$
The constant $k=\frac{1}{s+1}$ is the best possible in the second inequality in (\[1-2a\]).
Let $f\in Q(I)$, $a,b\in I$ with $a<b$ and $f\in L[a,b].$ Then$$f\left( \frac{a+b}{2}\right) \leq \frac{4}{b-a}\dint\limits_{a}^{b}f(x)dx.$$
Let $f\in P(I)$, $a,b\in I$ with $a<b$ and $f\in L[a,b].$ Then$$f\left( \frac{a+b}{2}\right) \leq \frac{2}{b-a}\dint\limits_{a}^{b}f(x)dx%
\leq 2\left[ f(a)+f(b)\right] .$$
In [@GL85], Dragomir et. al. proved two inequalities of Hadamard type for classes of Godunova-Levin functions and $P-$functions.
In [@SSY08], Sarikaya et. al. established a new Hadamard-type inequality for $h-$convex functions.
Let $f\in SX(h,I)$, $a,b\in I$ with $a<b$ and $f\in L([a,b])$. Then$$\frac{1}{2h\left( \frac{1}{2}\right) }f\left( \frac{a+b}{2}\right) \leq
\frac{1}{b-a}\dint\limits_{a}^{b}f(x)dx\leq \left[ f(a)+f(b)\right]
\dint\limits_{0}^{1}h(t)dt. \label{1-2}$$
The following inequality is well known in the literature as Simpson’s inequality .
Let $f:\left[ a,b\right] \mathbb{\rightarrow R}$ be a four times continuously differentiable mapping on $\left( a,b\right) $ and $\left\Vert
f^{(4)}\right\Vert _{\infty }=\underset{x\in \left( a,b\right) }{\sup }%
\left\vert f^{(4)}(x)\right\vert <\infty .$ Then the following inequality holds:$$\left\vert \frac{1}{3}\left[ \frac{f(a)+f(b)}{2}+2f\left( \frac{a+b}{2}%
\right) \right] -\frac{1}{b-a}\dint\limits_{a}^{b}f(x)dx\right\vert \leq
\frac{1}{2880}\left\Vert f^{(4)}\right\Vert _{\infty }\left( b-a\right) ^{2}.$$ In recent years many authors have studied error estimations for Simpson’s inequality; for refinements, counterparts, generalizations and new Simpson’s type inequalities, see [@ADD09; @SA11; @SSO10; @SSO10a].
In [@I12], Iscan obtained a new generalization of some integral inequalities for differentiable convex mapping which are connected Simpson’s, midpoint and trapezoid inequalities, and he used the following lemma to prove this.
\[2.1\]Let $f:I\subseteq \mathbb{R\rightarrow R}$ be a differentiable mapping on $I^{\circ }$ such that $f^{\prime }\in L[a,b]$, where $a,b\in I$ with $a<b$ and $\alpha ,\lambda \in \left[ 0,1\right] $. Then the following equality holds:$$\begin{aligned}
&&\lambda \left( \alpha f(a)+\left( 1-\alpha \right) f(b)\right) +\left(
1-\lambda \right) f(\alpha a+\left( 1-\alpha \right) b)-\frac{1}{b-a}%
\dint\limits_{a}^{b}f(x)dx \\
&=&\left( b-a\right) \left[ \dint\limits_{0}^{1-\alpha }\left( t-\alpha
\lambda \right) f^{\prime }\left( tb+(1-t)a\right) dt\right. \\
&&\left. +\dint\limits_{1-\alpha }^{1}\left( t-1+\lambda \left( 1-\alpha
\right) \right) f^{\prime }\left( tb+(1-t)a\right) dt\right] .\end{aligned}$$
The main inequality in [@I12], pointed out, is as follows.
Let $f:I\subseteq \mathbb{R\rightarrow R}$ be a differentiable mapping on $%
I^{\circ }$ such that $f^{\prime }\in L[a,b]$, where $a,b\in I^{\circ }$ with $a<b$ and $\alpha ,\lambda \in \left[ 0,1\right] $. If $\left\vert
f^{\prime }\right\vert ^{q}$ is convex on $[a,b]$, $q\geq 1,$ then the following inequality holds:$$\begin{aligned}
&&\left\vert \lambda \left( \alpha f(a)+\left( 1-\alpha \right) f(b)\right)
+\left( 1-\lambda \right) f(\alpha a+\left( 1-\alpha \right) b)-\frac{1}{b-a}%
\dint\limits_{a}^{b}f(x)dx\right\vert \label{1-3} \\
&\leq &\left\{
\begin{array}{cc}
\begin{array}{c}
\left( b-a\right) \left\{ \gamma _{2}^{1-\frac{1}{q}}\left( \mu
_{1}\left\vert f^{\prime }(b)\right\vert ^{q}+\mu _{2}\left\vert f^{\prime
}(a)\right\vert ^{q}\right) ^{\frac{1}{q}}\right. \\
\left. +\upsilon _{2}^{1-\frac{1}{q}}\left( \eta _{3}\left\vert f^{\prime
}(b)\right\vert ^{q}+\eta _{4}\left\vert f^{\prime }(a)\right\vert
^{q}\right) ^{\frac{1}{q}}\right\} ,%
\end{array}
& \alpha \lambda \leq 1-\alpha \leq 1-\lambda \left( 1-\alpha \right) \\
\begin{array}{c}
\left( b-a\right) \left\{ \gamma _{2}^{1-\frac{1}{q}}\left( \mu
_{1}\left\vert f^{\prime }(b)\right\vert ^{q}+\mu _{2}\left\vert f^{\prime
}(a)\right\vert ^{q}\right) ^{\frac{1}{q}}\right. \\
\left. +\upsilon _{1}^{1-\frac{1}{q}}\left( \eta _{1}\left\vert f^{\prime
}(b)\right\vert ^{q}+\eta _{2}\left\vert f^{\prime }(a)\right\vert
^{q}\right) ^{\frac{1}{q}}\right\} ,%
\end{array}
& \alpha \lambda \leq 1-\lambda \left( 1-\alpha \right) \leq 1-\alpha \\
\begin{array}{c}
\left( b-a\right) \left\{ \gamma _{1}^{1-\frac{1}{q}}\left( \mu
_{3}\left\vert f^{\prime }(b)\right\vert ^{q}+\mu _{4}\left\vert f^{\prime
}(a)\right\vert ^{q}\right) ^{\frac{1}{q}}\right. \\
\left. +\upsilon _{2}^{1-\frac{1}{q}}\left( \eta _{3}\left\vert f^{\prime
}(b)\right\vert ^{q}+\eta _{4}\left\vert f^{\prime }(a)\right\vert
^{q}\right) ^{\frac{1}{q}}\right\} ,%
\end{array}
& 1-\alpha \leq \alpha \lambda \leq 1-\lambda \left( 1-\alpha \right)%
\end{array}%
\right. \notag\end{aligned}$$where $$\gamma _{1}=\left( 1-\alpha \right) \left[ \alpha \lambda -\frac{\left(
1-\alpha \right) }{2}\right] ,\ \gamma _{2}=\left( \alpha \lambda \right)
^{2}-\gamma _{1}\ ,$$$$\begin{aligned}
\upsilon _{1} &=&\frac{1-\left( 1-\alpha \right) ^{2}}{2}-\alpha \left[
1-\lambda \left( 1-\alpha \right) \right] , \\
\upsilon _{2} &=&\frac{1+\left( 1-\alpha \right) ^{2}}{2}-\left( \lambda
+1\right) \left( 1-\alpha \right) \left[ 1-\lambda \left( 1-\alpha \right) %
\right] ,\end{aligned}$$$$\begin{aligned}
\mu _{1} &=&\frac{\left( \alpha \lambda \right) ^{3}+\left( 1-\alpha \right)
^{3}}{3}-\alpha \lambda \frac{\left( 1-\alpha \right) ^{2}}{2},\ \\
\mu _{2} &=&\frac{1+\alpha ^{3}+\left( 1-\alpha \lambda \right) ^{3}}{3}-%
\frac{\left( 1-\alpha \lambda \right) }{2}\left( 1+\alpha ^{2}\right) , \\
\mu _{3} &=&\alpha \lambda \frac{\left( 1-\alpha \right) ^{2}}{2}-\frac{%
\left( 1-\alpha \right) ^{3}}{3}, \\
\mu _{4} &=&\frac{\left( \alpha \lambda -1\right) \left( 1-\alpha
^{2}\right) }{2}+\frac{1-\alpha ^{3}}{3},\end{aligned}$$$$\begin{aligned}
\eta _{1} &=&\frac{1-\left( 1-\alpha \right) ^{3}}{3}-\frac{\left[ 1-\lambda
\left( 1-\alpha \right) \right] }{2}\alpha \left( 2-\alpha \right) ,\ \\
\eta _{2} &=&\frac{\lambda \left( 1-\alpha \right) \alpha ^{2}}{2}-\frac{%
\alpha ^{3}}{3},\end{aligned}$$$$\begin{aligned}
\eta _{3} &=&\frac{\left[ 1-\lambda \left( 1-\alpha \right) \right] ^{3}}{3}-%
\frac{\left[ 1-\lambda \left( 1-\alpha \right) \right] }{2}\left( 1+\left(
1-\alpha \right) ^{2}\right) +\frac{1+\left( 1-\alpha \right) ^{3}}{3}, \\
\eta _{4} &=&\frac{\left[ \lambda \left( 1-\alpha \right) \right] ^{3}}{3}-%
\frac{\lambda \left( 1-\alpha \right) \alpha ^{2}}{2}+\frac{\alpha ^{3}}{3}.\end{aligned}$$
In [@ADK11] Alomari et al. obtained the following inequalities of the left-hand side of Hermite-Hadamard’s inequality for s-convex mappings.
Let $f:I\subseteq \lbrack 0,\infty )\rightarrow \mathbb{R}$ be a differentiable mapping on $I^{\circ }$, such that $f^{\prime }\in L[a,b]$, where $a,b\in I$ with $a<b$. If $|f^{\prime }|^{q},\ q\geq 1,$ is $s-$convex on $[a,b]$, for some fixed $s\in (0,1]$, then the following inequality holds:$$\begin{aligned}
&&\left\vert f\left( \frac{a+b}{2}\right) -\frac{1}{b-a}\dint%
\limits_{a}^{b}f(x)dx\right\vert \notag \\
&\leq &\frac{b-a}{8}\left( \frac{2}{(s+1)(s+2)}\right) ^{\frac{1}{q}}\left[
\left\{ \left( 2^{1-s}+1\right) \left\vert f^{\prime }(b)\right\vert
^{q}+2^{1-s}\left\vert f^{\prime }(a)\right\vert ^{q}\right\} ^{\frac{1}{q}%
}\right. \notag \\
&&\left. +\left\{ \left( 2^{1-s}+1\right) \left\vert f^{\prime
}(a)\right\vert ^{q}+2^{1-s}\left\vert f^{\prime }(b)\right\vert
^{q}\right\} ^{\frac{1}{q}}\right] . \label{1-4}\end{aligned}$$
Let $f:I\subseteq \lbrack 0,\infty )\rightarrow \mathbb{R}$ be a differentiable mapping on $I^{\circ }$, such that $f^{\prime }\in L[a,b]$, where $a,b\in I$ with $a<b$. If $|f^{\prime }|^{\frac{p}{p-1}},\ p>1,$ is $%
s- $convex on $[a,b]$, for some fixed $s\in (0,1]$, then the following inequality holds:$$\begin{aligned}
\left\vert f\left( \frac{a+b}{2}\right) -\frac{1}{b-a}\dint%
\limits_{a}^{b}f(x)dx\right\vert &\leq &\left( \frac{b-a}{4}\right) \left(
\frac{1}{p+1}\right) ^{\frac{1}{p}}\left( \frac{1}{s+1}\right) ^{\frac{2}{q}}
\notag \\
&&\times \left[ \left( \left( 2^{1-s}+s+1\right) \left\vert f^{\prime
}\left( a\right) \right\vert ^{q}+2^{1-s}\left\vert f^{\prime }\left(
b\right) \right\vert ^{q}\right) ^{\frac{1}{q}}\right. \notag \\
&&+\left. \left( \left( 2^{1-s}+s+1\right) \left\vert f^{\prime }\left(
b\right) \right\vert ^{q}+2^{1-s}\left\vert f^{\prime }\left( a\right)
\right\vert ^{q}\right) ^{\frac{1}{q}}\right] , \label{1-4a}\end{aligned}$$where $p$ is the conjugate of $q$, $q=p/(p-1).$
In [@SSO10], Sarikaya et al. obtained a new upper bound for the right-hand side of Simpson’s inequality for $s-$convex mapping as follows:
Let $f:I\subseteq \lbrack 0,\infty )\rightarrow \mathbb{R}$ be a differentiable mapping on $I^{\circ }$, such that $f^{\prime }\in L[a,b]$, where $a,b\in I^{\circ }$ with $a<b$. If $|f^{\prime }|^{q},\ $ is $s-$convex on $[a,b]$, for some fixed $s\in (0,1]$ and $q>1,$ then the following inequality holds:$$\begin{aligned}
&&\left\vert \frac{1}{6}\left[ f(a)+4f\left( \frac{a+b}{2}\right) +f(b)%
\right] -\frac{1}{b-a}\dint\limits_{a}^{b}f(x)dx\right\vert \leq \frac{b-a}{%
12}\left( \frac{1+2^{p+1}}{3\left( p+1\right) }\right) ^{\frac{1}{p}}
\label{1-5} \\
&&\times \left\{ \left( \frac{\left\vert f^{\prime }\left( \frac{a+b}{2}%
\right) \right\vert ^{q}+\left\vert f^{\prime }\left( a\right) \right\vert
^{q}}{s+1}\right) ^{\frac{1}{q}}+\left( \frac{\left\vert f^{\prime }\left(
\frac{a+b}{2}\right) \right\vert ^{q}+\left\vert f^{\prime }\left( b\right)
\right\vert ^{q}}{s+1}\right) ^{\frac{1}{q}}\right\} , \notag\end{aligned}$$where $\frac{1}{p}+\frac{1}{q}=1.$
In [@KBOP07], Kirmaci et al. proved the following trapezoid inequality:
Let $f:I\subseteq \lbrack 0,\infty )\rightarrow \mathbb{R}$ be a differentiable mapping on $I^{\circ }$, such that $f^{\prime }\in L[a,b]$, where $a,b\in I^{\circ }$, $a<b$. If $|f^{\prime }|^{q},\ $ is $s-$convex on $[a,b]$, for some fixed $s\in (0,1)$ and $q>1,$ then$$\begin{aligned}
&&\left\vert \frac{f\left( a\right) +f\left( b\right) }{2}-\frac{1}{b-a}%
\dint\limits_{a}^{b}f(x)dx\right\vert \leq \frac{b-a}{2}\left( \frac{q-1}{%
2\left( 2q-1\right) }\right) ^{\frac{q-1}{q}}\left( \frac{1}{s+1}\right) ^{%
\frac{1}{q}} \label{1-6} \\
&&\times \left\{ \left( \left\vert f^{\prime }\left( \frac{a+b}{2}\right)
\right\vert ^{q}+\left\vert f^{\prime }\left( a\right) \right\vert
^{q}\right) ^{\frac{1}{q}}+\left( \left\vert f^{\prime }\left( \frac{a+b}{2}%
\right) \right\vert ^{q}+\left\vert f^{\prime }\left( b\right) \right\vert
^{q}\right) ^{\frac{1}{q}}\right\} . \notag\end{aligned}$$
Main results
============
The following theorems give a new result of integral inequalities for $h-$convex functions. In the sequel of the paper $I$ and $J$ are intervals in $%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$, $(0,1)\subset J$ and $h$ and $f$ are real non-negative functions defined on $J$ and $I$, respectively and $h$ $\in L[0,1],\ h\neq 0.$
\[2.2\]Let $f:I\subseteq \mathbb{R\rightarrow R}$ be a differentiable mapping on $I^{\circ }$ such that $f^{\prime }\in L[a,b]$, where $a,b\in
I^{\circ }$ with $a<b$ and $\alpha ,\lambda \in \left[ 0,1\right] $. If $%
\left\vert f^{\prime }\right\vert ^{q}$ is $h-$convex on $[a,b]$, $q\geq 1,$ then the following inequality holds:$$\begin{aligned}
&&\left\vert \lambda \left( \alpha f(a)+\left( 1-\alpha \right) f(b)\right)
+\left( 1-\lambda \right) f(\alpha a+\left( 1-\alpha \right) b)-\frac{1}{b-a}%
\dint\limits_{a}^{b}f(x)dx\right\vert \label{2-2} \\
&\leq &\left\{
\begin{array}{cc}
\left( b-a\right) \left[ \gamma _{2}^{1-\frac{1}{q}}A^{\frac{1}{q}}+\upsilon
_{2}^{1-\frac{1}{q}}B^{\frac{1}{q}}\right] & \alpha \lambda \leq 1-\alpha
\leq 1-\lambda \left( 1-\alpha \right) \\
\left( b-a\right) \left[ \gamma _{2}^{1-\frac{1}{q}}A^{\frac{1}{q}}+\upsilon
_{1}^{1-\frac{1}{q}}B^{\frac{1}{q}}\right] & \alpha \lambda \leq 1-\lambda
\left( 1-\alpha \right) \leq 1-\alpha \\
\left( b-a\right) \left[ \gamma _{1}^{1-\frac{1}{q}}A^{\frac{1}{q}}+\upsilon
_{2}^{1-\frac{1}{q}}B^{\frac{1}{q}}\right] & 1-\alpha \leq \alpha \lambda
\leq 1-\lambda \left( 1-\alpha \right)%
\end{array}%
\right. \notag\end{aligned}$$where $$\gamma _{1}=\left( 1-\alpha \right) \left[ \alpha \lambda -\frac{\left(
1-\alpha \right) }{2}\right] ,\ \gamma _{2}=\left( \alpha \lambda \right)
^{2}-\gamma _{1}\ , \label{2-2a}$$$$\begin{aligned}
\upsilon _{1} &=&\frac{1-\left( 1-\alpha \right) ^{2}}{2}-\alpha \left[
1-\lambda \left( 1-\alpha \right) \right] , \label{2-2b} \\
\upsilon _{2} &=&\frac{1+\left( 1-\alpha \right) ^{2}}{2}-\left( \lambda
+1\right) \left( 1-\alpha \right) \left[ 1-\lambda \left( 1-\alpha \right) %
\right] , \notag\end{aligned}$$$$\begin{aligned}
A &=&\left\vert f^{\prime }(b)\right\vert ^{q}\dint\limits_{0}^{1-\alpha
}\left\vert t-\alpha \lambda \right\vert h(t)dt+\left\vert f^{\prime
}(a)\right\vert ^{q}\dint\limits_{0}^{1-\alpha }\left\vert t-\alpha \lambda
\right\vert h(1-t)dt, \\
B &=&\left\vert f^{\prime }(b)\right\vert ^{q}\dint\limits_{1-\alpha
}^{1}\left\vert t-1+\lambda \left( 1-\alpha \right) \right\vert
h(t)dt+\left\vert f^{\prime }(a)\right\vert ^{q}\dint\limits_{1-\alpha
}^{1}\left\vert t-1+\lambda \left( 1-\alpha \right) \right\vert h(1-t)dt\end{aligned}$$
Suppose that $q\geq 1.$ From Lemma \[2.1\] and using the well known power mean inequality, we have$$\begin{aligned}
&&\left\vert \lambda \left( \alpha f(a)+\left( 1-\alpha \right) f(b)\right)
+\left( 1-\lambda \right) f(\alpha a+\left( 1-\alpha \right) b)-\frac{1}{b-a}%
\dint\limits_{a}^{b}f(x)dx\right\vert \\
&\leq &\left( b-a\right) \left[ \dint\limits_{0}^{1-\alpha }\left\vert
t-\alpha \lambda \right\vert \left\vert f^{\prime }\left( tb+(1-t)a\right)
\right\vert dt+\dint\limits_{1-\alpha }^{1}\left\vert t-1+\lambda \left(
1-\alpha \right) \right\vert \left\vert f^{\prime }\left( tb+(1-t)a\right)
\right\vert dt\right] \\
&\leq &\left( b-a\right) \left\{ \left( \dint\limits_{0}^{1-\alpha
}\left\vert t-\alpha \lambda \right\vert dt\right) ^{1-\frac{1}{q}}\left(
\dint\limits_{0}^{1-\alpha }\left\vert t-\alpha \lambda \right\vert
\left\vert f^{\prime }\left( tb+(1-t)a\right) \right\vert ^{q}dt\right) ^{%
\frac{1}{q}}\right.\end{aligned}$$$$\left. +\left( \dint\limits_{1-\alpha }^{1}\left\vert t-1+\lambda \left(
1-\alpha \right) \right\vert dt\right) ^{1-\frac{1}{q}}\left(
\dint\limits_{1-\alpha }^{1}\left\vert t-1+\lambda \left( 1-\alpha \right)
\right\vert \left\vert f^{\prime }\left( tb+(1-t)a\right) \right\vert
^{q}dt\right) ^{\frac{1}{q}}\right\} \label{2-3}$$Consider $$I_{1}=\dint\limits_{0}^{1-\alpha }\left\vert t-\alpha \lambda \right\vert
\left\vert f^{\prime }\left( tb+(1-t)a\right) \right\vert ^{q}dt,\ \
I_{2}=\dint\limits_{1-\alpha }^{1}\left\vert t-1+\lambda \left( 1-\alpha
\right) \right\vert \left\vert f^{\prime }\left( tb+(1-t)a\right)
\right\vert ^{q}dt$$
Since $\left\vert f^{\prime }\right\vert ^{q}$ is $h-$convex on $[a,b],$$$I_{1}\leq \left\vert f^{\prime }(b)\right\vert
^{q}\dint\limits_{0}^{1-\alpha }\left\vert t-\alpha \lambda \right\vert
h(t)dt+\left\vert f^{\prime }(a)\right\vert ^{q}\dint\limits_{0}^{1-\alpha
}\left\vert t-\alpha \lambda \right\vert h(1-t)dt. \label{2-4}$$Similarly$$I_{2}\leq \left\vert f^{\prime }(b)\right\vert ^{q}\dint\limits_{1-\alpha
}^{1}\left\vert t-1+\lambda \left( 1-\alpha \right) \right\vert
h(t)dt+\left\vert f^{\prime }(a)\right\vert ^{q}\dint\limits_{1-\alpha
}^{1}\left\vert t-1+\lambda \left( 1-\alpha \right) \right\vert h(1-t)dt.
\label{2-5}$$Additionally, by simple computation$$\dint\limits_{0}^{1-\alpha }\left\vert t-\alpha \lambda \right\vert
dt=\left\{
\begin{array}{cc}
\gamma _{2}, & \alpha \lambda \leq 1-\alpha \\
\gamma _{1}, & \alpha \lambda \geq 1-\alpha%
\end{array}%
\right. , \label{2-6}$$$$\gamma _{1}=\left( 1-\alpha \right) \left[ \alpha \lambda -\frac{\left(
1-\alpha \right) }{2}\right] ,\ \gamma _{2}=\left( \alpha \lambda \right)
^{2}-\gamma _{1}\ ,$$$$\dint\limits_{1-\alpha }^{1}\left\vert t-1+\lambda \left( 1-\alpha \right)
\right\vert dt=\left\{
\begin{array}{cc}
\upsilon _{1}, & 1-\lambda \left( 1-\alpha \right) \leq 1-\alpha \\
\upsilon _{2}, & 1-\lambda \left( 1-\alpha \right) \geq 1-\alpha%
\end{array}%
\right. , \label{2-7}$$$$\begin{aligned}
\upsilon _{1} &=&\frac{1-\left( 1-\alpha \right) ^{2}}{2}-\alpha \left[
1-\lambda \left( 1-\alpha \right) \right] , \\
\upsilon _{2} &=&\frac{1+\left( 1-\alpha \right) ^{2}}{2}-\left( \lambda
+1\right) \left( 1-\alpha \right) \left[ 1-\lambda \left( 1-\alpha \right) %
\right] .\end{aligned}$$Thus, using (\[2-4\]) (\[2-7\]) in (\[2-3\]), we obtain the inequality (\[2-2\]). This completes the proof.
Under the assumptions of Theorem \[2.2\] with $q=1,$ we have $$\begin{aligned}
&&\left\vert \lambda \left( \alpha f(a)+\left( 1-\alpha \right) f(b)\right)
+\left( 1-\lambda \right) f(\alpha a+\left( 1-\alpha \right) b)-\frac{1}{b-a}%
\dint\limits_{a}^{b}f(x)dx\right\vert \\
&\leq &\left( b-a\right) \left\{ \left\vert f^{\prime }(b)\right\vert \left[
\dint\limits_{0}^{1-\alpha }\left\vert t-\alpha \lambda \right\vert
h(t)dt+\dint\limits_{1-\alpha }^{1}\left\vert t-1+\lambda \left( 1-\alpha
\right) \right\vert h(t)dt\right] \right. \\
&&\left. \left\vert f^{\prime }(a)\right\vert \left[ \dint\limits_{0}^{1-%
\alpha }\left\vert t-\alpha \lambda \right\vert
h(1-t)dt+\dint\limits_{1-\alpha }^{1}\left\vert t-1+\lambda \left( 1-\alpha
\right) \right\vert h(1-t)dt\right] \right\} .\end{aligned}$$
\[2.2a\]Under the assumptions of Theorem \[2.2\] with $I\subseteq %
\left[ 0,\infty \right) ,$ $h(t)=t^{s},\ s\in \left( 0,1\right] $, we have$$\begin{aligned}
&&\left\vert \lambda \left( \alpha f(a)+\left( 1-\alpha \right) f(b)\right)
+\left( 1-\lambda \right) f(\alpha a+\left( 1-\alpha \right) b)-\frac{1}{b-a}%
\dint\limits_{a}^{b}f(x)dx\right\vert \label{2-8} \\
&\leq &\left\{
\begin{array}{cc}
\begin{array}{c}
\left( b-a\right) \left[ \gamma _{2}^{1-\frac{1}{q}}\left( \mu _{1}^{\ast
}\left\vert f^{\prime }(b)\right\vert ^{q}+\mu _{2}^{\ast }\left\vert
f^{\prime }(a)\right\vert ^{q}\right) ^{\frac{1}{q}}\right. \\
+\left. \upsilon _{2}^{1-\frac{1}{q}}\left( \eta _{3}^{\ast }\left\vert
f^{\prime }(b)\right\vert ^{q}+\eta _{4}^{\ast }\left\vert f^{\prime
}(a)\right\vert ^{q}\right) ^{\frac{1}{q}}\right]%
\end{array}%
, & \alpha \lambda \leq 1-\alpha \leq 1-\lambda \left( 1-\alpha \right) \\
\begin{array}{c}
\left( b-a\right) \left[ \gamma _{2}^{1-\frac{1}{q}}\left( \mu _{1}^{\ast
}\left\vert f^{\prime }(b)\right\vert ^{q}+\mu _{2}^{\ast }\left\vert
f^{\prime }(a)\right\vert ^{q}\right) ^{\frac{1}{q}}\right. \\
+\left. \upsilon _{2}^{1-\frac{1}{q}}\left( \eta _{1}^{\ast }\left\vert
f^{\prime }(b)\right\vert ^{q}+\eta _{2}^{\ast }\left\vert f^{\prime
}(a)\right\vert ^{q}\right) ^{\frac{1}{q}}\right]%
\end{array}%
, & \alpha \lambda \leq 1-\lambda \left( 1-\alpha \right) \leq 1-\alpha \\
\begin{array}{c}
\left( b-a\right) \left[ \gamma _{2}^{1-\frac{1}{q}}\left( \mu _{3}^{\ast
}\left\vert f^{\prime }(b)\right\vert ^{q}+\mu _{4}^{\ast }\left\vert
f^{\prime }(a)\right\vert ^{q}\right) ^{\frac{1}{q}}\right. \\
+\left. \upsilon _{2}^{1-\frac{1}{q}}\left( \eta _{3}^{\ast }\left\vert
f^{\prime }(b)\right\vert ^{q}+\eta _{4}^{\ast }\left\vert f^{\prime
}(a)\right\vert ^{q}\right) ^{\frac{1}{q}}\right]%
\end{array}%
, & 1-\alpha \leq \alpha \lambda \leq 1-\lambda \left( 1-\alpha \right)%
\end{array}%
\right. , \notag\end{aligned}$$where $\gamma _{1},\ \gamma _{2},\ \nu _{1}$ and $\nu _{2}\ $ are defined as in (\[2-2a\])-(\[2-2b\]) and $$\begin{aligned}
\mu _{1}^{\ast } &=&\left( \alpha \lambda \right) ^{s+2}\frac{2}{\left(
s+1\right) \left( s+2\right) }-\left( \alpha \lambda \right) \frac{\left(
1-\alpha \right) ^{s+1}}{s+1}+\frac{\left( 1-\alpha \right) ^{s+2}}{s+2}, \\
\mu _{2}^{\ast } &=&\left( 1-\alpha \lambda \right) ^{s+2}\frac{2}{\left(
s+1\right) \left( s+2\right) }-\frac{\left( 1-\alpha \lambda \right) \left(
1+\alpha ^{s+1}\right) }{s+1}+\frac{1+\alpha ^{s+2}}{s+2}, \\
\mu _{3}^{\ast } &=&\left( \alpha \lambda \right) \frac{\left( 1-\alpha
\right) ^{s+1}}{s+1}-\frac{\left( 1-\alpha \right) ^{s+2}}{s+2}, \\
\mu _{4}^{\ast } &=&\frac{\left( \alpha \lambda -1\right) \left( 1-\alpha
^{s+1}\right) }{s+1}+\frac{1-\alpha ^{s+2}}{s+2},\end{aligned}$$$$\begin{aligned}
\eta _{1}^{\ast } &=&\frac{1-\left( 1-\alpha \right) ^{s+2}}{s+2}-\frac{%
\left[ 1-\lambda \left( 1-\alpha \right) \right] }{s+1}\left[ 1-\left(
1-\alpha \right) ^{s+1}\right] , \\
\eta _{2}^{\ast } &=&\frac{\lambda \left( 1-\alpha \right) \alpha ^{s+1}}{s+1%
}-\frac{\alpha ^{s+2}}{s+2}, \\
\eta _{3}^{\ast } &=&\frac{2\left[ 1-\lambda \left( 1-\alpha \right) \right]
^{s+2}}{\left( s+1\right) \left( s+2\right) }-\frac{\left[ 1+\left( 1-\alpha
\right) ^{s+1}\right] \left[ 1-\lambda \left( 1-\alpha \right) \right] }{s+1}%
+\frac{1+\left( 1-\alpha \right) ^{s+2}}{s+2}, \\
\eta _{4}^{\ast } &=&\left[ \lambda \left( 1-\alpha \right) \right] ^{s+2}%
\frac{2}{\left( s+1\right) \left( s+2\right) }-\lambda \left( 1-\alpha
\right) \frac{\alpha ^{s+1}}{s+1}+\frac{\alpha ^{s+2}}{s+2}.\end{aligned}$$
Let the assumptions of Theorem \[2.2\] hold. Then for $h(t)=t$ the inequality (\[2-2\]) reduced to the inequality (\[1-3\]).
Under the assumptions of Theorem \[2.2\] with $h(t)=1$, we have$$\begin{aligned}
&&\left\vert \lambda \left( \alpha f(a)+\left( 1-\alpha \right) f(b)\right)
+\left( 1-\lambda \right) f(\alpha a+\left( 1-\alpha \right) b)-\frac{1}{b-a}%
\dint\limits_{a}^{b}f(x)dx\right\vert \\
&\leq &\left( b-a\right) \left( \left\vert f^{\prime }(b)\right\vert
^{q}+\left\vert f^{\prime }(a)\right\vert ^{q}\right) ^{\frac{1}{q}}\times
\left\{
\begin{array}{cc}
\gamma _{2}+\nu _{2} & \alpha \lambda \leq 1-\alpha \leq 1-\lambda \left(
1-\alpha \right) \\
\gamma _{2}+\nu _{1} & \alpha \lambda \leq 1-\lambda \left( 1-\alpha \right)
\leq 1-\alpha \\
\gamma _{1}+\nu _{2} & 1-\alpha \leq \alpha \lambda \leq 1-\lambda \left(
1-\alpha \right)%
\end{array}%
\right. ,\end{aligned}$$where $\gamma _{1},\ \gamma _{2},\ \nu _{1}$ and $\nu _{2}\ $ are defined as in (\[2-2a\])-(\[2-2b\]).
In Corollary \[2.2a\] , if we take $\alpha =\frac{1}{2}$ and $\lambda =%
\frac{1}{3},$ then we have the following Simpson type inequality$$\left\vert \frac{1}{6}\left[ f(a)+4f\left( \frac{a+b}{2}\right) +f(b)\right]
-\frac{1}{b-a}\dint\limits_{a}^{b}f(x)dx\right\vert \leq \frac{b-a}{2}\left(
\frac{5}{36}\right) ^{1-\frac{1}{q}} \label{2-9}$$$$\begin{aligned}
&&\times \left\{ \left( \frac{(2s+1)3^{s+1}+2}{3\times 6^{s+1}(s+1)(s+2)}%
\left\vert f^{\prime }(b)\right\vert ^{q}+\frac{2\times
5^{s+2}+(s-4)6^{s+1}-(2s+7)3^{s+1}}{3\times 6^{s+1}(s+1)(s+2)}\left\vert
f^{\prime }(a)\right\vert ^{q}\right) ^{\frac{1}{q}}\right. \notag \\
&&\left. +\left( \frac{2\times 5^{s+2}+(s-4)6^{s+1}-(2s+7)3^{s+1}}{%
3.6^{s+1}(s+1)(s+2)}\left\vert f^{\prime }(b)\right\vert ^{q}+\frac{%
(2s+1)3^{s+1}+2}{3\times 6^{s+1}(s+1)(s+2)}\left\vert f^{\prime
}(a)\right\vert ^{q}\right) ^{\frac{1}{q}}\right\} , \notag\end{aligned}$$which is the same of the inequality in [@SSO10 Theorem 10] .
In Corollary \[2.2a\] , if we take $\alpha =\frac{1}{2}$ and $\lambda =0,$ then we have following midpoint inequality$$\begin{aligned}
&&\left\vert f\left( \frac{a+b}{2}\right) -\frac{1}{b-a}\dint%
\limits_{a}^{b}f(x)dx\right\vert \leq \frac{b-a}{8}\left( \frac{2}{(s+1)(s+2)%
}\right) ^{\frac{1}{q}} \label{2-10} \\
&&\times \left\{ \left( \frac{2^{1-s}\left( s+1\right) \left\vert f^{\prime
}(b)\right\vert ^{q}}{2}+\frac{2^{1-s}\left( 2^{s+2}-s-3\right) \left\vert
f^{\prime }(a)\right\vert ^{q}}{2}\right) ^{\frac{1}{q}}\right. \notag \\
&&\left. +\left( \frac{2^{1-s}\left( s+1\right) \left\vert f^{\prime
}(a)\right\vert ^{q}}{2}+\frac{2^{1-s}\left( 2^{s+2}-s-3\right) \left\vert
f^{\prime }(b)\right\vert ^{q}}{2}\right) ^{\frac{1}{q}}\right\} . \notag\end{aligned}$$We note that the obtained midpoint inequality (\[2-10\]) is better than the inequality (\[1-4\]). Because $\frac{s+1}{2}\leq 1$ and $\frac{%
2^{s+2}-s-3}{2}\leq \frac{2^{1-s}+1}{2^{1-s}}.$
In Corollary \[2.2a\] , if we take $\alpha =\frac{1}{2}$ , and $\lambda
=1, $ then we get the following trapezoid inequality$$\begin{aligned}
&&\left\vert \frac{f\left( a\right) +f\left( b\right) }{2}-\frac{1}{b-a}%
\dint\limits_{a}^{b}f(x)dx\right\vert \leq \frac{b-a}{8}\left( \frac{2^{1-s}%
}{(s+1)(s+2)}\right) ^{\frac{1}{q}} \\
&&\times \left\{ \left( \left\vert f^{\prime }(b)\right\vert ^{q}+\left\vert
f^{\prime }(a)\right\vert ^{q}\left( 2^{s+1}+1\right) \right) ^{\frac{1}{q}%
}+\left( \left\vert f^{\prime }(a)\right\vert ^{q}+\left\vert f^{\prime
}(b)\right\vert ^{q}\left( 2^{s+1}+1\right) \right) ^{\frac{1}{q}}\right\}\end{aligned}$$
Using Lemma \[2.1\] we shall give another result for convex functions as follows.
\[2.3\]Let $f:I\subseteq \mathbb{R\rightarrow R}$ be a differentiable mapping on $I^{\circ }$ such that $f^{\prime }\in L[a,b]$, where $a,b\in
I^{\circ }$ with $a<b$ and $\alpha ,\lambda \in \left[ 0,1\right] $. If $%
\left\vert f^{\prime }\right\vert ^{q}$ is $h-$convex on $[a,b]$, $q>1,$ then the following inequality holds:$$\left\vert \lambda \left( \alpha f(a)+\left( 1-\alpha \right) f(b)\right)
+\left( 1-\lambda \right) f(\alpha a+\left( 1-\alpha \right) b)-\frac{1}{b-a}%
\dint\limits_{a}^{b}f(x)dx\right\vert \leq \left( b-a\right) \label{2-12}$$$$\times \left( \frac{1}{p+1}\right) ^{\frac{1}{p}}\left(
\dint\limits_{0}^{1}h(t)dt\right) ^{\frac{1}{q}}.\left\{
\begin{array}{cc}
\left[ \varepsilon _{1}^{\frac{1}{p}}C^{\frac{1}{q}}+\varepsilon _{3}^{\frac{%
1}{p}}D^{\frac{1}{q}}\right] , & \alpha \lambda \leq 1-\alpha \leq 1-\lambda
\left( 1-\alpha \right) \\
\left[ \varepsilon _{1}^{\frac{1}{p}}C^{\frac{1}{q}}+\varepsilon _{4}^{\frac{%
1}{p}}D^{\frac{1}{q}}\right] , & \alpha \lambda \leq 1-\lambda \left(
1-\alpha \right) \leq 1-\alpha \\
\left[ \varepsilon _{2}^{\frac{1}{p}}C^{\frac{1}{q}}+\varepsilon _{3}^{\frac{%
1}{p}}D^{\frac{1}{q}}\right] , & 1-\alpha \leq \alpha \lambda \leq 1-\lambda
\left( 1-\alpha \right)%
\end{array}%
\right.$$where $$\begin{aligned}
C &=&\left( 1-\alpha \right) \left[ \left\vert f^{\prime }\left( \left(
1-\alpha \right) b+\alpha a\right) \right\vert ^{q}+\left\vert f^{\prime
}\left( a\right) \right\vert ^{q}\right] , \label{2-12a} \\
\ D &=&\alpha \left[ \left\vert f^{\prime }\left( \left( 1-\alpha \right)
b+\alpha a\right) \right\vert ^{q}+\left\vert f^{\prime }\left( b\right)
\right\vert ^{q}\right] , \notag\end{aligned}$$$$\begin{aligned}
\varepsilon _{1} &=&\left( \alpha \lambda \right) ^{p+1}+\left( 1-\alpha
-\alpha \lambda \right) ^{p+1},\ \varepsilon _{2}=\left( \alpha \lambda
\right) ^{p+1}-\left( \alpha \lambda -1+\alpha \right) ^{p+1}, \notag \\
\varepsilon _{3} &=&\left[ \lambda \left( 1-\alpha \right) \right] ^{p+1}+%
\left[ \alpha -\lambda \left( 1-\alpha \right) \right] ^{p+1},\ \varepsilon
_{4}=\left[ \lambda \left( 1-\alpha \right) \right] ^{p+1}-\left[ \lambda
\left( 1-\alpha \right) -\alpha \right] ^{p+1}, \notag\end{aligned}$$and $\frac{1}{p}+\frac{1}{q}=1.$
From Lemma \[2.1\] and by Hölder’s integral inequality, we have$$\begin{aligned}
&&\left\vert \lambda \left( \alpha f(a)+\left( 1-\alpha \right) f(b)\right)
+\left( 1-\lambda \right) f(\alpha a+\left( 1-\alpha \right) b)-\frac{1}{b-a}%
\dint\limits_{a}^{b}f(x)dx\right\vert \\
&\leq &\left( b-a\right) \left[ \dint\limits_{0}^{1-\alpha }\left\vert
t-\alpha \lambda \right\vert \left\vert f^{\prime }\left( tb+(1-t)a\right)
\right\vert dt+\dint\limits_{1-\alpha }^{1}\left\vert t-1+\lambda \left(
1-\alpha \right) \right\vert \left\vert f^{\prime }\left( tb+(1-t)a\right)
\right\vert dt\right] \\
&\leq &\left( b-a\right) \left\{ \left( \dint\limits_{0}^{1-\alpha
}\left\vert t-\alpha \lambda \right\vert ^{p}dt\right) ^{\frac{1}{p}}\left(
\dint\limits_{0}^{1-\alpha }\left\vert f^{\prime }\left( tb+(1-t)a\right)
\right\vert ^{q}dt\right) ^{\frac{1}{q}}\right.\end{aligned}$$$$+\left. \left( \dint\limits_{1-\alpha }^{1}\left\vert t-1+\lambda \left(
1-\alpha \right) \right\vert ^{p}dt\right) ^{\frac{1}{p}}\left(
\dint\limits_{1-\alpha }^{1}\left\vert f^{\prime }\left( tb+(1-t)a\right)
\right\vert ^{q}dt\right) ^{\frac{1}{q}}\right\} . \label{2-13}$$Since $\left\vert f^{\prime }\right\vert ^{q}$ is $h-$convex on $[a,b],$ for $\alpha \in \left[ 0,1\right) $ by the inequality (\[1-2\]), we get $$\begin{aligned}
\dint\limits_{0}^{1-\alpha }\left\vert f^{\prime }\left( tb+(1-t)a\right)
\right\vert ^{q}dt &=&\left( 1-\alpha \right) \left[ \frac{1}{\left(
1-\alpha \right) \left( b-a\right) }\dint\limits_{a}^{\left( 1-\alpha
\right) b+\alpha a}\left\vert f^{\prime }\left( x\right) \right\vert ^{q}dx%
\right] \notag \\
&\leq &\left( 1-\alpha \right) \left[ \left\vert f^{\prime }\left( \left(
1-\alpha \right) b+\alpha a\right) \right\vert ^{q}+\left\vert f^{\prime
}\left( a\right) \right\vert ^{q}\right] \dint\limits_{0}^{1}h(t)dt.
\label{2-14}\end{aligned}$$The inequality (\[2-14\]) holds for $\alpha =1$ too. Similarly, for $%
\alpha \in \left( 0,1\right] $ by the inequality (\[1-2\]), we have $$\begin{aligned}
\dint\limits_{1-\alpha }^{1}\left\vert f^{\prime }\left( tb+(1-t)a\right)
\right\vert ^{q}dt &=&\alpha \left[ \frac{1}{\alpha \left( b-a\right) }%
\dint\limits_{\left( 1-\alpha \right) b+\alpha a}^{b}\left\vert f^{\prime
}\left( x\right) \right\vert ^{q}dx\right] \notag \\
&\leq &\alpha \left[ \left\vert f^{\prime }\left( \left( 1-\alpha \right)
b+\alpha a\right) \right\vert ^{q}+\left\vert f^{\prime }\left( b\right)
\right\vert ^{q}\right] \dint\limits_{0}^{1}h(t)dt. \label{2-15}\end{aligned}$$The inequality (\[2-15\]) holds for $\alpha =0$ too. By simple computation$$\dint\limits_{0}^{1-\alpha }\left\vert t-\alpha \lambda \right\vert
^{p}dt=\left\{
\begin{array}{cc}
\frac{\left( \alpha \lambda \right) ^{p+1}+\left( 1-\alpha -\alpha \lambda
\right) ^{p+1}}{p+1}, & \alpha \lambda \leq 1-\alpha \\
\frac{\left( \alpha \lambda \right) ^{p+1}-\left( \alpha \lambda -1+\alpha
\right) ^{p+1}}{p+1}, & \alpha \lambda \geq 1-\alpha%
\end{array}%
\right. , \label{2-16}$$and$$\dint\limits_{1-\alpha }^{1}\left\vert t-1+\lambda \left( 1-\alpha \right)
\right\vert ^{p}dt=\left\{
\begin{array}{cc}
\frac{\left[ \lambda \left( 1-\alpha \right) \right] ^{p+1}+\left[ \alpha
-\lambda \left( 1-\alpha \right) \right] ^{p+1}}{p+1}, & 1-\alpha \leq
1-\lambda \left( 1-\alpha \right) \\
\frac{\left[ \lambda \left( 1-\alpha \right) \right] ^{p+1}-\left[ \lambda
\left( 1-\alpha \right) -\alpha \right] ^{p+1}}{p+1}, & 1-\alpha \geq
1-\lambda \left( 1-\alpha \right)%
\end{array}%
\right. , \label{2-17}$$thus, using (\[2-14\])-(\[2-17\]) in (\[2-13\]), we obtain the inequality (\[2-12\]). This completes the proof.
\[2.4\]Under the assumptions of Theorem \[2.3\] with $I\subseteq \left[
0,\infty \right) ,\ h(t)=t^{s},\ s\in \left( 0,1\right] $, we have$$\left\vert \lambda \left( \alpha f(a)+\left( 1-\alpha \right) f(b)\right)
+\left( 1-\lambda \right) f(\alpha a+\left( 1-\alpha \right) b)-\frac{1}{b-a}%
\dint\limits_{a}^{b}f(x)dx\right\vert \leq \left( b-a\right) \label{2-12b}$$$$\times \left( \frac{1}{p+1}\right) ^{\frac{1}{p}}\left( \frac{1}{s+1}\right)
^{\frac{1}{q}}.\left\{
\begin{array}{cc}
\left[ \varepsilon _{1}^{\frac{1}{p}}C^{\frac{1}{q}}+\varepsilon _{3}^{\frac{%
1}{p}}D^{\frac{1}{q}}\right] , & \alpha \lambda \leq 1-\alpha \leq 1-\lambda
\left( 1-\alpha \right) \\
\left[ \varepsilon _{1}^{\frac{1}{p}}C^{\frac{1}{q}}+\varepsilon _{4}^{\frac{%
1}{p}}D^{\frac{1}{q}}\right] , & \alpha \lambda \leq 1-\lambda \left(
1-\alpha \right) \leq 1-\alpha \\
\left[ \varepsilon _{2}^{\frac{1}{p}}C^{\frac{1}{q}}+\varepsilon _{3}^{\frac{%
1}{p}}D^{\frac{1}{q}}\right] , & 1-\alpha \leq \alpha \lambda \leq 1-\lambda
\left( 1-\alpha \right)%
\end{array}%
\right. ,$$where $\varepsilon _{1},\ \varepsilon _{2},\ \varepsilon _{3},\ \varepsilon
_{4},\ C$ and $D$ are defined as in (\[2-12a\]).
Under the assumptions of Theorem \[2.3\] with $h(t)=t$, we have$$\left\vert \lambda \left( \alpha f(a)+\left( 1-\alpha \right) f(b)\right)
+\left( 1-\lambda \right) f(\alpha a+\left( 1-\alpha \right) b)-\frac{1}{b-a}%
\dint\limits_{a}^{b}f(x)dx\right\vert \leq \left( b-a\right)$$$$\times \left( \frac{1}{p+1}\right) ^{\frac{1}{p}}\left( \frac{1}{2}\right) ^{%
\frac{1}{q}}.\left\{
\begin{array}{cc}
\left[ \varepsilon _{1}^{\frac{1}{p}}C^{\frac{1}{q}}+\varepsilon _{3}^{\frac{%
1}{p}}D^{\frac{1}{q}}\right] , & \alpha \lambda \leq 1-\alpha \leq 1-\lambda
\left( 1-\alpha \right) \\
\left[ \varepsilon _{1}^{\frac{1}{p}}C^{\frac{1}{q}}+\varepsilon _{4}^{\frac{%
1}{p}}D^{\frac{1}{q}}\right] , & \alpha \lambda \leq 1-\lambda \left(
1-\alpha \right) \leq 1-\alpha \\
\left[ \varepsilon _{2}^{\frac{1}{p}}C^{\frac{1}{q}}+\varepsilon _{3}^{\frac{%
1}{p}}D^{\frac{1}{q}}\right] , & 1-\alpha \leq \alpha \lambda \leq 1-\lambda
\left( 1-\alpha \right)%
\end{array}%
\right. ,$$where $\varepsilon _{1},\ \varepsilon _{2},\ \varepsilon _{3},\ \varepsilon
_{4},\ C$ and $D$ are defined as in (\[2-12a\]).
Under the assumptions of Theorem \[2.3\] with $h(t)=1$, we have$$\left\vert \lambda \left( \alpha f(a)+\left( 1-\alpha \right) f(b)\right)
+\left( 1-\lambda \right) f(\alpha a+\left( 1-\alpha \right) b)-\frac{1}{b-a}%
\dint\limits_{a}^{b}f(x)dx\right\vert \leq \left( b-a\right)$$$$\times \left( \frac{1}{p+1}\right) ^{\frac{1}{p}}.\left\{
\begin{array}{cc}
\left[ \varepsilon _{1}^{\frac{1}{p}}C^{\frac{1}{q}}+\varepsilon _{3}^{\frac{%
1}{p}}D^{\frac{1}{q}}\right] , & \alpha \lambda \leq 1-\alpha \leq 1-\lambda
\left( 1-\alpha \right) \\
\left[ \varepsilon _{1}^{\frac{1}{p}}C^{\frac{1}{q}}+\varepsilon _{4}^{\frac{%
1}{p}}D^{\frac{1}{q}}\right] , & \alpha \lambda \leq 1-\lambda \left(
1-\alpha \right) \leq 1-\alpha \\
\left[ \varepsilon _{2}^{\frac{1}{p}}C^{\frac{1}{q}}+\varepsilon _{3}^{\frac{%
1}{p}}D^{\frac{1}{q}}\right] , & 1-\alpha \leq \alpha \lambda \leq 1-\lambda
\left( 1-\alpha \right)%
\end{array}%
\right. ,$$where $\varepsilon _{1},\ \varepsilon _{2},\ \varepsilon _{3},\ \varepsilon
_{4},\ C$ and $D$ are defined as in (\[2-12a\]).
In Corollary \[2.4\], if we take $\alpha =\frac{1}{2}$ and $\lambda =\frac{%
1}{3}$, then we have the following Simpson type inequality $$\left\vert \frac{1}{6}\left[ f(a)+4f\left( \frac{a+b}{2}\right) +f(b)\right]
-\frac{1}{b-a}\dint\limits_{a}^{b}f(x)dx\right\vert \label{2-18}$$$$\leq \frac{b-a}{12}\left( \frac{1+2^{p+1}}{3\left( p+1\right) }\right) ^{%
\frac{1}{p}}\left\{ \left( \frac{\left\vert f^{\prime }\left( \frac{a+b}{2}%
\right) \right\vert ^{q}+\left\vert f^{\prime }\left( a\right) \right\vert
^{q}}{s+1}\right) ^{\frac{1}{q}}+\left( \frac{\left\vert f^{\prime }\left(
\frac{a+b}{2}\right) \right\vert ^{q}+\left\vert f^{\prime }\left( b\right)
\right\vert ^{q}}{s+1}\right) ^{\frac{1}{q}}\right\} ,$$which is the same of the inequality (\[1-5\]).
In Corollary \[2.4\], if we take $\alpha =\frac{1}{2}$ and $\lambda =0,$ then we have the following midpoint inequality$$\begin{aligned}
&&\left\vert f\left( \frac{a+b}{2}\right) -\frac{1}{b-a}\dint%
\limits_{a}^{b}f(x)dx\right\vert \\
&\leq &\frac{b-a}{4}\left( \frac{1}{p+1}\right) ^{\frac{1}{p}}\left\{ \left(
\frac{\left\vert f^{\prime }\left( \frac{a+b}{2}\right) \right\vert
^{q}+\left\vert f^{\prime }\left( a\right) \right\vert ^{q}}{s+1}\right) ^{%
\frac{1}{q}}+\left( \frac{\left\vert f^{\prime }\left( \frac{a+b}{2}\right)
\right\vert ^{q}+\left\vert f^{\prime }\left( b\right) \right\vert ^{q}}{s+1}%
\right) ^{\frac{1}{q}}\right\} .\end{aligned}$$We note that by inequality $$2^{s-1}\left\vert f^{\prime }\left( \frac{a+b}{2}\right) \right\vert
^{q}\leq \frac{\left\vert f^{\prime }\left( a\right) \right\vert
^{q}+\left\vert f^{\prime }\left( b\right) \right\vert ^{q}}{s+1}$$we have$$\begin{aligned}
\left\vert f\left( \frac{a+b}{2}\right) -\frac{1}{b-a}\dint%
\limits_{a}^{b}f(x)dx\right\vert &\leq &\left( \frac{b-a}{4}\right) \left(
\frac{1}{p+1}\right) ^{\frac{1}{p}}\left( \frac{1}{s+1}\right) ^{\frac{2}{q}}
\\
&&\times \left[ \left( \left( 2^{1-s}+s+1\right) \left\vert f^{\prime
}\left( a\right) \right\vert ^{q}+2^{1-s}\left\vert f^{\prime }\left(
b\right) \right\vert ^{q}\right) ^{\frac{1}{q}}\right. \\
&&+\left. \left( \left( 2^{1-s}+s+1\right) \left\vert f^{\prime }\left(
b\right) \right\vert ^{q}+2^{1-s}\left\vert f^{\prime }\left( a\right)
\right\vert ^{q}\right) ^{\frac{1}{q}}\right] ,\end{aligned}$$which is the same of the inequality (\[1-4a\]).
In Corollary \[2.4\], if we take $\alpha =\frac{1}{2}$ and $\lambda =1,$ then we have the following trapezoid inequality$$\begin{aligned}
&&\left\vert \frac{f\left( a\right) +f\left( b\right) }{2}-\frac{1}{b-a}%
\dint\limits_{a}^{b}f(x)dx\right\vert \leq \frac{b-a}{4}\left( \frac{1}{p+1}%
\right) ^{\frac{1}{p}} \label{2-19} \\
&&\times \left\{ \left( \frac{\left\vert f^{\prime }\left( \frac{a+b}{2}%
\right) \right\vert ^{q}+\left\vert f^{\prime }\left( a\right) \right\vert
^{q}}{s+1}\right) ^{\frac{1}{q}}+\left( \frac{\left\vert f^{\prime }\left(
\frac{a+b}{2}\right) \right\vert ^{q}+\left\vert f^{\prime }\left( b\right)
\right\vert ^{q}}{s+1}\right) ^{\frac{1}{q}}\right\} . \notag\end{aligned}$$We note that the obtained midpoint inequality (\[2-19\]) is better than the inequality (\[1-6\]).
\[2.5\]Let $f:I\subseteq \mathbb{R\rightarrow R}$ be a differentiable mapping on $I^{\circ }$ such that $f^{\prime }\in L[a,b]$, where $a,b\in
I^{\circ }$ with $a<b$ and $\alpha ,\lambda \in \left[ 0,1\right] $. If $%
\left\vert f^{\prime }\right\vert ^{q}$ is $h-$concave on $[a,b]$, $q>1,$ then the following inequality holds:$$\left\vert \lambda \left( \alpha f(a)+\left( 1-\alpha \right) f(b)\right)
+\left( 1-\lambda \right) f(\alpha a+\left( 1-\alpha \right) b)-\frac{1}{b-a}%
\dint\limits_{a}^{b}f(x)dx\right\vert \leq \left( b-a\right) \label{2-20}$$$$\times \left( \frac{1}{2h\left( \frac{1}{2}\right) }\right) ^{\frac{1}{q}%
}\left( \frac{1}{p+1}\right) ^{\frac{1}{p}}.\left\{
\begin{array}{cc}
\left[ \varepsilon _{1}^{\frac{1}{p}}E^{\frac{1}{q}}+\varepsilon _{3}^{\frac{%
1}{p}}F^{\frac{1}{q}}\right] , & \alpha \lambda \leq 1-\alpha \leq 1-\lambda
\left( 1-\alpha \right) \\
\left[ \varepsilon _{1}^{\frac{1}{p}}E^{\frac{1}{q}}+\varepsilon _{4}^{\frac{%
1}{p}}F^{\frac{1}{q}}\right] , & \alpha \lambda \leq 1-\lambda \left(
1-\alpha \right) \leq 1-\alpha \\
\left[ \varepsilon _{2}^{\frac{1}{p}}E^{\frac{1}{q}}+\varepsilon _{3}^{\frac{%
1}{p}}F^{\frac{1}{q}}\right] , & 1-\alpha \leq \alpha \lambda \leq 1-\lambda
\left( 1-\alpha \right)%
\end{array}%
\right. ,$$where$$E=\left( 1-\alpha \right) \left\vert f^{\prime }\left( \frac{\left( 1-\alpha
\right) b+\left( 1+\alpha \right) a}{2}\right) \right\vert ^{q},\ F=\alpha
\left\vert f^{\prime }\left( \frac{\left( 2-\alpha \right) b+\alpha a}{2}%
\right) \right\vert ^{q},$$and $\varepsilon _{1},\ \varepsilon _{2},\ \varepsilon _{3},\ \varepsilon
_{4}$ are defined as in (\[2-12a\]).
We proceed similarly as in the proof Theorem \[2.3\]. Since $\left\vert
f^{\prime }\right\vert ^{q}$ is $h-$concave on $[a,b],$ for $\alpha \in %
\left[ 0,1\right) $ by the inequality (\[1-2\]), we get$$\begin{aligned}
\dint\limits_{0}^{1-\alpha }\left\vert f^{\prime }\left( tb+(1-t)a\right)
\right\vert ^{q}dt &=&\left( 1-\alpha \right) \left[ \frac{1}{\left(
1-\alpha \right) \left( b-a\right) }\dint\limits_{a}^{\left( 1-\alpha
\right) b+\alpha a}\left\vert f^{\prime }\left( x\right) \right\vert ^{q}dx%
\right] \notag \\
&\leq &\frac{\left( 1-\alpha \right) }{2h\left( \frac{1}{2}\right) }%
\left\vert f^{\prime }\left( \frac{\left( 1-\alpha \right) b+\left( 1+\alpha
\right) a}{2}\right) \right\vert ^{q} \label{2-21}\end{aligned}$$The inequality (\[2-21\]) holds for $\alpha =1$ too. Similarly, for $%
\alpha \in \left( 0,1\right] $ by the inequality (\[1-2\]), we have$$\begin{aligned}
\dint\limits_{1-\alpha }^{1}\left\vert f^{\prime }\left( tb+(1-t)a\right)
\right\vert ^{q}dt &=&\alpha \left[ \frac{1}{\alpha \left( b-a\right) }%
\dint\limits_{\left( 1-\alpha \right) b+\alpha a}^{b}\left\vert f^{\prime
}\left( x\right) \right\vert ^{q}dx\right] \notag \\
&\leq &\frac{\alpha }{2h\left( \frac{1}{2}\right) }\left\vert f^{\prime
}\left( \frac{\left( 2-\alpha \right) b+\alpha a}{2}\right) \right\vert ^{q}
\label{2-22}\end{aligned}$$The inequality (\[2-22\]) holds for $\alpha =0$ too. Thus, using ([2-16]{}),(\[2-17\]),(\[2-21\])and (\[2-22\]) in (\[2-13\]), we obtain the inequality (\[2-20\]). This completes the proof.
\[2.6\]Under the assumptions of Theorem \[2.5\] with $I\subseteq \left[
0,\infty \right) ,\ h(t)=t^{s},\ s\in \left( 0,1\right] $, we have$$\left\vert \lambda \left( \alpha f(a)+\left( 1-\alpha \right) f(b)\right)
+\left( 1-\lambda \right) f(\alpha a+\left( 1-\alpha \right) b)-\frac{1}{b-a}%
\dint\limits_{a}^{b}f(x)dx\right\vert \leq \left( b-a\right)$$$$\times 2^{\frac{s-1}{q}}\left( \frac{1}{p+1}\right) ^{\frac{1}{p}}.\left\{
\begin{array}{cc}
\left[ \varepsilon _{1}^{\frac{1}{p}}E^{\frac{1}{q}}+\varepsilon _{3}^{\frac{%
1}{p}}F^{\frac{1}{q}}\right] , & \alpha \lambda \leq 1-\alpha \leq 1-\lambda
\left( 1-\alpha \right) \\
\left[ \varepsilon _{1}^{\frac{1}{p}}E^{\frac{1}{q}}+\varepsilon _{4}^{\frac{%
1}{p}}F^{\frac{1}{q}}\right] , & \alpha \lambda \leq 1-\lambda \left(
1-\alpha \right) \leq 1-\alpha \\
\left[ \varepsilon _{2}^{\frac{1}{p}}E^{\frac{1}{q}}+\varepsilon _{3}^{\frac{%
1}{p}}F^{\frac{1}{q}}\right] , & 1-\alpha \leq \alpha \lambda \leq 1-\lambda
\left( 1-\alpha \right)%
\end{array}%
\right. ,$$where $\varepsilon _{1},\ \varepsilon _{2},\ \varepsilon _{3},\ \varepsilon
_{4},\ E$ and $F$ are defined as in Theorem \[2.5\].
\[2.7\]Under the assumptions of Theorem \[2.5\] with $h(t)=t$, we have$$\left\vert \lambda \left( \alpha f(a)+\left( 1-\alpha \right) f(b)\right)
+\left( 1-\lambda \right) f(\alpha a+\left( 1-\alpha \right) b)-\frac{1}{b-a}%
\dint\limits_{a}^{b}f(x)dx\right\vert \leq \left( b-a\right)$$$$\times \left( \frac{1}{p+1}\right) ^{\frac{1}{p}}.\left\{
\begin{array}{cc}
\left[ \varepsilon _{1}^{\frac{1}{p}}E^{\frac{1}{q}}+\varepsilon _{3}^{\frac{%
1}{p}}F^{\frac{1}{q}}\right] , & \alpha \lambda \leq 1-\alpha \leq 1-\lambda
\left( 1-\alpha \right) \\
\left[ \varepsilon _{1}^{\frac{1}{p}}E^{\frac{1}{q}}+\varepsilon _{4}^{\frac{%
1}{p}}F^{\frac{1}{q}}\right] , & \alpha \lambda \leq 1-\lambda \left(
1-\alpha \right) \leq 1-\alpha \\
\left[ \varepsilon _{2}^{\frac{1}{p}}E^{\frac{1}{q}}+\varepsilon _{3}^{\frac{%
1}{p}}F^{\frac{1}{q}}\right] , & 1-\alpha \leq \alpha \lambda \leq 1-\lambda
\left( 1-\alpha \right)%
\end{array}%
\right. ,$$where $\varepsilon _{1},\ \varepsilon _{2},\ \varepsilon _{3},\ \varepsilon
_{4},\ E$ and $F$ are defined as in Theorem \[2.5\].
Under the assumptions of Theorem \[2.5\] with $h(t)=1$, we have$$\left\vert \lambda \left( \alpha f(a)+\left( 1-\alpha \right) f(b)\right)
+\left( 1-\lambda \right) f(\alpha a+\left( 1-\alpha \right) b)-\frac{1}{b-a}%
\dint\limits_{a}^{b}f(x)dx\right\vert \leq \left( b-a\right)$$$$\times 2^{-\frac{1}{q}}\left( \frac{1}{p+1}\right) ^{\frac{1}{p}}.\left\{
\begin{array}{cc}
\left[ \varepsilon _{1}^{\frac{1}{p}}E^{\frac{1}{q}}+\varepsilon _{3}^{\frac{%
1}{p}}F^{\frac{1}{q}}\right] , & \alpha \lambda \leq 1-\alpha \leq 1-\lambda
\left( 1-\alpha \right) \\
\left[ \varepsilon _{1}^{\frac{1}{p}}E^{\frac{1}{q}}+\varepsilon _{4}^{\frac{%
1}{p}}F^{\frac{1}{q}}\right] , & \alpha \lambda \leq 1-\lambda \left(
1-\alpha \right) \leq 1-\alpha \\
\left[ \varepsilon _{2}^{\frac{1}{p}}E^{\frac{1}{q}}+\varepsilon _{3}^{\frac{%
1}{p}}F^{\frac{1}{q}}\right] , & 1-\alpha \leq \alpha \lambda \leq 1-\lambda
\left( 1-\alpha \right)%
\end{array}%
\right. ,$$where $\varepsilon _{1},\ \varepsilon _{2},\ \varepsilon _{3},\ \varepsilon
_{4},\ E$ and $F$ are defined as in Theorem \[2.5\].
Under the assumptions of Theorem \[2.5\] with $h(t)=\frac{1}{t}$, $t\in
\left( 0,1\right) ,\ $we have$$\left\vert \lambda \left( f^{\prime }\alpha f(a)+\left( 1-\alpha \right)
f(b)\right) +\left( 1-\lambda \right) f(\alpha a+\left( 1-\alpha \right) b)-%
\frac{1}{b-a}\dint\limits_{a}^{b}f(x)dx\right\vert \leq \left( b-a\right)$$$$\times 4^{-\frac{1}{q}}\left( \frac{1}{p+1}\right) ^{\frac{1}{p}}.\left\{
\begin{array}{cc}
\left[ \varepsilon _{1}^{\frac{1}{p}}E^{\frac{1}{q}}+\varepsilon _{3}^{\frac{%
1}{p}}F^{\frac{1}{q}}\right] , & \alpha \lambda \leq 1-\alpha \leq 1-\lambda
\left( 1-\alpha \right) \\
\left[ \varepsilon _{1}^{\frac{1}{p}}E^{\frac{1}{q}}+\varepsilon _{4}^{\frac{%
1}{p}}F^{\frac{1}{q}}\right] , & \alpha \lambda \leq 1-\lambda \left(
1-\alpha \right) \leq 1-\alpha \\
\left[ \varepsilon _{2}^{\frac{1}{p}}E^{\frac{1}{q}}+\varepsilon _{3}^{\frac{%
1}{p}}F^{\frac{1}{q}}\right] , & 1-\alpha \leq \alpha \lambda \leq 1-\lambda
\left( 1-\alpha \right)%
\end{array}%
\right. ,$$where $\varepsilon _{1},\ \varepsilon _{2},\ \varepsilon _{3},\ \varepsilon
_{4},\ E$ and $F$ are defined as in Theorem \[2.5\].
In Corollary \[2.6\], if we take $\alpha =\frac{1}{2}$ and $\lambda =1,$ then we have the following trapezoid inequality$$\begin{aligned}
&&\left\vert \frac{f\left( a\right) +f\left( b\right) }{2}-\frac{1}{b-a}%
\dint\limits_{a}^{b}f(x)dx\right\vert \\
&\leq &\frac{b-a}{4}\left( \frac{1}{p+1}\right) ^{\frac{1}{p}}\times \left(
\frac{1}{2}\right) ^{\frac{1-s}{q}}\left[ \left\vert f^{\prime }\left( \frac{%
3b+a}{4}\right) \right\vert +\left\vert f^{\prime }\left( \frac{3a+b}{4}%
\right) \right\vert \right]\end{aligned}$$which is the same of the inequality in [@P10 Theorem 8 (i)].
In Corollary \[2.6\], if we take $\alpha =\frac{1}{2}$ and $\lambda =0,$ then we have the following midpoint inequality$$\begin{aligned}
&&\left\vert f\left( \frac{a+b}{2}\right) -\frac{1}{b-a}\dint%
\limits_{a}^{b}f(x)dx\right\vert \\
&\leq &\frac{b-a}{4}\left( \frac{1}{p+1}\right) ^{\frac{1}{p}}\times \left(
\frac{1}{2}\right) ^{\frac{1-s}{q}}\left[ \left\vert f^{\prime }\left( \frac{%
3b+a}{4}\right) \right\vert +\left\vert f^{\prime }\left( \frac{3a+b}{4}%
\right) \right\vert \right]\end{aligned}$$which is the same of the inequality in [@P10 Theorem 8 (ii)].
In Corollary \[2.7\], if we take $\alpha =\frac{1}{2}$ and $\lambda =1,$ then we have the following trapezoid inequality$$\begin{aligned}
&&\left\vert \frac{f\left( a\right) +f\left( b\right) }{2}-\frac{1}{b-a}%
\dint\limits_{a}^{b}f(x)dx\right\vert \label{2-23} \\
&\leq &\frac{b-a}{4}\left( \frac{1}{p+1}\right) ^{\frac{1}{p}}\left[
\left\vert f^{\prime }\left( \frac{3b+a}{4}\right) \right\vert +\left\vert
f^{\prime }\left( \frac{3a+b}{4}\right) \right\vert \right] \notag\end{aligned}$$which is the same of the inequality in [@KBOP07 Theorem 2].
In Corollary \[2.7\], if we take $\alpha =\frac{1}{2}$ and $\lambda =0,$ then we have the following trapezoid inequality$$\begin{aligned}
&&\left\vert f\left( \frac{a+b}{2}\right) -\frac{1}{b-a}\dint%
\limits_{a}^{b}f(x)dx\right\vert \label{2-24} \\
&\leq &\frac{b-a}{4}\left( \frac{1}{p+1}\right) ^{\frac{1}{p}}\left[
\left\vert f^{\prime }\left( \frac{3b+a}{4}\right) \right\vert +\left\vert
f^{\prime }\left( \frac{3a+b}{4}\right) \right\vert \right] \notag\end{aligned}$$which is the same of the inequality in [@ADK11 Theorem 2.5].
In Corollary \[2.7\], since $\left\vert f^{\prime }\right\vert ^{q},\ q>1,$ is concave on $\left[ a,b\right] ,$ using the power mean inequality, we have$$\begin{aligned}
\left\vert f^{\prime }\left( \lambda x+\left( 1-\lambda \right) y\right)
\right\vert ^{q} &\geq &\lambda \left\vert f^{\prime }\left( x\right)
\right\vert ^{q}+\left( 1-\lambda \right) \left\vert f^{\prime }\left(
y\right) \right\vert ^{q} \\
&\geq &\left( \lambda \left\vert f^{\prime }\left( x\right) \right\vert
+\left( 1-\lambda \right) \left\vert f^{\prime }\left( y\right) \right\vert
\right) ^{q},\end{aligned}$$$\forall x,y\in \left[ a,b\right] $ and $\lambda \in \left[ 0,1\right] .$ Hence$$\left\vert f^{\prime }\left( \lambda x+\left( 1-\lambda \right) y\right)
\right\vert \geq \lambda \left\vert f^{\prime }\left( x\right) \right\vert
+\left( 1-\lambda \right) \left\vert f^{\prime }\left( y\right) \right\vert$$so $\left\vert f^{\prime }\right\vert $ is also concave. Then by the inequality (\[1-1\]), we have$$\left\vert f^{\prime }\left( \frac{3b+a}{4}\right) \right\vert +\left\vert
f^{\prime }\left( \frac{3a+b}{4}\right) \right\vert \leq 2\left\vert
f^{\prime }\left( \frac{a+b}{2}\right) \right\vert . \label{2-25}$$Thus, using the inequality (\[2-25\]) in (\[2-23\]) and (\[2-24\]) we get$$\begin{aligned}
\left\vert \frac{f\left( a\right) +f\left( b\right) }{2}-\frac{1}{b-a}%
\dint\limits_{a}^{b}f(x)dx\right\vert &\leq &\frac{b-a}{2}\left( \frac{1}{p+1%
}\right) ^{\frac{1}{p}}\left\vert f^{\prime }\left( \frac{a+b}{2}\right)
\right\vert , \\
\left\vert f\left( \frac{a+b}{2}\right) -\frac{1}{b-a}\dint%
\limits_{a}^{b}f(x)dx\right\vert &\leq &\frac{b-a}{2}\left( \frac{1}{p+1}%
\right) ^{\frac{1}{p}}\left\vert f^{\prime }\left( \frac{a+b}{2}\right)
\right\vert .\end{aligned}$$
Some applications for special means
===================================
Let us recall the following special means of arbitrary real numbers $a,b$ with $a\neq b$ and $\alpha \in \left[ 0,1\right] :$
1. The weighted arithmetic mean$$A_{\alpha }\left( a,b\right) :=\alpha a+(1-\alpha )b,~a,b\in
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
.$$
2. The unweighted arithmetic mean$$A\left( a,b\right) :=\frac{a+b}{2},~a,b\in
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
.$$
3. The Logarithmic mean$$L\left( a,b\right) :=\frac{b-a}{\ln \left\vert b\right\vert -\ln \left\vert
a\right\vert },\ \ \left\vert a\right\vert \neq \left\vert b\right\vert ,\
ab\neq 0.$$
4. Then $p-$Logarithmic mean$$L_{p}\left( a,b\right) :=\ \left( \frac{b^{p+1}-a^{p+1}}{(p+1)(b-a)}\right)
^{\frac{1}{p}}\ ,\ p\in
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
\backslash \left\{ -1,0\right\} ,\ a,b>0.$$
From known Example 1 in [@HM94], we may find that for any $s\in \left(
0,1\right) $ and $\beta >0,$ $f:\left[ 0,\infty \right) \rightarrow \left[
0,\infty \right) $, $f(t)=\beta t^{s},\ f\in K_{s}^{2}.$
Now, using the resuls of Section 2, some new inequalities are derived for the above means.
Let $a,b\in
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$ with $0<a<b,\ q\geq 1$ and $s\in \left( 0,\frac{1}{q}\right) $we have the following inequality:$$\begin{aligned}
&&\left\vert \lambda A_{\alpha }\left( a^{s+1},b^{s+1}\right) +\left(
1-\lambda \right) A_{\alpha }^{s+1}\left( a,b\right) -L_{s+1}^{s+1}\left(
a,b\right) \right\vert \\
&\leq &\left\{
\begin{array}{cc}
\begin{array}{c}
\left( b-a\right) \left( s+1\right) \left\{ \gamma _{2}^{1-\frac{1}{q}%
}\left( \mu _{1}^{\ast }b^{sq}+\mu _{2}^{\ast }a^{sq}\right) ^{\frac{1}{q}%
}\right. \\
\left. +\upsilon _{2}^{1-\frac{1}{q}}\left( \eta _{3}^{\ast }b^{sq}+\eta
_{4}^{\ast }a^{sq}\right) ^{\frac{1}{q}}\right\} ,%
\end{array}
& \alpha \lambda \leq 1-\alpha \leq 1-\lambda \left( 1-\alpha \right) \\
\begin{array}{c}
\left( b-a\right) \left( s+1\right) \left\{ \gamma _{2}^{1-\frac{1}{q}%
}\left( \mu _{1}^{\ast }b^{sq}+\mu _{2}^{\ast }a^{sq}\right) ^{\frac{1}{q}%
}\right. \\
\left. +\upsilon _{1}^{1-\frac{1}{q}}\left( \eta _{1}^{\ast }b^{sq}+\eta
_{2}^{\ast }a^{sq}\right) ^{\frac{1}{q}}\right\} ,%
\end{array}
& \alpha \lambda \leq 1-\lambda \left( 1-\alpha \right) \leq 1-\alpha \\
\begin{array}{c}
\left( b-a\right) \left( s+1\right) \left\{ \gamma _{1}^{1-\frac{1}{q}%
}\left( \mu _{3}^{\ast }b^{sq}+\mu _{4}^{\ast }a^{sq}\right) ^{\frac{1}{q}%
}\right. \\
\left. +\upsilon _{2}^{1-\frac{1}{q}}\left( \eta _{3}^{\ast }b^{sq}+\eta
_{4}^{\ast }a^{sq}\right) ^{\frac{1}{q}}\right\} ,%
\end{array}
& 1-\alpha \leq \alpha \lambda \leq 1-\lambda \left( 1-\alpha \right)%
\end{array}%
\right. ,\end{aligned}$$where $\gamma _{1},\ \gamma _{2},\ \upsilon _{1},\ \ \upsilon _{2},$ $\mu
_{1}^{\ast },\ \mu _{2}^{\ast },\ \mu _{3}^{\ast },\ \mu _{4}^{\ast },\ \eta
_{1}^{\ast },\ \eta _{2}^{\ast },\ \eta _{3}^{\ast },\ \eta _{4}^{\ast }$ numbers are defined as in Corollary \[2.2a\].
The assertion follows from applied the inequality (\[2-8\]) to the function $f(t)=t^{s+1}$,$\ t\in \left[ a,b\right] $ and $s\in \left( 0,\frac{%
1}{q}\right) ,$ which implies that $f^{\prime }(t)=(s+1)t^{s}$,$\ \ t\in %
\left[ a,b\right] $ and $\left\vert f^{\prime }(t)\right\vert
^{q}=(s+1)^{q}t^{qs}$,$\ \ t\in \left[ a,b\right] $ is a $s-$convex function in the second sense since $qs\in \left( 0,1\right) $ and $(s+1)^{q}>0.$
Let $a,b\in
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$ with $0<a<b,\ p,q>1,\ \frac{1}{p}+\frac{1}{q}=1$ and $s\in \left( 0,%
\frac{1}{q}\right) $we have the following inequality:$$\left\vert \lambda A_{\alpha }\left( a^{s+1},b^{s+1}\right) +\left(
1-\lambda \right) A_{\alpha }^{s+1}\left( a,b\right) -L_{s+1}^{s+1}\left(
a,b\right) \right\vert \leq \left( b-a\right) \left( \frac{1}{p+1}\right) ^{%
\frac{1}{p}}\left( s+1\right) ^{1-\frac{1}{q}}$$$$\times \left\{
\begin{array}{cc}
\left[ \left( 1-\alpha \right) ^{\frac{1}{q}}\varepsilon _{1}^{\frac{1}{p}%
}\theta _{1}+\alpha ^{\frac{1}{q}}\varepsilon _{3}^{\frac{1}{p}}\theta _{2}%
\right] , & \alpha \lambda \leq 1-\alpha \leq 1-\lambda \left( 1-\alpha
\right) \\
\left[ \left( 1-\alpha \right) ^{\frac{1}{q}}\varepsilon _{1}^{\frac{1}{p}%
}\theta _{1}+\alpha ^{\frac{1}{q}}\varepsilon _{4}^{\frac{1}{p}}\theta _{2}%
\right] , & \alpha \lambda \leq 1-\lambda \left( 1-\alpha \right) \leq
1-\alpha \\
\left[ \left( 1-\alpha \right) ^{\frac{1}{q}}\varepsilon _{2}^{\frac{1}{p}%
}\theta _{1}+\alpha ^{\frac{1}{q}}\varepsilon _{3}^{\frac{1}{p}}\theta _{2}%
\right] , & 1-\alpha \leq \alpha \lambda \leq 1-\lambda \left( 1-\alpha
\right)%
\end{array}%
\right. ,$$where $$\theta _{1}=A_{\alpha }^{sq}\left( a,b\right) +a^{sq},\ \theta
_{2}=A_{\alpha }^{sq}\left( a,b\right) +b^{sq},$$and $\varepsilon _{1},\ \varepsilon _{2},\ \varepsilon _{3},\ \varepsilon
_{4}$ numbers are defined as in Corollary \[2.4\].
The assertion follows from applied the inequality (\[2-12b\]) to the function $f(t)=t^{s+1}$,$\ t\in \left[ a,b\right] $ and $s\in \left( 0,\frac{%
1}{q}\right) ,$ which implies that $f^{\prime }(t)=(s+1)t^{s}$,$\ \ t\in %
\left[ a,b\right] $ and $\left\vert f^{\prime }(t)\right\vert
^{q}=(s+1)^{q}t^{qs}$,$\ \ t\in \left[ a,b\right] $ is a $s-$convex function in the second sense since $qs\in \left( 0,1\right) $ and $(s+1)^{q}>0.$
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
We study a weighted online bipartite matching problem: $G(V_1, V_2, E)$ is a weighted bipartite graph where $V_1$ is known beforehand and the vertices of $V_2$ arrive online. The goal is to match vertices of $V_2$ as they arrive to vertices in $V_1$, so as to maximize the sum of weights of edges in the matching. If assignments to $V_1$ cannot be changed, no bounded competitive ratio is achievable. We study the weighted online matching problem with [*free disposal*]{}, where vertices in $V_1$ can be assigned multiple times, but only get credit for the maximum weight edge assigned to them over the course of the algorithm. For this problem, the greedy algorithm is $0.5$-competitive and determining whether a better competitive ratio is achievable is a well known open problem.
We identify an interesting special case where the edge weights are decomposable as the product of two factors, one corresponding to each end point of the edge. This is analogous to the well studied related machines model in the scheduling literature, although the objective functions are different. For this case of decomposable edge weights, we design a [0.5664]{} competitive randomized algorithm in complete bipartite graphs. We show that such instances with decomposable weights are non-trivial by establishing upper bounds of 0.618 for deterministic and $0.8$ for randomized algorithms.
A tight competitive ratio of $1-1/e \approx 0.632$ was known previously for both the 0-1 case as well as the case where edge weights depend on the offline vertices only, but for these cases, reassignments cannot change the quality of the solution. Beating 0.5 for weighted matching where reassignments are necessary has been a significant challenge. We thus give the first online algorithm with competitive ratio strictly better than 0.5 for a non-trivial case of weighted matching with free disposal.
author:
- 'Moses Charikar[^1]'
- 'Monika Henzinger[^2]'
- 'Huy L. Nguyên[^3]'
bibliography:
- '../matching.bib'
title: Online Bipartite Matching with Decomposable Weights
---
### Acknowledgments.
MC was supported by NSF awards CCF 0832797, AF 0916218 and a Google research award. MH’s support: The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement no. 340506 and the Austrian Science Fund (FWF) grant P23499-N23. HN was supported by NSF awards CCF 0832797, AF 0916218, a Google research award, and a Gordon Wu fellowship.
[^1]: Princeton University, USA, `[email protected]`
[^2]: University of Vienna, Austria, `[email protected]`
[^3]: Princeton University, USA, `[email protected]`
|
{
"pile_set_name": "ArXiv"
}
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---
abstract: 'We define new higher-order Alexander modules $\mathcal{A}_n(C)$ and higher-order degrees $\delta_n(C)$ which are invariants of the algebraic planar curve $C$. These come from analyzing the module structure of the homology of certain solvable covers of the complement of the curve $C$. These invariants are in the spirit of those developed by T. Cochran in [@C] and S. Harvey in [@H] and [@Har], which were used to study knots, 3-manifolds, and finitely presented groups, respectively. We show that for curves in general position at infinity, the higher-order degrees are finite. This provides new obstructions on the type of groups that can arise as fundamental groups of complements to affine curves in general position at infinity.'
address:
- 'C. Leidy: Department of Mathematics, University of Pennsylvania, 209 S 33rd St., Philadelphia, PA, 19104-6395, USA.'
- 'L. Maxim : Department of Mathematics, University of Illinois at Chicago, 851 S Morgan Street, Chicago, IL 60607, USA.'
author:
- Constance Leidy
- Laurentiu Maxim
title: 'Higher-order Alexander invariants of plane algebraic curves'
---
Introduction
============
The study of plane singular curves is a subject going back to the work of Zariski, who observed that the position of singularities has an influence on the topology of the curve, and that this phenomena can be detected by the fundamental group of the complement. However, the fundamental group of a plane curve complement is in general highly non-commutative, and thus difficult to handle. It is therefore natural to look for invariants of the fundamental group that capture information about the topology of the curve, such as Alexander-type invariants associated to various covering spaces of the curve complement. By analogy with the classical theory of knots and links in a $3$-sphere, Libgober developed invariants of the total linking number infinite cyclic cover in [@Li0; @Li2; @Li22] and those of the universal abelian cover in [@Li3; @Li4]. In this paper, we consider certain solvable covers of the curve complement, and their associated Alexander invariants.
Using techniques developed by T. Cochran, K. Orr, and P. Teichner in [@COT], S. Harvey (in [@H]) and T. Cochran (in [@C]) defined higher-order Alexander modules and higher-order degrees associated to 3-manifolds and knots, respectively. Harvey (in [@Har]) has constructed similar invariants associated to finitely presented groups, in general. Our new invariants for planar curves are constructed in the same way as these.
Let $C$ be a reduced curve in ${\mathbb{C}}^2$, and consider ${{\mathcal U}}$, the complement of $C$ in ${\mathbb{C}}^2$, with $G=\pi_1({{\mathcal U}})$. The multivariable Alexander invariant, studied in [@Li3], [@Li4] (but see also [@DM]), is defined by considering the universal abelian covering space of ${{\mathcal U}}$ corresponding to the map $G \to G/[G,G] \cong {\mathbb{Z}}^s$, where $s$ is the number of irreducible components of the curve. We continue this construction by taking iterated universal torsion-free abelian covers of ${{\mathcal U}}$ corresponding to the map $G \to G/G^{(n+1)}_r
\equiv \Gamma_n$, where $G^{(n)}_r$ is the $n^{th}$-term in the rational derived series of $G$, (defined in §2 below). We define the higher-order Alexander modules of the plane curve complement to be $\mathcal{A}^{{\mathbb{Z}}}_n(C)=H_1({{\mathcal U}};{\mathbb{Z}}\Gamma_n)$. The following is a corollary to Theorem \[alg-geo-bound\]:
If $C$ is a reduced curve in ${\mathbb{C}}^2$, that is in general position at infinity, $\mathcal{A}^{{\mathbb{Z}}}_n(C)$ is a torsion ${\mathbb{Z}}\Gamma_n$-module.
Furthermore, we consider some skew Laurent polynomial rings ${\mathbb{K}}_n[t^{\pm1}]$, which are obtained from ${\mathbb{Z}}\Gamma_n$ by inverting the non-zero elements of a particular subring. The advantage of using ${\mathbb{K}}_n[t^{\pm1}]$ coefficients instead of ${\mathbb{Z}}\Gamma_n$ coefficients is that the former is a principal ideal domain. We define the higher-order degree of $C$ to be $\delta_n(C)=\text{rk}_{{\mathbb{K}}_n} H_1({{\mathcal U}};{\mathbb{K}}_n[t^{\pm 1}])$.
Even though, in principal, the higher-order degrees may be computed by means of Fox free calculus (cf. [@H], $\S 6$), the calculations are tedious as they depend on a presentation of the fundamental group of the curve complement. However, in the case that the curve is in general position at infinity, we find a bound on the higher-order degrees. In particular, we have the following theorem:
Suppose $C$ is a degree $d$ curve in ${\mathbb{C}}^2$, such that its projective completion $\bar{C}$ is transverse to the line at infinity. If $C$ has singularities $c_k$, $1 \leq k \leq l$, then $$\label{bound} \delta_n(C) \leq \Sigma_{k=1}^l \left(\mu(C,c_k) + 2 n_k\right)
+ 2g + d - l,$$ where $\mu(C,c_k)$ is the Milnor number associated to the singularity germ at $c_k$, $n_k$ is the number of branches through the singularity $c_k$, and $g$ is the genus of the normalized curve.
We also can find a bound on the higher-order degrees of the curve in terms of “local” degrees, $\bar{\delta_n^k}$, for each singularity $c_k$ of $C$. The latter were defined and studied by Harvey ([@H]).
If $C$ satisfies the assumptions of the previous theorem, then $$\delta_n(C) \leq \Sigma_{k=1}^l (\bar{\delta}_n^k + 2 n_k)+ 2g + d,$$ where $\bar{\delta}_n^k=\bar{\delta}_n(X_k)$ is Harvey’s invariant of the link complement $X_k$ associated to the singularity $c_k$.
We view Theorem \[top-bound\] as an analogue of the divisibility properties for the infinite cyclic Alexander polynomial of the complement as shown in [@Li2].
For irreducible curves, regardless of the position of the line at infinity, the higher-order degrees are finite and thus the higher-order Alexander modules are torsion. However, if the line at infinity is not transverse to the irreducible curve $C$, then the upper bounds mentioned above will also include the contribution of the singular points at infinity (similar to [@Li], Thm. 4.3).
To complete the analogy with the case of Alexander polynomials of the infinite cyclic cover of the complement, we also provide an upper bound on $\delta_n(C)$ by the corresponding higher-order Alexander invariant of the link at infinity (see Theorem \[infinity\]). For a curve of degree $d$, in general position at infinity, this is an uniform bound equal to $d(d-2)$.
The authors would like to thank Tim Cochran, Shelly Harvey, Anatoly Libgober and Julius Shaneson for many helpful conversations about this project.
Rational derived series of a group; PTFA groups
===============================================
In this section, we review the definitions and basic constructions that we will need from [@H] and [@C]. More details can be found in these sources.
We begin by recalling the definition of the rational derived series $\{G_r^{(i)}\}$ associated to any group $G$.
Let $G_r ^{(0)}=G$. For $n \geq 1$, define the $n^{th}$ term of the rational derived series of $G$ by: $$G_r ^{(n)}=\{g \in G_r ^{(n-1)} | g^k \in
[G_r ^{(n-1)},G_r ^{(n-1)}], \ \text{for some} \ k \in {\mathbb{Z}}-\{0\}
\}.$$
We denote by $\Gamma_n$ the quotient $G/G_r^{(n+1)}$. Since $G_r^{(n)}$ is a normal subgroup of $G_r^{(i)}$ for $0 \leq i \leq
n$ ([@H], Lemma 3.2), it follows that $\Gamma_n$ is a group.
The use of rational derived series, as opposed to the usual derived series $\{G^{(n)}\}$, is necessary to avoid zero divisors in the group ring ${\mathbb{Z}}\Gamma_n$. However, if $G$ is a knot group or a free group, the rational derived series and the derived series coincide ([@H], p. 902). If $G$ is a finite group then $G_r^{(n)}=G$, hence $\Gamma_n=\{1\}$ for all $n \geq 0$.
The rational derived series is defined in such a way that the successive quotients $G_r^{(n)}/G_r^{(n+1)}$ are ${\mathbb{Z}}$-torsion-free and abelian. In fact ([@H], Lemma 3.5): $$\label{quot}
G_r^{(n)}/G_r^{(n+1)} \cong \left(G_r^{(n)}/[G_r^{(n)}, G_r^{(n)}]
\right)/\{{\mathbb{Z}}-\text{torsion}\}.$$ If $G=\pi_1(X)$ this says that $G_r^{(n)}/G_r^{(n+1)} \cong
H_1(X_{\Gamma_{n-1}})/\{{\mathbb{Z}}-\text{torsion}\}$, where $X_{\Gamma_{n-1}}$ is the regular $\Gamma_{n-1}$-cover of $X$. In particular, $G/G_r^{(1)}=G_r^{(0)}/G_r^{(1)} \cong
H_1(X)/\{{\mathbb{Z}}-\text{torsion}\} \cong {\mathbb{Z}}^{\beta_1(X)}$.
\[rm1\] If $G$ is the fundamental group of a link complement in $S^3$ or that of a plane curve complement, then $G_r^{(1)}=G^{(1)}$ (since there is no torsion in the first homology of the complement).
A group $\Gamma$ is poly-torsion-free-abelian (PTFA) if it admits a normal series of subgroups such that each of the successive quotients of the series is torsion-free abelian.
\[rm3\]We collect the following facts about PTFA groups:
\(1) Any subgroup of a PTFA group is a PTFA group.
\(2) If $\Gamma$ is PTFA, then ${\mathbb{Z}}\Gamma$ is a right (and left) Ore domain (i.e., has no zero divisors and ${\mathbb{Z}}\Gamma - \{0\}$ is a right divisor set). Thus it embeds in its classical right ring of quotients $\mathcal{K}$, a skew field ([@C], Prop. 3.2).
\(3) If $R$ is an Ore domain and $S$ is a right divisor set, then $RS^{-1}$ is flat as a left $R$-module. In particular, $\mathcal{K}$ is a flat ${\mathbb{Z}}\Gamma$-module ([@Ste], Prop. II.3.5).
\(4) Every module over $\mathcal{K}$ is a free module ([@Ste], Prop. I.2.3). Such modules have a well-defined rank $\text{rk}_{\mathcal{K}}$ which is additive on short exact sequences.
If $A$ is a module over an Ore domain $R$, then the rank of $A$ is defined as: $rk(A)=\text{rk}_{\mathcal{K}}(A \otimes_R
\mathcal{K})$. In particular, $A$ is a torsion $R$-module if and only if $A \otimes_R \mathcal{K}=0$, where $\mathcal{K}$ is the quotient field of $R$.
([@H], Cor. 3.6) For any group $G$, $\Gamma_n=G/G_r^{(n+1)}$ is PTFA. Thus it embeds in its classical right ring of quotients, $\mathcal{K}_n$.
Suppose $X$ is a topological space that has the homotopy type of a connected CW-complex. Let $\Gamma$ be any group and $\phi:\pi_1(X)
\to \Gamma$ be a homomorphism. We denote by $X_{\Gamma}$ the regular $\Gamma$-cover of $X$ associated to $\phi$. If $\phi$ is surjective, this is the covering space associated to $\ker{\phi}$. (For details about the case where $\phi$ is not surjective, we refer the reader to §3 of [@C].) Let $C(X_{\Gamma};{\mathbb{Z}})$ be the ${\mathbb{Z}}\Gamma$-free cellular chain complex for $X_{\Gamma}$ obtained by lifting the cell structure of $X$. If $\mathcal{M}$ is a ${\mathbb{Z}}\Gamma$-bimodule, then define: $$H_{\ast}(X;\mathcal{M})=H_{\ast}(C(X_{\Gamma};{\mathbb{Z}}) \otimes_{{\mathbb{Z}}\Gamma} \mathcal{M})$$ as a right ${\mathbb{Z}}\Gamma$-module.
([@COT], Prop. 2.9)\[zero\] Let $X$ be a connected CW-complex and $\Gamma$ a PTFA group. If $\phi : \pi_1(X) \to \Gamma$ is a non-trivial coefficient system, then $H_0(X;{\mathbb{Z}}\Gamma)$ is a torsion ${\mathbb{Z}}\Gamma$-module.
([@C], Prop. 3.10) \[torsion\] Let $X$ be a connected CW-complex and $\Gamma$ be a PTFA group. Suppose $\pi_1(X)$ is finitely generated and $\phi : \pi_1(X) \to
\Gamma$ is a non-trivial coefficient system. Then $rk\left(
H_1(X;{\mathbb{Z}}\Gamma)\right) \leq \beta_1(X) -1$. In particular, if $\beta_1(X)=1$ then $H_1(X;{\mathbb{Z}}\Gamma)$ is a torsion ${\mathbb{Z}}\Gamma$-module.
Definitions of new invariants
=============================
Let $C$ be a reduced curve in ${\mathbb{C}}^2$, defined by the equation: $f=f_1\cdots f_s =0$, where $f_i$ are the irreducible factors of $f$, and let $C_i=\{f_i=0\}$ denote the irreducible components of $C$. Embed ${\mathbb{C}}^2$ in ${\mathbb{C}}{\mathbb{P}}^2$ by adding the plane at infinity, $H$, and let $\bar{C}$ be the projective curve in ${\mathbb{C}}{\mathbb{P}}^2$ defined by the homogenization $f^h$ of $f$. We let $\bar{C}_i=\{f_i^h=0\}$, $i=1,\cdots,s$, be the corresponding irreducible components of $\bar{C}$. Let $\mathcal{U}$ be the complement $\mathbb{CP}^{2} - (\bar{C} \cup
H)$. (Alternatively, ${{\mathcal U}}$ may be regarded as the complement of the curve $C$ in the affine space ${\mathbb{C}}^{2}$.) Then $H_1 (\mathcal{U})
\cong \mathbb{Z}^s$ ([@Di1], (4.1.3), (4.1.4)), generated by the meridian loops $\gamma_i$ about the non-singular part of each irreducible component $\bar{C}_i$, for $i=1,\cdots, s$. If $\gamma_\infty$ denotes the meridian about the line at infinity, then in $H_1(\mathcal{U})$ there is a relation: $\gamma_\infty +
\sum {d_i \gamma_i} = 0$, where $d_i=deg(V_i)$.
Higher-order Alexander modules
------------------------------
We let $G=\pi_1({{\mathcal U}})$, $\Gamma_n=G/G_r^{(n+1)}$, and $\mathcal{K}_n$ be the classical right ring of quotients of ${\mathbb{Z}}\Gamma_n$.
We define the higher-order Alexander modules of the plane curve to be: $$\mathcal{A}^{{\mathbb{Z}}}_n(C)=H_1({{\mathcal U}};{\mathbb{Z}}\Gamma_n)=H_1({{\mathcal U}}_{\Gamma_n};{\mathbb{Z}})$$ where ${{\mathcal U}}_{\Gamma_n}$ is the covering of ${{\mathcal U}}$ corresponding to the subgroup $G_r^{(n+1)}$. That is, $\mathcal{A}^{{\mathbb{Z}}}_n(C)=G_r^{(n+1)}/[G_r^{(n+1)},G_r^{(n+1)}]$ as a right ${\mathbb{Z}}\Gamma_n$-module.
The $n^{th}$ order rank of (the complement of) $C$ is: $$r_n(C)=\text{rk}_{\mathcal{K}_n}H_1({{\mathcal U}};\mathcal{K}_n)$$
\[rm2\] (1) Note that $\mathcal{A}^{{\mathbb{Z}}}_0(C)=G_r^{(1)}/[G_r^{(1)},G_r^{(1)}]=G'/G''$, by Remark \[rm1\]. This is just the universal abelian invariant of the complement.
\(2) $\mathcal{A}^{{\mathbb{Z}}}_n(C) / \{{\mathbb{Z}}-\text{torsion}\} = G^{(n+1)}_r /
G^{(n+2)}_r$.
\(3) If $C$ is irreducible, then $\beta_1({{\mathcal U}})=1$. By Proposition \[torsion\], it follows that $\mathcal{A}^{{\mathbb{Z}}}_n(C)$ is a torsion module.
In Corollary \[finite\], we show that under the assumption of transversality at infinity, the module $\mathcal{A}^{{\mathbb{Z}}}_n(C)$ is a torsion ${\mathbb{Z}}\Gamma_n$-module. Therefore, $r_n(C)=0$.
Since $\mathcal{U}$ is a $2$-dimensional affine variety, it has the homotopy type of a $2$-dimensional CW-complex. Thus the modules $H_k({{\mathcal U}};{\mathbb{Z}}\Gamma_n)$ are trivial for $k
> 2$ and $H_{2}({{\mathcal U}};{\mathbb{Z}}\Gamma_n)$ is a torsion-free ${\mathbb{Z}}\Gamma_n$-module. Moreover, we will show that in our setting, the rank of $H_{2}({{\mathcal U}};{\mathbb{Z}}\Gamma_n)$ is equal to the Euler characteristic of the complement, ${{\mathcal U}}$.
\[trivial\]Assume that the universal abelian Alexander module of the complement is *trivial*, i.e. $\mathcal{A}^{{\mathbb{Z}}}_0(C)=0$. (Note that this is the case if $G$ is abelian or finite.) Then all higher-order Alexander modules $\mathcal{A}^{{\mathbb{Z}}}_n(C)=0$, for $n
\geq 1$, are also trivial. Indeed, by Remark \[rm1\], $G'=G_r^{(1)}$ and $\mathcal{A}^{{\mathbb{Z}}}_0(C)=G'/G''$. It follows that $G^{(n)}=G'=G_r^{(1)}$, for all $n \geq 1$. From the definition of the rational derived series, it is now easy to see that $G_r^{(n)}=G'$ for all $n \geq1$. Therefore $\mathcal{A}^{{\mathbb{Z}}}_n(C)
\cong G_r^{(n+1)}/[G_r^{(n+1)},G_r^{(n+1)}] \cong G'/G'' =0$, for all $n \geq 0$.
\[ex\]
\(1) If $C$ is a non-singular curve in general position at infinity, then $\pi_1({{\mathcal U}}) \cong {\mathbb{Z}}$ ([@Li]), hence abelian. By the above remark, it follows that $\mathcal{A}^{{\mathbb{Z}}}_n(C) = 0$, for all $n \geq
0$.(2) Suppose ${{\mathcal U}}$ is the complement in ${\mathbb{C}}^2$ of a union of two lines. Then $\pi_1({{\mathcal U}})$ is ${\mathbb{Z}}^2$. Hence $\mathcal{A}^{{\mathbb{Z}}}_n(C) = 0$ for all $n \geq 0$.(3) If $\bar{C}$ is a reduced curve having only nodes as its singularities (i.e., locally at each singular point, $\bar{C}$ looks like $x^2-y^2=0$), then it is known that $\pi_1({\mathbb{C}}{\mathbb{P}}^2-\bar{C})$ is abelian (e.g., see [@O]), thus has trivial commutator subgroup. Under the assumption that the line at infinity is generic, this implies that the commutator subgroup of $\pi_1({{\mathcal U}})$ is also trivial ([@O], Lemma 2), so $\pi_1({{\mathcal U}})$ is abelian. Now from Remark \[trivial\] it follows that $\mathcal{A}^{{\mathbb{Z}}}_n(C) = 0$, for all $n \geq 0$.
Localized higher-order Alexander modules
----------------------------------------
In this section we define some skew Laurent polynomial rings ${\mathbb{K}}_n[t^{\pm1}]$, which are obtained from ${\mathbb{Z}}\Gamma_n$ by inverting the non-zero elements of a particular subring described below. This construction is used in [@H] and [@C] and is described in algebraic generality in [@Har]. We refer to those sources for the background definitions.
Recall our notations: $G =\pi_1({{\mathcal U}})$, $\Gamma_n=G/G_r^{(n+1)}$ and $\mathcal{K}_n$ is the classical right ring of quotients of ${\mathbb{Z}}\Gamma_n$. Let $\psi \in H^1(G;{\mathbb{Z}}) \cong \text{Hom}_{{\mathbb{Z}}}(G,{\mathbb{Z}})$ be the primitive class representing the linking number homomorphism $G \overset{\psi}{\to} {\mathbb{Z}}$, $\alpha \mapsto \text{lk}(\alpha,C)$. Since the commutator subgroup of $G$ is in the kernel of $\psi$, it follows that $\psi$ induces a well-defined epimorphism $\bar{\psi} :
\Gamma_n \to {\mathbb{Z}}$. Let $\bar{\Gamma}_n$ be the kernel of $\bar{\psi}$. Since $\bar{\Gamma}_n$ is a subgroup of $\Gamma_n$, by Remark \[rm3\], $\bar{\Gamma}_n$ is a PTFA group. Thus ${\mathbb{Z}}\bar{\Gamma}_n$ is an Ore domain and $S_n={\mathbb{Z}}\bar{\Gamma}_n -
\{0\}$ is a right divisor set of ${\mathbb{Z}}\bar{\Gamma}_n$. Let ${\mathbb{K}}_n=({\mathbb{Z}}\bar{\Gamma}_n)S_n ^{-1}$ be the right ring of quotients of ${\mathbb{Z}}\bar{\Gamma}_n$, and set $R_n=({\mathbb{Z}}\Gamma_n)S_n^{-1}$.
If we *choose* a $t \in \Gamma_n$ such that $\bar{\psi}(t)=1$, this yields a splitting $\phi : {\mathbb{Z}}\to \Gamma_n$ of $\bar{\psi}$. As in Prop. 4.5 of [@H], the embedding ${\mathbb{Z}}\bar{\Gamma}_n
\hookrightarrow {\mathbb{K}}_n$ extends to an isomorphism $R_n
\overset{\cong}{\to} {\mathbb{K}}_n[t^{\pm 1}]$. (However this isomorphism depends on the choice of splitting!) It follows that $R_n$ is a non-commutative principal left and right ideal domain, since this is known to be true for any skew Laurent polynomial rings with coefficients in a skew field ([@C], Prop. 4.5). Also note that by Remark \[rm3\], $R_n$ is a flat left ${\mathbb{Z}}\Gamma_n$-module.
The $n^{th}$-order localized Alexander module of the curve $C$ is defined to be $\mathcal{A}_n(C)=H_1({{\mathcal U}};R_n)$, viewed as a right $R_n$-module. If we choose a splitting $\phi$ to identify $R_n$ with ${\mathbb{K}}_n[t^{\pm 1}]$, we define $\mathcal{A}^{\phi}_n=H_1({{\mathcal U}};{\mathbb{K}}_n[t^{\pm 1}])$.
The $n^{th}$-order degree of $C$ is defined to be: $$\delta_n(C)=\text{rk}_{{\mathbb{K}}_n} \mathcal{A}_n(C)$$
For any choice of $\phi$, $\text{rk}_{{\mathbb{K}}_n} \mathcal{A}_n(C) = \text{rk}_{{\mathbb{K}}_n}
\mathcal{A}^{\phi}_n(C)$. So although the module $\mathcal{A}^{\phi}_n(C)$ depends on the splitting, the rank of the module does not.
The degrees $\delta_n(C)$ are integral invariants of the fundamental group $G$ of the complement. Indeed, we have ([@Har], §1): $$\label{inv}
\delta_n(C)=rk_{{\mathbb{K}}_n} \left( {G^{(n+1)}_r}/[G^{(n+1)}_r,G^{(n+1)}_r]
\otimes_{{\mathbb{Z}}\bar{\Gamma}_n} {\mathbb{K}}_n \right)$$
Furthermore, for any choice of splitting $\phi$, since ${\mathbb{K}}_n[t^{\pm
1}]$ is a principal ideal domain, there exist some nonzero $p_i(t)
\in {\mathbb{K}}_n[t^{\pm 1}]$, $i=1,\cdots,m$, such that: $$\mathcal{A}^{\phi}_n(C) \cong \left( \oplus_{i=1}^m \frac{{\mathbb{K}}_n[t^{\pm 1}]}{p_i(t){\mathbb{K}}_n[t^{\pm 1}]} \right) \oplus {\mathbb{K}}_n[t^{\pm 1}]^{r_n(C)}$$ Therefore, $\delta_n(C)$ is finite if and only if one of the equivalent statements is true:
1. $r_n(C)=\text{rk}_{\mathcal{K}_n}H_1({{\mathcal U}};\mathcal{K}_n)=0$.
2. $\mathcal{A}_n(C)$ is a torsion $R_n$-module.
3. For any $\phi$, $\mathcal{A}^{\phi}_n(C)$ is a torsion ${\mathbb{K}}_n[t^{\pm 1}]$-module.
4. $\mathcal{A}^{{\mathbb{Z}}}_n(C)$ is a torsion ${\mathbb{Z}}\Gamma_n$-module.
If this is the case, then $\delta_n(C)$ is the sum of the degrees of polynomials $p_i(t)$.
The invariant $\delta_n(C)$ is difficult to calculate, in general. However, the special case of weighted homogeneous affine curves is well understood:
\[wh\] Suppose $C$ is defined by a weighted homogeneous polynomial $f(x,y)=0$ in ${\mathbb{C}}^2$, and assume that either $n >0$ or $\beta_1({{\mathcal U}})>1$. Then we have: $$\label{whp}
\delta_n(C)=\mu(C,0)-1,$$ where $\mu(C,0)$ is the Milnor number associated to the singularity germ at the origin. If $\beta_1({{\mathcal U}})=1$, then $\delta_0(C)=\mu(C,0)$.
The key observation here is the existence of a global Milnor fibration ([@Di1], (3.1.12)): $$F=\{f=1\} \hookrightarrow
{{\mathcal U}}={\mathbb{C}}^2-C \overset{f}{\to} {\mathbb{C}}^{\ast},$$ and the fact that $F$ is homotopy equivalent to the infinite cyclic cover of ${{\mathcal U}}$ corresponding to the kernel of the total linking number homomorphism, $\psi$. The $\Gamma_n$-cover of ${{\mathcal U}}$ factors through the infinite cyclic cover corresponding to $\psi$, which is homotopy equivalent to $F$. It follows that there is an isomorphism of ${\mathbb{K}}_n$-modules: $$H_{\ast}({{\mathcal U}};R_n) \cong H_{\ast}(F;{\mathbb{K}}_n).$$ In particular, $$\delta_n(C)=\text{rk}_{{\mathbb{K}}_n}H_{1}(F;{\mathbb{K}}_n).$$ Since $F$ has the homotopy type of a $1$-dimensional CW complex, $H_{2}(F;{\mathbb{K}}_n)=0$. Moreover, if either $n>0$ or $\beta_1({{\mathcal U}})>1$, the coefficient system $\pi_1(F) \to \bar{\Gamma}_n$ is non-trivial. Hence, by Proposition \[zero\], $H_0(F;{\mathbb{K}}_n)=0$. It follows, in this case, that $$\delta_n(C)=-\chi(F)=\mu(C,0)-1.$$ On the other hand, if $\beta_1({{\mathcal U}})=1$, then $\text{rk}_{{\mathbb{K}}_0}H_0(F;{\mathbb{K}}_0)=\text{rk}_{{\mathbb{Q}}}H_0(F;{\mathbb{Q}})=1$. Hence, if $\beta_1({{\mathcal U}})=1$, then $\delta_0(C)=1-\chi(F)=\mu(C,0)$.
\[cusp\]Since $f(x)=x^3-y^2$ is a weighted homogeneous polynomial, if $C$ is the curve defined by $f=0$, it follows from Proposition \[wh\], that $\delta_0(C)=2$ and $\delta_n(C)=1$ for $n>0$.
\[fib\]Due to the existence of Milnor fibrations for hypersurface singularity germs, formula (\[whp\]) holds for the case of any algebraic link, by replacing ${{\mathcal U}}$ by the link complement and $\delta_n(C)$ by Harvey’s invariant of the algebraic link. For a more general discussion on fibered $3$-manifolds, see [@H], Prop. 8.4, 8.5.
As noted in [@H], $\S6$ and $\S 8$, the higher-order Alexander invariants $r_n(C)$ and $\delta_n(C)$ can be computed from a presentation of the fundamental group of the curve complement, by means of Fox free calculus.
Upper bounds on the higher order degree of a curve complement
=============================================================
In this section, we find upper bounds for $\delta_n(C)$. In Theorem \[alg-geo-bound\], we find an upper bound in terms of the Milnor number of each singularity. In Theorem \[top-bound\], we find an upper bound in terms of the Harvey’s invariants, $\bar{\delta}_n$, associated to each of the singular points of $C$. This result is analogous to the divisibility properties for the infinite cyclic Alexander polynomial of the complement (e.g., see [@Li1], [@Li], [@Li2], [@M]). As a corollary to these theorems, we have that, if $C$ is a curve in general position at infinity, then $\delta_n(C)$ is finite, and therefore $\mathcal{A}_n(C)$ is a torsion ${\mathbb{Z}}\Gamma_n$-module. We also give an upper bound for $\delta_n(C)$ in terms of the higher-order degrees of the link at infinity.
\[alg-geo-bound\] Suppose $C$ is a degree $d$ curve in ${\mathbb{C}}^2$, such that its projective completion $\bar{C}$ is transverse to the line at infinity, $H$. If $C$ has singularities $c_k$, $1 \leq k \leq l$, then $$\label{bound} \delta_n(C) \leq \Sigma_{k=1}^l \left(\mu(C,c_k) + 2 n_k\right)
+ 2g + d - l,$$ where $\mu(C,c_k)$ is the Milnor number associated to the singularity germ at $c_k$, $n_k$ is the number of branches through the singularity $c_k$, and $g$ is the genus of the normalized curve.
Before proving Theorem \[alg-geo-bound\], we state an immediate corollary.
\[finite\] If $C$ is a plane curve in general position at infinity, then $\delta_n(C) <\infty$, i.e., $\mathcal{A}^{{\mathbb{Z}}}_n(C)$ is a torsion ${\mathbb{Z}}\Gamma_n$-module.
Note that the upper bound in (\[bound\]) is independent of $n$.
If $C$ is an irreducible curve, then independently of the position of the line at infinity, we have that $\beta_1({{\mathcal U}})=1$. By Proposition \[torsion\], it follows that $\mathcal{A}^{{\mathbb{Z}}}_n(C)$ is a torsion module. However, if the curve $C$ is not in general position at infinity, then the upper bound on $\delta_n(C)$ also includes the contribution of the ‘singularities at infinity’ (similar to Thm 4.3 of [@Li]).
We first reduce the problem to the study of the boundary, $X$, of a regular neighborhood of $C$ in ${\mathbb{C}}^2$. In order to do this, let $N(\bar{C})$ be a regular neighborhood of $\bar{C}$ inside ${\mathbb{C}}{\mathbb{P}}^2$ and note that, due to the transversality assumption, the complement $N(\bar{C})-(\bar{C} \cup H)$ can be identified with $N(C)-C$, where $N(C)$ is a regular neighborhood of $C$ in ${\mathbb{C}}^2$. But $N(C)-C$ deformation retracts to $X=\partial N(C)$. Now by the Lefschetz hyperplane section theorem ([@Di1], p. 25), it follows that the inclusion map induces a group epimorphism $$\pi_1(X) \twoheadrightarrow \pi_1(U)$$ (the argument used here is similar to the one used in the proof of Thm. 4.3 of [@Li]). It follows that $\pi_1(X_{\Gamma_n}) \twoheadrightarrow
\pi_1(U_{\Gamma_n})$. Hence, $H_1(X;{\mathbb{Z}}\Gamma_n) \twoheadrightarrow
H_1({{\mathcal U}};{\mathbb{Z}}\Gamma_n)$. Since $R_n$ is a flat ${\mathbb{Z}}\Gamma_n$-module, there is an $R_n$-module epimorphism $H_1(X;R_n) \twoheadrightarrow H_1({{\mathcal U}};R_n)$. Therefore, we have: $$\delta_n(C) = \text{rk}_{{\mathbb{K}}_n} H_1({{\mathcal U}};R_n) \leq \text{rk}_{{\mathbb{K}}_n} H_1(X;R_n).$$ Hence, it suffices to show that $\text{rk}_{{\mathbb{K}}_n} H_1(X;R_n)$ is finite. This will follow by a Mayer-Vietoris sequence argument.
Let $F$ be the (abstract) surface obtained from $C$ by removing disks $D_1 \cup \cdots \cup D_{n_k}$ around each singular point $c_k$ of $C$. Let $N=F \times S^1$. The boundary of $N$ is a union of disjoint tori $T^k_1 \cup \cdots \cup T^k _{n_k}$ for $k=1,\cdots, l$, where $l$ is the number of singular points of $C$. For each singular point $c_k$ of $C$ we let $(S^3_k, L_k)$ be the link pair of $c_k$, and denote by $X_k$ the link exterior, $S^3_k-L_k$. Then $X$ is obtained from $N$ by gluing the link exteriors $X_k$ along the tori $T^k_i$ for $i=1,\cdots,n_k$: $$X=N \cup_{\sqcup_i{T^k_i}}(\sqcup_{k=1}^l X_k).$$ The gluing map sends each longitude of $L_k$ to the restriction of a section in $N$, and each meridian to a fiber of $N$.
We consider the Mayer-Vietoris sequence in homology associated to the above cover of $X$ and with coefficients in $R_n$: $$\begin{aligned} \cdots \to \oplus_{k,i} H_1(T^k_i;R_n) \overset{\Psi}{\to} H_1(N;R_n)
\oplus \left( \oplus_{k=1}^l H_1(X_k;R_n) \right) \to H_1(X;R_n) \\
\to \oplus_{k,i} H_0(T^k_i;R_n) \to H_0(N;R_n) \oplus \left(
\oplus_{k=1}^l H_0(X_k;R_n) \right) \to H_0(X;R_n) \to 0
\end{aligned}$$ From Remark \[rm3\], we have:$$\begin{aligned}
\label{rankeq}
\text{rk}_{{\mathbb{K}}_n} H_1(X;R_n) = \text{rk}_{{\mathbb{K}}_n} H_1(N;R_n) +
\Sigma_{k=1}^l \text{rk}_{{\mathbb{K}}_n} H_1(X_k;R_n) - \Sigma_{k,i}
\text{rk}_{{\mathbb{K}}_n} H_1(T^k_i;R_n) \notag \\ + \text{rk}_{{\mathbb{K}}_n}
\ker(\Psi)
+ \Sigma_{k,i} \text{rk}_{{\mathbb{K}}_n}
H_0(T^k_i;R_n) - \text{rk}_{{\mathbb{K}}_n} H_0(N;R_n) \\ - \Sigma_{k=1}^l
\text{rk}_{{\mathbb{K}}_n} H_0(X_k;R_n) + \text{rk}_{{\mathbb{K}}_n}
H_0(X;R_n).\notag\end{aligned}$$
Recall that, for each singular point $c_k$ of $C$, the coefficient system $R_n$ on $X_k$ is induced by the following composition of maps: $${\mathbb{Z}}\pi_1(X_k) \to {\mathbb{Z}}\pi_1(X) \to {\mathbb{Z}}\pi_1({{\mathcal U}}) \to {\mathbb{Z}}\Gamma_n \to R_n.$$ Since each $X_k$ fibers over $S^1$ with Milnor fiber $F_k$, the $\Gamma_n$-cover of $X_k$ factors through the infinite cyclic cover of $X_k$ which is homeomorphic to $F_k \times {\mathbb{R}}$. Therefore we have the following isomorphisms of ${\mathbb{K}}_n$-modules: $$H_*(X_k;R_n) \cong H_*(F_k;{\mathbb{K}}_n).$$ Since $F_k$ has the homotopy type of a wedge of circles, $H_2(F_k;{\mathbb{K}}_n)=0$. Therefore, $$\chi(F_k)=-\text{rk}_{{\mathbb{K}}_n} H_1(F_k;{\mathbb{K}}_n) + \text{rk}_{{\mathbb{K}}_n} H_0(F_k;{\mathbb{K}}_n)
= -\text{rk}_{{\mathbb{K}}_n} H_1(X_k;R_n) + \text{rk}_{{\mathbb{K}}_n} H_0(X_k;R_n).$$ Similar, since $N=F \times S^1$, the $\Gamma_n$-cover of $N$ factors through the infinite cyclic cover of $N$ which is homeomorphic to $F
\times {\mathbb{R}}$. So if $F_n$ denotes the corresponding $\Gamma_n$-cover of $F$, then $F_n$ is a non-compact surface and we have $H_2(F;{\mathbb{K}}_n)=0$. Therefore: $$\chi(F)=-\text{rk}_{{\mathbb{K}}_n} H_1(F;{\mathbb{K}}_n) + \text{rk}_{{\mathbb{K}}_n}
H_0(F;{\mathbb{K}}_n) = -\text{rk}_{{\mathbb{K}}_n} H_1(N;R_n) + \text{rk}_{{\mathbb{K}}_n}
H_0(N;R_n).$$ Finally, for each $k$ and $i$, we have: $$0=\chi(S^1)=-\text{rk}_{{\mathbb{K}}_n} H_1(S^1;{\mathbb{K}}_n) + \text{rk}_{{\mathbb{K}}_n}
H_0(S^1;{\mathbb{K}}_n) = -\text{rk}_{{\mathbb{K}}_n} H_1(T^k_i;R_n) + \text{rk}_{{\mathbb{K}}_n}
H_0(T^k_i;R_n).$$
Now we can rewrite equation (\[rankeq\]) as follows: $$\label{rankeq2}\text{rk}_{{\mathbb{K}}_n} H_1(X;R_n) = -\Sigma_{k=1}^l \chi(F_k) -
\chi(F)
+ \text{rk}_{{\mathbb{K}}_n} \ker(\Psi) + \text{rk}_{{\mathbb{K}}_n}
H_0(X;R_n).$$
Since $\pi_1(X) \twoheadrightarrow \pi(U) \twoheadrightarrow
\Gamma_n$ is an epimorphism, it follows that the $\Gamma_n$-cover of $X$ is connected, thus yielding that $\text{rk}_{{\mathbb{K}}_n}H_0(X;R_n)=1$.
Since $\Psi : \oplus_{k,i} H_1(T^k_i;R_n) \to H_1(N;R_n)$, it follows that $\text{rk}_{{\mathbb{K}}_n} \ker(\Psi) \leq \Sigma_{k,i}
\text{rk}_{{\mathbb{K}}_n} H_1(T^k_i;R_n)$. For each $k$ and $i$, we have that: $$\text{rk}_{{\mathbb{K}}_n} H_1(T^k_i;R_n)=\text{rk}_{{\mathbb{K}}_n}
H_0(T^k_i;R_n)=\text{rk}_{{\mathbb{K}}_n} H_0(S^1;{\mathbb{K}}_n) \leq 1,$$ since $S^1$ is connected. Therefore, $\text{rk}_{{\mathbb{K}}_n} \ker(\Psi)$ is less than or equal to the number of tori, which is $\Sigma_{k=1}^l
n_k$ where $n_k$ is the number of branches through the singularity $c_k$. From equation (\[rankeq2\]) we have the following: $$\label{rank3}\text{rk}_{{\mathbb{K}}_n} H_1(X;R_n) \leq \Sigma_{k=1}^l (-\chi(F_k) +
n_k) - \chi(F) + 1.$$
Furthermore, $-\chi(F_k) = \mu(C,c_k)-1$ and $-\chi(F) \leq 2g +
\sum_k {n_k} + d - 1$, where $g$ is the genus of the normalized curve and $d$ is the degree of the curve, i.e. the number of ‘punctures at infinity’. It follows that: $$\delta_n(C) \leq \text{rk}_{{\mathbb{K}}_n} H_1(X;R_n) \leq \Sigma_{k=1}^l \left(\mu(C,c_k) + 2 n_k\right) +
2g + d - l.$$
\[top-bound\] Suppose $C$ is a degree $d$ curve in ${\mathbb{C}}^2$, such that its projective completion $\bar{C}$ is transverse to the line at infinity, $H$. If $C$ has singularities $c_k$, $1 \leq k \leq l$, then $$\delta_n(C) \leq \Sigma_{k=1}^l (\bar{\delta}_n^k + 2 n_k)+ 2g + d,$$ where $\bar{\delta}_n^k=\bar{\delta}_n(X_k)$ is Harvey’s invariant of the link complement $X_k$ associated to the singularity $c_k$, $n_k$ is the number of branches through the singularity $c_k$, and $g$ is the genus of the normalized curve.
We have equation (\[rank3\]) in the above proof: $$\text{rk}_{{\mathbb{K}}_n} H_1(X;R_n) \leq \Sigma_{k=1}^l (-\chi(F_k) + n_k)
- \chi(F) + 1.$$ Furthermore, $-\chi(F) \leq 2g + \sum_k {n_k} + d
- 1$. From Prop. 8.4 of [@H], we have $$\bar{\delta}_n^k=\bar{\delta}_n(X_k)=\begin{cases} -\chi(F_k) & n \neq
0 \text{ or } \beta_1(X_k) \neq 1 \\1-\chi(F_k) & n=0 \text{ and }
\beta_1(X_k)=1.\end{cases}$$ In particular, $-\chi(F_k) \leq
\bar{\delta}_n^k$, which proves the theorem.
We can also give a topological estimate for the rank of the torsion-free ${\mathbb{Z}}\Gamma_n$-module $H_2({{\mathcal U}};{\mathbb{Z}}\Gamma_n)$:
The rank of the torsion-free ${\mathbb{Z}}\Gamma_n$-module $H_2({{\mathcal U}};{\mathbb{Z}}\Gamma_n)$ is equal to the Euler characteristic $\chi({{\mathcal U}})$ of the curve complement.
Let $\mathcal{C}$ be the equivariant complex $$0 \to C_2 \to C_1 \to
C_1 \to 0$$ of free ${\mathbb{Z}}\Gamma_n$-modules, obtained by lifting the cell structure of ${{\mathcal U}}$ to ${{\mathcal U}}_{\Gamma_n}$, the $\Gamma_n$-covering of ${{\mathcal U}}$. Then $\chi(\mathcal{C})=\chi({{\mathcal U}})$. On the other hand, $$\chi(\mathcal{C})=\sum_{i=0}^2 (-1)^i
\text{rk}_{\mathcal{K}_n}H_i(\mathcal{C} \otimes_{{\mathbb{Z}}\Gamma_n}
\mathcal{K}_n) = \sum_{i=0}^2 (-1)^i rk H_i(\mathcal{C}).$$ Therefore, by Prop. \[zero\] and Cor. \[finite\], it follows that $\chi(\mathcal{C})= rk H_2({{\mathcal U}};{\mathbb{Z}}\Gamma_n)$ and the claim follows.
We will end this section by relating the higher-order degrees of the curve $C$ to the higher-order degrees of the link at infinity. Let $S^3_{\infty}$ be a sphere in ${\mathbb{C}}^2$ of a sufficiently large radius, or equivalently, the boundary of a small tubular neighborhood of the hyperplane $H$ at infinity. Let $C_{\infty}=S^3_{\infty} \cap C$ be the link of $C$ at infinity, and denote by $X_{\infty}$ its complement $S^3_{\infty} - C_{\infty}$. Let $G_{\infty}$ denote the fundamental group of $X_{\infty}$. We define $\delta_n^{\infty}$ to be the ${\mathbb{K}}_n$-rank of $H_1(X_{\infty}; R_n)$ , where the coefficient system is induced via the composition of maps: $${\mathbb{Z}}G_{\infty} \to
{\mathbb{Z}}G \to {\mathbb{Z}}\Gamma_n \to R_n.$$
We are now ready to prove the following theorem, similar in flavor to results on the infinite cyclic and universal abelian Alexander invariants (see [@Li1], [@Li], [@Li2], [@DM], [@M]):
\[infinity\] $$\label{inf}
\delta_n(C) \leq \delta_n^{\infty}.$$
We note that there is a group epimorphism $G_{\infty}
\twoheadrightarrow G$. Indeed, $X_{\infty}$ is homotopy equivalent to $N(H)-(\bar{C} \cup H)$, where $N(H)$ is a tubular neighborhood of $H$ in ${\mathbb{C}}{\mathbb{P}}^2$ whose boundary is $S^3_{\infty}$. If $L$ is a generic line in ${\mathbb{C}}{\mathbb{P}}^2$, which can be assumed to be contained in $N(H)$, then by the Lefschetz theorem, it follows that the composition $$\pi_1(L- L \cap(\bar{C} \cup H)) \to \pi_1(N(H)-(\bar{C} \cup H)) \to \pi_1
({\mathbb{C}}{\mathbb{P}}^2 - (\bar{C} \cup H))$$ is surjective, thus proving our claim (this is the same argument as the one used in [@Li], Thm. 4.5).
It follows that there is a ${\mathbb{Z}}\Gamma_n$-module epimorphism $$H_1(X_{\infty};{\mathbb{Z}}\Gamma_n) \twoheadrightarrow H_1({{\mathcal U}};{\mathbb{Z}}\Gamma_n).$$ Since $R_n$ is a flat ${\mathbb{Z}}\Gamma_n$-module, we also get a $R_n$-module epimorphism: $$H_1(X_{\infty};R_n) \twoheadrightarrow H_1({{\mathcal U}};R_n).$$ This proves the inequality (\[inf\]).
For a curve in general position at infinity, this yields a uniform upper bound on the higher-order degrees of the curve, which is independent of the local type of singularities and the number of singular points of the curve:
\[bestb\] If $C$ is a curve of degree $d$, in general position at infinity, then: $$\label{deg}
\delta_n(C) \leq d(d-2) \ , \ \ \text{for all n}.$$
The claim follows by noting that if $C$ is transversal to the line at infinity, then $C_{\infty}$ is the Hopf link on $d_1+\cdots
+d_s=d$ components (i.e., the union of fibers of the Hopf fibration), thus an algebraic link. By the argument used in the proof of Proposition \[wh\], it follows that $\delta_n^{\infty}=\mu_{\infty}-1$, where $\mu_{\infty}$ is the Milnor number associated to the link at infinity. On the other hand, $\mu_{\infty}$ is the degree of the Alexander polynomial of the link at infinity, so it is equal to $d(d-2)+1$ (cf. [@Li2]). The inequality (\[deg\]) follows now from Theorem \[infinity\].
Examples
========
In this section, we calculate the higher-order degrees for some of the classical examples of irreducible curves, including general cuspidal curves, Zariski’s sextics with $6$ cusps, Oka’s curves, and branched loci of generic projections.
We begin with the following:
\[fact\] Let $C \subset {\mathbb{C}}^2$ be an irreducible affine curve. Let $G=\pi_1({\mathbb{C}}^2-C)$, and denote by $\Delta_C(t)$ the Alexander polynomial of the curve complement. If $\Delta_C(t)=1$, then $\delta_n(C)=0$ for all $n$. Moreover, in this case, $\mathcal{A}^{{\mathbb{Z}}}_n(C) \cong \mathcal{A}^{{\mathbb{Z}}}_0(C)$ as ${\mathbb{Z}}[G/G']$-modules, for all $n$.
As $C$ is an irreducible affine curve, we have $G/G' \cong
{\mathbb{Z}}$. Hence $G' \cong G'_r$. The Alexander polynomial $\Delta_C(t)$ is the order of the infinite cyclic (and universal abelian) Alexander module of the complement, that is $G'/G'' \otimes {\mathbb{Q}}$, regarded as a ${\mathbb{Q}}[{\mathbb{Z}}]$-module under the action of the covering transformations group $G/G'$ (cf. [@Li0; @Li2; @Li22]). Since $C$ is irreducible, the infinite cyclic Alexander module is a torsion ${\mathbb{Q}}[t,t^{-1}]$-module, regardless of the position of the line at infinity (cf. [@Li0]). So its order $\Delta_C(t)$ is well-defined. Moreover, $\Delta_C(t)$ can be normalized so that $\Delta_C(1)=1$ ([@Li0]).
The triviality of the Alexander polynomial means that the universal abelian module $G'/G'' \otimes {\mathbb{Q}}$ is trivial, i.e. ${G'}/{G''}$ is a torsion abelian group. By $(\ref{quot})$, we obtain: $${G_r'}/{G''_r} \cong ({G_r'}/[G'_r,G'_r])/{\{{\mathbb{Z}}-\text{torsion}\}} \cong
({G'}/{G''})/{\{{\mathbb{Z}}-\text{torsion}\}} \cong 0.$$ Hence $G''_r \cong
G'_r=G'$. It follows by induction that $G_r^{(n)} = G'$, for all $n
>0$. Therefore, for any $n$, $$\mathcal{A}^{{\mathbb{Z}}}_n(C)={G_r^{(n+1)}}/[G_r^{(n+1)},G_r^{(n+1)}] \cong
G'/G''=\mathcal{A}^{{\mathbb{Z}}}_0(C).$$ Now recall that the higher-order degrees of $C$ may be defined by (\[inv\]): $$\delta_n(C)=\text{dim}_{{\mathbb{K}}_n} \left(
{G^{(n+1)}_r}/[G^{(n+1)}_r,G^{(n+1)}_r] \otimes_{{\mathbb{Z}}\bar{\Gamma}_n}
{\mathbb{K}}_n \right),$$ where $\bar \Gamma_n$ is the kernel of $\Gamma_n
\overset{\bar{\psi}}{\to} {\mathbb{Z}}=G/G'$. The map $\bar {\psi}$ is induced by the total linking number homomorphism, which in our setting is just the abelianization map $G \to G/G'={\mathbb{Z}}$. It follows that for all $n$ we have: $\Gamma_n=G/{G^{(n+1)}_r}=G/G'$, $\bar
\Gamma_n=G'/{G^{(n+1)}_r} \cong 0$, and ${\mathbb{K}}_n\cong {\mathbb{Q}}$. Therefore, for all $n$, $$\delta_n(C)=\text{dim}_{{\mathbb{Q}}} \left( {G'}/{G''} \otimes_{{\mathbb{Z}}} {\mathbb{Q}}\right)=0.$$
Note that if the commutator subgroup $G'$ is either perfect (i.e. $G'=G''$) or a torsion group, then the Alexander polynomial of the irreducible curve $C$ is trivial. In particular, this is the case if $G$ is abelian. The following examples deal with each of these cases.
Let $\bar C \subset {\mathbb{C}}{\mathbb{P}}^2$ be an irreducible curve of degree $d$ which has $a$ cusps (these are locally defined by the equation $x^2=y^3$) and $b$ nodes as the only singularities. If $d > 6a+2b$, then by a result of Nori (cf. [@No], but see also [@Li22]), it follows that $\pi_1({\mathbb{C}}{\mathbb{P}}^2- \bar C)$ is abelian. If we choose a generic line $H$ ‘at infinity’ and set $C=\bar C -H$, then as in \[ex\] it follows that $\pi_1({\mathbb{C}}^2-C)$ is also abelian. Hence all higher-order degrees of $C$ vanish.
Let $\bar C \subset {\mathbb{C}}{\mathbb{P}}^2$ be a degree $d$ irreducible cuspidal curve, i.e. it admits as singularities only nodes and cusps. Choose a generic line $H \subset {\mathbb{C}}{\mathbb{P}}^2$, and set $C:= \bar C -H$ and $G=\pi_1({\mathbb{C}}^2-C)$. If $d \not\equiv 0 \ (\text{mod} \ 6)$, then all higher-order degrees of $C$ vanish.
This follows from Proposition \[fact\] combined with Libgober’s divisibility results for the Alexander polynomial of a curve complement (see for instance [@Li2], Theorem 4.1), by noting that the Alexander polynomial of a cusp is $t^2-t+1$, that of a node is $t-1$, and using the fact that the Alexander polynomial of an irreducible curve $C$ can be normalized so that $\Delta_C(1)=1$; moreover, all zeros of $\Delta_C(t)$ are roots of unity of order $d$.
Here is a more concrete example:
\[3cusp\] [ Zariski’s three-cuspidal quartic. ]{}
Let $\bar C \subset {\mathbb{C}}{\mathbb{P}}^2$ be a quartic curve with three cusps as its only singularities. Choose as above a generic line, $H$, and set $C=\bar C -H$. Then the fundamental group of the affine complement is given by: $$G= \pi_1({\mathbb{C}}^2 - C)= \langle a, b \ | \ aba=bab, a^2=b^2 \rangle.$$ It is easy to see (using for example a Redemeister-Shreier process, see [@MKS]) that $G' \cong {\mathbb{Z}}/3{\mathbb{Z}}$. It follows by Proposition \[fact\] that $\delta_n(C)=0$, for all $n$. Moreover, the integral higher Alexander modules are given by: $\mathcal{A}^{{\mathbb{Z}}}_n(C)={\mathbb{Z}}/3{\mathbb{Z}}$, for all $n$.
If $\bar C \subset {\mathbb{C}}{\mathbb{P}}^2$ is an irreducible quartic curve, but not a three-cuspidal quartic, then the fundamental group $\pi_1({\mathbb{C}}{\mathbb{P}}^2- \bar C)$ is abelian (cf. [@Di1], Proposition 4.3). If $H$ is a generic line, and $C= \bar
C -H$, then by [@O], Lemma 2, it follows that $\pi_1({\mathbb{C}}^2-C)$ is also abelian. Thus all higher-order degrees of such a curve vanish. Based on this observation and the previous example, it follows that the higher-order degrees of any irreducible quartic curve are all zero.
In what follows, we give examples of curves having (some) non-trivial higher-order degrees. The key observation in these examples is the fact that the higher-order degrees of an affine curve are invariants of the fundamental group of the complement, see (\[inv\]).
\[sex\] [ Sextics with six cusps. ]{}
\(a) Let $\bar C \subset {\mathbb{C}}{\mathbb{P}}^2$ be a curve of degree $6$ with $6$ cusps on a conic. Fix a generic line, $H$, and set $C=\bar C -H$. Then $\pi_1({\mathbb{C}}^2 - C)=\pi_1({\mathbb{C}}{\mathbb{P}}^2- \bar C \cup H)$ is isomorphic to the fundamental group of the trefoil knot, and has Alexander polynomial $t^2-t+1$ (see [@Li0], §7). By Remark \[fib\], the higher-order degrees of $C$ are the same as Cochran-Harvey higher-order degrees for the trefoil knot, i.e. $\delta_0(C)=2$, and $\delta_n(C)=1$ for all $n>0$.
\(b) Let $\bar C \subset {\mathbb{C}}{\mathbb{P}}^2$ be a curve of degree $6$ with $6$ cusps as its only singular points, but this time we assume that the six cusps are not on a conic. Then $\pi_1({\mathbb{C}}{\mathbb{P}}^2-\bar C)$ is abelian, (isomorphic to ${\mathbb{Z}}_2 \times {\mathbb{Z}}_3$). Assuming the line $H$ as above is generic and setting $C=\bar C -H$, this implies that $\pi_1({\mathbb{C}}^2- C)$ is abelian as well. Therefore, $\delta_n(C)=0$ for all $n \geq 0$.
From (a) and (b) we see that the higher-order order degrees of a curve, at any level $n$, are also sensitive to the position of singular points. An interesting open problem is to find Zariski pairs that are distinguished by some $\delta_k$, but not distinguished by any $\delta_n$ for $n < k$.
[ Oka’s curves. ]{}
M. Oka [@Ok] has constructed the curves $\bar C_{p,q} \subset
{\mathbb{C}}{\mathbb{P}}^2$ ($p,q$ - relatively prime), with $pq$ singular points locally defined by $$x^p + y^q = 0,$$ such that $\pi_1({\mathbb{C}}{\mathbb{P}}^2- \bar C_{p,q})={\mathbb{Z}}_p \ast
{\mathbb{Z}}_q$. In fact, the curve $\bar C_{p,q}$ is defined by the equation: $$(x^p+y^p)^q +(y^q+z^q)^p=0.$$ Fix a generic line $H \subset {\mathbb{C}}{\mathbb{P}}^2$, and set $C_{p,q} = \bar
C_{p,q}-H$. Then $\pi_1({\mathbb{C}}^2-C_{p,q})=\pi_1({\mathbb{C}}{\mathbb{P}}^2- \bar C_{p,q}
\cup H)$ is isomorphic to the fundamental group of the torus knot of type $(p,q)$. The associated Alexander polynomial is (see for instance [@Li0], §7): $$\Delta(t)=\frac{(t^{pq}-1)(t-1)}{(t^p-1)(t^q-1)}.$$ By Remark \[fib\] and Proposition \[wh\], we obtain: $\delta_0(C_{p,q})=\text{deg} \Delta(t)=(p-1)(q-1)$, and $\delta_n(C_{p,q})=pq-p-q$ for all $n
> 0$.
[ Branching curves of generic projections. Braid groups. ]{}
Let $V_k$ be a degree $k$ non-singular surface in ${\mathbb{C}}{\mathbb{P}}^3$ and $\alpha : V_k \to {\mathbb{C}}{\mathbb{P}}^2$ be a generic projection. If $\bar C_k
\subset {\mathbb{C}}{\mathbb{P}}^2$ denotes the branching locus of $\alpha$, then $\bar
C_k$ is an irreducible curve of degree $k(k-1)$ with $k(k-1)(k-2)(k-3)/2$ nodes and $k(k-1)(k-2)$ cusps. In the case $k=3$, one obtains as branching locus the six-cuspidal sextic with all cusps on a conic.
If $C_k$ is the affine curve obtained from $\bar C_k$ by removing the intersection with a generic line, then B. Moishezon ([@Mo]) showed that $\pi_1({\mathbb{C}}^2-C_k)$ is Artin’s braid group on $k$ strands, $B_k$. The Reidemeister-Schreier process ([@MKS]) leads to the explicit computation of $B_k'/{B_k''}$ (cf. [@GL]). For $k \geq
5$, $B_k'/{B_k''} = 0$, hence $C_k$ has a trivial Alexander polynomial. By Proposition \[fact\] we obtain that $\delta_n(C_k)=0$, for all $n \geq 0$. For $k=3$, $B_3$ is the fundamental group of the trefoil knot, so by Example \[sex\](a) we obtain: $\delta_0(C_3)=2$ and $\delta_n(C_3)=1$ for all $n>0$.
The case $k=4$ requires more work. Here we will only calculate $\delta_0$ and $\delta_1$ of the corresponding curve $C_4$. The Alexander polynomial of $C_4$ is $t^2-t+1$ (see for example [@Li2]), thus $\delta_0(C_4)=2$. A presentation for the braid group on four strands is: $$B_4=\langle \sigma_1, \sigma_2, \sigma_3 | \sigma_1\sigma_3 = \sigma_3\sigma_1,
\sigma_1\sigma_2\sigma_1 = \sigma_2\sigma_1\sigma_2,
\sigma_2\sigma_3\sigma_2 = \sigma_3\sigma_2\sigma_3\>\rangle.$$ By using Reidemeister-Schreier techniques (see for instance, [@MKS]), we can obtain a presentation for $B_4'$. (This was calculated in [@GL].) $$B_4'=\langle p,q,a,b,c | pap^{-1}=b, pbp^{-1}=b^2c, qaq^{-1}=c, qbq^{-1}=c^3a^{-1}c, c=a^{-1}b \rangle,$$ where $p=\sigma_2\sigma_1^{-1}$, $q=\sigma_1\sigma_2\sigma_1^{-2}$, $a=\sigma_3\sigma_1^{-1}$, $b=\sigma_2\sigma_1^{-1}\sigma_3\sigma_2^{-1}$, and $c=\sigma_1\sigma_2\sigma_1^{-2}\sigma_3\sigma_1\sigma_2^{-1}\sigma_1^{-1}$. Then, $B_4'/B_4'' \cong {\mathbb{Z}}\oplus {\mathbb{Z}}$, generated by $p$ and $q$. Notice that since $B_4'/B_4''$ is torsion-free, $(B_4)''_r = B_4''$. Hence by (\[inv\]), we have: $$\delta_1 = \text{rk}_{{\mathbb{K}}_1}\left(B_4''/B_4'''\otimes_{{\mathbb{Z}}\bar{\Gamma}_1} {\mathbb{K}}_1\right),$$ where $\bar{\Gamma}_1=\ker(\bar{\psi}:B_4/B_4'' \to
B_4/B_4')=B_4'/B_4''$. Therefore, we must understand $B_4''/B_4'''$ as a ${\mathbb{Z}}[p^{\pm1},q^{\pm1}]$-module, and then determine the rank as a ${\mathbb{Q}}(p,q)$-vector space.
Again using Reidemeister-Schreier techniques, we calculate a group presentation for $B_4''$: $$\begin{aligned}
B_4''=\langle \rho_{i,j}, \alpha_{i,j}
| \rho_{i,0}=1, \rho_{i,j}\alpha_{i+1,j}\rho_{i,j}^{-1} =
\alpha_{i,j}\alpha_{i,j+1},
\rho_{i,j}\alpha_{i+1,j+1}\rho_{i,j}^{-1} = \alpha_{i,j}
\left(\alpha_{i,j+1}\right)^2, \\
\alpha_{i,j+2} =
\left(\alpha_{i,j+1}\right)^2\alpha_{i,j}^{-1}\alpha_{i,j+1}
\rangle,\end{aligned}$$ where $\rho_{i,j}=p^i q^j p q^{-j}
p^{-(i+1)}$ and $\alpha_{i,j}=p^i q^j a q^{-j} p^{-i}$. Notice that $p$ and $q$ act on $B_4''$ by conjugation. Furthermore, $p*(q*\gamma)=\rho_{0,1}^{-1}(q*(p*\gamma))\rho_{0,1}$, for all $\gamma \in B_4''$. Hence although the actions of $p$ and $q$ do not commute in $B_4''$, they do commute in $B_4''/B_4'''$. In particular, $B_4''/B_4'''$ is indeed a ${\mathbb{Z}}[p^{\pm1},q^{\pm1}]$-module.
We have the following presentation for $B_4''/B_4'''$ as an abelian group: $$B_4''/B_4'''=\langle \rho_{i,j}, \alpha_{i,j} |
\rho_{i,0}=0, \alpha_{i+1,j}=\alpha_{i,j}+\alpha_{i,j+1},
\alpha_{i+1,j+1}=\alpha_{i,j}+2\alpha_{i,j+1},\alpha_{i,j+2}=3\alpha_{i,j+1}-\alpha_{i,j}
\rangle.$$ To get a presentation as a ${\mathbb{Z}}[p^{\pm1},q^{\pm1}]$-module, we note that: $$\begin{aligned}
\rho_{i,j}&=&\prod_{k=1}^j (p^i q^{j-k} * \rho_{0,1}), \text{ for }
j
\geq 1, \\
\rho_{i,j}&=&\prod_{k=1}^{-j} (p^i q^{j-1+k} * \rho_{0,1}^{-1}),
\text{ for } j \leq -1, \\
\rho_{i,0}&=&0, \\
\alpha_{i,j}&=&p^i q^j * \alpha_{0,0}, \text{ for all }
i,j\in{\mathbb{Z}}.\end{aligned}$$ Therefore, as a ${\mathbb{Z}}[p^{\pm1},q^{\pm1}]$-module, $B_4''/B_4'''$ is generated by $\rho_{0,1}$ and $\alpha_{0,0}$. Furthermore, $\rho_{0,1}$ generates a free submodule, while $\alpha_{0,0}$ generates a torsion submodule. Hence the rank as a ${\mathbb{Q}}(p,q)$-vector space is 1. Therefore, $\delta_1(C_4)=1$.
*Note.* The background material on the constructions mentioned in this example are beautifully explained in Libgober’s papers [@Li2] and [@Li22]. In particular, the latter contains a summary of Moishezon’s results [@Mo].
Concluding Remarks
==================
Although in geometric problems the fundamental group of complements to projective curves plays a central role, by switching to the affine setting (i.e. by removing also a generic line) no essential information is lost. Indeed, the two groups are related by the central extension $$0 \to {\mathbb{Z}}\to \pi_1({\mathbb{C}}{\mathbb{P}}^2-(\bar{C} \cup H)) \to
\pi_1({\mathbb{C}}{\mathbb{P}}^2 - \bar {C}) \to 0.$$
Our finiteness result on the higher-order degrees provides new obstructions on the type of groups that can arise as fundamental groups of complements to affine curves in general position at infinity. Note that for a general group, one does not expect the higher-order degrees $\delta_n$ to be finite. For instance, for a free group with at least 2 generators the free ranks $r_n$ are positive (cf. [@H], Example 8.2) therefore $\delta_n$ is infinite.
Similar obstructions were previously obtained by Libgober. For example, from the study of the total linking number infinite cyclic cover of the complement ([@Li0], [@Li2]), it follows that the Alexander polynomial of the (affine) curve is cyclotomic. More precisely, this polynomial divides the product of the local Alexander polynomials at the singular points, and its zeros are roots of unity of order $d=\text{deg} (\bar C)$. Note also that the fundamental group of the affine complement maps onto ${\mathbb{Z}}/d$ (this can be seen from the above central extension). More obstructions were derived by Libgober from the study of the universal abelian cover of the affine complement ([@Li4], [@Li3]): for example, the support of the universal abelian module is contained in the zero-set of the polynomial $ (\prod_{i=1}^s t_i^{d_i})-1=0$, where $d_i$ are the degrees of the irreducible components of the curve.
Our obstructions come from analyzing the solvable coverings associated to the rational derived series of the fundamental group of the affine complement. It would be interesting to understand how the higher-order degrees are related to (or influenced by) the invariants of the infinite cyclic or universal abelian covers of the complement. Proposition \[fact\] already provides such a relation. In connection with the universal abelian cover, A. Libgober told us that he proved the following result: if the codimension (in the character torus) of support of the universal abelian Alexander module is greater than $1$, then $\delta_0(C)=0$. Of course, this assumption can only be satisfied if the curve is reducible, with at least $2$ components, but it is an interesting problem to understand for what type of curves this condition holds.
[10]{}
Cochran, T., *Noncommutative knot theory*, Algebraic & Geometric Topology, Volume 4 (2004), 347-398.
Cochran, T., Orr, K., Teichner, P. *Knot concordance, Whitney towers and $L^2$-signatures*, Annals of Mathematics, 157 (2003), 433-519.
Dimca, A., *Singularities and Topology of Hypersurfaces*, Universitext, Springer-Verlag, 1992.
Dimca, A., *Sheaves in Topology*, Universitext, Springer-Verlag, 2004.
Dimca, A., Maxim, L., *Multivariable Alexander invariants of hypersurface complements*, arXiv: math.AT/0506324.
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Harvey, S., *Higher-order polynomial invariants of $3$-manifolds giving lower bounds for the Thurston norm*, Topology, Volume 44 (2005), Issue 5, 895-945.
Harvey, S., *Monotonicity of degrees of generalized Alexander polynomials of groups and $3$-manifolds* arXiv:math.GT/0501190.
Libgober, A., *Alexander polynomials of plane algebraic curves and cyclic multiple planes*, Duke Math. J., 49(1982), 833-851
Libgober, A., *Homotopy groups of the complements to singular hypersurfaces*, Bulletin of the AMS, 13(1), 1985.
Libgober, A., *Homotopy groups of the complements to singular hypersurfaces, II*, Annals of Mathematics, 139(1994), 117-144.
Libgober, A., *Alexander invariants of plane algebraic curves*, Singularities, Proc. Symp. Pure Math., Vol. 40(2), 1983, 135-143.
Libgober, A., *Fundamental groups of the complements to plane singular curves.* Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), 29-45, Proc. Sympos. Pure Math., 46, Part 2, Amer. Math. Soc., Providence, RI, 1987.
Libgober, A., *On the homology of finite abelian covers*, Topology and its applications, **43** (1992) 157-166.
Libgober, A., *Characteristic varieties of algebraic curves* arXiv: math.AG/9801070, in: C.Ciliberto et al.(eds), Applications of Algebraic Geometry to Coding Theory, Physics and Computation, 215-254, Kluwer, 2001.
Magnus, W., Karrass, A., Solitar, D., *Combinatorial group theory: Presentations of groups in terms of generators and relations*, Dover Publications, Inc., New York, 1976. xii+444 pp.
Maxim, L., *Intersection homology and Alexander modules of hypersurface complements*, Comm. Math. Helv. (to appear).
Moishezon, B., *Stable branch curves and braid monodromies*, Lecture Notes in Mathematics, vol. 862, 107-193.
Nori, M., *Zariski’s conjecture and related problems.* Ann. Sci. ´ École Norm. Sup. (4) 16 (1983), no. 2, 305–344.
Oka, M., *Some plane curves whose complements have nonabelian fundamental groups*, Math. Ann. 218 (1978), 55-65.
Oka, M., *A survey on Alexander polynomials of plane curves*, Singularités Franco-Japonaises, 209–232, Sémin. Congr., 10, Soc. Math. France, Paris, 2005
Stenström, B., *Rings of Quotients*, Springer-Verlag, New York, 1975.
|
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"pile_set_name": "ArXiv"
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|
---
abstract: 'Given a $*$-homomorphism $\sigma: C(M)\to \mathscr{L}(\mathcal{H})$ on a Hilbert space $\mathcal{H}$ for a compact metric space $M$, a projection $P$ onto a subspace $\mathcal{P}$ in $\mathcal{H}$ is said to be essentially normal relative to $\sigma$ if $[\sigma(\varphi),P]\in \mathcal{K}$ for $\varphi\in C(M)$, where $\mathcal{K}$ is the ideal of compact operators on $\mathcal{H}$. In this note we consider two notions of span for essentially normal projections $P$ and $Q$, and investigate when they are also essentially normal. First, we show the representation theorem for two projections, and relate these results to Arveson’s conjecture for the closure of homogenous polynomial ideals on the Drury-Arveson space. Finally, we consider the relation between the relative position of two essentially normal projections and the $K$ homology elements defined for them.'
address:
- 'Department of Mathematics, Texas A&M University, College Station, TX 77843, USA'
- 'School of Mathematical Sciences, Fudan University, Shanghai, 200433, P. R. China'
author:
- 'Ronald G. Douglas'
- Kai Wang
title: Some Remarks On Essentially Normal Submodules
---
[^1]
Introduction
============
Spurred by a question of Arveson [@Ar1; @Ar2; @Ar3; @Ar4], several researchers have been considering when certain submodules of various Hilbert modules of holomorphic functions on the unit ball in $\mathbb{C}^n$ are essentially normal. In particular, Guo and the second author showed in [@GW] that the closure of a principal homogenous polynomial ideal in the Drury-Arveson space in $\mathbb{B}^n$ is essentially normal. More recently, the authors have shown in [@DW] that the closure of all principal polynomial ideals in the Bergman module on the unit ball are essentially normal. Other results have been obtained by Arveson [@Ar5], Douglas [@Dou; @Dou1], the first author and Sarkar [@DS], Eschmeier [@Esch], Kennedy [@Ke], and Shalit [@Sha].
The Arveson conjecture concerns the closure of an arbitrary homogeneous polynomial ideal which, in general, is not singly generated. For the case of $n=1$, one knows that a pure hyponormal operator submodule is essentially normal if it is finitely generated. The basis on which this result depends is the Berger-Shaw Theorem [@BS].
For ideals that are not principal, or singly generated, the results in the several variable case are few. Guo in [@G] firstly proved Arveson’s conjecture in case of the dimension $n=2$. Guo and the second author established in [@GW2; @GW] essential normality when $n=3$ or the dimension of the zero variety of the homogeneous ideal is one or less, the opposite extreme, more or less, of the case of principal ideals. There is also a result of Shalit [@Sha] which holds for ideals having a “very nice” basis relative to the norm. More recently, Kennedy [@Ke] extended that result in another direction, considering when the linear span of the closures of polynomial ideals is closed. He gives some examples, but it would appear that not all non-principal ideals are covered by this result. One should note that these results when the linear span of two essentially normal submodules is closed is implicit in the work of Arveson [@Ar5 Theorem 4.4].
In this note, we explore a more general version of the question of when the linear span of two essentially normal submodules is also essentially normal. We show that this result contains one aspect of the results of Shalit and Kennedy.
Our work does not depend on the special nature of the submodules; that is, we do not assume any connection with any underlying algebraic structure, only the fact that the linear span is closed.
There is more than one sense of the span of two submodules relevant in this context: the first is the obvious one defined to be the closure of the linear span of two submodules $\mathcal{P}$ and $\mathcal{Q}$, while the second one considers the span modulo the ideal of compact operators. If $P$ and $Q$ denote the orthogonal projections onto $\mathcal{P}$ and $\mathcal{Q}$, respectively, then we will show that this notion makes sense if $0$ is an isolated point in the essential spectrum of $P+Q$.
We consider the first notion in Section 2, and the results obtained are based on the structure theorem for two projections. The latter notion of the essential span is taken up in Section 3. We apply these results to the context of Arveson’s conjecture and raise some questions. In particular, we assume that there is a $*$-homomorphism $\sigma$ of $C(M)$ for some compact metric space $M$ and the projections essentially commute with the range of $\sigma$. Finally, in Section 4 we observe that an essentially normal projection determines an element of the odd $K$-homology group for some compact subset of $M$ and consider the relation of the K-homology elements defined by two essentially normal submodules and their sum.
Refinement of the Two Projection Representation {#sec1}
===============================================
Our results in this section are based on refinements of the structure theorem for two projections [@Br; @H].
Let $P$ and $Q$ be two projections on the Hilbert space $\mathcal{H}$. Then there exist operators $S_1 : \mathcal{P} \to \mathcal{P}, S_2 : \mathcal{P}^\perp \to \mathcal{P}^\perp$ and $X: \mathcal{P}^\perp
\to \mathcal{P}$, where $\mathcal{P}={\text{ran }} {P}$ with $0_{\mathcal{P}}\leq {S_1} \leq I_{\mathcal{P}}, {0_{\mathcal{P}^\perp}} \leq{S_2}\leq I_{\mathcal{P}^\perp}$ and $ \|{X}\| \thinspace \leq 1$, such that $$P=
\left(
\begin{array}{cc}
I_{\mathcal{P}} & 0 \\0 & 0_{\mathcal{P}^\perp}
\end{array}
\right)
\text{ and } Q=
\left(
\begin{array}{cc}
S_1 & X \\X^* & S_2
\end{array}
\right).$$ Moreover, if we set $\mathcal{P}=\mathcal{P}_1 \oplus \mathcal{P}_2 \oplus \mathcal{P}_3$ and $\mathcal{P}^\perp=\mathcal{Q}_1 \oplus \mathcal{Q}_2 \oplus \mathcal{Q}_3$, where $\mathcal{P}_1=\{x \in \mathcal{P}: \thinspace S_1{x}=0\}, \mathcal{P}_2=\{x \in \mathcal{P}: \thinspace S_1{x}=x\}$, $\mathcal{P}_3=\mathcal{P}\ominus(\mathcal{P}_1\oplus\mathcal{P}_2)$, $\mathcal{Q}_2=\{x \in \mathcal{P}^\perp: \thinspace S_2{x}=x\}$, $\mathcal{Q}_3=\{x \in \mathcal{P}^\perp: \thinspace S_2{x}=0\}$, and $\mathcal{Q}_1=\mathcal{P}^\perp\ominus(\mathcal{Q}_2\oplus\mathcal{Q}_3)$, then we have $$\begin{aligned}
S_1 & =
\left(
\begin{array}{ccc}
0_{\mathcal{P}_1} & 0 & 0 \\
0 & I_{\mathcal{P}_2} & 0 \\
0 & 0 & {S'_1}
\end{array}
\right) \in \mathscr{L}(\mathcal{P}_1 \oplus \mathcal{P}_2 \oplus \mathcal{P}_3) \text{ with } S'_1\in \mathscr{L}(\mathcal{P}_3),\\
S_2 & =
\left(
\begin{array}{ccc}
{S'_2} & 0 & 0 \\
0 & I_{\mathcal{Q}_2} & 0 \\
0 & 0 & 0_{\mathcal{Q}_3}
\end{array}
\right) \in \mathscr{L}(\mathcal{Q}_1 \oplus \mathcal{Q}_2 \oplus \mathcal{Q}_3) \text{ with } S'_2\in \mathscr{L}(\mathcal{Q}_1), \text{ and}\end{aligned}$$ $$\begin{aligned}
X & =
\left(
\begin{array}{ccc}
0 & 0 & 0 \\
0 & 0 & 0 \\
X' & 0 & 0
\end{array}
\right) \in \mathscr{L}(\mathcal{Q}_1 \oplus \mathcal{Q}_2 \oplus \mathcal{Q}_3, \mathcal{P}_1 \oplus \mathcal{P}_2 \oplus \mathcal{P}_3) \text{ with } X'\in \mathscr{L}(\mathcal{Q}_1, \mathcal{P}_3).\end{aligned}$$ These results are all straightforward.
Further, using matrix computations and the fact that $Q^2=Q=Q^\ast$, one shows that there exists an isometry $V$ from $\mathcal{Q}_1$ onto $\mathcal{P}_3$ such that $V^* S'_1 V=I_{\mathcal{Q}_1}-S'_2$. We refer the reader to [@H] for a detailed argument. Therefore, we have derived the standard model for two projections.
Two projections $P$ and $Q$ on a Hilbert space $\mathcal{H}$ are determined by
\(1) a decomposition $\mathcal{H}=\mathcal{H}_0 \oplus \mathcal{H}_1 \oplus \mathcal{H}' \oplus \mathcal{H}' \oplus \mathcal{H}_2 \oplus \mathcal{H}_3$, and
\(2) a positive contraction $S \in \mathcal{L}(\mathcal{H}')$ with $\{0, 1\}$ not in its point spectrum.
In this case, one has $$P=
\left(
\begin{array}{cccccc}
I_{\mathcal{H}_0} & 0 & 0 & 0 & 0 & 0\\
0 & I_{\mathcal{H}_1} & 0 & 0 & 0 & 0 \\
0 & 0 & I_{\mathcal{H}'} & 0 & 0 & 0 \\
0 & 0 & 0 & 0_{\mathcal{H}'} & 0 & 0 \\
0 & 0 & 0 & 0 & 0_{\mathcal{H}_2} & 0 \\
0 & 0 & 0 & 0 & 0 & 0_{\mathcal{H}_3} \\
\end{array}
\right) \text{ and }
Q=
\left(
\begin{array}{cccccc}
0_{\mathcal{H}_0} & 0 & 0 & 0 & 0 & 0\\
0 & I_{\mathcal{H}_1} & 0 & 0 & 0 & 0\\
0 & 0 & S & X & 0 & 0\\
0 & 0 & X & I_{\mathcal{H}'}-S & 0 & 0 \\
0 & 0 & 0 & 0 & I_{\mathcal{H}_2} & 0 \\
0 & 0 & 0 & 0 & 0 & 0_{\mathcal{H}_3}
\end{array}
\right),$$ where $X=\sqrt{S(I_{\mathcal{H}'}-S)} \in \mathcal{L}(\mathcal{H}')$.
Again, the representation results follow from standard matrix computations.
The following question happens frequently in many concrete problems in operator theory.
When is $\mathcal{P}+\mathcal{Q}$ closed in $\mathcal{H}$, where $\mathcal{P}=\text{ran }P$ and $\mathcal{Q}=\text{ran }Q$?
Note we have:
$$\begin{aligned}
\mathcal{P} & =
\left\{
\left(
\begin{array}{c}
x_0 \\
x_1 \\
x' \\
0 \\
0 \\
0
\end{array}
\right)
: x_0 \in \mathcal{H}_0, x_1 \in \mathcal{H}_1 , x' \in \mathcal{H}' \right\} \text{ and }\end{aligned}$$
$$\begin{aligned}
\mathcal{Q} & =
\left\{
\left(
\begin{array}{c}
0 \\
x_1 \\
Sx'+Xy' \\
Xx'+(I_{\mathcal{H}'}-S)y' \\
x_2 \\
0
\end{array}
\right): x_1 \in \mathcal{H}_1 , x', y' \in \mathcal{H}' , x_2 \in \mathcal{H}_2 \right\}.\end{aligned}$$
Therefore, $$\mathcal{P}+\mathcal{Q}=
\left\{
\left(
\begin{array}{c}
x_0 \\
x_1 \\
x' \\
z \\
x_2 \\
0
\end{array}
\right) : x_0 \in \mathcal{H}_0 , x_1 \in \mathcal{H}_1, x' \in \mathcal{H}', z \in \text{ ran} X +\text{ ran}(I_{\mathcal{H}'}-S), x_2 \in \mathcal{H}_2 \right\}.$$ This implies that $\mathcal{P}+\mathcal{Q}$ is closed if and only if $\text{ran }X+ \text{ ran}(I_{\mathcal{H}'}-S)$ is closed. Since $X=\sqrt{S(I_{\mathcal{H}'}-S)}$, we have that $\text{ran }X \subseteq \text{ran}(I_{\mathcal{H}'}-S)^{\frac{1}{2}} $. Moreover, by the spectral theorem for the positive contraction $S$, one sees that $\sqrt{S}+\sqrt{I_{\mathcal{H}'}-S}$ is invertible on $\mathcal{H}'$. This implies that $\text{ran }X +\text{ran}(I_{\mathcal{H}'}-S) \supseteq \text{ran}(I_{\mathcal{H}'}-S)^{\frac{1}{2}}$. Therefore, $\text{ran }X+\text{ran}(I_{\mathcal{H}'}-S)$ is closed if and only if $\text{ran}(I_{\mathcal{H}'}-S)^{\frac{1}{2}}$ is closed. Since $1$ is not in the point spectrum of $S$, it follows from the spectral theorem that $\text{ran}(I_{\mathcal{H}'}-S)^{\frac{1}{2}}$ is closed if and only if $1$ is not in the spectrum of $S$. Hence we have the following result.
For two projections $P$ and $Q$ on the Hilbert space $\mathcal{H}$ with $\mathcal{P}=\textrm{ ran} P$ and $\mathcal{Q}=\textrm{ ran} Q$, the linear span $\mathcal{P}+\mathcal{Q}$ is closed if and only if $1\notin\sigma(S)$, or equivalently, $\sigma(PQP)\cap (\varepsilon,1)=\phi$ for some $0<\varepsilon<1$, where $S$ is the same as in Theorem 2.1. Moreover, in the case that $\mathcal{R}=\mathcal{P}+\mathcal{Q}$ is closed, the projection $R$ onto $\mathcal{R}$ has the form $$R=
\left(
\begin{array}{cccccc}
I_{\mathcal{H}_0} & 0 & 0 & 0 & 0 & 0\\
0 & I_{\mathcal{H}_1} & 0 & 0 & 0 & 0 \\
0 & 0 & I_{\mathcal{H}'} & 0 & 0 & 0 \\
0 & 0 & 0 & I_{\mathcal{H}'} & 0 & 0 \\
0 & 0 & 0 & 0 & I_{\mathcal{H}_2} & 0 \\
0 & 0 & 0 & 0 & 0 & 0_{\mathcal{H}_3} \\
\end{array}
\right)$$
Only the last statement remains to be proved and that follows from the fact that $1 \notin \sigma( S)$ if and only if $0\notin\sigma( I_{\mathcal{H}'}-S)$, which implies $\text{ ran} (I_{\mathcal{H}'}-S)=\mathcal{H}'$.
A nearly immediate consequence of the representation theorem is the following characterization of the $C^*$-algebra $\mathcal{A}(P,Q,I)$ generated by projections $P$, $Q$ and the identity operator on the Hilbert space $\mathcal{H} $. This is usually attributed to Dixmier [@Di].
Let $P$ and $Q$ be the projections onto the subspaces $\mathcal{P}$ and $\mathcal{Q}$ of the Hilbert space $\mathcal{H}$, respectively. If $\mathcal{P}\cap \mathcal{Q}=\mathcal{P}^\perp\cap \mathcal{Q}=\mathcal{P}\cap \mathcal{Q}^\perp=\mathcal{P}^\perp\cap \mathcal{Q}^\perp=\{0\},$ then $ \mathcal{A}(P,Q,I)$ is $*-$algebaically isomorphic to a \*-subalgebra $\mathcal{C} $ of $M_2(C(M)), $ where $M=\sigma(PQP)$, $M_2(C(M))$ denotes the algebra of two by two matrices with entries in $C(M)$ and $$\mathcal{C}=\{\left(
\begin{array}{cc }
\phi_{11} & \phi_{12} \\
\phi_{21} & \phi_{22}
\end{array}
\right)\in M_2(C(M)):\phi_{12}(i)=\phi_{21}(i)=0, \text{if } i=0,1 \text{ and } i\in M\}.$$
Applying the spectral theorem to the operators $I_{\mathcal{P}}$ and $S$, one obtains the correspondence from which the result follows: $$P=
\left(
\begin{array}{cc }
I_{\mathcal{P}} & 0 \\
0 & 0
\end{array}
\right) \thicksim \left(\begin{array}{cc }
1 & 0 \\
0 & 0
\end{array}
\right)\in M_2(C(M))\text{ and }$$ $$Q=\left(
\begin{array}{cc }
S & \sqrt{S(1-S)} \\
\sqrt{S(1-S)} & 1-S
\end{array}
\right)\thicksim \left(\begin{array}{cc }
\chi & \sqrt{\chi(1-\chi)} \\
\sqrt{\chi(1-\chi)} & 1-\chi
\end{array}
\right)\in M_2(C(M)),$$ where $1$ and $\chi$ denote the functions on $M$ defined by $1(x)=1$ and $\chi (x)=x$ for $x\in M$. The fact that the functions $\phi_{12}$ and $\phi_{21}$ in the definition of $\mathcal{C}$ vanish at $0,1\in M$ follows from the fact that the function $\sqrt{\chi (1-\chi)}$ does.
We now use the characterization of the $C^*$-algebra generated by two projections to get our first result on the essential normality of the projection onto the linear span when it is closed.
For two projections $P$ and $Q$ on the Hilbert space $\mathcal{H}$, if $\mathcal{R}=\text{ran}P+\text{ran}Q$ is closed, then the $C^*$-algebra $\mathcal{A}(P,Q,I_\mathcal{H})$ generated by $P, Q$ and the identity operator $I_\mathcal{H}$ contains the projection $R$ onto the subspace $\mathcal{R}$.
Using a direct matrix computation, one sees that the operator $
P+(I-P)Q(I-P)$, which is in the $C^*$-algebra $\mathcal{A}(P,Q,I)$, has the form $$P+(I-P)Q(I-P)=
\left(
\begin{array}{cccccc}
I_{\mathcal{H}_0} & 0 & 0 & 0 & 0 & 0\\
0 & I_{\mathcal{H}_1} & 0 & 0 & 0 & 0 \\
0 & 0 & I_{\mathcal{H}'} & 0 & 0 & 0 \\
0 & 0 & 0 & I_{\mathcal{H}'}-S & 0 & 0 \\
0 & 0 & 0 & 0 & I_{\mathcal{H}_2} & 0 \\
0 & 0 & 0 & 0 & 0 & 0_{\mathcal{H}_3} \\
\end{array}
\right).$$ This implies that $\sigma( P+(I-P)Q(I-P))\subseteq \{0\}\cup [ \varepsilon,1] $ for some $0<\varepsilon<1$. Since $[\varepsilon,1]\cap \sigma( P+(I-P)Q(I-P))$ is an open and closed subset of the spectrum $\sigma( P+(I-P)Q(I-P))$, it follows from the spectral theorem that the spectral projection ${\mathbf{1}}_{[\varepsilon,1]} (P+(I-P)Q(I-P))$ is in $\mathcal{A}(P,Q,I)$, which leads to the desired result since $\mathbf{1}_{[\varepsilon,1]} (P+(I-P)Q(I-P)) = {R}$.
We now relate the representation result to a question in the context of Arveson’s conjecture. We will provide a more precise statement in Section 4.
Suppose $\sigma:\thinspace C(M)\to \mathcal{L}(\mathcal{H})$ is a $\ast$-homomorphism for some compact metric space $M$, and $P,Q$ are projections on the Hilbert space $\mathcal{H}$ such that the commutators $[\sigma(\varphi),P]\in \mathcal{K}$ and $[\sigma(\varphi),Q]\in \mathcal{K}$ for $\varphi \in C(M)$, where $\mathcal{K}$ denotes the ideal of compact operators on $\mathcal{H}$. If $\mathcal{R}=\text{ran}P+\text{ran}Q $ is closed and $R$ is the projection onto $\mathcal{R}$, then $[\sigma(\varphi),R]\in \mathcal{K}$ for $\varphi \in C(M)$.
Using an elementary $C^*$-algebra argument, one shows that $[\sigma(\varphi),T]\in \mathcal{K}$ for any operator $T\in \mathcal{A}(P,Q,I)$. Combining this fact with Theorem 2.5, one obtains the desired result.
With the same hypotheses, the projection $\widetilde{R}$ onto $\mathcal{P}\cap \mathcal{Q}$ essentially commutes with the range of $\sigma$.
This is an immediate consequence of Theorem 1 in [@Dou1] and the exact sequence $$0\to\widetilde{R} \stackrel{i}{\to} \mathcal{P}\oplus {\mathcal{Q}}\stackrel{j}{ \to} \mathcal{R}\to 0,$$ where $i(r)=(r,-r)$, and $j(p,q)=p+q$ for $r\in \widetilde{R},p\in \mathcal{P},q\in\mathcal{Q}$.
In both the theorem and corollary, $C(M)$ can be replaced by any $C$\*-subalgebra of $\mathscr{L}(\mathcal{H})$.
These results are related to a theorem of Arveson [@Ar5 Theorem 4.4] and the more recent work of Kennedy [@Ke] in which essential normality is replaced by $p$-essential normality, where the commutators are assumed to be in the Schatten $p-$class for $1\leq p< \infty$. If one examines the proof of Theorem 2.5 more closely, the preceding arguments can be refined to obtain analogous results for $p$-essential normality. Basically, this is true because the functional calculus which yields the spectral projection $\mathbf{1}_{[\varepsilon,1]} (P+(I-P)Q(I-P))$ can be approximated on a neighborhood of the spectrum with analytic functions.
We can extend these results somewhat using the following reduction which is essentially algebraic.
Suppose $\mathcal{P}$ and $\mathcal{Q}$ are subspaces of the Hilbert space $\mathcal{H}$ and $\mathcal{R}^\sharp$ is a subspace of $\mathcal{P}\cap\mathcal{Q}$. Then $\mathcal{P}+\mathcal{Q}$ is closed if and only if $\mathcal{P}/ {\mathcal{R}^\sharp}+ \mathcal{Q}/ {\mathcal{R}^\sharp}$ is closed in $\mathcal{H}/ {\mathcal{R}^\sharp}$.
It follows from the fact that $ \mathcal{P}/ {\mathcal{R}^\sharp}+ \mathcal{Q}/ {\mathcal{R}^\sharp}= (\mathcal{P}+ \mathcal{Q})/ {\mathcal{R}^\sharp} $ and the fact that for any linear manifold $\mathcal{L}$ containing $\mathcal{R}^\sharp$, $\mathcal{L}/ \mathcal{R}^\sharp$ is closed if and only if $\mathcal{L}$ is closed.
With the same hypotheses, the closeness of $\mathcal{P}/ {\mathcal{R}^\sharp}+\mathcal{Q}/ {\mathcal{R}^\sharp}$ is equivalent to the closeness of $\mathcal{P}/ (\mathcal{P}\cap \mathcal{Q})+\mathcal{Q}/ (\mathcal{P}\cap \mathcal{Q}) $.
Both of these statements are equivalent to $\mathcal{P}+\mathcal{Q}$ being closed in $\mathcal{H}$.
Essential Span of Subspaces {#sec3}
===========================
The following question and corresponding result are important for considering the notion of essential span in this section.
When do two projections $P$ and $Q$ on a Hilbert space $\mathcal{H}$ almost commute; that is, when is $[P,Q]\in \mathcal{K}(\mathcal{H})?$
Using the representation theorem for $P$ and $Q$ above, we see that $[P,Q]\in \mathcal{K}$ if and only if $X\in \mathcal{K}$ and so we have the following result.
For projections $P$ and $Q$ onto subspaces $\mathcal{P}$ and $\mathcal{Q}$, respectively, on a Hilbert space $\mathcal{H}, \thinspace [P,Q]\in \mathcal{K}$ if and only if ${\sigma_e}(S)\subset \{0,1\}$. Moreover, $PQ\in\mathcal{K}$ if and only if $S\in\mathcal{K}$ and $\dim \mathcal{P}\cap \mathcal{Q} < \infty$ in the representation appearing in Theorem 2.1.
The proof follows from a matrix calculation in the above representation theorem which shows that $[P,Q]\in\mathcal{K}$ if and only if $X=\sqrt{S(I_{\mathcal{H}'}-S)}$ is compact. For $PQ\in\mathcal{K}$, it is necessary and sufficient for $S$ and $I_{\mathcal{H}_1}$ to be compact.
If $P$ and $Q$ are projections on the Hilbert space $\mathcal{H}$, then another notion of the span of the ranges of $P$ and $Q$ is relevant when considering questions of essential normality, which involves the images of $P$ and $Q$ in the Calkin algebra. If $0$ is an isolated point in the essential spectrum, ${\sigma_e}(P+Q)$, of $P+Q$, or $0\notin {\sigma_e}(P+Q)$, then the image in the Calkin algebra of the spectral projection, ${P} \bigvee_e {Q}$, for $[\varepsilon, \infty]$, where $(0, \varepsilon)\cap {\sigma_e}(P+Q)=\phi$, can be thought of as the “essential span” of $\text{ran}P$ and $\text{ran}Q$. (Note that the image of this spectral projection in the Calkin algebra does not depend on $\varepsilon$ whenever $(0, \varepsilon)\cap {\sigma_e}(P+Q)=\phi$.) One result related to this notion is the following.
If $[P,Q]\in\mathcal{K}$, then $0$ is isolated in ${\sigma_e}(P+Q)$. Moreover, if $P$ and $Q$ almost commute with a $C^*$-algebra $ \mathfrak{A}$, then so does any projection on $\mathcal{H}$ with the image ${P} \bigvee_e {Q}$ in the Calkin algebra.
Considering the standard model for two projection in Section 2, one sees that $[P,Q]\in\mathcal{K}$ implies that $X$ is compact and ${\sigma_e}(P+Q)\subseteq \{0,1,2\}$. This implies that $[\varepsilon,2]$ is an open and closed subset of ${\sigma }(P+Q)$ for some $0<\varepsilon<1 $ and hence $\mathbf{1}_{[\varepsilon,2]} (P+ Q )$ is in $\mathcal{A}(P,Q,I)$, where $\mathcal{A}(P,Q,I)$ is the $C^*$-algebra generated by $P,Q$ and the identity operator $I$. Therefore, its image in the Calkin algebra, ${P} \bigvee_e {Q}$, commutes with the image of $ \mathfrak{A}$, which completes the proof.
One thing one needs to be clear on is that the image of ${P} \bigvee {Q}$ in the Calkin algebra and ${P} \bigvee_e {Q}$ are not necessarily the same. Consider, for example, the subspaces $\mathcal{P}=\overline{span}\{ e_n \oplus 0: n \in \mathbb{N}\}$ in $\ell^2 \oplus \ell^2$ and $\mathcal{Q}=\overline{span}\{ e_n \oplus \frac{1}{n}e_n: n \in \mathbb{N}\}$. These subspaces have the images of $\pi(P \bigvee Q)$ and $P \bigvee_e Q$ in the Calkin algebra which are the images of the projections onto $\ell^2 \oplus \ell^2$ and $\ell^2 \oplus (0)$, respectively. Note that in this case $\mathcal{P}+\mathcal{Q}$ is not closed, which is the key as the following result shows.
Let $P,Q$ be the projections onto the subspaces $\mathcal{P}$ and $\mathcal{Q}$ of a Hilbert space $\mathcal{H}$, respectively. Then $\mathcal{P}+\mathcal{Q}$ is closed if and only if $\pi( {P}\bigvee {Q})= {P}\bigvee_e {Q} $.
We first suppose that $\mathcal{P}+\mathcal{Q}$ is closed. Using the notation in Theorem 2.1, we have that $$P+Q=
\left(
\begin{array}{cccccc}
I_{\mathcal{H}_0} & 0 & 0 & 0 & 0 & 0\\
0 & 2I_{\mathcal{H}_1} & 0 & 0 & 0 & 0\\
0 & 0 & I_{\mathcal{H}'}+S & \sqrt{S(I_{\mathcal{H}'}-S)} & 0 & 0\\
0 & 0 & \sqrt{S(I_{\mathcal{H}'}-S)} & I_{\mathcal{H}'}-S & 0 & 0 \\
0 & 0 & 0 & 0 & I_{\mathcal{H}_2} & 0 \\
0 & 0 & 0 & 0 & 0 & 0_{\mathcal{H}_3}
\end{array}
\right).$$ By Theorem 2.3 we know that $1\notin \sigma(S)$ when $\mathcal{P}+\mathcal{Q}$ is closed. Applying the spectral theorem to the operator $S$, one obtains that $0$ is isolated in $\sigma(P+Q)$. This implies that the notion ${P}\bigvee_e {Q} $ makes sense and, in fact, it is the image in the Calkin algebra of the projection onto $\text{ran}(P+Q)$. Moreover, by the above representation of $P+Q$, one sees that $$\text{ran}(P+Q)=\mathcal{H}_0\oplus\mathcal{H}_1\oplus \mathcal{H}'\oplus\mathcal{H}'\oplus \mathcal{H}_2 =\mathcal{P}+\mathcal{Q}.$$ It follows that ${P}\bigvee_e {Q} $ is the image of the projection onto $\mathcal{P}+\mathcal{Q}$.
On the other hand, in case that $\pi( {P}\bigvee {Q})= {P}\bigvee_e {Q} $, there exists $0<\varepsilon<1$ such that $(0,\varepsilon)\cap \sigma_e(P+Q)=\phi$ and ${P}\bigvee {Q}-\mathbf{1}_{[\varepsilon,\infty]} (P+ Q )$ is a finite dimensional projection. Applying the spectral theorem for $S$ to the matrix representation of $P+ Q $, one sees that the spectral projection of $S$ for $(1-\varepsilon,1)$ is also a finite dimensional projection. Combing this fact with that $1$ is not in the point spectrum of $S$, we have that $1\notin \sigma(S)$, which leads to the desired result using Theorem 2.3.
While it seems inconceivable that $[p]+[q]$ is always closed for polynomials $p$ and $q$ in $\mathbb{C}[z_1,\cdots,z_n]$; here $[\cdot]$ denotes the closure in the Hardy, Bergman or Drury-Arveson modules on the unit ball, it seems quite possible that the projections onto $[p]$ and $[q]$ always almost commute. One thing making the answering of such a question difficult is the fact that $[p]\cap [q]$ is always large containing $[pq]$. One possible way to circumvent this problem might be to consider the quotient modules $[p]^\perp$ and $[q]^\perp$. We’ll have something more to say about them in the next section.
Another possibility to handle the fact that $[p]\cap [q]$ is large might be to use Theorem 2.10 to reduce the matter to $[p]/([p]\cap [q])$ and $[q]/([p]\cap [q]).$ In this case, $[p]/([p]\cap [q])$ and $[q]/([p]\cap [q]) $ are semi-invariant modules. We will obtain a result using this approach in the following section.
Locality of Essentially Normal Projection
=========================================
Let $M$ be a compact metric space and $\sigma: C(M)\to \mathscr{L}(\mathcal{H})$ be a $*$-homomorphism for a Hilbert space $\mathcal{H}$. We say that a projection $P$ on $\mathcal{H}$ is essentially normal relative to $\sigma$ if $[\sigma(\varphi),P]\in\mathcal{K}$ for any $\varphi\in C(M)$. This implies that the map $\sigma_P: \varphi\to \pi(P\sigma(\varphi)P)\in\mathscr{Q}(\mathcal{H}) $ into the Calkin algebra $\mathscr{Q}(\mathcal{H})=\mathscr{L}(\mathcal{H})/\mathcal{K}(\mathcal{H})$ is a $*-$homomorphism. Hence, there exists a compact subset $M_P$ of $M$ such that the following diagram commutes: $$\begin{array}{ccc}
C(M) & \stackrel{\sigma}{\longrightarrow} & \mathscr{L}(\mathcal{H}) \\
\downarrow & & \downarrow \\
C(M_P) & \stackrel{\hat{\sigma}_P}{\longrightarrow} & \mathscr{Q}(P\mathcal{H}) \\
\end{array}.$$ Here the vertical arrow on the left is defined by restriction; that is, $\varphi\to \varphi|_{M_P}$, and the one on the right is the compression to $\text{ran } P$ followed by the map onto the Calkin algebra. Therefore, using [@BDF], one knows that $(\sigma, P)$ defines an element $[\sigma,P]\in K_1(M_ {P})$. An interesting question concerns the relation of elements $[\sigma,P]$ and $[\sigma,Q]$ for two essentially normal projections $P$ and $Q$ relative to $\sigma$.
Now this relationship can’t be too simple. In particular, consider the representation $\tau$ of $C(clos \mathbb{B}^n)$ in $L^2(\mathbb{B}^n)$ and the projection $P$ onto the Bergman space $L^2_a(\mathbb{B}^n)$. For $p\in \mathbb{C}[z_1,\cdots,z_n]$, one knows [@DW] that the projection $Q_p$ of $L^2(\mathbb{B}^n)$ onto the closure $[p]$ of the ideal $(p)$ in $ \mathbb{C}[z_1,\cdots,z_n]$ generated by $p$ is essentially normal; that is, $[\tau(\varphi),Q_p]\in\mathcal{K}$ for $\varphi\in C(clos \mathbb{B}^n)$. Further, we have that $R_p=P-Q_p $ is also essentially normal and $M_{[\tau,R_p]}\subseteq Z(p)\cap \partial \mathbb{B}^n$, where $Z(p)$ is the zero variety of the polynomial $p$. It follows that the image of $[\tau, R_p]\in K_1(\partial \mathbb{B}^n)$ is zero since $Z(p)\cap \partial \mathbb{B}^n$ is a proper subset of $ \partial \mathbb{B}^n $. Therefore, one has $[\tau,P]=[\tau, Q_p]$ for every polynomial $p\in\mathbb{C}[z_1,\cdots,z_n]$. Hence, there is a great variety of essentially normal projections defining the same element in $K_1(\partial \mathbb{B}^n).$
However, we do have a result for what happens at the opposite extreme.
Suppose that $P$ and $Q$ are essentially normal projections on the Hilbert space $\mathcal{H}$ for the $*-$homomorphism $\sigma: C(M)\to \mathscr{L}(\mathcal{H})$ for some compact space $M$. If $M_P\cap M_Q=\phi$, then $PQ\in \mathcal{K}.$
By the assumption that $P$ and $Q$ are essentially normal relative to $\sigma$, one sees that the operator $PQ$ almost intertwines the two representations $\sigma|_{C(M_P)}$ and $\sigma|_{C(M_Q)}$; that is, one has that $P \sigma(\varphi)P (PQ)-(PQ) Q\sigma(\varphi)Q\in \mathcal {K}$ for $\varphi\in C(M)$. Thus, in the Calkin algebra, if $\varphi\in C(M)$ satisfies $\varphi|_{M_P}\equiv 1$ and $\varphi|_{M_Q}\equiv 0$, we obtain that $\pi (Q \sigma(\varphi)Q)\pi(PQ)=\pi(PQ)\pi(P\sigma(\varphi)P)$. But, $\pi (Q \sigma(\varphi)Q)=0$ and $\pi (P \sigma(\varphi)P)=\pi(P)$, this means that $\pi(PQ P)=0$, which implies $PQ\in\mathcal{K}$ and completes the proof.
We can use this theorem to obtain a partial result concerning the relation of the projections onto $[p]$ and $[q]$ for $p,q\in\mathbb{C}[z_1,\cdots,z_n]$.
For two polynomials $p,q\in\mathbb{C}[z_1,\cdots,z_n]$, let $P$ and $Q$ be the projections onto the submodule $\mathcal{P}=[p],\mathcal{Q}=[q]$ on $L^2_a(\mathbb{B}^n)$, respectively. If $p,q$ satisfy $Z(p)\cap Z(q) \cap \partial \mathbb{B}^n=\phi$, then we have that $[P,Q]\in \mathcal{K}$.
Note that $I-P,I-Q$ are the projections onto the quotient modules $\mathcal{P}^\perp$ and $\mathcal{Q}^\perp$, respectively. Using the notation in the above, by [@DW] we know that $M_{I-P}\subseteq Z(p)\cap \partial \mathbb{B}^n$ and $M_{I-Q}\subseteq Z(q)\cap \partial \mathbb{B}^n$. It follows from the hypothesis that $M_{I-P}\cap M_{I-Q}=\phi$. By Theorem 4.1 we have that $(I-P)(I-Q) $ and $(I-Q)(I-P) $ are compact. Therefore, one sees that $[P,Q]\in\mathcal{K}$, which completes the proof.
We can extend this result using Theorem 2.10 as follows.
For polynomials $p,q,r\in\mathbb{C}[z_1,\cdots,z_n]$ with $Z(p)\cap Z(q)\cap\partial \mathbb{B}^n=\phi$, let $\mathcal{P}=[pr]$ and $\mathcal{Q}=[qr]$ be the submodules in $L_a^2(\mathbb{B}^n)$. Then one has $[P,Q]\in\mathcal{K}$, where $P,Q$ are the projections onto $\mathcal{P}$ and $\mathcal{Q}$, respectively.
One can generalize the argument in [@DW] to show that $$M_{[pr]^\perp/[r]^\perp}\subseteq Z(p)\cap \partial \mathbb{B}^n \, and \, M_{[qr]^\perp/[r]^\perp}\subseteq Z(q)\cap \partial \mathbb{B}^n.$$ By Theorem 4.1, this implies that $(R-P)(R-Q)$ and $(R-Q)(R-P) $ are compact, where $R$ is the projection onto the submodule $[r].$ This means that $[P,Q]=[R-P,R-Q]\in\mathcal{K}$, which completes the proof.
Another example of the application of the notion of the locality of essentially normal projection is the following result which is more or less the opposite situation of the previous theorem.
Assume that $\sigma: C(M) \to \mathscr{Q}(\mathcal{H})$ is a $*-$homomorphism on the Hilbert space $\mathcal{H}$ for a compact metric space $M$, and $P$ and $Q$ are two essentially normal projections such that $\mathcal{P}\cap \mathcal{Q}^\perp=\mathcal{P}^\perp \cap \mathcal{Q}=\{0\}$, where $\mathcal{P}=\text{ran} P$ and $\mathcal{Q}=\text{ran}Q$. If $\mathcal{P}+\mathcal{Q}^\perp$ is closed, then $M_P=M_Q$ and $[\hat{\sigma}_P]=[\hat{\sigma}_Q]\in K_1(M_P)$.
In the representation theorem for $P,Q$, the spaces $\mathcal{H}_0$ and $\mathcal{H}_2$ are $\{0\}$ by assumption and we can write $\mathcal{P}=\mathcal{H}_1\oplus \mathcal{P}' $ and $\mathcal{Q}=\mathcal{H}_1\oplus \mathcal{Q}'$ corresponding to ${P}'=P-I_{\mathcal{H}_1}$ and ${Q}'=Q-I_{\mathcal{H}_1}$. As in the proof of Theorem 4.1, the image $\pi(P'Q')$ of $P'Q'$ in the Calkin algebra intertwines the operators $\pi (P' \sigma(\varphi) P')$ and $\pi (Q' \sigma(\varphi) Q')$. Using Theorem 2.3 and the assumption $\mathcal{P}+\mathcal{Q}^\perp$ is closed, we have $0\notin\sigma(S)=\sigma(P'Q'P')$. Combining this with the fact $\ker P'Q' =\mathcal{P}'^\perp \cap \mathcal{Q}'=\{0\}$, one sees that $P'Q':\mathcal{Q}'\to\mathcal{P}'$ is invertible. Therefore, using the polar decomposition in the Calkin algebra, one sees that $M_P=M_Q$ and that the $K_1$ elements are equal.
There would seem to be a limit to what can be concluded about the $K_1$ element. If $k\in K_1(M_P)$ for some essentially normal projection $P$ on the Hilbert space $\mathcal{H}$ with a $*$-homomorphism $\sigma: C(M)\to \mathscr{L}(\mathcal{H})$, then there exists an essentially normal projection $Q\leq P$ such that $[\sigma, Q]=k\in K_1(M_P)$.
[10]{}
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[^1]: The second author was supported by NSFC, the Department of Mathematics at Texas A&M University and Laboratory of Mathematics for Nonlinear Science at Fudan University.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
Domain Adaptation (DA) transfers a learning model from a labeled source domain to an unlabeled target domain which follows different distributions. There are a variety of DA scenarios subject to label sets and domain configurations, including closed-set and partial-set DA, as well as multi-source and multi-target DA. It is notable that existing DA methods are generally designed only for a specific scenario, and may underperform for scenarios they are not tailored to. Towards a versatile DA method, a more universal inductive bias other than the domain alignment should be explored. In this paper, we delve into a missing piece of existing methods: **class confusion**, the tendency that a classifier confuses the predictions between the correct and ambiguous classes for target examples. We unveil that less class confusion explicitly indicates more class discriminability and implicitly implies more domain transferability in all the above scenarios.
Based on the more universal inductive bias, we propose a general loss function: **Minimum Class Confusion (MCC)**. It can be characterized by (1) a non-adversarial DA method without explicitly deploying domain alignment, enjoying fast convergence speed (about $3 \times$ faster than mainstream adversarial methods); (2) a versatile approach that can handle Closed-Set, Partial-Set, Multi-Source, and Multi-Target DA, outperforming the state-of-the-art methods in these scenarios, especially on the largest and hardest dataset to date ($7.25\%$ on DomainNet). In addition, it can also be used as a general regularizer that is orthogonal and complementary to a variety of existing DA methods, accelerating convergence and pushing those readily competitive methods to a stronger level. We will release our code for reproducibility.
author:
- |
Ying Jin, Ximei Wang, Mingsheng Long (), and Jianmin Wang\
School of Software, BNRist, Tsinghua University, China\
[{jiny18,wxm17}@mails.tsinghua.edu.cn, {mingsheng,jimwang}@tsinghua.edu.cn]{}
bibliography:
- 'egbib.bib'
title: |
Less Confusion More Transferable:\
Minimum Class Confusion for Versatile Domain Adaptation
---
Introduction
============
Deep Neural Network (DNN) excels at learning discriminative representations from a large set of labeled data, leading to unprecedented successes in a wide range of machine learning tasks [@DeCAF14; @Oquab2014Learning; @he2016Resnet]. However, DNN often suffers from the scarcity of labeled data in real-world applications. Such a dilemma gives rise to Domain Adaptation (DA) [@pan2010survey], an important technology that explores effective algorithms to transfer knowledge from a readily labeled dataset to an unlabeled one in the presence of distribution shift.
![The four Domain Adaptation (DA) scenarios studied in this paper: (1) Unsupervised Domain Adaptation (UDA), a close-set scenario where the source and target share the same label set; (2) Partial Domain Adaptation (PDA), a partial-set scenario where the source label set subsumes the target label set; (3) Multi-Source Domain Adaptation (MSDA); (4) Multi-Target Domain Adaptation (MTDA). MCC is a *versatile* method towards all four DA scenarios.[]{data-label="versatile"}](./pic/circle.pdf){width="0.9\linewidth"}
Early DA methods strived to bridge the gap in feature distributions between the source and target domains. These methods either learn invariant features [@pan2010survey; @Gong12GFK] across domains or reweight instances [@Huang06SampleBias; @Gong13ConnectingDots] to highlight source instances that are more relevant to the target domain. Recently, DNNs have been playing a main role in domain adaptation. As DNNs can learn transferable features [@Oquab2014Learning; @yosinski2014transferable; @zhao19on], a variety of methods based on DNNs were developed, pushing domain adaptation to a dramatically more effective level.
Many methods [@Tzeng14DDC; @Long15DAN; @Long16RTN; @Long17JAN; @AFN2019; @Tzeng15SDT; @DANN2016; @ADDA2017; @Pei18MADA; @Long18CDAN; @wang2019tada; @MDD2019] have been proposed for the closed-set scenario called Unsupervised DA (UDA), where there is only one source and one target domain, with identical label set. Meanwhile, there also exist other highly practical scenarios, such as Partial DA (PDA) [@SAN2018; @IWAN2018] where the source label set subsumes the target one, Multi-Source DA (MSDA) [@MDAN2018; @xu2018deep] with multiple source domains, and Multi-Target DA (MTDA) [@Peng2019DADA] with multiple target domains. As existing UDA methods cannot be directly applied to these challenging scenarios, various methods [@SAN2018; @Cao2017PADA; @IWAN2018; @xu2018deep; @Peng2019DADA] have been proposed for each specific scenario. With well-designed architectures or losses, these methods work quite well in their own scenarios.
However, in practical applications, complicated data acquired in the real-world makes it difficult to confirm the label sets and domain configurations. Therefore, we may be stuck in choosing a proper method tailored to the *right* DA scenario. The most ideal solution to escape from this dilemma is a *versatile* DA method that can handle all the above scenarios. Unfortunately, existing DA methods are generally designed only for a specific scenario and may underperform for scenarios they are not tailored to. For instance, PADA [@Cao2017PADA], a classic PDA method, excels at selecting out outlier classes, but suffers from internal domain shift in MSDA and MTDA and underperforms in these scenarios, while DADA [@Peng2019DADA], an outstanding method tailored to MTDA, cannot be directly applied to PDA or MSDA. Therefore, incapable of being directly applied into other scenarios, existing DA methods are not versatile enough to meet the practical requirements.
Towards a versatile DA method, a more universal inductive bias other than the domain alignment should be explored. In this paper, we delved into the error matrices of the target domain and found that the classifier trained on the source domain may confuse to distinguish the correct class from a similar class, such as car and truck. As shown in Figure \[ErrorMatrix:target\], the probability that a source-only model misclassifies cars as trucks on the target domain is over $25\%$. Further, we analyzed the error matrices in other DA scenarios and drew the same conclusion. These findings give us a fresh perspective to tackle domain adaptation: **class confusion**, the tendency that a classifier confuses the predictions between the correct and ambiguous classes for target examples. Further, we unveil that *less class confusion* explicitly indicates more class discriminability and implicitly implies more domain transferability in all the above scenarios. However, we still need to face a new challenge that the ground-truth class confusion needs to be calculated based on the labels in the target domain, which is inaccessible in UDA. Fortunately, an instance weighted inner product of the classifier predictions with their transposes naturally reveal the confusion relationship between different classes. Therefore, we can define class confusion from this perspective, enabling it to be computed just based on the classifier predictions. To this end, we propose a novel loss function: **Minimum Class Confusion (MCC)**, which can be characterized as a novel and versatile DA approach without explicitly deploying feature alignment, enjoying fast convergence speed. In addition, it can also be used as a general regularizer that is orthogonal and complementary to various existing DA methods, further accelerating and improving those readily competitive methods. Our contributions are summarized as follows:
- We unveil that class confusion is a common missing piece of existing DA methods and discover that less class confusion implies more transferability.
- We propose a novel loss function: Minimum Class Confusion (MCC), which is versatile to handle four different DA scenarios, including closed-set, partial-set, multi-source, and multi-target.
- We conduct extensive experiments on four standard datasets, and demonstrate that MCC can outperform the state-of-the-art methods in four DA scenarios, especially on the largest and hardest dataset to date ($7.25\%$ on DomainNet). It also enjoys an obvious (about $3 \times$) faster convergence speed than mainstream DA methods.
Related Work {#relate}
============
**Unsupervised Domain Adaptation (UDA).** Most existing domain adaptation researches focus on UDA, giving birth to many competitive methods. Most mainstream UDA methods based on DNNs can be classified into two categories: **(1)** Moment Matching and **(2)** Adversarial Training.
**(1) Moment Matching** methods aim to minimize the distribution discrepancy between the two domains. Deep Coral [@coral2016] align second-order statistics of the two distributions. DDC [@Tzeng14DDC] and DAN [@Long15DAN] utilizes Maximum Mean Discrepancy [@MKMMD2012], JAN [@Long17JAN] leverages Joint Maximum Mean Discrepancy. SWD [@SWD2019] introduces Sliced Wasserstein Distance and CAN [@CAN2019] uses Contrastive domain discrepancy.
**(2) Adversarial Training** methods borrow the spirit of Generative Adversarial Network [@Goodfellow14GAN], aiming at learning domain invariant features in an adversarial manner. DANN [@DANN2016] introduces a domain discriminator to distinguish source and target features, while the feature extractor strives to fool the domain discriminator. ADDA [@ADDA2017], MADA [@Pei18MADA] and MCD [@MCD2018] extends such architecture to multiple feature extractors and classifiers. Akin to Conditional Generative Adversarial Networks [@CGAN2014], CDAN [@Long18CDAN] proposes to align features in a conditional adversarial manner. CyCADA [@Hoffman18CyCADA] adapts features in both pixel and feature level. TADA [@wang2019tada] proposes a transferable attention mechanism. SymNet [@symDA2019] introduces a symmetric classifier, and DTA [@DTA2019] learns discriminative features with adversarial dropout.
Recently, **other** novel methods are proposed to tackle domain adaptation from new perspectives. For instance, SE [@SE2018] is based on teacher-student [@MeanTea] model. TPN [@TPN2019] introduces a prototypical network. TAT [@TAT2019] proposes a novel transferable adversarial training method. BSP [@BSP2019] penalizes the largest singular values of features in order to boost feature discriminability. AFN [@AFN2019] unveils that those features with larger norm are more transferable and enlarges feature norm. Some methods [@DIRT; @ProDA2018; @CBST; @Zou2019] also utilize self-training or pseudo label. These methods enlighten the road of domain adaptation from new perspectives.
**Partial Domain Adaptation (PDA).** In PDA, the target label space is a subspace of source label space, deteriorating negative transfer [@pan2010survey]. SAN [@SAN2018], IWAN [@IWAN2018], PADA [@Cao2017PADA] and ETN [@ETN2019] introduces various weighting mechanisms to select out outlier classes in the source domain, while AFN [@AFN2019] enlarges feature norms to alleviate negative transfer.
**Multi-Source Domain Adaptation (MSDA).** In MSDA, there are multiple source domains that may be significantly different. MDAN [@MDAN2018] provides solid theoretical insights for MSDA. Deep Cocktail Network [@xu2018deep] (DCTN) introduces a k-way domain classifier while $\rm M^{3}SDA$ [@peng2018moment] proposed a moment-matching model for MSDA.
**Multi-Target Domain Adaptation (MTDA).** In MTDA, multiple target domains are included. DADA [@Peng2019DADA] enjoys strong performance in this task by learning domain-invariant features with well-designed network architecture and losses.
In this paper, we aim at proposing a *versatile* method for all the four scenarios above and compare its performance with these state-of-the-art methods respectively.
Approach
========
In this paper, we aim at proposing a novel loss function, Minimum Class Confusion (MCC), as a versatile approach to four domain adaptation scenarios. **(1)** Unsupervised Domain Adaptation (**UDA**) [@DANN2016], the standard scenario, constitutes a labeled source domain $\mathcal{S} = \{(\mathbf{x}_s^i,{\bf y}_s^i)\}_{i=1}^{n_s}$ and an unlabeled target domain $\mathcal{T} = \{{\mathbf{x}}_t^i\}_{i=1}^{n_t}$, where ${\bf x}^i$ is an example and ${\bf y}^i$ is the associated label. **(2)** Partial Domain Adaptation (**PDA**) [@Cao2017PADA] extends UDA by letting the source domain labeled set subsume the target domain label set. **(3)** Multi-Source Domain Adaptation (**MSDA**)[@peng2018moment] extends UDA by expanding to $S$ labeled source domains $\{\mathcal{S}_{1}, \mathcal{S}_{2},..., \mathcal{S}_{S}\}$. **(4)** Multi-Target Domain Adaptation (**MTDA**) [@Peng2019DADA] extends UDA by expanding to $T$ unlabeled target domains $\{\mathcal{T}_{1}, \mathcal{T}_{2},..., \mathcal{T}_{T}\}$. Hereafter, we denote by ${\bf a}_{i\cdot}$, ${\bf a}_{\cdot j}$ and $A_{ij}$ the $i$-row, the $j$-th column and the $ij$-th entry of matrix ${\bf A}$ respectively.
{width="0.85\linewidth"}
Minimum Class Confusion
-----------------------
To minimize class confusion, we need to find out some proper criteria to measure the pairwise class confusion on the target domain. Different from previous methods such as CORAL [@coral2016] that place focus on features, we explore the classifier predictions. First, we denote the classifier output of the target domain as ${\widehat{\bf Y}}_t = G(F(\mathbf{X}_t)) \in \mathbb{R }^{B \times |{\cal {C}}|}$, where $B$ is the batch size of the target data, $|{\cal {C}}|$ is the number of source classes, $F$ is the feature extractor and $G$ is the classifier. We focus on the classification predictions $\widehat{\bf Y}$ and omit the domain subscript $t$ for clarity. The probability $\widehat{Y}_{ij}$ that the $i$-[th]{} instance belongs to the $j$-[th]{} class is given by $$\label{softmax}
\centering
{\widehat Y_{ij}} = \frac{{\exp \left( {{Z_{ij}}/T} \right)}}{{\sum\nolimits_{j' = 1}^{|{\mathcal{C}}|} {\exp \left( {{Z_{ij'}}/T} \right)} }},$$ where $Z_{ij}$ is the logit output of the classifier layer (before the softmax function) and $T$ is the temperature [@hinton2015distilling] for scaling. Obviously, Eq. boils down to the vanilla softmax function when $T=1$. Hence, $\widehat{Y}_{ij}$ reveals the relationship between the $i$-[th]{} instance and the $j$-[th]{} class. A natural question arises: Can we quantify the class confusion relationship by using $\widehat{\bf Y}$? This paper gives a positive answer.
**First,** we note that the examples in the target domain are not equally important for computing class confusion. Those examples with higher certainty in class predictions given by the classifier are more reliable and should contribute more to the pairwise class confusion. We use the *entropy* functional $H(p) \triangleq - {\mathbb{E}_p}\log p$ in information theory as an uncertainty measure of distribution $p$. The entropy (uncertainty) $H(\widehat{\bf y}_{i\cdot})$ of predicting the $i$-[th]{} example by the classifier is defined as $$\centering
H(\widehat{\bf y}_{i\cdot})= - { \sum _{j=1 }^{ |{\cal {C}}| }{ { \widehat { Y } }_{ ij }\log{ \widehat { Y } }_{ ij } } }.$$ While the entropy is a measure of uncertainty, what we want is a probability distribution that places a larger probability on the examples with larger certainty of class predictions. A *de facto* transformation to probability is the softmax function $${W_{ii}} = \frac{{B\left( {1 + \exp ( { - H( {{{{\widehat{\bf y}}}_{i \cdot }}} )} )} \right)}}{{\sum\limits_{i' = 1}^B {\left( {1 + \exp ( { - H( {{{{\widehat{\bf y}}}_{i' \cdot }}} )} )} \right)} }},$$ where $W_{ii}$ is the probability quantifying the importance of the $i$-th example for computing the class confusion, and $\mathbf{W}$ is the corresponding diagonal matrix. Note that we take the opposite value of the entropy to reflect the *certainty*. Laplace Smoothing [@Laplace2008] (i.e. adding a constant $1$ to each addend of the softmax function) is used to form a heavier-tailed weight distribution, which is suitable for highlighting more certain examples as well as avoiding overly penalizing the others. For better manipulation, the probability over the examples in each batch of size $B$ is rescaled to sum up to $B$ such that the average weight for each example is $1$.
**Second,** we recall that $\widehat{Y}_{ij}$ reveals the relationship between the $i$-[th]{} example and the $j$-[th]{} class. By prioritizing on the examples with more certain class predictions, we define the pairwise *class confusion* between two classes $j$ and $j'$ as $$\label{eq:CC}
{{\mathbf{C}}_{jj'}} = {\widehat{\mathbf{y}}}_{ \cdot j}^{\sf T}{\mathbf{W}}{{\widehat{\mathbf{y}}}_{ \cdot j'}}.$$ Lets delve into the definition of the class confusion in Eq. . Note that ${\widehat{\mathbf{y}}_{ \cdot j}}$ denotes the probabilities of the $B$ examples in each batch to come from the $j$-th class. The class confusion is defined as the inner product between ${\widehat{\mathbf{y}}_{ \cdot j}}$ and ${\widehat{\mathbf{y}}_{ \cdot j'}}$ weighted by ${\bf W}$, the certainties of class predictions for the $B$ examples. So it measures the possibility of simultaneously classifying the $B$ examples into the $j$-th and the $j'$-th classes.
The batch-based definition of the class confusion in Eq. is native for the mini-batch SGD optimization. However, when the number of classes is large, it will run into a severe class imbalance in each batch. To tackle this problem, we adopt a normalization technique widely used in Random Walk [@vonLuxburg2007]: $$\label{eq:norm}
{{{\widetilde{\mathbf C}}}_{jj'}} = \frac{{{{\mathbf{C}}_{jj'}}}}{{\sum\nolimits_{{j''} = 1}^{|{\mathcal{C}}|} {{{\mathbf{C}}_{j{j''}}}} }}.$$ Taking the idea of Random Walk, the normalized class confusion in Eq. has a neat interpretation: It is probable to walk from one class to another (resulting in wrong classification) if the two classes have a high class confusion.
**Finally,** lets come back to designing the final loss. Recall that ${{{\widetilde{\mathbf C}}}_{jj'}}$ well measures the confusion between classes $j$ and $j'$. We only need to minimize the cross-class confusion, i.e. $j\ne j'$. In other words, the ideal situation is that no examples are ambiguously classified into two classes at the same time. We define the **Minimum Class Confusion (MCC)** loss as $$\label{eq:MCC}
{L_{{\rm{MCC}}}} ( {{{\widehat {\mathbf{Y}}}_t}} ) = \sum\limits_{j = 1}^{|{\mathcal{C}}|} {\sum\limits_{j' \ne j}^{|{\mathcal{C}}|} {\left| {{{{\widetilde{\mathbf C}}}_{jj'}}} \right|} }.$$ Since the class confusion in Eq. has been normalized, minimizing the *between-class* confusion in Eq. readily implies that the *within-class* confusion is maximized. Eq. is a universal loss that is pluggable to all existing approaches.
Less Confusion More Transferable {#sec:bias}
--------------------------------
Based on the proposed Minimum Class Confusion (MCC) loss function, we further elaborate a more *universal* inductive bias towards designing versatile domain adaptation methods: *less confusion, more discriminable, and more transferable*.
- It is intuitively reasonable that when the classes are less confused, they are more easily discriminated.
- Our key observation is that, if the labels are available in both source and target domains, we can train a joint network with the cross-entropy loss. Then the features learned by the network from both domains will be *implicitly* aligned with the class supervision (supported by the oracle results of A-distance in Figure \[tsne\]), not requiring *explicit* feature alignment. This is a strong inductive bias of deep networks with *builtin* transferability.
- Since the labels are unavailable in the target domain, we only add the cross-entropy loss in the source domain, and impose our MCC loss in the target domain to boost the discriminability. Based on the builtin transferability of deep networks, the representations of the same class from the source and target domains will be *implicitly* aligned by MCC, thereby boosting the transferability.
The above propositions will be justified in the empirical studies (Section \[sec:exp\]). We want to emphasize that the inductive bias of *less confusion* in this work is more universal than that of *domain alignment* in prior work. As discussed in Section \[relate\], many prior methods *explicitly* align features from the source and target domains, facing the risk of deteriorating the feature discriminability and impeding the transferability [@BSP2019]. Further, the inductive bias of *less confusion* is general and applicable to a variety of domain adaptation scenarios, while that of *domain alignment* will suffer when the domains cannot be aligned naturally (e.g. the partial-set DA scenarios).
Versatile Approach to Domain Adaptation
---------------------------------------
The main motivation of the MCC loss is to design a versatile approach to a variety of domain adaptation scenarios. As elaborated above, by combining the cross-entropy loss on the source domain labeled data and the MCC loss on the target domain unlabeled data, the deep network will be guided to *explicitly* improve the discriminability of the target data and *implicitly* boost the transferability across domains. Hence, there is no need for explicit feature alignment.
Denote by ${\widehat {\mathbf{y}}_s} = G(F({{\mathbf{x}}_s}))$ the class prediction for each source domain instance ${{\mathbf{x}}_s}$, by ${\widehat {\mathbf{Y}}_t} = G(F({{\mathbf{X}}_t}))$ the class predictions for a batch (size $B$) of target domain instances ${{\mathbf{X}}_t}$. The versatile approach (termed also by **MCC**) proposed for a variety of domain adaptation scenarios is formulated as $$\label{Method}
\mathop {\min }\limits_{F,G} \, {\mathbb{E}_{({{\mathbf{x}}_s},{{\mathbf{y}}_s}) \in \mathcal{S}}}{L_{{\rm{CE}}}}\left( {{{\widehat {\mathbf{y}}}_s},{{\mathbf{y}}_s}} \right) + \mu \, {\mathbb{E}_{{{\mathbf{X}}_t} \subset \mathcal{T}}}{L_{{\rm{MCC}}}}( {{{\widehat {\mathbf{Y}}}_t}} ),$$ where $L_{\rm CE}$ is the cross-entropy loss, $\mu$ is a hyper-parameter for the importance of the MCC loss. With the joint loss, the feature extractor $F$ and the classifier $G$ of the deep adaptation model are trained end-to-end by back-propagation.
The deep adaptation model in Eq. , without any extra modifications or techniques, is *versatile enough* to tackle the four typical domain adaptation scenarios.
- **Unsupervised Domain Adaptation (UDA).** Eq. is formulated natively for this vanilla scenario.
- **Partial Domain Adaptation (PDA).** Since no explicit domain alignment is deployed, there is no worry about the misalignment between source *outlier* classes and target classes as [@Cao2017PADA]. Hence, Eq. can also be directly applied to PDA. In PDA, the source label set subsumes the target label set, and the MCC loss is computed on ${{{\widehat {\mathbf{Y}}}_t}} \in \mathbb{R}^{B \times |\mathcal{C}|}$ ($|\mathcal{C}|$ is the number of source classes). However, compared to the confusion between the target classes, the confusion between the source outlier classes on the target domain will be negligible in the MCC loss.
- **Multi-Source Domain Adaptation (MSDA).** Prior methods of MSDA consider multiple source domains as different domains, capturing the internal source domain shifts, and a simple merge of all source domains proves fragile. However, based on the *builtin* transferability in Section \[sec:bias\], we can safely merge $S$ source domains as $\mathcal{S} \leftarrow {\mathcal{S}_1} \cup \cdots \cup {\mathcal{S}_S}$ to enable *implicit* domain alignment, yielding a much simpler but effective MSDA approach.
- **Multi-Target Domain Adaptation (MTDA).** Based on similar idea as MSDA, also note that the MCC loss is the same for different target domains, we can safely merge $T$ target domains as $\mathcal{T} \leftarrow {\mathcal{T}_1} \cup \cdots \cup {\mathcal{T}_T}$ to enable *implicit* domain alignment.
We will show by empirical studies that Eq. , without extra modifications, is simple and effective for all scenarios.
Regularization to Existing DA Methods
-------------------------------------
Since the MCC loss is defined on the target domain with the *less confusion* inductive bias, which is different from the widely used *domain alignment* inductive bias, our method is naturally orthogonal and complementary to the previous methods, pushing those readily competitive methods to a stronger level. The MCC loss in Eq. can be serving as a regularization term pluggable into existing methods.
Take the domain alignment framework [@DANN2016] based on the adversarial training as an example, integrating the MCC loss: $$\centering
\label{regularization}
\begin{aligned}
\mathop {\min }\limits_{F,G} \mathop {\max }\limits_D \; & {\mathbb{E}_{({{\mathbf{x}}_s},{{\mathbf{y}}_s}) \in \mathcal{S}}}{L_{{\rm{CE}}}}\left( {{{\widehat {\mathbf{y}}}_s},{{\mathbf{y}}_s}} \right) + \mu \, {\mathbb{E}_{{{\mathbf{X}}_t} \subset \mathcal{T}}}{L_{{\rm{MCC}}}}({\widehat {\mathbf{Y}}_t}) \\
- \, \lambda \, & {\mathbb{E}_{{\mathbf{x}} \in \mathcal{S} \cup \mathcal{T}}}{L_{{\rm{CE}}}}(D( {\widehat {\mathbf{f}}} ),{\bf d}),
\end{aligned}$$ where the last equation stands for the domain discriminator $D$ that strives to distinguish the source from the target, and ${\bf d}$ is the domain label, $\widehat{\mathbf{f}} = F(\mathbf{x})$ is the features learned to confuse the domain discriminator. The overall framework is a *minimax* game between two players $F$ and $D$ with $\lambda$ to be the balancing hyper-parameter. Note that the classifier $G$ is not involved in the adversarial training, hence it is easy to directly integrate the MCC loss into the framework. The MCC loss can also be readily integrated into other representative frameworks e.g. moment matching [@Long15DAN] and large norm [@AFN2019].
Experiments {#sec:exp}
===========
We evaluate MCC with many state-of-the-art transfer learning methods on MTDA, MSDA, PDA and UDA scenarios. We will release our code for reproducibility.
Setup
-----
We used four real-world datasets: (1) *Office-31* [@Saenko10Office]: a classical domain adaptation dataset with 31 categories and 3 domains: Amazon (**A**), Webcam (**W**) and DSLR (**D**); (2) *Office-Home* [@Venkateswara17Officehome]: a more difficult dataset with 65 categories and 4 domains: Art (**A**), Clip Art (**C**), Prodcut (**P**) and Real World (**R**). The domain gap of Office-Home is significantly larger than that of Office-31; (3) *VisDA-2017* [@peng2017VisDA]: a simulation-to-real dataset with 12 categories and more than 280,000 images, and (4) *DomainNet* [@peng2018moment]: the largest and hardest domain adaptation dataset till now, with approximately 0.6 million images from 345 categories and 6 domains: Clipart (**c**), Infograph (**i**), Painting (**p**), Quickdraw (**q**), Real (**r**) and Sketch (**s**).
Our methods were implemented based on **PyTorch**. Deep Embedded Validation (DEV) [@you19icml] was conducted to select hyper-parameters. Then, we set $\mu=1.0$ in all experiments, which generally works well as the value of MCC is comparable to cross-entropy loss. For a fair comparison, we report the results of other algorithms according to the original paper. We run each experiment for $5$ times.
Results
-------
**Multi-Target Domain Adaptation.** The performance of MCC in MTDA is evaluated on DomainNet, the most difficult dataset to date. As shown in Table \[table:domainnet-tgt\], many competitive methods are not effective in this challenging dataset. However, our simple yet effective method outperforms the current state-of-the-art method DADA [@Peng2019DADA] by a big margin ($7.3\%$).
**Multi-Source Domain Adaptation.** When evaluated our method in MSDA, we adopt the *source combine* strategy for MCC and compare it with existing DA algorithms that are specifically designed for MSDA on DomainNet . As shown in Table \[table:domainnet-src\], [source combine]{} strategy is fragile for MSDA as many mainstream DA methods suffer from negative transfer. However, with such a naive strategy, MCC can significantly outperform $\rm M^{3}SDA$ [@peng2018moment], the state-of-the-art method tailored to MSDA scenario, by a big margin ($5.0\%$).
**Partial Domain Adaptation.** Since the existence of outlier source classes, PDA is known as a challenging scenario. For a fair comparison, we follow the setting in PADA [@Cao2017PADA] and AFN [@AFN2019], where the first $25$ categories in alphabetic order are taken as the target domain. As shown in Table \[table:office-home-pda\], our method on Office-Home can also outperform AFN [@AFN2019] which is the strongest PDA method to date.
Method (S:) c: i: p: q: r: s: Avg
------------------------ ---------- ---------- ---------- ---------- ---------- ---------- ---------- --
ResNet [@he2016Resnet] 25.6 16.8 25.8 9.2 20.6 22.3 20.1
SE [@SE2018] 21.3 8.5 14.5 13.8 16.0 19.7 15.6
MCD [@MCD2018] 25.1 19.1 27.0 10.4 20.2 22.5 20.7
DADA [@Peng2019DADA] 26.1 20.0 26.5 12.9 20.7 22.8 21.5
**MCC** **33.6** **30.0** **32.4** **13.5** **28.0** **35.3** **28.8**
: Accuracy (%) on [DomainNet]{} for [MTDA]{} (ResNet-101).[]{data-label="table:domainnet-tgt"}
[lccccccc]{} Method (:T) &:c & :i & :p & :q & :r & :s & Avg\
ResNet [@he2016Resnet] & 47.6 & 13.0 & 38.1 & 13.3 & 51.9 & 33.7 & 32.9\
DAN [@Long15DAN] & 45.4 & 12.8 & 36.2 & 15.3 & 48.6 & 34.0 & 32.1\
RTN [@Long16RTN] & 44.2 & 12.6 & 35.3 & 14.6 & 48.4 & 31.7 & 31.1\
JAN [@Long17JAN] & 40.9 & 11.1 & 35.4 & 12.1 & 45.8 & 32.3 & 29.6\
ADDA [@ADDA2017] & 47.5 & 11.4 & 36.7 & 14.7 & 49.1 & 33.5 & 32.2\
SE [@SE2018] & 24.7 & 3.9 & 12.7 & 7.1 & 22.8 & 9.1 & 16.1\
MCD [@MCD2018] & 54.3 & 22.1 & 45.7 & 7.6 & 58.4 & 43.5 & 38.5\
DCTN [@xu2018deep] & 48.6 & 23.5 & 48.8 & 7.2 & 53.5 & 47.3 & 38.2\
$\rm M^{3}SDA$ [@peng2018moment] & 58.6 & 26.0 & 52.3 & 6.3 & 62.7 & 49.5 & 42.6\
**MCC** & **63.0** & **29.8** & **56.4** & **15.6** & **66.4** & **54.1** & **47.6**\
[l\*[14]{}Cc]{} Method (S:T) & A:C & A:P & A:R & C:A & C:P & C:R & P:A & P:C & P:R & R:A & R:C & R:P & Avg\
ResNet [@he2016Resnet] & 38.6 & 60.8 & 75.2 & 39.9 & 48.1 & 52.9 & 49.7 & 30.9 & 70.8 & 65.4 & 41.8 & 70.4 & 53.7\
DAN [@Long15DAN] & 44.4 & 61.8 & 74.5 & 41.8 & 45.2 & 54.1 & 46.9 & 38.1 & 68.4 & 64.4 & 51.5 & 74.3 & 56.3\
JAN [@Long17JAN] & 45.9 & 61.2 & 68.9 & 50.4 & 59.7 & 61.0 & 45.8 & 43.4 & 70.3 & 63.9 & 52.4 & 76.8 & 58.3\
PADA [@Cao2017PADA] & 51.2 & 67.0 & 78.7 & 52.2 & 53.8 & 59.0 & 52.6 & 43.2 & 78.8 & 73.7 & 56.6 & 77.1 & 62.0\
AFN [@AFN2019] & **58.9** & 76.3 & 81.4 & 70.4 & **73.0** & 77.8 & 72.4 & 55.3 & 80.4 & 75.8 & 60.4 & 79.9 & 71.8\
**MCC** & 57.5 & **82.0** & **86.4** & **70.7** & 70.6 & **78.2** & **76.5** & **61.7** & **86.5** & **82.0** & **64.5** & **84.0** & **75.1**\
[l\*[14]{}Cc]{} Method & plane & bcybl & bus & car & horse & knife & mcyle & person & plant & sktbrd & train & truck & mean\
ResNet [@he2016Resnet] & 55.1& 53.3 & 61.9 & 59.1 & 80.6 & 17.9 & 79.7 & 31.2 & 81.0 & 26.5 & 73.5 & 8.5 & 52.4\
MinEnt [@Grandvalet2005Semi] & 80.3 & 75.5 & 75.8 & 48.3 & 77.9 & 27.3 & 69.7 & 40.2 & 46.5 & 46.6 & 79.3 & 16.0 & 57.0\
DANN [@DANN2016] & 81.9 & 77.7 & 82.8 & 44.3 & 81.2 & 29.5 & 65.1 & 28.6 & 51.9 & 54.6 & 82.8 & 7.8 & 57.4\
DAN [@Long15DAN] & 87.1 & 63.0 & 76.5 & 42.0 & 90.3 & 42.9 & 85.9 & 53.1 & 49.7 & 36.3 & 85.8 & 20.7 & 61.1\
MCD [@MCD2018] & 87.0 & 60.9 & 83.7 & 64.0 & 88.9 & 79.6 & 84.7 & 76.9 & 88.6 & 40.3 & 83.0 & 25.8 & 71.9\
CDAN [@Long18CDAN]& 85.2 & 66.9 & 83.0 & 50.8 & 84.2 & 74.9 & 88.1 & 74.5 & 83.4 & 76.0 & 81.9 & 38.0 & 73.9\
ADR [@ADR2018] & 87.8 & 79.5 & 83.7 & 65.3 & 92.3 & 61.8 & 88.9 & 73.2 & 87.8 & 60.0 & 85.5 & 32.3 & 74.8\
AFN [@AFN2019] & 93.6 & 61.3 & 84.1 & 70.6 & 94.1 & 79.0 & 91.8 & 79.6 & 89.9 & 55.6 & 89.0 & 24.4 & 76.1\
**MCC** & 88.1 & 80.3 & 80.5 & 71.5 & 90.1 & 93.2 & 85.0 & 71.6 & 89.4 & 73.8 & 85.0 & 36.9 &**78.8**\
**Unsupervised Domain Adaptation.** We evaluate MCC in UDA on standard benchmark datasets. [(1)]{} *Visda-2017*. As reported in Table \[table:VisDA\], when applied as a domain adaptation method, MCC surpass the state-of-the-art UDA algorithms. [(2)]{} *Office-31*. As shown in Table \[table:office31\] (standard deviation is reported in the supplementary material), MCC can outperform all the other algorithms. It is noteworthy that MCC does not induce any additional learnable parameters, while other algorithms may involve complicated network architectures with extra parameters and training skills.
Analyses
--------
**A General Regularizer.** In addition, MCC can be used as a regularization term for various DA methods. We apply it to mainstream domain adaptation methods, and compare its performance with entropy minimization (MinEnt) [@Grandvalet2005Semi] and Batch Spectral Penalization (BSP) [@BSP2019]. As shown in Table \[table:VisDA-reg\] and Table \[table:office31-reg\] (standard deviation is reported in the supplementary material), MCC implies larger improvements than MinEnt and BSP to various kinds of DA methods. It is noteworthy that MCC can push the accuracy of CDAN to a higher level of over $80\%$ on Visda-2017.
**Convergence Speed.** We show the accuracy curve of the whole training procedure in Figure \[fig:converge-lambda\]. MCC enjoys faster convergence speed. Besides, when used as a regularization term for existing domain adaptation methods, MCC can also largely accelerate convergence. Totally, both MinEnt and BSP take approximately $10000$ iterations to converge, while MCC takes about $2500$ iterations, which is about $3 \times$ faster.
[l\*[14]{}Cc]{} Method & plane & bcybl & bus & car & horse & knife & mcyle & persn & plant & sktb & train & truck & mean\
DANN [@DANN2016] & 81.9 & 77.7 & 82.8 & 44.3 & 81.2 & 29.5 & 65.1 & 28.6 & 51.9 & 54.6 & 82.8 & 7.8 & 57.4\
DANN + MinEnt [@Grandvalet2005Semi] & 87.4 & 55.0 & 75.3 & **63.8** & 87.4 & 43.6 & **89.3** & 72.5 & 82.9 & **78.6** & 85.6 & 27.4 & 70.7\
DANN + BSP [@BSP2019] & **92.2** & 72.5 & **83.8** & 47.5 & 87.0 & 54.0 & 86.8 & 72.4 & 80.6 & 66.9 & 84.5 & 37.1 & 72.1\
**DANN + MCC** & 90.4 & **79.8** & 72.3 & 55.1 & **90.5** & **86.8** & 86.6 & **80.0** & **94.2** & 76.9 & **90.0** & **49.6** & **79.4**\
CDAN [@Long18CDAN]& 85.2 & 66.9 & **83.0** & 50.8 & 84.2 & 74.9 & 88.1 & 74.5 & 83.4 & 76.0 & 81.9 & 38 & 73.9\
CDAN + MinEnt [@Grandvalet2005Semi] & 90.5 & 65.8 & 79.1 & 62.2 & 89.8 & 28.7 & **92.8** & 75.4& 86.8 & 65.3 & 85.2 & 35.3 & 71.4\
CDAN + BSP [@BSP2019] & 92.4 & 61.0 & 81.0 & 57.5 & 89.0 & **80.6** & 90.1 & 77.0 & 84.2 & 77.9 & 82.1 & 38.4 & 75.9\
**CDAN + MCC** & **94.5** & **80.8** & 78.4 & **65.3** & **90.6** & 79.4 & 87.5 & **82.2** & **94.7** & **81.0**& **86.0** & **44.6** & **80.4**\
\[table:office31\]
[l\*[7]{}c]{} Method & A:W & D:W & W:D & A:D & D:A & W:A & Avg\
ResNet [@he2016Resnet] & 68.4 & 96.7 & 99.3 & 68.9 & 62.5 & 60.7 & 76.1\
DAN [@Long15DAN] & 80.5 & 97.1 & 99.6 & 78.6 & 63.6 & 62.8 & 80.4\
RTN [@Long16RTN] & 84.5 & 96.8 & 99.4 & 77.5 & 66.2 & 64.8 & 81.6\
DANN [@DANN2016] & 82.0 & 96.9 & 99.1 & 79.7 & 68.2 & 67.4 & 82.2\
ADDA [@ADDA2017]& 86.2 & 96.2 & 98.4 & 77.8 & 69.5 & 68.9 & 82.9\
JAN [@Long17JAN] & 85.4 & 97.4 & 99.8 & 84.7 & 68.6 & 70.0 & 84.3\
MADA [@Pei18MADA] & 90.0 & 97.4 & 99.6 & 87.8 & 70.3 & 66.4 & 85.2\
MinEnt [@Grandvalet2005Semi] & 92.5 & 98.0 & 99.8 & 92.6 & 70.3 & 63.1 & 86.1\
SimNet [@Pedro17SimNet] & 88.6 & 98.2 & 99.7 & 85.3 & **73.4** & 71.6 & 86.2\
GTA [@Swami17GTA] & 89.5 & 97.9 & 99.8 & 87.7 & 72.8 & 71.4 & 86.5\
CDAN [@Long18CDAN]& 94.1 & **98.6** & **100.0** & 92.9 & 71.0 & 69.3 & 87.7\
AFN [@AFN2019] & 88.8 & 98.4 & 99.8 & 87.7 & 69.8 & 69.7 & 85.7\
**MCC** & **95.4** & **98.6** & **100.0** & **95.6** & 72.6 & **73.9** & **89.4**\
**Synthetic Dataset.** We explore the performance of MCC on Two Moon, whose target samples are generated by rotating the source samples by $30^{\circ}$. We plot the decision boundary of the classifiers to compare the performance of MCC and MinEnt. As shown in Figure \[twomoon\_MinEnt\] and \[twomoon\_MCC\], vanilla MinEnt fails to attain a correct decision boundary, while MCC can classify most examples correctly. In addition, when applied to DANN as a regularization term, MinEnt still works poorly but MCC generates a satisfying decision boundary.
\[fig:converge-lambda\]
**Feature Discriminability.** Ben-David *et al.* [@ben2010theory] derived the expected error $\mathcal{E}_\mathcal{T}(h)$ of a hypothesis $h$ on the target domain ${\mathcal{E}_\mathcal{T}}(h) \le {\mathcal{E}_\mathcal{S}}(h) + \frac{1}{2}{d_{\mathcal{H}\Delta \mathcal{H}}}(\mathcal{S},\mathcal{T}) + \epsilon_{ideal} $ by three terms: **(a)** expected error of $h$ on the source domain, $\mathcal{E}_\mathcal{S}(h)$; **(b)** the A-distance ${d_{\mathcal{H}\Delta \mathcal{H}}}(\mathcal{S},\mathcal{T})$, a measure of domain discrepancy; and **(c)** the error $\epsilon_{ideal}$ of the ideal joint hypothesis $h^*$ on both source and target domains. BSP [@BSP2019] states out that $\epsilon_{ideal}$ represents the feature discriminability. As Figure \[fig:converge-lambda\] shows, the $\epsilon_{ideal}$ value of MCC is lower than that of mainstream DA methods. When used as a regularizer, MCC can attain a lower $\epsilon_{ideal}$ value than MinEnt [@Grandvalet2005Semi] and BSP [@BSP2019], revealing that MCC can further enhance feature discriminability.
\[hyper\]
**Hyperparameter Sensitivity.** Temperature scaling $T$ and the trade-off $\mu$ of regularization term are the only two hyperparameters of MCC and MinEnt when applying them to existing DA methods. We take hyper-parameters around the optimal hyper-parameter $[T^*, \mu^*]$ for each regularizer to test its hyperparameter sensitivity. Consider the task $A \rightarrow W$ on Office-31, through DEV [@you19icml] we can find that the optimal hyperparameters $[T^*, \mu^*]$ for MinEnt are $[1.0, 0.25]$ and those for MCC are $[2.5, 1.0]$. Thus, we can take $T \in \{0.5, 1.0, 1.5, 2.0\}, \mu \in \{0, 0.25, 0.5, 0.75\}$ for MinEnt and $T \in \{2.0, 2.5, 3.0, 3.5\}, \mu \in \{0.5, 1.0, 1.5, 2.0\}$ for MCC.
[l\*[7]{}c]{} Method & A:W & D:W & W:D & A:D & D:A & W:A & Avg\
DANN [@DANN2016] & 82.0 & 96.9 & 99.1 & 79.7 & 68.2 & 67.4 & 82.2\
+ MinEnt [@Grandvalet2005Semi] & 91.7 & 98.3 & **100.0** & 87.9 & 68.8 & 68.1 & 85.8\
+ BSP [@BSP2019] & 93.0 & 98.0 & **100.0** & 90.0 & 71.9 & 73.0 & 87.7\
**+ MCC** & **95.6** & **98.6** & 99.3 & **93.8** & **74.0** & **75.0** & **89.4**\
CDAN [@Long18CDAN]& 94.1 & **98.6** & **100.0** & 92.9 & 71.0 & 69.3 & 87.7\
+ MinEnt [@Grandvalet2005Semi] & 91.7 & 98.5 & **100.0** & 90.4 & 72.3 & 69.5 & 87.1\
+ BSP [@BSP2019] & 93.3 & 98.2 & **100.0** & 93.0 & **73.6** & 72.6 & 88.5\
**+ MCC**& **94.7** & **98.6** & **100.0** & **95.0** & 73.0 & **73.6** & **89.2**\
AFN [@AFN2019] & 88.8 & 98.4 & 99.8 & 87.7 & 69.8 & 69.7 & 85.7\
+ MinEnt [@Grandvalet2005Semi] &90.3 & **98.7** & **100.0** & 92.1 & 73.4 & 71.2 & 87.6\
+ BSP [@BSP2019] & 89.7 & 98.0 & 99.8 & 91.0 & 71.4 & 71.4 & 86.9\
**+ MCC** &**95.4** & 98.6 & **100.0** & **96.0** & **74.6** & **75.2** & **90.0**\
As shown in Figure \[hyper\], MCC is less sensitive to hyper-parameters, while the performance of MinEnt collides under some parameters near the optimal ones. **Feature Transferability.** As shown in Figure \[tsne\], MCC has lower A-distance [@ben2010theory], which is close to the oracle one (supervised learning on both domains). We also visualize features of the last *fc*-layer of ResNet-50 using t-SNE [@JMLR08tsne] embedding. As shown in Figure \[tsne\], when applied as a DA method on PDA, MCC can learn highly discriminable features and has sharp class boundaries. Additionally, without explicitly deploying feature alignment, MCC can make features much more transferable across domains. More visualization results are included in the supplementary material.
\[tsne\]
Conclusion
==========
In this paper, we unveil that less class confusion implies more transferability, which is a general discovery for Versatile Domain Adaptation. To this end, we propose a novel loss function: Minimum Class Confusion (MCC). MCC can be applied as a versatile domain adaptation method that can handle various DA scenarios. Extensive empirical results prove that our method can outperform state-of-the-art methods in four DA scenarios respectively, enjoying faster convergence. Meanwhile, MCC can also be used as a general regularizer for existing domain adaptation methods, further improving performance and accelerating training.
|
{
"pile_set_name": "ArXiv"
}
|
=1
Introduction
============
=1 The dressing chain [@Sha; @ShaYa; @ves_shab] appeared in the application of Darboux transformations to the Schödinger (Sturm–Liouville) equation, which is the spectral problem for the Korteweg–de Vries (KdV) equation. A detailed study concerning the integrability properties as well as solutions of the model were presented in [@ves_shab]. In particular, the authors proved that the dressing chain with a periodic constraint in odd dimensions is completely integrable in the sense of Liouville–Arnold.
The dressing chain can also be obtained as an auto-Bäcklund transformation of the modified KdV (mKdV) equation via the celebrated Miura transformation between KdV and mKdV [@Miu], and a symmetry of mKdV. The two ways of deriving the dressing chain are not unrelated, as the Miura transformation itself can be derived from factorisation of the Schrödinger equation [@FG].
Both the Darboux and Bäcklund approach can be seen as a discretisation process, and the two methods have been applied to other equations. In particular, in the discrete setting, Spiridonov and Zhedanov [@SZ] considered a tri-diagonal discrete Schrödinger equation, for which discrete Darboux transformations gave rise to two equivalent systems: the discrete time Toda lattice and a system they called the discrete dressing chain [@SZ equations (5.30) and (5.31)]. As the discrete Schrödinger equation considered in [@SZ] is the spectral problem for the Toda lattice [@MatSal] one could refer to these systems as dressing chains of the Toda lattice. Starting from the Volterra equation and using two discrete Miura transformations, Levi and Yamilov obtained an integrable lattice equation which they regard as a direct analogue of the dressing chain [@LY equation (31)], cf. [@GHY]. In our context, we would refer to that equation as an auto-Bäcklund transformation for a modified Volterra equation.
In general, for a given integrable equation, one can ask the following questions, see Fig. \[sch\]:
1. Is there a Miura or, more generally, a Bäcklund transformation to a modified equation which has a symmetry? This then gives rise to an auto-Bäcklund transformation of the modified equation, which discretises the equation.
2. Is there an associated spectral problem, whose factorisation yields a dressing chain?
3. Does the Bäcklund transformation (1) arise in the factorisation of the spectral problem (2)?
4. Does the auto-Bäcklund transformation for the modified equation coincide with the dressing chain of the equation?
5. Can the original equation be recovered by appropriate continuum limits?
6. Is the auto-Bäcklund transformation/dressing chain integrable?
In this paper, our starting point is the dressing chain. We factorise its discrete spectral problem, which itself is an exact discretisation, cf. [@ZPZ], of the (continuous) Schrödinger equation. It turns out that the [*discrete*]{} dressing chain (of the dressing chain) coincides with the (non-autonomous) lattice Korteweg–de Vries (lKdV) equation. By studying a related Lax representation we identify a Bäcklund transformation to a modified dressing chain which admits a symmetry. The derived auto-Bäcklund transformation is again given by the lKdV equation.
In analogy to the continuous case, cf. [@ves_shab], we study the (0,$n$)-periodic reduction[^1] of the (discrete) dressing chain of the dressing chain (a.k.a. the lKdV equation), which is a two-valued correspondence (i.e., multi-valued map). We provide explicit formulas for its two branches, and establish linear growth of multi-valuedness. Moreover, we prove (in odd dimensions) that the map is Liouville integrable with respect to a quadratic Poisson structure of Lotka–Volterra type.
Background {#sodc}
==========
We clarify Fig. \[sch\] by succinctly providing some details for the KdV equation. We hope it also makes clear to the reader that how the dressing chain is related to the KdV equation is completely analogous to how the lattice KdV equation is related to the dressing chain.
The KdV equation $u_t=u_{xxx}-6uu_x$ arises as the compatibility condition, $L_t=[L,M]$, for the system of linear equations $L\phi =\lambda \phi$, $\phi_t=M\phi$ where $L$ is the Schrödinger operator $L=-D^2+u$, $M=4D^3-3(uD+Du)$, and $\lambda$ is a spectral parameter. One can check that if $v$ satisfies the mKdV equation $v_t=v_{xxx}-6(v^2+\alpha)v_x$, then $u$ given by the Miura transformation $$\begin{gathered}
\label{Miu}
u=v_x+v^2+\alpha\end{gathered}$$ satisfies the KdV equation. As the mKdV equation is invariant under $v\rightarrow-v$ another Miura transformation is given by $$\begin{gathered}
\label{Miub}
\bar{u}=-v_x+v^2+\alpha.\end{gathered}$$ Combining the two equations (\[Miu\]) and (\[Miub\]) yields an auto-Bäcklund transformation for the mKdV equation, $$\begin{gathered}
(\bar{v}+v)_x=v^2-\bar{v}^2+\alpha-\bar{\alpha},\end{gathered}$$ which coincides with the dressing chain [@ves_shab]. A related chain, which is an auto-Bäcklund transformation for the potential KdV equation, was already written down by Wahlquist and Estabrook [@WE], who used Bianchi’s permutability theorem to show that it generates hierarchies of solutions due to a nonlinear superposition principle. More general auto-Bäcklund transformations (and their interpretation as differential-difference equations) were given in [@Levi2; @Levi]. Auto-Bäcklund transformations for differential-difference equations are lattice equations, and some examples were presented in [@GY; @LY].
Darboux transformations for differential and difference equations (also known as dressing transformations) are maps of the functions and the coefficients that preserve the form of the equations [@Darb1]. They can be obtained by factorisation of operators, cf. [@BC; @FG; @IH; @Schr1; @Schr2], and they provide an effective way to construct exact solutions of a wide range of integrable equations (see, e.g., the monograph [@Darb2]).
Recall that the Schrödinger operator $L$ can be decomposed as $$\begin{gathered}
L=-(D+v)(D-v)+\alpha ,\end{gathered}$$ subject to the constraint (\[Miu\]). Darboux [@Darb1] showed that under the transformation $$\begin{gathered}
\label{tip}
\phi\mapsto \widetilde{\phi}=(D-v) \phi ,\end{gathered}$$and $L \mapsto \widetilde{L}$, where (interchanging the two factors in the decomposition) $$\begin{gathered}
\widetilde{L} =-(D-v)(D+v)+\alpha\end{gathered}$$ the form of the Schödinger equation is unchanged: $\widetilde{L} \widetilde{\phi}=\lambda \widetilde{\phi}$. The $~\widetilde{}~$ operation characterises a Darboux transformation for the Schrödinger equation $L$, if $\widetilde{L}$ is still a Schrödinger operator, i.e., $\widetilde{L} =-D^2+\widetilde{u}$ with $\widetilde{u} = -v_x+v^2+\alpha$, cf. equation . Iterated Darboux transformations result in the dependency of the functions $u$ and $\phi$ on shifts in the $~\widetilde{}~$ direction, and $\alpha$ becomes a lattice parameter. Eliminating $u,\widetilde{u}$ in the above decompositions yields the dressing chain $$\begin{gathered}
\label{dc}
(\widetilde{v}+v)_x=v^2-\widetilde{v}^2+\alpha-\widetilde{\alpha}.\end{gathered}$$Denoting $v=v_i$, $\widetilde{v}=v_{i+1}$, etc., and adding a periodic constraint, i.e., $v_{i+n}=v_{i}$ and $\alpha_{i+n}=\alpha_{i}$, one gets the finite dimensional systems of ordinary differential equations $$\begin{gathered}
\label{pdc}
(v_{i+1}+v_i)_x=v_i^2-v_{i+1}^2+\alpha_i-\alpha_{i+1} , \qquad 1\leq i\leq n,\end{gathered}$$ which was shown to be completely integrable for odd $n$ [@ves_shab].
Dressing the dressing chain
===========================
By eliminating the $x$-derivatives in the Schrödinger equation $L\phi=\lambda\phi$, using equation (\[tip\]), one obtains the discrete Schrödinger equation $$\begin{gathered}
\label{dSe}
K\phi=\lambda\phi,\qquad K=-T^2+h T+\alpha\end{gathered}$$ where $$\begin{gathered}
\label{hv}
h=-v-\widetilde{v},\end{gathered}$$ and $T\colon z\rightarrow\widetilde{z}$ represents a shift operator. The discrete Schrödinger operator $K$ is the dual, with respect to (\[tip\]), to the continuous operator $L$, cf. [@shabat1]. We note that the compatibility condition $K_x=[N,K]$, with $N=T+v$, provides a Lax representation for the dressing chain (\[dc\]).
The operator $K$ can be decomposed as, cf. [@ZPZ], $$\begin{gathered}
K = -(T+f) (T-g) - \beta .\end{gathered}$$ Here $\beta$ does not depend on the $~\widetilde{}~$ direction (as $\alpha$ does) but will depend on another discrete direction introduced below. In order that such a decomposition holds, one needs $$\begin{gathered}
\label{h}
h=\widetilde{ g}-f\end{gathered}$$ and $$\begin{gathered}
\label{fg}
f g=\alpha + \beta.\end{gathered}$$ Eliminating $f$ leads to $(h-\widetilde{ g}) g+\alpha+\beta=0$. This can be solved by posing $g = \widetilde{\psi}\psi^{-1}$, where $\psi$ is a special solution of with $\lambda = -\beta$. Now that $f$ and $g$ are well defined, we can apply the usual tactics (interchanging the two factors in the decomposition) to generate a Darboux transformation for . With $$\begin{gathered}
\label{hap}
\widehat{\phi}=G \phi,\qquad G=T-g ,\end{gathered}$$ and $$\begin{gathered}
\widehat{K}=-(T-g)(T+f)-\beta\end{gathered}$$ we have $\widehat{K} \widehat{\phi}=\lambda \widehat{\phi}$. Letting $\widehat{K} =- T^2+\widehat{h} T+\alpha$ imposes another constraint $$\begin{gathered}
\label{hh}
\widehat{h}= g-\widetilde{f},\end{gathered}$$ which together with and yields the non-autonomous lattice KdV equation (lKdV) $$\begin{gathered}
\label{dkdv}
{\widetilde{f}}-\widehat{f}=\frac{\alpha+\beta}{f}-\frac{\widetilde{\alpha}+\widehat{\beta}}{\widehat{{\widetilde{f}}}} ,\end{gathered}$$ or, in terms of $g$, $$\begin{gathered}
\label{dkdvg}
g-\widehat{\widetilde{g}}=\frac{\widetilde{\alpha}+\beta}{\widetilde{g}}-\frac{\alpha+\widehat{\beta}}{\widehat{g}} .\end{gathered}$$ Shifts in the $~\widehat{}~$ direction correspond to a second discrete direction, created by iterated Darboux transformations, and the parameter $\beta$ varies in this direction. The linear system of equations (\[dSe\]) and (\[hap\]) provides a Lax representation for the lKdV equation, $\widehat{K}G=\widetilde{G}K$.
In the light of Fig. \[sch\], equation (\[dkdv\]), or (\[dkdvg\]), is the dressing chain of the dressing chain. In analogy to the continuous case, we will consider a periodic reduction in the $~\widehat{}~$ direction, i.e., $f_{i+n}=f_i$ and $\beta_{i+n}=\beta_i$, and we take $\widetilde{\alpha}=\alpha$ to be a constant. The finite dimensional system of difference equations we will study is $$\begin{gathered}
\label{dpdc}
\widetilde{f_i}-f_{i+1}=\frac{\alpha+\beta_i}{f_i}-\frac{\alpha+\beta_{i+1}}{{\widetilde{f}}_{i+1}} ,\qquad 1\leq i\leq n.\end{gathered}$$As we make explicitly in Section \[sc\], it gives rise to a two-valued correspondence.
It would also be justified to refer to the above system (\[dkdv\]) as [*the discrete dressing chain*]{}, since its continuum limit coincides with . Using equations and , one can express $f=\widehat{\widetilde{w}}-w$, $g=\widehat{w}-\widetilde{w}$. Substituting them into gives the lattice potential KdV equation $(\widehat{\widetilde{w}}-w)(\widehat{w}-\widetilde{w})=\alpha+\beta$, whose continuum limit with respect to the $~\widehat{ }~$ direction is [@HNJ1; @Nij1] $$\begin{gathered}
\label{pDC}
(\widetilde{w}+w)_x = (\widetilde{w}-w)^2 + \alpha .\end{gathered}$$ From , and the above expressions for $f$ and $g$, one obtains $v=w-\widetilde{w}$ which relates the [*potential dressing chain*]{} to the dressing chain . We will refer to the $(0,n)$-reduction of the lKdV equation (\[dpdc\]) as the $n$-dimensional discrete dressing chain.
The modified dressing chain
===========================
One next wonders if the discrete dressing chain (or ) is the auto-Bäcklund transformation of a modified dressing chain. This is indeed the case. The Lax equation $G_x=\widehat{N}G-GN$ gives rise to the system $g(\widehat{v}-v)-g_x=\widetilde{v}-\widehat{v}-g+\widetilde{g}=0$, which together with $(h=)-v-\widetilde{v}=\widetilde{g}-f$ yields $$\begin{gathered}
\label{fbt}
v=\frac12 \left(f-g-\frac{g_x}{g}\right),\qquad \widetilde{v}=\frac12 \left(f+g+\frac{g_x}{g}\right)-\widetilde{g}\end{gathered}$$ (as well as $\widehat{v}=\frac12 \big(f-g+\frac{g_x}{g}\big))$. The system (\[fbt\]) provides a Bäcklund transformation, cf. [@HNJ1 Definition 2.1.1] between the dressing chain and the following equation $$\begin{gathered}
\label{gtx}
{\widetilde{g}}+\frac{\widetilde{\alpha}+\beta}{{\widetilde{g}}}-\frac{{\widetilde{g}}_x}{{\widetilde{g}}}=g+\frac{\alpha+\beta}{g}+\frac{g_x}{g} ,\end{gathered}$$ which we will refer to as [*the modified dressing chain*]{}. The modified dressing chain (\[gtx\]) admits the symmetry $$\begin{gathered}
\sigma(g,\widetilde{g},x)=(\widetilde{g},g,-x).\end{gathered}$$ Applying this symmetry to the right hand sides of (\[fbt\]) and transforming the left hand sides by $(v,\widetilde{v}) \rightarrow (\widetilde{\bar{v}},\bar{v})$, we obtain another Bäcklund transformation $$\begin{gathered}
\widetilde{\bar{v}}=\frac12 \left(\widetilde{f}-\widetilde{g}+\frac{\widetilde{g}_x}{\widetilde{g}}\right),\qquad
\bar{v}=\frac12 \left(\widetilde{f}+\widetilde{g}-\frac{\widetilde{g}_x}{\widetilde{g}}\right)-g.\end{gathered}$$ Combining the two Bäcklund transformations $\big(\bar{v}+\widetilde{\bar{v}}=\overline{v+\widetilde{v}}\big)$ we obtain $\widetilde{f}-g=\bar{f}-\bar{\widetilde{g}}$, which shows that the lKdV equation is an auto-Bäcklund transformation for the modified dressing chain (\[gtx\]).
Explicit formulas for the $\boldsymbol{n}$-dimensional discrete dressing\
chain, and linear growth of multivaluedness {#sc}
=========================================================================
In this section we consider the $(0,n)$-reduction of the lattice KdV equation, which is a two-valued correspondence. We give explicit formulas for both branches ($M,N$), and prove that $MNM=N$. The latter implies that the $l$-th iteration of the correspondence is $2l$-valued, cf. [@KQ Section 6.2].
In the finite reduction , without loss of generality, we set $\alpha = 0$ since it can be absorbed into the parameters $\beta_j$. Having fixed $n\in\mathbb N$ and taking $i\in{\mathcal{I}}={\left\{1,2,\ldots,n\right\}}$ subject to the periodic boundary conditions $f_{n+i}=f_i$, $\beta_{n+i}=\beta_i$ for all $i\in\mathbb {\mathcal{I}}$, the system of equations reads $$\begin{aligned}
\label{ddc}
\begin{split}
E_1\colon \quad & {\widetilde{f}}_1 +\frac{\beta_2}{{\widetilde{f}}_{2}} = f_{2} +\frac{\beta_1}{f_{1}} , \\
E_2\colon \quad & {\widetilde{f}}_2 +\frac{\beta_3}{{\widetilde{f}}_{3}} = f_{3} +\frac{\beta_2}{f_{2}} , \\
& \qquad \vdots \quad \qquad \\
E_n\colon \quad & {\widetilde{f}}_n +\frac{\beta_1}{{\widetilde{f}}_{1}} = f_{1} +\frac{\beta_n}{f_{n}} .
\end{split}\end{aligned}$$
These equations define a two-valued correspondence on $\mathbb R^n$. One solution of the system is given by $$\begin{gathered}
\label{os}
{\widetilde{f}}_i=\frac{\beta_i}{f_i} ,\qquad i\in {\mathcal{I}}\end{gathered}$$ (which is ${\widetilde{f}}_i=g_i$, cf. (\[fg\])). This defines a map $$\begin{gathered}
N\colon \ (f_1,f_2,\ldots,f_n)\mapsto \left(\frac{\beta_1}{f_1},\frac{\beta_2}{f_2},\ldots,\frac{\beta_n}{f_n}\right),\end{gathered}$$ which is an involution. The other solution of the system gives rise to a more intriguing map on $\mathbb R^n$, which will be denoted by $M$. We next provide explicit formulas for $M$ and for its inverse.
Consider the finite version of system defined by choosing $n\in\mathbb N$ and restricting $i\in{\mathcal{I}}$ subject to the open boundary condition $f_{n+1}=\beta_{n+1}=0$. The resulting system then takes the form $$\begin{gathered}
{\widetilde{f}}_i +\frac{\beta_{i+1}}{{\widetilde{f}}_{i+1}} =f_{i+1} +\frac{\beta_i}{f_{i}} , \qquad \text{for } i=1,2,\ldots, n-1, \qquad \text{and} \qquad {\widetilde{f}}_n =\frac{\beta_n}{f_{n}},\end{gathered}$$ whose unique solution is given by the involution .
In order to describe the nontrivial solution of we introduce some notation. With and $k\in{\mathcal{I}}$ we consider $\mathbb R^k$ with coordinates $f_1, f_2, \ldots, f_k$. We fix the parameters $\beta_1, \beta_2, \ldots, \beta_n$, and define functions $F_k\colon \mathbb R^{k}\rightarrow \mathbb R$ by $F_1(f_1)=1$ and $$\begin{gathered}
\label{functionF}
F_k=F_k(f_1, f_2, \ldots, f_k)=f_1f_2^2f_3^2\cdots f_{k-1}^2f_k ,\qquad k>1.\end{gathered}$$ We also define a function $G:\mathbb R^{2n} \rightarrow \mathbb R$ by $$\begin{aligned}
\label{functionG}
\begin{split}
G&=G(f_1, f_2, \ldots, f_n,\beta_1,\beta_2,\ldots,\beta_n)\\
&=\sum_{i=0}^{n-1}
\left( \prod_{j=1}^i\beta_j \cdot F_{n-i}(f_{i+1}, f_{i+2}, \ldots, f_n)\right)\\
&=F_n(f_1, f_2, \ldots, f_n)+\beta_1F_{n-1}(f_2, f_3, \ldots, f_n)+\dots+\beta_1\beta_2\cdots\beta_{n-1}.
\end{split}\end{aligned}$$ For $n=3$ the function $G$ reads $$\begin{gathered}
G=f_1f_2^2f_3+\beta_1f_2f_3+\beta_1\beta_2.\end{gathered}$$ We will make use of the following cyclic permutation $\tau\colon {\mathcal{I}}\rightarrow{\mathcal{I}}$ $$\begin{gathered}
\tau_i= \begin{cases}
i+1, &\text{if } i< n, \\
1 , &\text{if } i= n,
\end{cases}\end{gathered}$$ and of the involution $\sigma\colon {\mathcal{I}}\rightarrow{\mathcal{I}}$ defined by $$\begin{gathered}
\sigma_i=n+1-i.\end{gathered}$$ Simply stated, the permutation $\tau$ is a shift modulo $n$, and $\sigma$ is to reverse the elements of ${\mathcal{I}}$. By some abuse of notation we will write, for any function $H$ depending on the variables $f_1, $ $f_2, \ldots, f_n$ and the parameters $\beta_1,\beta_2,\ldots,\beta_n$, $$\begin{aligned}
\tau H(f_1, f_2, \ldots, f_n,\beta_1,\beta_2,\ldots,\beta_n)&=H(f_{\tau_1}, f_{\tau_2}, \ldots, f_{\tau_n},\beta_{\tau_1},\beta_{\tau_2},\ldots,\beta_{\tau_n})\\
&=H(f_2, f_3, \ldots, f_1,\beta_2,\beta_3,\ldots,\beta_1) ,\end{aligned}$$ and similarly $$\begin{aligned}
\sigma H(f_1, f_2, \ldots, f_n,\beta_1,\beta_2,\ldots,\beta_n)&= H(f_{\sigma_1}, f_{\sigma_2}, \ldots, f_{\sigma_n},\beta_{\sigma_1},\beta_{\sigma_2},\ldots,\beta_{\sigma_n})\\
&=H(f_n, f_{n-1}, \ldots, f_1,\beta_n,\beta_{n-1},\ldots,\beta_1) .\end{aligned}$$ For example, for $n=3$ we have $$\begin{gathered}
\tau G=f_1f_2f_3^2+\beta_2f_1f_3+\beta_2\beta_3 ,\qquad \sigma G=f_1f_2^2f_3+\beta_3f_1f_2+\beta_2\beta_3 .\end{gathered}$$ A useful property of the above-defined functions is that, for any $n$, the expression $$\begin{gathered}
\label{fixedpoint}
f_1f_nG-\beta_1\tau G=f_1^2f_2^2\cdots f_n^2-\beta_1\beta_2\cdots\beta_n\end{gathered}$$ is invariant under $\tau$ and $\sigma$. The following formula, which can be easily proved, is also useful. For any function $H$ depending on $f_1, f_2, \ldots, f_n$, $\beta_1,\beta_2,\ldots,\beta_n$, we have $\tau\sigma\tau H=\sigma H$, which implies, for all $i$, $$\begin{gathered}
\label{prod_tau_sigma}
\sigma\tau^{i} H=\tau^{n-i}\sigma H .\end{gathered}$$
We now define functions $$\begin{gathered}
\label{nontrivial_map}
M_i=f_{i-1}\frac{\tau^{i-1}G}{\tau^iG},\end{gathered}$$ where the indices are considered modulo $n$ and in the set ${\mathcal{I}}$, in particular $$\begin{gathered}
M_1=f_n\dfrac{G}{\tau G}.\end{gathered}$$
The map $M\colon (f_1,f_2,\ldots,f_n)\mapsto(M_1,M_2,\ldots,M_n)$, where $M_i$ is defined by , is a solution of the system .
By definition , we have $M_i=\tau M_{i-1}$ for all $i$ and a similar property holds for the equations in (each equation is obtained by applying $\tau$ to the previous one). Hence, it is enough to show that the functions $M_i$ satisfy the equation $E_1$. Taking ${\widetilde{f}}_1=M_1$ and ${\widetilde{f}}_2=M_2$, we have to verify $$\begin{aligned}
M_1 +\frac{\beta_2}{M_2} = f_{2} +\frac{\beta_1}{f_{1}} \iff& f_n\frac{G}{\tau G} +\frac{\beta_2}{f_1}\frac{\tau^2 G}{\tau G} = f_2+\frac{\beta_1}{f_1} \\
\iff& f_1f_nG+\beta_2\tau^2 G = (f_1f_2+\beta_1)\tau G \\
\iff& f_1f_nG-\beta_1\tau G= \tau(f_1f_n G-\beta_1\tau G)\end{aligned}$$ or equivalently that $f_1f_nG-\beta_1\tau G$ is fixed under $\tau$, which holds due to .
The maps $M$ and $N$ satisfy the following relation.
\[lem\]The map $N$ is a reversing symmetry of $M$.
The statement entails $$\begin{gathered}
MN=NM^{-1}.\end{gathered}$$ Applying the involution $\sigma$ to all indices in system , and interchanging $f_i\leftrightarrow {\widetilde{f}}_i$ one observes that $$\begin{gathered}
E_j\big(f_i,{\widetilde{f}}_i\big)=E_{n-j}\big({\widetilde{f}}_{\sigma_i},f_{\sigma_i}\big),\end{gathered}$$ for $1\leq j< n$ and $E_n\big(f_i,{\widetilde{f}}_i\big)=E_n\big({\widetilde{f}}_{\sigma_i},f_{\sigma_i}\big)$. If we write the nontrivial solution as ${\widetilde{f}}_j=M_j(f_i)$, then we also have $f_{\sigma_j}=M_j \big({\widetilde{f}}_{\sigma_i}\big)$, and applying $\sigma$ to the $j$ index (which just enumerates the functions), one has $$\begin{gathered}
f_j=M_{\sigma_j}\big({\widetilde{f}}_{\sigma_i}\big)=M_j^{-1}\big({\widetilde{f}}_i\big).\end{gathered}$$ Thus the inverse map is $$\begin{gathered}
M^{-1}=\sigma M \sigma,\end{gathered}$$ which as a function of the $f_i$ has components $$\begin{gathered}
M^{-1}_j = \sigma \left( f_{\sigma_j-1} \frac{ \tau^{\sigma_j-1}G}{\tau^{\sigma_j}G}\right) = f_{j+1} \frac{\sigma \tau^{n-j}G}{\sigma\tau^{n-j+1}G} = f_{j+1} \frac{\tau^{j}\sigma G}{\tau^{j-1}\sigma G}.\end{gathered}$$ The latter formula is obtained using . It follows immediately that $M^{-1}_{j+1}=\tau M^{-1}_j$. Therefore it is enough to show that $(MN)_1=\big(NM^{-1}\big)_1$, that is $$\begin{gathered}
\label{to_be_proved}
\frac{\beta_n G\big(\frac{\beta_1}{f_1},\ldots,\frac{\beta_n}{f_n}\big)}{f_n \tau G\big(\frac{\beta_1}{f_1},\ldots,\frac{\beta_n}{f_n}\big)}
= \frac{\beta_1\sigma G(f_1,\ldots,f_n)}{f_2 \tau\sigma G(f_1,\ldots,f_n)}.\end{gathered}$$ Using the formulas and , one has $$\begin{aligned}
G\left(\frac{\beta_1}{f_1},\ldots,\frac{\beta_n}{f_n}\right)&=
\sum_{i=0}^{n-1} \left(\prod_{j=1}^i\beta_j\!\right)F_{n-i}\left(\frac{\beta_{i+1}}{f_{i+1}},\ldots,\frac{\beta_n}{f_n}\right) =\sum_{i=0}^{n-1}\left(\prod_{j=1}^i\beta_j\!\right)\frac{F_{n-i}(\beta_{i+1},\ldots,\beta_n)}{F_{n-i}(f_{i+1},\ldots,f_n)}\\
&=\frac{\prod\limits_{j=1}^{n-1}\beta_j}{F_n(f_1,\ldots,f_n)}\sum_{i=0}^{n-1}\left(\prod_{j=i+2}^{n}\beta_j\right)F_{i+1}(f_1,\ldots,f_{i+1}) ,\end{aligned}$$ and similarly $$\begin{gathered}
\tau G\left(\frac{\beta_1}{f_1},\ldots,\frac{\beta_n}{f_n}\right) = \frac{\prod\limits_{j=1}^{n-1}\beta_{j+1}}{F_n(f_2,\ldots,f_n,f_1)}
\sum_{i=0}^{n-1}\left(\prod_{j=i+2}^{n}\beta_{j+1}\right)F_{i+1}(f_2,\ldots,f_{i+2}),\\
\tau\sigma G(f_1,\ldots,f_n) =\sum_{i=0}^{n-1} \left(\prod_{j=1}^i\beta_{n+2-j}\right)F_{n-i}(f_{i+2},f_{i+1},\ldots,f_2) ,\end{gathered}$$ and $$\begin{gathered}
\sigma G(f_1,\ldots,f_n) =\tau\sigma\tau G(f_1,\ldots,f_n) =\sum_{i=0}^{n-1}\left(\prod_{j=1}^i\beta_{n+1-j}\right)F_{n-i}(f_{i+1},f_{i},\ldots,f_1) .\end{gathered}$$ Combining all these leads to .
As a corollary of Lemma \[lem\], using the fact that $N$ is an involution, it follows that $MN$ and $NM$ are involutions, and hence $M$ can be written as a composition of two involutions, $M=(MN)N=N(NM)$. Furthermore, Lemma \[lem\] implies the relations: $MNM=NNN$ and $MNN=NNM$. These relations are also satisfied by the branches of the quotient-difference $(n, 0)$-correspondence. In [@KQ Section 6.2] it is proved that the $l$-th iteration of such a correspondence is $2l$-valued.
Complete integrability of the odd-dimensional discrete\
dressing chain {#fs}
=======================================================
In this section we show that the correspondence $(M,N)$ is Liouville integrable with respect to a quadratic Poisson structure which is of Lotka–Volterra type. Our main result is the following theorem.
\[thr:ddc\_int\]For odd $n$, the correspondence defined by is Liouville integrable.
Before proving the theorem, we introduce the relevant Poisson structures and provide some of their basic properties.
Lotka–Volterra Poisson structures
---------------------------------
Lotka–Volterra Poisson structures are homogeneous quadratic, and they are defined on $\mathbb R^n$ by the formulas$$\begin{gathered}
\label{LVPB}
{\left\{x_i,x_j\right\}}_q=A_{i,j}x_ix_j,\qquad i,j\in{\mathcal{I}},\end{gathered}$$ where $A$ is a constant skew-symmetric matrix.
The rank of this Poisson structure is equal, at a generic point, to the rank of the constant matrix $A$. Each null-vector of the matrix $A$ is associated to a Casimir of the corresponding Poisson bracket. If $\textbf{v}=(v_1,v_2,\ldots,v_n)$ is such that $\textbf{v}A=0$, then the function $\prod\limits_{i=1}^nx_i^{v_i}$ is a Casimir of the Poisson bracket. Two linearly independent null-vectors correspond to two functionally independent Casimirs (for a proof see [@polpoisson Example 8.14]).
In what follows we consider the Lotka–Volterra structures where $A$ is the $n\times n$ skew-symmetric matrix with its upper triangular part defined by $$\begin{gathered}
\label{eq:Bogo_mat}
A_{i,j}=(-1)^{j-i+1} ,\qquad 1\leq i<j \leq n .\end{gathered}$$ The rank of the matrix $A$ is $n$ when $n$ is even and $n-1$ when $n$ is odd with the null-vector $\textbf{v}=(1,1,\ldots,1)$; a Casimir of the corresponding Poisson structure is the function $x_1x_2\cdots x_n$. With $H=x_1+x_2+\cdots+x_n$, the Hamiltonian vector field $\{\cdot, H\}_q$ defines a system of differential equations which, up to a simple change of variables, is isomorphic to the Bogoyavlenskij lattice [@bog2; @bog3; @damianou_evr_kass_van; @pol2014].
A simpler Poisson structure is the constant Poisson structure defined by the brackets $$\begin{gathered}
\label{CPB}
{\left\{x_i,x_j\right\}}_c=B_{i,j} ,\end{gathered}$$ where $B$ is a constant $n\times n$ skew-symmetric matrix. After some tedious but straightforward calculations (see [@PPP2017 Proposition 3]), one proves that the Poisson structures and are compatible if and only if $$\begin{gathered}
\label{eq:expl_mat_B}
\begin{cases}
B_{i,j}=0 & \text{for all } |j-i|>1, \text{ when } n \text{ is even,}\\
B_{i,j}=0 & \text{for all } 1<|j-i|<n-1, \text{ when } n \text{ is odd.}
\end{cases}\end{gathered}$$
For $n$ odd, let ${\left\{\cdot ,\cdot\right\}}_{\bf{b}}={\left\{\cdot ,\cdot\right\}}_q+{\left\{\cdot ,\cdot\right\}}_c$ denote the sum of the brackets defined by (\[LVPB\]), (\[eq:Bogo\_mat\]) and (\[CPB\]), (\[eq:expl\_mat\_B\]) where the subscript $\bf{b}$ is the vector ${\bf b}=(b_{1,2},b_{2,3},\ldots,b_{n-1,n}, b_{1,n})$, i.e., the non-zero elements of the matrix $B$. With $H=x_1+x_2+\cdots+x_n$, the Hamiltonian vector field $\{\cdot, H\}_{\bf{b}}$ defines a system of differential equations which can be transformed to the dressing chain . The Poisson structure ${\left\{\cdot ,\cdot\right\}}_{\bf b}$ is of rank $n-1$ but with a more complicated Casimir than the product $x_1x_2\cdots x_n$ (see [@PPP2017; @ForHon] for an explicit construction of this Casimir).
Complete integrability
----------------------
We start by showing that the maps $M$ and $N$, defined in Section \[sc\], preserve the Poisson structure ${\left\{\cdot ,\cdot\right\}}_{\bf b}={\left\{\cdot ,\cdot\right\}}_q+{\left\{\cdot ,\cdot\right\}}_c$ with ${\bf b}=(-\beta_2,-\beta_3,\ldots,-\beta_n,\beta_1)$. To this end we define two additional maps $\phi,\psi\colon \mathbb R^n\rightarrow\mathbb R^n$, $$\begin{aligned}
\phi(f_1,f_2,\ldots,f_n)&=(f_1+g_2,f_2+g_3,\ldots,f_n+g_1) ,\\
\psi(f_1,f_2,\ldots,f_n)&=(f_2+g_1,f_3+g_2,\ldots,f_1+g_n) .\end{aligned}$$
\[poissonmaps\] For $n$ odd, the maps $M$, $N$, $\phi$ and $\psi$ are Poisson maps as follows
1. $\phi\colon (\mathbb R^n,{\left\{\cdot ,\cdot\right\}}_q)\rightarrow (\mathbb R^n,{\left\{\cdot ,\cdot\right\}}_{{\bf b}})$;
2. $\psi\colon (\mathbb R^n,{\left\{\cdot ,\cdot\right\}}_q)\rightarrow (\mathbb R^n,{\left\{\cdot ,\cdot\right\}}_{{\bf b}})$;
3. $M\colon (\mathbb R^n,{\left\{\cdot ,\cdot\right\}}_q)\rightarrow (\mathbb R^n,{\left\{\cdot ,\cdot\right\}}_q)$;
4. $N\colon (\mathbb R^n,{\left\{\cdot ,\cdot\right\}}_q)\rightarrow (\mathbb R^n,{\left\{\cdot ,\cdot\right\}}_q)$.
The proof of items $(1)$, $(2)$ and $(4)$ follows from straightforward computations. For example, for any $i=1,2,\ldots, n$ $$\begin{aligned}
{\left\{f_i,f_{i+1}\right\}}_{\bf b}\circ \phi& =(f_i+g_{i+1})(f_{i+1}+g_{i+2})-
\beta_{i+1}=f_if_{i+1}+f_ig_{i+2}+g_{i+1}g_{i+2}\\
& =\{f_i+g_{i+1},f_{i+1}+g_{i+2}\}_q ,\\
{\left\{f_i,f_{i+1}\right\}}_{\bf b}\circ \psi& =(f_{i+1}+g_i)(f_{i+2}+g_{i+1})-\beta_{i+1}=f_{i+1}f_{i+2}+g_if_{i+2}+g_ig_{i+1}\\
&={\left\{f_{i+1}+g_i,f_{i+2}+g_{i+1}\right\}}_q ,\end{aligned}$$ where the indices are considered modulo $n$ and in the set ${\mathcal{I}}$. Item (3) follows from items (1) and (2) combined with $$\begin{gathered}
\phi\circ M=\psi \qquad \text{and} \qquad \phi\circ N=\psi,\end{gathered}$$ which are a consequence of .
The involution $N$ preserves any Poisson structure of Lotka–Volterra form. It follows that item (4), of the previous proposition, is (trivially) true for all $n\in\mathbb N$. However, this is not the case for items $(1)-(3)$ as for even $n$ the equations in the previous proof do not hold for $i=n$. Note, for even $n$ the bracket ${\left\{\cdot ,\cdot\right\}}_{{\bf b}}$ is not Poisson, see condition .
To derive sufficiently many independent invariants for the map , we employ a matrix version of the Lax representation. Let $\widetilde{\Phi_i}={\mathcal{V}}_i \Phi_i$, $\widehat{\Phi_i}=\Phi_{i+1}={\mathcal{G}}_i \Phi_i$, where $$\begin{gathered}
{\mathcal{V}}_i=
\begin{pmatrix}
0 & 1 \\
-\lambda & \widetilde{g_i}-f_i
\end{pmatrix},
\qquad
{\mathcal{G}}_i=
\begin{pmatrix}
-g_i & 1 \\
-\lambda & -f_i
\end{pmatrix}\end{gathered}$$ and $\Phi=\big(\phi,\widetilde{\phi}\big)^T$. The compatibility condition is now written (modulo the periodic condition $n+j=j$) as ${\mathcal{V}}_{i+1} {\mathcal{G}}_i=\widetilde{{\mathcal{G}}}_i {\mathcal{V}}_i$. Then one has $$\begin{gathered}
\widetilde{{\mathcal{G}}}_n \widetilde{{\mathcal{G}}}_{n-1} \cdots \widetilde{{\mathcal{G}}}_1={\mathcal{V}}_1 {\mathcal{G}}_n {\mathcal{G}}_{n-1} \cdots {\mathcal{G}}_1 {\mathcal{V}}_1^{-1} ,\end{gathered}$$ which implies that the (monodromy) matrix ${\mathcal{L}}={\mathcal{G}}_n {\mathcal{G}}_{n-1} \cdots {\mathcal{G}}_1$ is isospectral. The eigenvalues of ${\mathcal{L}}$ (and therefore its trace) are invariants of our map .
Using a decomposition of the Lax matrix ${\mathcal{G}}_i$ similar to the one used in [@PPP2017; @ves_shab] (explained in a more general manner in [@tran]), we are able to calculate the trace of ${\mathcal{L}}$ and provide a succinct formula for the invariants. Futhermore, using the results of [@ves_shab] we show that these invariants are in involution with respect to the Poisson bracket with matrix and therefore complete the proof of Theorem \[thr:ddc\_int\].
Let $r=\frac{n-1}{2}\in\mathbb{N}$. There exist functions $I_0, I_1, \ldots, I_r$, which are polynomials of the variables $$\begin{gathered}
x_1=-(g_2+f_1),\quad x_2=-(g_3+f_2),\quad \ldots,\quad x_n=-(g_1+f_n)\end{gathered}$$ of degree $\deg(I_i)=2i+1$, such that $\operatorname{tr}({\mathcal{L}})=\sum\limits_{i=0}^r I_i\lambda^i .$ With $$\begin{gathered}
D_i=\frac{\partial^2}{\partial x_i \partial x_{i+1}},\qquad D=\sum_{i\in{\mathcal{I}}} D_i ,\end{gathered}$$ where $x_{n+1}=x_1$ is understood, we have $$\begin{gathered}
I_0=\prod_{i\in{\mathcal{I}}} (1-\beta_i D_i) \prod_{j\in{\mathcal{I}}} x_j , \qquad I_j=\frac{(-1)^j}{j!}D^j I_{0} .\end{gathered}$$ Furthermore, these functions are in involution with respect to the Poisson bracket ${\left\{\cdot ,\cdot\right\}}_q$.
We decompose the matrix ${\mathcal{G}}_i$ as follows $$\begin{gathered}
{\mathcal{G}}_i={\mathcal{A}}^i_i-(\lambda+\beta_i){\mathcal{K}},\end{gathered}$$ where $$\begin{gathered}
{\mathcal{K}}=
\begin{pmatrix}
0 & 0 \\
1 & 0
\end{pmatrix}, \qquad
{\mathcal{A}}^i_j=\begin{pmatrix}
-1 \\
f_i
\end{pmatrix}
\cdot \begin{pmatrix}
g_j & -1
\end{pmatrix}.\end{gathered}$$This structure simplifies the computations. For example we can immediately verify the following properties $$\begin{gathered}
{\mathcal{K}}^2=\textbf{0},\qquad {\mathcal{A}}^i_j {\mathcal{A}}^k_l=-(g_j+f_k) {\mathcal{A}}^i_l,\qquad {\mathcal{A}}^i_j {\mathcal{K}}{\mathcal{A}}^k_l={\mathcal{A}}^i_l,\qquad {\mathcal{K}}{\mathcal{A}}^i_j {\mathcal{K}}={\mathcal{K}}.\end{gathered}$$ The monodromy matrix is, with $y_i=-(\lambda+\beta_i)$, in the form $$\begin{aligned}
{\mathcal{L}}&=\big({\mathcal{A}}^n_n+y_n{\mathcal{K}}\big)\big({\mathcal{A}}^{n-1}_{n-1}+y_{n-1}{\mathcal{K}}\big)\cdots\big({\mathcal{A}}^1_1+y_1{\mathcal{K}}\big)\\
&={\mathcal{A}}^n_n{\mathcal{A}}^{n-1}_{n-1}\cdots {\mathcal{A}}^1_1 + \sum_{i\in{\mathcal{I}}} y_i{\mathcal{A}}^n_n{\mathcal{A}}^{n-1}_{n-1}\cdots {\mathcal{A}}^{i+1}_{i+1}{\mathcal{K}}{\mathcal{A}}^{i-1}_{i-1}\cdots {\mathcal{A}}^1_1 + \sum_{i,j\in{\mathcal{I}}} \cdots\\
&=x_nx_{n-1}\cdots x_2{\mathcal{A}}^n_1 + \sum_{i\in{\mathcal{I}}} y_i x_nx_{n-1}\cdots x_{i+2}x_{i-1}\cdots x_1{\mathcal{A}}^n_1 + \sum_{i,j\in{\mathcal{I}}} \cdots,\end{aligned}$$ and its trace $$\begin{aligned}
\operatorname{tr}({\mathcal{L}})&=\prod_{i\in{\mathcal{I}}} x_i + \sum_{i\in{\mathcal{I}}}\frac{y_i}{x_{i+1}x_{i}}\prod_{j\in{\mathcal{I}}} x_j + \sum_{i>j+1\in{\mathcal{I}}}\frac{y_i}{x_{i+1}x_{i}} \frac{y_j}{x_{j+1}x_{j}} \prod_{k\in{\mathcal{I}}} x_j + \cdots \\
&=\prod_{i\in{\mathcal{I}}} \left(1+y_iD_i\right) \prod_{j\in{\mathcal{I}}}x_i=\sum_{i=0}^k I_i\lambda^i.\end{aligned}$$ Substituting $\lambda=0$ yields the expression for $I_0$, i.e., $$\begin{gathered}
\label{I0}
I_0=(1-\beta_1D_1)(1-\beta_2D_2)\cdots(1-\beta_nD_n)\prod_{i\in{\mathcal{I}}} x_i ,\end{gathered}$$ while $$\begin{aligned}
I_1&=\sum_{j\in{\mathcal{I}}}(1-\beta_1D_1)(1-\beta_2D_2)\cdots(-D_j)\cdots(1-\beta_nD_n)\prod_{i\in{\mathcal{I}}} x_i ,\\
I_2&=\sum_{j<k\in{\mathcal{I}}}(1-\beta_1D_1)(1-\beta_2D_2)\cdots(-D_j)\cdots(-D_k)\cdots(1-\beta_nD_n)\prod_{i\in{\mathcal{I}}} x_i ,\\
&\vdots\end{aligned}$$ Applying $D$ to $I_0$ gives a sum of similar products where the $j$-th term $(1-\beta_jD_j)$ is replaced by $D_j$, which is equal to $-I_1$. Similarly, applying this operator $k$ times yields $(-1)^k I_k$ $k!$ times, namely once for each permutation of $k$ indices. This proves the first part of the proposition.
To prove the second part of the proposition, we first note that the invariants $I_i$ (as functions of $x_i$), as obtained from the trace of ${\mathcal{L}}$, coincide with the invariants that Veselov and Shabat provided for the continuous dressing chain [@ves_shab]. In that paper, using the Lenard–Magri scheme, they proved that the invariants $I_i$ are in involution with respect to the Poisson bracket ${\left\{\cdot ,\cdot\right\}}_{\bf b}={\left\{\cdot ,\cdot\right\}}_q+{\left\{\cdot ,\cdot\right\}}_c$. Therefore, in order to prove that the invariants $I_i$ (considered as functions of $f_i$) are in involution with respect to ${\left\{\cdot ,\cdot\right\}}_q$, it suffices to show that the map $$\begin{gathered}
\mathbb R^n \rightarrow \mathbb R^n, \qquad (f_1, f_2, \ldots, f_n)\mapsto (x_1, x_2, \ldots, x_n)\end{gathered}$$ is a Poisson map between $(\mathbb R^n,{\left\{\cdot ,\cdot\right\}}_q)$ and $(\mathbb R^n,{\left\{\cdot ,\cdot\right\}}_{\bf b})$. This is precisely item $(1)$ of Lemma \[poissonmaps\].
The expression for $I_0$ (\[I0\]) in terms of $f_i$, $g_i$, and $\beta_i$ is quite cumbersome, e.g., for $n=3$, $$\begin{gathered}
I_0 =-f_{{1}}f_{{2}}f_{{3}}-f_{{1}}f_{{2}}g_{{1}}-f_{{1}}f_{{3}}g_{{3}}-f_{{1}}g_{{1}}g_{{3}}-f_{{2}}f_{{3}}g_{{2}}-f_{{2}}g_{{1}}g_{{2}}-f_{{3}}g_{{2}}g_{{3}}\\
\hphantom{I_0 =}{} -g_{{1}}g_{{2}}g_{{3}}+\beta_{{1}}f_{{2}}+\beta_{{3}}f_{{1}}+\beta_{{1}}g_{
{3}}+\beta_{{2}}f_{{3}}+\beta_{{2}}g_{{1}}+\beta_{{3}}g_{{2}} .\end{gathered}$$ However, when imposing the relation $f_ig_i=\beta_i$, the expression simplifies drastically, and we have $I_0=-(f_1f_2f_3+g_1g_2g_3)$. The fact that a similar expression can be obtained for any $n$ can be seen from $$\begin{gathered}
{\mathcal{L}}\mid_{\lambda=0}=\prod_{i=1}^n\begin{pmatrix}-g_i&1\\0&-f_i\end{pmatrix}=(-1)^n
\begin{pmatrix}\displaystyle \prod_{i=1}^n g_i&\displaystyle -\sum_{i=1}^n\prod_{j=1}^{i-1}f_i\prod_{k=i+1}^{n}g_i\\0 &\displaystyle\prod_{i=1}^nf_i\end{pmatrix}.\end{gathered}$$
For $n$ even the map $M$ is anti-volume preserving and for odd $n$ it is volume preserving. The map $N$ is measure preserving when $n$ is even and anti-measure preserving when $n$ is odd. The density of the measure is $\prod\limits_{i=1}^n\frac{1}{f_i}$.
Acknowledgements {#acknowledgements .unnumbered}
----------------
This work was supported by the Australian Research Council, by the China Strategy Implementation Grant Program of La Trobe University, by the NSFC (No. 11601312) and by the Shanghai Young Eastern Scholar program (2016-2019).
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[^1]: The ($n$,0)-periodic reduction gives rise to the same maps, up to a minus sign.
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{
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[email protected]\
Abstract {#abstract .unnumbered}
========
A formal symplectic structure on $R \times M$ is constructed for the unsteady flow of an incompressible viscous fluid on a three dimensional domain $M$. The evolution equation for the helicity density is expressed via the divergence of the associated Liouville vector field that generates symplectic dilation. For an inviscid fluid this equation reduces to a conservation law. As an application the symplectic dilation is used to generate Hamiltonian automorphisms of the symplectic structure which are then related to the symmetries of the velocity field.
The helicity which is first discovered in [@skr66] has been recognized to be an important ingradient of the problem of relationship between invariants of fluid motion and the topological structure of the vorticity field [@FRI]-[@ark92]. For three-dimensional flows its ergodic and topological interpretations were introduced and investigated in [@mof69]-[@gik92]. It has also been studied in the context of Noether theorems [@mor61]-[@pad98]. Kinematical aspects of helicity invariants in connection with the particle relabelling symmetries were discussed in [@hg99].
In this work, we shall show that there is also a dynamical content of the helicity density in the sense that the information contained in the Eulerian dynamical equations can be represented in the framework of symplectic geometry by a current vector field governing the dynamics of helicity. More precisely, starting from the Navier-Stokes equations of incompressible fluids we shall construct helicity four-vector whose divergence will define the time-evolution of helicity density. The dynamical properties of the fluid, such as viscosity, are implicit in this vector field. The evolution equation for the helicity density reduces to a conservation law for inviscid Euler flows. For fluid dynamical content of this work we shall refer to [@FRI] and the necessary mathematical background can be found in [@via]-[@LM].
The Navier-Stokes equation for a viscous incompressible fluid in a bounded domain $M \subset R^3$ is $${\partial {\bf v} \over \partial t} + {\bf v} \cdot \nabla {\bf v}
= - \nabla p +\nu \nabla^2 {\bf v} \label{euler}$$ where ${\bf v}$ is the divergence-free velocity field tangent to the boundary of $M$, $p$ is the pressure per unit density and $\nu$ is the kinematic viscosity. The identity ${\bf v} \cdot \nabla {\bf v} =
{1 \over 2} \nabla | {\bf v}|^{2} - {\bf v} \times (\nabla \times {\bf v} )$ can be used to bring the equation (\[euler\]) into the form $${\partial {\bf v} \over \partial t} -
{\bf v} \times (\nabla \times {\bf v}) = \nu \nabla^2 {\bf v}
- \nabla (p + {1 \over 2} v^{2} ) \label{reuler}$$ and in terms of the vorticity field ${\bf w} \equiv \nabla \times {\bf v}$ this gives $${\partial {\bf w} \over \partial t} -
\nabla \times ( {\bf v} \times {\bf w} )
= \nu \nabla^2 {\bf w} \;. \label{weq}$$ For a fluid with a potential $\varphi$ and velocity field ${\bf v}$ the densities $${\cal H} = {1 \over 2} {\bf v} \cdot \nabla \times {\bf v} \;,\;\;\;
{\cal H}_w = {1 \over 2} {\bf w} \cdot \nabla \times {\bf w} \;,\;\;\;
q= {\bf w} \cdot \nabla \varphi =w(\varphi)$$ will be called helicity, vortical helicity and potential vorticity, respectively.
For a velocity field satisfying Eqs.(\[euler\]) and for $q \neq 2 \nu {\cal H}_w$ the two-form $$\Omega_{\nu} = - (\nabla \varphi + {\bf v} \times {\bf w}
- \nu \nabla \times {\bf w} ) \cdot d{\bf x} \wedge dt
+ {\bf w} \cdot (d{\bf x} \wedge d{\bf x}) \label{sympd}$$ is symplectic on $I \times M$ where $I$ is an open interval in $R$. Moreover, it is exact, $\Omega_{\nu} = d \theta$ with the Liouville (or canonical) one-form $$\theta = - ( \varphi + p + {1 \over 2} v^{2} )
\, dt + {\bf v} \cdot d{\bf x} \label{cone2}$$ which is independent of the viscosity $\nu$.
[**Proof:**]{} $\Omega_{\nu}$ is closed by Eq. (\[weq\]) and the divergence-free property of the vorticity field. The non-degeneracy follows from the recognition that the density in the symplectic volume $\Omega_{\nu} \wedge \Omega_{\nu} /2$ is the function $q-2\nu {\cal H}_w$ which is assumed to be non-zero. The exactness can be verified using Eq. (\[reuler\]). $\bullet$
For an arbitrary smooth function $f$ of $(t,x)$ the unique Hamiltonian vector field $X_f$ defined by the symplectic two-form (\[sympd\]) via $i(X_f)(\Omega_{\nu})= -df$ is given by $$X_{f}={1 \over q-2 \nu {\cal H}_{w} }
[ - w(f) ({\partial \over \partial t}+v)
+{df \over dt} w + (( \nabla \varphi -\nu \nabla \times {\bf w})
\times \nabla f) \cdot \nabla ] \;.$$ Here, $d/dt$ denotes the convective derivative $\partial_t+ {\bf v}
\cdot \nabla$ which, viewed as a vector field on $I \times M$, is not Hamiltonian. In fact, with the notation $v \equiv {\bf v} \cdot
\nabla$, one can check that the one-form $$i(\partial_t+v)(\Omega_{\nu})=(\nabla \varphi - \nu \nabla \times {\bf w})
\cdot (d {\bf x} - {\bf v} dt)$$ is not closed and hence $\partial_t+v$ is not even locally Hamiltonian.
Next proposition describes invariantly the connection between the symplectic structure (\[sympd\]) and the the helicity density.
\[ee\] The identity $$d (\theta \wedge \Omega_{\nu} ) -
\Omega_{\nu} \wedge \Omega_{\nu} \equiv 0 \label{helc}$$ gives the equation $${ \partial {\cal H} \over \partial t} + \nabla \cdot
( {\cal H} {\bf v} + {1 \over 2} ( p - {1 \over 2} {\bf v}^2 ) {\bf w} )
= {\nu \over 2} {\bf v} \cdot \nabla^2 {\bf w} - \nu {\cal H}_w \label{hel}$$ for the evolution of helicity density.
[**Proof:**]{} We have $\Omega_{\nu} \wedge \Omega_{\nu} = -2(q-2\nu {\cal H}_w)
dx \wedge dy \wedge dz \wedge dt$ and we compute $$\nabla \cdot [ \varphi {\bf w} + {\bf v} \times ( \nabla \varphi -
\nu \nabla \times {\bf w}) ] =
2q - 2 \nu {\cal H}_w - \nu {\bf v} \cdot \nabla^2 {\bf w}$$ for the derivative of certain terms in the expression $$\begin{aligned}
& & \theta_{\nu} \wedge \Omega_{\nu} =
2{\cal H} \, dx \wedge dy \wedge dz - \nonumber \\
& & \;\;\;\; [ ( \varphi +p - {1 \over 2} v^2 ) {\bf w}
+ 2{\cal H} {\bf v}
+ {\bf v} \times ( \nabla \varphi -
\nu \nabla \times {\bf w}) ] \cdot
d{\bf x} \wedge d{\bf x} \wedge dt \;\;\;\end{aligned}$$ for the three-form. Putting them together in the identity (\[helc\]) we obtain Eq. (\[hel\]). Upon integration, the term $\nu {\bf v} \cdot \nabla^2 {\bf w} /2$ in Eq. (\[hel\]) gives the integral of $- \nu {\cal H}_w$ and one obtains the usual expression for the time change of total helicity as given in, for example, Ref. [@FRI]. $\bullet$
Note that the helicity flux in Eq. (\[hel\]) is independent of the function $\varphi$ which we have introduced by hand to make the symplectic form non-degenerate.
Using the invariant description (\[helc\]) of the evolution of helicity density, we shall introduce a current vector $J_{\nu}$ and show that it is an infinitesimal symplectic dilation of $\Omega_{\nu}$. $J_{\nu}$ will be defined as the one-dimensional kernel of the three-form $\theta \wedge \Omega_{\nu}$. Since the symplectic two-form is nondegenerate, it can be obtained as the unique solution of $$i(J_{\nu})(\Omega_{\nu} \wedge \Omega_{\nu} /2)=
\theta \wedge \Omega_{\nu} \;, \label{defj}$$ that is, as the dual of the three-form $\theta \wedge \Omega_{\nu}$ with respect to the symplectic volume. We find $$J_{\nu} = {1 \over q-2 \nu {\cal H}_{w} } [ 2{\cal H} (\partial_t + v)
+ ( \varphi +p - {1 \over 2} v^2 ) w
+ {\bf v} \times ( \nabla \varphi -
\nu \nabla \times {\bf w}) \cdot \nabla ]$$ as the expression for the helicity current.
$J_{\nu}$ is a vector field of divergence $2$ with respect to the symplectic volume and it is an infinitesimal symplectic dilation for $\Omega_{\nu}$. The evolution of helicity density ${\cal H}$ can be described by the identity $$div_{\Omega_{\nu}}(J_{\nu}) -2 \equiv 0 \;. \label{divj}$$
[**Proof:**]{} The exterior derivative of Eq. (\[defj\]) gives $$\begin{aligned}
d i(J_{\nu})(\Omega_{\nu} \wedge \Omega_{\nu} /2)&=&
{\cal L}_{J_{\nu}}( \Omega_{\nu}\wedge \Omega_{\nu}/2) \equiv
div_{\Omega_{\nu}}(J_{\nu}) \, \Omega_{\nu}\wedge \Omega_{\nu}/2 \\
&=& d (\theta \wedge \Omega_{\nu} )
\; = \; \Omega_{\nu}\wedge \Omega_{\nu}\end{aligned}$$ where we used the identity ${\cal L}_{J} = i(J) \circ d + d \circ i(J)$ in the first equality and the second equality is the definition of the divergence. We see that $J_{\nu}$ is a vector field whose divergence is $2$. From the last equality, we conclude that the equation (\[divj\]) is equivalent to Eq. (\[hel\]) describing the evolution of helicity density. $J_{\nu}$ is the unique vector field satisfying $$i(J_{\nu})(\Omega_{\nu}) = \theta \label{teta}$$ and it follows from this that $J_{\nu}$ fulfills the condition $${\cal L}_{J_{\nu}} (\Omega_{\nu}) = d i(J_{\nu}) (\Omega_{\nu})
= d \theta = \Omega_{\nu} \label{ome}$$ of being an infinitesimal symplectic dilation for $\Omega_{\nu}$ [@alan]. $J_{\nu}$ is also called to be the Liouville vector field of $\Omega_{\nu}$ [@LM]. $\bullet$
We observed that the local existence of a Hamiltonian function for $\partial_t+v$ is being prevented by the viscosity term [@hg97]. Moreover, the viscosity term causes the helicity not to be conserved. We shall now show that, for the case of inviscid incompressible fluids described by the Euler equation, namely Eq. (\[euler\]) with $\nu=0$, $\partial_t+v$ is Hamiltonian and that the helicity density ${\cal H}$ is conserved. To this end, we assume that the scalar field $\varphi$ is advected by the fluid motion $${\partial \varphi \over \partial t} +
{\bf v} \cdot \nabla \varphi =0 \label{he}$$ and that the potential vorticity $ q \neq 0$.
[@hg97] Let $v$ and $\varphi$ satisfy Eq. (\[euler\]) with $\nu=0$ and Eq.(\[he\]), respectively. Then, the suspended velocity field $\partial_{t}+v$ on $I \times M$ and $q^{-1}w$ are Hamiltonian vector fields for the exact symplectic two-form $$\Omega_{0} = - (\nabla \varphi + {\bf v} \times {\bf w})
\cdot d{\bf x} \wedge dt + {\bf w} \cdot
(d{\bf x} \wedge d{\bf x}) = d \theta \label{sympo}$$ with the Hamiltonian functions $\varphi$ and $t$, respectively. The evolution equation (\[hel\]) reduces to the conservation law in divergence form for the helicity density.
[**Proof:**]{} Using Eq. (\[he\]) $\partial_{t}+v$ can be written in Hamiltonian form $i(\partial_t+v)(\Omega_0)= - d\varphi$. More generally, the Hamiltonian vector field with the symplectic two-form (\[sympo\]) for an arbitrary function $f$ on $I \times M$ is given by $$X_{f}={1 \over q } [ - w(f)
({\partial \over \partial t}+v) +{df \over dt} w
+ (\nabla \varphi \times \nabla f) \cdot \nabla ]$$ which clearly reduces to $\partial_t+v$ for $f=\varphi$ and to $q^{-1}w$ for $f=t$. The conservation of helicity density is obvious. $\bullet$
For the inviscid flow of the Euler equation the helicity current takes the form $$J_{0} ={1 \over q} [ 2{\cal H} (\partial_t + v)
+ ( \varphi +p - {1 \over 2} v^2 ) w
+ {\bf v} \times \nabla \varphi \cdot \nabla ]$$ while the canonical one-form remains to be the same. That means, the difference between the dynamics of fluid motion with $\nu =0$ and $\nu \neq 0$ is contained in the helicity current. Thus, the dynamical content of the helicity is encoded in its current and this, in turn, is connected with the symplectic structure on $I \times M$ which was constructed as a consequence of the Eulerian dynamical equations.
The realization of dynamics of fluid motion in the symplectic framework is useful in the study of the geometry of the motion on $M$ and of the hypersurfaces in $I \times M$ defined by the time-dependent Lagrangian invariants, that is, the invariants of the velocity field. The present framework also provides geometric tools for the investigation of scaling properties of the fluid motion because the action by the Lie derivative of helicity current on tensorial objects corresponds to infinitesimal scaling transformations [@LM]. Leaving the discussions of these issues elsewhere, we shall conclude this work with an application to the symmetry structure of the velocity field which is also related to the results presented in [@hg98].
Let $X_f$ be a Hamiltonian vector field for $\Omega_{\nu}$. Then, the vector fields $({\cal L}_{J_{\nu}})^k(X_f), \; k=0,1,2,...$ are infinitesimal Hamiltonian automorphisms of $\Omega_{\nu}$.
[**Proof:**]{} The symplectic two-form is invariant under the flows of Hamiltonian vector fields because ${\cal L}_{X_f}(\Omega_{\nu}) = d i(X_f)(\Omega_{\nu}) = d^2f \equiv 0$ where we used the identity ${\cal L}_{X}=i(X) \circ d+d \circ i(X)$ for the Lie derivative, $d \Omega_{\nu} =0$ and the Hamilton’s equations $i(X_f)(\Omega_{\nu}) = -df$. It then follows from the identity $${\cal L}_{[ J_{\nu},X_f ]}=
{\cal L}_{J_{\nu}} \circ {\cal L}_{X_f} -
{\cal L}_{X_f} \circ {\cal L}_{J_{\nu}} \label{iden}$$ evaluated on $\Omega_{\nu}$ that $[ J_{\nu},X_f ]$ also leaves $\Omega_{\nu}$ invariant. Replacing $X_f$ with $[ J_{\nu},X_f ]$ in Eq. (\[iden\]) we see that one can generate an infinite hierarchy of invariants of the symplectic two-form $\Omega_{\nu}$. To see that these are Hamiltonian vector fields we compute $$\begin{aligned}
i([ J_{\nu},X_f ])(\Omega_{\nu}) &=&
{\cal L}_{J_{\nu}} ( i(X_f) (\Omega_{\nu})) -
i(X_f) ( {\cal L}_{J_{\nu}} (\Omega_{\nu})) \label{jx} \\
&=& - d ( J_{\nu}(f) -f) \end{aligned}$$ where we used Eq. (\[ome\]). Thus, $[ J_{\nu},X_f ]$ is Hamiltonian with the function $J_{\nu}(f) -f$. By induction one can find similarly that $({\cal L}_{J_{\nu}})^2(X_f)$ is Hamiltonian with $(J_{\nu})^2(f)-2J_{\nu}(f)+f$ and so on. Interchanging $J_{\nu}$ and $X_f$ in the identity (\[jx\]) we also obtain $i(X_f)(\theta) = J_{\nu}(f) $. $\bullet$
In particular, we let $\nu=0$, $f=t$ so that $X_t=q^{-1}w$ and consider the infinitesimal Hamiltonian automorphisms $({\cal L}_{J_{0}})^k(q^{-1}w), \; k=0,1,2,...$ of $\Omega_0$. The identity (\[iden\]) evaluated on the vector field $\partial_t+v$ gives $${\cal L}_{[ J_{0},q^{-1}w ]}(\partial_t+v)=
- {\cal L}_{q^{-1}w} ([ J_{0},\partial_t+v ]) \label{sym}$$ where the vector field $[ J_{0},\partial_t+v ]$ is, by proposition (5), Hamiltonian with the function $J_{0}(\varphi) -\varphi = p-v^2/2$. By the Lie algebra isomorphism $[X_f,X_g]= X_{ \{ f,g \} }$ defined by the symplectic structure $\Omega_0$, the right hand side of Eq. (\[sym\]) is a Hamiltonian vector field with the function $$\{ t,p-{1 \over 2}v^2 \} = {1 \over q} w(p-{1 \over 2}v^2) \;.
\label{ff}$$ On the level surfaces defined by the constant values of the function (\[ff\]) we have $[[ J_{0},q^{-1}w ], \partial_t+v]= 0$ In fact, if we restrict to the constant values of the function $p-v^2/2$ the hierarchy of Hamiltonian automorphisms of $\Omega_0$ can be identified as the infinitesimal symmetries of the velocity field. This can be seen by replacing $q^{-1}w $ with $[ J_{0},q^{-1}w ]$ in Eq. (\[sym\]). We thus proved that
For the Euler flow, the hierarchy of infinitesimal Hamiltonian automorphisms $({\cal L}_{J_{0}})^k(q^{-1}w), \; k=0,1,2,...$ of $\Omega_0$ generate infinitesimal time-dependent symmetries of the velocity field on the level surfaces $p-v^2/2 = constant$.
As a matter of fact, the function $p-v^2/2$ is related, in Ref. [@pam], to the invariance under particle relabelling symmetries of the Lagrangian density of the variational formulation of the Euler equation.
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{
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---
abstract: 'Super-high resolution laser-based angle-resolved photoemission spectroscopy measurements have been carried out on a heavily overdoped (Bi,Pb)$_2$Sr$_2$CuO$_{6+\delta}$ ($T_c\sim 5$ K) superconductor. Taking advantage of the high-precision data on the subtle change of the quasi-particle dispersion at different temperatures, we develop a general procedure to determine the bare band dispersion and extract the bosonic spectral function quantitatively. Our results show unambiguously that the $\sim$70 meV nodal kink is due to the electron coupling with the multiple phonon modes, with a large mass enhancement factor $\lambda \sim 0.42$ even in the heavily over-doped regime.'
author:
- 'Lin Zhao$^{1}$, Jing Wang$^{2}$'
- 'Junren Shi$^{2}$'
- 'Wentao Zhang$^{1}$, Haiyun Liu$^{1}$, Jianqiao Meng$^{1}$, Guodong Liu$^{1}$, Xiaoli Dong$^{1}$, Wei Lu$^{1}$, Guiling Wang$^3$, Yong Zhu$^3$, Xiaoyang Wang$^{3}$, Qinjun Peng$^{3}$, Zhimin Wang$^{3}$, Shenjin Zhang$^{3}$, Feng Yang$^{3}$, Chuangtian Chen$^{3}$, Zuyan Xu$^{3}$'
- 'X. J. Zhou$^{1,}$'
date: 'January 31, 2010'
title: 'Quantitative Determination of Eliashberg Function and Evidence of Strong Electron Coupling with Multiple Phonon Modes in Heavily Over-doped (Bi,Pb)$_2$Sr$_2$CuO$_{6+\delta}$'
---
The interaction of electrons with phonons and other collective excitations (bosons) in solids dictates fundamental physical properties of materials [@GGrimvall]. In a Fermi liquid picture, such interaction can be described by the Eliashberg spectral function $\alpha^{2}F(\omega)$, which accounts for the bosonic modes involved in the coupling and their strengths. In the conventional superconductors, extraction of the Eliashberg function played a key role in pinning down the phonon as the glue for the electron pairing that gives rise to the BCS superconductivity [@JMRowell]. For high temperature cuprate superconductors and other complex compounds, one expects that the extraction of the Eliashberg function will also be important in understanding the exotic physical properties and the mechanism of high temperature superconductivity [@Tunneling; @Optical; @ADamascelli; @JunrenMEM; @XJZhouMEM; @NonBi2201].
The electron-boson coupling gives rise to the band renormalization and the change of the quasiparticle scattering rate, which can be described by the real part (Re$\Sigma(k, \omega)$) and the imaginary part (Im$\Sigma(k, \omega)$) of the electron self-energy $\Sigma(k,
\omega)$, respectively. With the dramatic improvement of resolution, angle-resolved photoemission spectroscopy (ARPES) has emerged as a powerful tool in probing such many-body effects [@ADamascelli]. Under the sudden approximation, it measures the single particle spectral function $A(k,\omega) =
({1}/{\pi}){\mathrm{Im}\Sigma(k,\omega)}/\{[\omega-\epsilon^0_{k}-\mathrm
{Re}\Sigma(k,\omega)]^{2}+[\mathrm{Im}\Sigma(k,\omega)]^{2}\}$, which is directly related to the electron self-energy [@ADamascelli]. ARPES has been extensively employed in investigating many-body effects in conventional metals [@BeSystem; @TValla], graphene [@EliGraphene] and complex materials [@ADamascelli; @Manganites]. However, a long-standing issue with the ARPES technique is the ambiguous “bare band" $\epsilon^0_{k}$[@DresdenTry], which hinders a quantitative determination of the electron self-energy $\mathrm{Re}\Sigma$(k,$\epsilon$)=$\epsilon$-$\epsilon^0_{k}~$ and the underlying bosonic spectral function $\alpha^2
F(\omega)$ [@JunrenMEM; @XJZhouMEM; @NonBi2201].
Coming specifically to high-$T_c$ cuprate superconductors, although there is a general consensus on the electron-boson coupling as the origin of the universally observed 50$\sim$80 meV dispersion kink along the (0,0)-($\pi$,$\pi$) nodal direction [@PVBogdanov; @ALanzara; @PJohnson; @AKaminski; @XJZhouNature; @AAKordyuk; @WTZhang1], it remains under debate on the nature of the boson(s) involved, mainly between phonons [@ALanzara] and magnetic origins [@AAKordyuk; @ARPESNeutron; @Takahashi]. Theoretical calculations suggest that the electron-phonon coupling in cuprates is too weak to account for the nodal kink [@Louie; @Stuttgart]. To eventually resolve the issue, it is crucial to extract the bosonic spectral function quantitatively so that a direct comparison between the experiments and the calculations could be possible [@JunrenMEM; @XJZhouMEM; @NonBi2201].
In this paper, we introduce a new approach which can unambiguously determine the bare quasi-particle dispersion, and hence enables the quantitative determination of the bosonic spectral function. The method utilizes the subtle change of the quasi-particle dispersion induced by the temperature. This is made possible by the super-high resolution laser-based ARPES measurements, which present data of the significantly higher quality feasible for a quantitative analysis of a modified maximum entropy method [@JunrenMEM]. By carrying out such a procedure on a heavily over-doped (Bi,Pb)$_2$Sr$_2$CuO$_{6+\delta}$ (Pb-Bi2201, T$_c$$\sim$5 K), we are able to determine the bare band and extract the bosonic spectral function quantitatively along the nodal direction. Our results present a conclusive evidence that the electron-boson coupling is strong (with a mass enhancement factor $\lambda\approx 0.42$) even in the heavily over-doped cuprate compounds, and it originates from the electron coupling to the multiple phonon modes. Moreover, our approach is general enough to be applicable for many other materials.
{width="1.0\columnwidth"}
The angle-resolved photoemission measurements have been carried out on our newly developed vacuum ultra-violet (VUV) laser-based ARPES system [@GDLiu]. The photon energy of the laser is 6.994 eV with a bandwidth of 0.26 meV. The overall instrumental energy resolution is set at 1 meV, which is significantly improved from 10$\sim$20 meV of previous measurements [@PVBogdanov; @ALanzara; @PJohnson; @AKaminski; @XJZhouNature; @AAKordyuk; @WTZhang1; @JunrenMEM; @XJZhouMEM; @NonBi2201]. The angular resolution is $\sim 0.3$ degree, corresponding to a momentum resolution $\sim$0.004 A$^{-1}$ at the photon energy of 6.994 eV, more than twice improved from 0.009 A$^{-1}$ at a usual photon energy of 21.2 eV for the same angular resolution. The heavily overdoped (Bi,Pb)$_2$Sr$_2$CuO$_{6+\delta}$ (Pb-Bi2201) single crystals with a $T_c\sim 5$ K were grown by the traveling-solvent floating zone method. The samples are cleaved [*in situ*]{} in vacuum with a base pressure better than 5 $\times$ 10$^{-11}$ Torr.
{width="1.00\columnwidth"}
Figure 1(a) shows the raw data of photoemission image taken on a heavily overdoped Pb-Bi2201 (T$_c$$\sim$5 K) sample along the (0,0)-($\pi$,$\pi$) nodal direction at 15 K. Clear kink is observed near 70 meV in the dispersion extracted from MDC (momentum distribution curve) fitting (Fig. 1(c)), with a corresponding drop in the MDC width (inset of Fig. 1(c)). In order to reveal the possible fine structures in the dispersion, we plot the energy difference between the measured dispersion and a featureless straight line connecting 0 ($E_F$) and $-0.3$ eV on the dispersion (Fig. 1(d)). A prominent peak shows up near $\sim$73 meV. In addition, in the energy difference between the measured dispersion and another straight line connecting 0 and $-0.07$ eV (Fig. 1e), one can also identify low energy features at $\sim 41$ meV and $\sim
17$ meV, signified by the slope changes. We note that these three features at $\sim$73meV, $\sim$41meV, and $\sim$17meV are robust and have been reproduced in several independent measurements. Moreover, a clear dip at 73 meV (Fig. 1(b)) can be observed on the EDCs (energy distribution curves) near the Fermi momentum ($k_F$).
The underlying bosonic spectral function $\alpha^2F(\omega)$ can be extracted from the measured real part of the electron self-energy [@JunrenMEM]. The real part of self-energy is related to $\alpha^2F(\omega)$ by: $$\mathrm{Re}\Sigma(\epsilon;T)= \int^{\infty}_0 \mathrm{d}\omega\,
K \left(\frac{\epsilon}{kT},\frac{\omega}{kT} \right) \alpha^2F(\omega)\, ,$$ where $K(y,y')=\int^{\infty}_{-\infty} dx\,f(x-y)2y'/(x^2-y'^2)$ with $f(x)$ being the Fermi distribution function [@GGrimvall]. In the ARPES measurements, the real part of the electron self-energy is determined from the measured MDC dispersion $\epsilon_k$ by Re$\Sigma(\epsilon_k)=\epsilon_k-\epsilon^0_{k}$, where $
\epsilon^0_k$ is the bare quasi-particle dispersion for an [*ad-hoc*]{} system with the electron-boson coupling turned off [@BareBNote]. Attempts have been made to invert the “effective" bosonic spectral function $\alpha^2F(\omega)$ from the measured MDC dispersion by assuming an empirical bare band of a quadratic form $\epsilon^0_k =
a(k-k_F)^{2}+b(k-k_F)^{2}$ [@JunrenMEM; @XJZhouMEM; @NonBi2201]. It was found that the qualitative features of the bosonic spectral function do not sensitively depend on the choice of the bare band. However, the arbitrariness in choosing the appropriate parameters for the bare band prevents a quantitative determination of the bosonic spectral function [@JunrenMEM; @XJZhouMEM; @NonBi2201]. Here we propose a general approach to circumvent this problem by making use of the self-energy change at different temperatures. We assume that $\alpha^2F(\omega)$ and the bare band have negligible temperature dependence over a relatively small temperature window, and the Eliashberg function can be extracted from the self-energy difference at different temperatures by: $$\begin{gathered}
\mathrm{Re}\Sigma(\epsilon;T_1)-\mathrm{Re}\Sigma(\epsilon;T_2)=
\int^\infty_0 \mathrm{d}\omega\, \alpha^2F(\omega) \\
\times \left[K\left(\frac{\epsilon}{kT_1},\frac{\omega}{kT_1}\right)
-K\left(\frac{\epsilon}{kT_2},\frac{\omega}{kT_2}\right)\right] \,
\label{DSigma}\end{gathered}$$ The successful application of the approach relies critically on the high quality of data, as it utilizes a small temperature-induced change between two dispersions subjected to noises. Such a stringent requirement on the data quality can now be met by the super-high resolution laser-ARPES measurements(Fig. 2). In this paper, we test our new approach on the heavily overdoped Pb-Bi2201, in which the electron-electron correlation is relatively weak, and Eq. (\[DSigma\]) is believed to be applicable.
Figure 3 shows the detailed procedure to extract the bare band dispersion and the bosonic spectral function. To obtain the bosonic spectral function $\alpha^2F(\omega)$ from Eq. (\[DSigma\]), one needs to know the difference of the real part of self-energy $\Delta\mathrm{Re}\Sigma(\epsilon) \equiv
\mathrm{Re}\Sigma(\epsilon;T_1)-\mathrm{Re}\Sigma(\epsilon;T_2)$ at two different temperatures, as schematically shown in Fig. 3(a). Here, we use two dispersions measured at 25 K and 100 K. Note that $\Delta\mathrm{Re}\Sigma(\epsilon)$ depends on the selection of the bare band dispersion. As the result, an iteration approach is needed, as detailed in the following: (1) Determine $\Delta\mathrm{Re}\Sigma(\epsilon)$ based on the current selection of the bare band dispersion. The initial value ($\Delta\mathrm{Re}\Sigma_0(\epsilon)$) is taken as the direct difference between the two dispersions, as schematically shown in Fig. 3(a). The result is shown in Fig. 3(c); (2) Extract the “bosonic spectral function" SF from $\Delta\mathrm{Re}\Sigma(\epsilon)$ by Eq. (\[DSigma\]), using a maximum entropy method (MEM) similar to that in Ref. \[\] but with the modified integral kernel. The result is shown in Fig. 3(d). Note that the significant change of the real part of the self-energy is limited to the energy up to $\sim 100$ meV, above which the self-energy difference fluctuates around zero and is most likely due to the data noise, as shown in Fig. 3(c). Therefore, we assume a maximum bosonic mode energy 100 meV when doing the MEM analysis; (3) Calculate the real part of self-energy at 25K from the extracted bosonic spectral function using Eq. (1). The result is shown in Fig. 3(d); (4) Determine the bare band dispersion by taking the difference between the measured dispersion and the calculated real part of the self-energy, both at 25K. The result is shown in Fig. 3(b). These steps iterate until the result converges. We find that the procedure converges quickly and only three iterations are needed for this analysis.
{width="0.96\columnwidth"}
Fig. 4 summarizes the final results of the bare band (Fig. 4(a)), Eliashberg spectral function (Fig. 4(b)), and electron self-energy (inset of Fig. 4(a)). Although the results are obtained by using the two representative dispersions at 25K and 100K, the self-energy calculated from the extracted Eliashberg function (Fig. 4(b)) at other temperatures (here we show 60K data) matches very well with the measured values, as shown in the inset of Fig. 4(a), indicating the internal consistency of our approach. The obtained bare band is close but less steeper than the one given by the LDA calculations [@NonBi2201PRB] (Fig. 4(a)), and is significantly different from the empirical one used previously (green line in Fig. 4a) [@NonBi2201].
Sharp features are clearly identified in the extracted Eliashberg spectral function (Fig. 4(b)). Five peaks at (14-17),(30-32),(41-44),(54-58) and (70-74) meV can be identified below 100 meV, and they are reproducible from several independent measurements. In fact, the features (14-17),(41-44) and (70-74) meV are clearly visible in the raw data (Fig. 1). These sharp features show reasonably good agreement with the phonon modes observed in the same material by infrared reflectance (Fig. 4(c)) [@IRBi2201] and Raman scattering (Fig. 4(d)) [@RamanBi2201]. Note that these optical measurements only see phonon modes with zero wavevector, and the full band of phonons will exhibit dispersion in the Brillouin zone, in particular for the higher energy phonon near 70$\sim$80 meV [@Neutron]. Considering the rather weak magnetic signal [@MagneticOverdoped], and the absence of magnetic mode at 15 K for the heavily overdoped Pb-Bi2201 sample (T$_c$$\sim$5 K), our present results leave no doubt that the $\sim 70$ meV dispersion kink is due to electron coupling with the multiple phonon modes.
![Extracted bare band dispersion, self-energy and Eliashberg spectral function of heavily over-doped Pb-Bi2201. (a) Measured nodal dispersion (empty blue circles) at 25 K and the extracted bare band (thick red line). For comparison, the bare band from the band structure calculations (dashed grey line) [@NonBi2201PRB] and the bare band used by previous maximum entropy method (thin green line) [@NonBi2201PRB] are also shown. The bottom-right inset shows the measured electron self-energy at different temperatures and their comparison with those calculated from the extracted Eliashberg spectral function. (b) Extracted Eliashberg functions $\alpha^{2}F(\omega)$. (c) Loss function from infrared measurements on optimally-doped (red line) and over-doped (blue line) Bi-2201 [@IRBi2201]. (d) Raman spectra of Bi2201 under different polarization geometries [@RamanBi2201]. ](LZhao_Fig4.eps){width="0.96\columnwidth"}
With the Eliashberg function extracted, it is possible to determine quantitatively the strength of the electron-boson coupling. We calculate the mass enhancement factor $\lambda$=2$\int^\infty_0\frac{d\omega}{\omega}\alpha^2F(\omega)$, and obtain $\lambda\approx 0.42$. As discussed above, the contribution is mainly from the electron-phonon coupling. Our study presents a conclusive evidence that the electron-phonon coupling is strong even in the heavily over-doped cuprate samples. It is much stronger than the value suggested by the first principles calculations (0.14$\sim$0.20) [@Louie]. Our present work thus asks for a re-examination of theoretical calculations on the strength of electron-phonon coupling in these compounds.
In summary, by taking advantage of the high quality data of the laser-ARPES measurements, we have developed a new approach to determine the bare band dispersion and extract Eliashberg spectral function quantitatively. The method is general and can be applied to many other materials, thus promoting ARPES technique to be quantitative in probing many-body effects. Our results on the heavily overdoped Pb-Bi2201 demonstrate unambiguously that the $\sim$70 meV nodal dispersion kink is due to electron coupling to the multiple phonons. In particular, we present a conclusive evidence that the electron-phonon coupling is strong in cuprates even in the heavily over-doped regime.
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|
{
"pile_set_name": "ArXiv"
}
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---
abstract: 'We propose and analyse a Raman spectroscopy technique for probing the properties of quantum degenerate bosons in the ground band of an optical lattice. Our formalism describes excitations to higher vibrational bands and is valid for deep lattices where a tight-binding approach can be applied to the describe the initial state of the system. In sufficiently deep lattices, localized states in higher vibrational bands play an important role in the system response, and shifts in resonant frequency of excitation are sensitive to the number of particles per site. We present numerical results of this formalism applied to the case of a uniform lattice deep in the Mott insulator regime.'
address: 'Jack Dodd Centre for Photonics and Ultra-Cold Atoms, Department of Physics, University of Otago, P.O. Box 56, Dunedin, New Zealand'
author:
- 'P. Blair Blakie'
bibliography:
- 'raman.bib'
title: Raman Spectroscopy of Mott insulator states in optical lattices
---
Introduction
============
The observation of the Mott-insulator state of a degenerate Bose gas in an optical lattice [@Greiner2002a] has opened an exciting new avenue for investigating strongly correlated condensed matter systems. Ensuing experiments (e.g. [@Greiner2002b; @Mandel2003a; @Widera2004a]) have investigated a wide variety of phenomena making optical lattices one of the leading systems for studying quantum atom optics.
In the initial experimental report, two key pieces of evidence were given for the Mott insulator state: the loss of phase coherence, and the appearance of a gap in the excitation spectrum. More recent experimental work has used Bragg spectroscopy [@Schori2004a; @Stoferle2004a] to study the equilibrium properties of this system. While this type of probing reveals the appearance of a gap in the excitation spectrum, experimental applications of this technique operate well-beyond the linear response regime, making direct comparison with theory difficult. [^1]
In this paper we propose and theoretically analyse a Raman spectroscopy scheme for probing Mott insulator states in the lattice system. The formalism we present is sufficiently general to include the excitation of atoms to higher vibrational bands where they should be easily discernible from the unscattered atoms in time-of-flight analysis. We apply this formalism to a uniform lattice and show how the correlated nature of the Mott insulator state can give rise to localized states in the excited bands when the lattice is sufficiently deep, and that the resonant frequency of these localized states is sensitive to the local filling factor in the optical lattice.
Formalism
=========
The system of interest, which we will refer to as system 1, is a degenerate collection of bosonic atoms populating the lowest vibrational band of an optical lattice well-characterized by the Bose-Hubbard Hamiltonian
$$\hat{H}_{1}=\epsilon_{0}\sum_{j}\hat{n}_{j}-J\sum_{\langle i,j\rangle}\hat{a}_{i}^{\dagger}\hat{a}_{j}+\frac{U_{11}\alpha_{w}}{2}\sum_{j}\hat{n}_{j}(\hat{n}_{j}-1),\label{eq:HBH}$$
where $\hat{a}_{j}$ is a bosonic operator that annihilates an atom from the Wannier state $w_{j}(\mathbf{x})$ centered on lattice site $j$, with $\hat{n}_{j}=\hat{a}_{j}^{\dagger}\hat{a}_{j}$ the respective number operator. The quantity $J$, known as the tunneling matrix element, characterizes the tunneling between lattice sites and is determined from band structure calculations [@Jaksch1998a; @Blakie2004a]. Interactions between particles are described by the matrix element $\alpha_{w}\equiv\int d^{3}\mathbf{x}\,|w_{j}(\mathbf{x})|^{4}$ and the coefficient $U_{11}=4\pi a_{11}\hbar^{2}/m,$ where $a_{11}$ is the s-wave scattering length for collisions between atoms in internal state $1$. We restrict our attention here to the translationally invariant system (i.e. neglect the influence of external trapping potentials in addition to the optical lattice), and the constant $\epsilon_{0}$ characterizes the mid-point energy of the ground band.
![\[cap:exptdiag\] Schematic diagram of the spectroscopic process. Atoms in the ground band of system 1 are transfered into the initially unoccupied system 2 by a two-photon Raman process through a virtual level $|v\rangle$. A residual energy difference $\hbar\omega_{0}$ and momentum kick $\hbar\mathbf{q}$ (not shown) are transfered to the atoms by each scattering event. ](Fig1.eps){width="9cm"}
In this work, we consider a scheme for probing the properties of system 1, by an internal state changing Raman transition, of the form used in Ref. [@Hagley1999a] to output couple an atom laser. A theory for this type of process in an optical lattices has also been proposed by Konabe *et al.* [@Konabe2004a][^2]. This is in contrast to a recent theory [@Menotti2003a; @Roth2004a; @Oosten2005a; @Rey2005a; @Batrouni2005a; @Pupillo2006a] and experiments [@Schori2004a; @Stoferle2004a] that have considered internal state preserving transitions to perform spectroscopy – known as Bragg scpectroscopy [@Blakie2002a]. A schematic diagram of this coupling is shown in Fig. \[cap:exptdiag\]. Following the formalism in Ref. [@blakie2003a] this type of coupling is described by an interaction term of the form$$\hat{V}=\frac{V_{p}}{2}\theta(t)\sum_{j}\int d^{3}\mathbf{x}\hat{\psi}_{2}^{\dagger}(\mathbf{x})\hat{a}_{j}w_{j}(\mathbf{x})e^{i(\mathbf{q}\cdot\mathbf{x}-\omega_{0}t)}+{\rm H.c},\label{eq:Hpert}$$ where $\hat{\psi}_{2}^{\dagger}$ is the field operator for bosonic atoms in the second internal state, and the quantities $\hbar\mathbf{q}$ and $\hbar\omega_{0}$, specify the momentum and energy transfer of the Raman process respectively. For simplicity we define $\hbar\omega_{0}$ to be the excess energy transferred over the internal state energy difference. We take the amplitude of the Raman coupling to be of strength $V_{p}$ beginning at $t=0,$ and of duration $T_{p}$. We will assume that the internal states 1 and 2 are different hyperfine ground states for which we can neglect any collisional spin evolution (see [@Widera2005a; @Hall1998a]). To arrange such a Raman probe experimentally, a pair of light fields (in addition to those used to create the lattice) with appropriately chosen polarizations to couple the states of interest will need to be applied to the atoms in the lattice. While such a probe could be focused down to address a few sites in a lattice, our theoretical development here is for the case where the Raman fields uniformly illuminate the system and have no spacial selectivity.
Our interest lies in the linear response regime, where only a small portion of the atoms are scattered into internal state 2. The evolution of these atoms in internal state 2, which we will refer to as system 2, is described by the Hamiltonian $$\begin{aligned}
\hat{H}_{2} & = & \int d^{3}\mathbf{x}\hat{\psi}_{2}^{\dagger}(\mathbf{x})(H_{{\rm sp}}+U_{12}\sum_{j}\hat{n}_{j}|w_{j}(\mathbf{x})|^{2})\hat{\psi}_{2}(\mathbf{x}),\label{eq:H2}\end{aligned}$$ where $$H_{{\rm sp}}=p^{2}/2m+V_{{\rm ext}}(\mathbf{x}),$$ is the single particle Hamiltonian with $V_{{\rm ext}}(\mathbf{x})$ the external potential (taken to be the same lattice potential experienced by atoms in system 1), and $U_{12}=4\pi a_{12}\hbar^{2}/m$ characterizes the interactions between particles in systems 1 and 2. Within the validity regime of linear response it is permissible to neglect interactions between atoms in system 2, as their density will remain low.
System 2 is initially in the vacuum state, and the spectroscopic signal to be measured is the number of atoms in system 2 after the Raman pulse. This is a convenient experimental observable as detection techniques can readily distinguish between atoms in different internal states [@Hall1998a]. Additionally when the coupling produces atoms in an excited vibrational band, the excited atoms can be differentiated by their behavior upon expansion from the lattice [@Greiner2001a; @Denschlag2002a]. These properties of the observable should enable the system response to be measured for small amounts of Raman excitation, allowing the system to be probed in the linear response regime.
The expression for the number of excited atoms, derived using linear response theory, is$$\left\langle \hat{N}_{2}\right\rangle =\frac{\pi V_{p}^{2}T_{p}}{2\hbar^{2}}\int_{-\infty}^{+\infty}d\omega R(\mathbf{q},\omega)\,\frac{2\sin^{2}([\omega-\omega_{0}]T_{p}/2)}{\pi T_{p}(\omega-\omega_{0})^{2}},\label{eq:LinRespforN2}$$ where we have introduced the correlation function
$$\begin{aligned}
R(\mathbf{q},\omega)=\frac{1}{2\pi}\int dt\int d^3\mathbf{x}\int d^3\mathbf{x}^\prime\,e^{i\mathbf{q}\cdot(\mathbf{x}^\prime-\mathbf{x})+i\omega t}\label{eq:RQWGen}\\ \times\sum_{ij}w_i^*(\mathbf{x})w_j(\mathbf{x}^\prime)\left\langle\hat{a}_i^\dagger(t)\hat{\psi}_2(\mathbf{x},t)\hat{\psi}_2^\dagger(\mathbf{x}^\prime,0)\hat{a}_j(0)\right\rangle_{\rm{Eq}}. \nonumber\end{aligned}$$
In this expression the time dependence of the operators is in an interaction picture with respect to unperturbed Hamiltonians $\hat{H}_{1}+\hat{H}_{2}$, and the expectation is taken on the initial equilibrium ensemble.
We will now show how to evaluate the correlation function $R(\mathbf{q},\omega)$, and how it relates to the properties of the atoms in system 1. We take the initial density matrix to be $\rho=\rho_{1}\otimes\rho_{2}^{{\rm vac}}$, where $\rho_{1}$ is the degenerate lattice state we are interested in probing and $\rho_{2}^{{\rm vac}}$ is the initial vacuum state for system 2. We can then show that without further approximation, $R(\mathbf{q},\omega)$ can be written in the form
$$\begin{aligned}
R(\mathbf{q},\omega)&=&\frac{1}{2\pi}\int dt\int d^{3}\mathbf{x}\int d^{3}\mathbf{x}^{\prime}e^{i\mathbf{q}\cdot(\mathbf{x}^{\prime}-\mathbf{x})+i\omega t}\label{eq:RQ}\\&& \sum_{Q\, ij}\left\langle Q\left|\hat{a}_{i}^{\dagger}(t)\hat{a}_{j}(0)\rho_{1}\right|Q\right\rangle \left\langle {\rm 0}_{2}\left|\hat{\psi}_{j}^{Q}(\mathbf{x},t)\hat{\psi}_{2}^{\dagger}(\mathbf{x}^{\prime},0)\right|0_{2}\right\rangle w_{i}^{*}(\mathbf{x})w_{j}(\mathbf{x}^{\prime}),\nonumber\end{aligned}$$
where the variable $Q$ represents a trace carried out over the number state basis for the operators $\{\hat{a}_{j}\}$, i.e. $Q\leftrightarrow|\ldots,n_{l-1}^{Q},n_{l}^{Q},\ldots\rangle$, and $|0_{2}\rangle$ represents the vacuum state of system 2. The operator $\hat{\psi}_{j}^{Q}(\mathbf{x},t)$ is $\hat{\psi}_{2}(\mathbf{x})$ evaluated in an interaction picture with respect to the Hamiltonian$$\hat{H}_{j}^{Q}\equiv\int d^{3}\mathbf{x}\hat{\psi}_{2}^{\dagger}(\mathbf{x})(H_{{\rm sp}}+U_{12}\sum_{l}(n_{l}^{Q}-\delta_{lj})|w_{l}(\mathbf{x})|^{2})\hat{\psi}_{2}(\mathbf{x}).\label{eq:HjQ}$$ We note that in deriving Eq. (\[eq:RQ\]) we have made use of the result $$e^{i\hat{H}_{2}t/\hbar}\hat{\psi}_{2}(\mathbf{x})e^{-i\hat{H}_{2}t/\hbar}\hat{a}_{j}(0)|Q\rangle=\hat{a}_{j}(0)|Q\rangle\hat{\psi}_{j}^{Q}(\mathbf{x},t),\label{eq:HiQresult}$$ where the $\delta_{lj}$ term in Eq. (\[eq:HjQ\]) arises from commuting $\hat{a}_{j}$ with $\exp(i\hat{H}_{2}t/\hbar)$, and we have made the replacement $\hat{n}_{l}\to n_{l}^{Q}$ as $|Q\rangle$ are number states.
We refer to $\hat{H}_{j}^{Q}$ as the $Q$-defect Hamiltonian for system 2, as it arises from the removal of a system 1 atom from site $j$ of the number state $Q$. As the system 1 atoms form an effective potential for those in system 2, the removal of an atom at site $j$ due to the Raman excitation creates a potential hole (i.e. the $n_{l}^{Q}-\delta_{lj}$ term in Eq. (\[eq:HjQ\])). This defect plays a key role in the response spectrum as we will demonstrate later. Since $\hat{H}_{j}^{Q}$ is a quadratic Hamiltonian it can be diagonalized by numerical methods to obtain its eigenvectors $\{\phi_{jm}^{Q}(\mathbf{x})\}$ and eigenvalues $\{\hbar\omega_{jm}^{Q}\}$, i.e. $\hbar\omega_{jm}^{Q}\phi_{jm}^{Q}(\mathbf{x})=\hat{H}_{jQ}\phi_{jm}^{Q}(\mathbf{x}),$ where $m$ is the quantum number specifying the state and $j$ labels the defect location. We note that in the limit $U_{12}\to0$, the $\phi_{jm}^{Q}$ reduce to the Bloch states of $H_{{\rm sp}}$ and $m$ becomes the quasimomentum and band index. We obtain $\hat{\psi}_{j}^{Q}(\mathbf{x},t)=\sum_{m}\phi_{jm}^{Q}(\mathbf{x})\hat{b}_{jm}^{Q}e^{-i\omega_{jm}^{Q}t},$ where **$\hat{b}_{jm}^{Q}$** is a bosonic annihilation operator and arrive at the expression $$R(\mathbf{q},\omega)=\frac{1}{2\pi}\int dt\, e^{i\omega t}\sum_{Q}\sum_{ijm}c_{ij}^{Q}(t)A_{mjj}^{Q}A_{mij}^{Q*}e^{-i\omega_{jm}^{Q}t},\label{eq:RQ2}$$ where we have defined $$\begin{aligned}
A_{mij}^{Q}&\equiv&\int d^{3}\mathbf{x}\,\phi_{jm}^{Q*}(\mathbf{x})e^{i\mathbf{q}\cdot\mathbf{x}}w_{i}(\mathbf{x}),\\
c_{ij}^{Q}(t)&\equiv&\langle Q|\hat{a}_{i}^{\dagger}(t)\hat{a}_{j}(0)\rho_{1}|Q\rangle.\end{aligned}$$
Eqs. (\[eq:RQ\]) and (\[eq:RQ2\]) represent the key results of this work, and we now briefly comment on the physical process they describe. Fundamentally, $R(\mathbf{q},\omega)$ characterizes the excitation spectrum of atoms from system 1 into system 2. In the context of ultra-cold gases, a result similar to our starting point [\[]{}Eq. (\[eq:LinRespforN2\])[\]]{} has been given by Luxat *et al.* [@Luxat2002a] for the case of a harmonically trapped Bose gas (also see [@Choi2000a; @Girardeau2001a]). However their treatment neglects interactions between atoms in different hyperfine states and assumes the single particle correlation functions for the atoms in systems 1 and 2 are independent. The extension of this theory to the optical lattice has been provided by Konabe *et al.* [@Konabe2004a]. For the current experiments with Rubidium atoms in a deep optical lattice these approximations are not tenable. In particular, as we noted previously, the lattice site from which the atom is excited acts as the localizing defect and accounting for correlations between the systems (as we have done in Eq. (\[eq:RQ\])) is essential.
We note that our formalism \[i.e. Eqs. (\[eq:RQ\]) and (\[eq:RQ2\])\] is quite general. As long as the dominant number states of the many-body state are known, e.g. through exact diagonalization or Matrix Product Decomposition techniques (e.g. esee [@Clark2004a]), then the Raman response can be determined. In the following sections we consider the application of our formalism for two special cases. The results we present in the next section are calculated for 1D systems for numerical convenience, though our interest is in the regime where the scattering between particles is three-dimensional and well-described by a contact interaction.
Single Site Limit
=================
The limiting case of a single tightly confining harmonic well of frequency $\omega_{{\rm ho}}$ is a useful approximation to a deep lattice in the regime where tunneling between sites can be neglected. This limit shows the main physical features of Raman spectroscopy and provides a useful approximation to the full solution. Assuming that interaction shifts are small compared to the oscillator energy $\hbar\omega_{{\rm ho}}$, we may approximate the modes of the system as being harmonic oscillator eigenstates $\{\varphi_{m}(\mathbf{x})\}$, with respective energies $\{\epsilon_{m}=\hbar\omega_{{\rm ho}}(m+1/2)\}$. For the single site case our formalism maps according to $$\begin{aligned}
w_{j}(\mathbf{x})&\to&\varphi_{0}(\mathbf{x}),\\
\phi_{jm}^{Q}(\mathbf{x})&\to&\varphi_{m}(\mathbf{x}),\\
A_{mij}^{Q}&\to& A_{m}\equiv\int d^{3}\mathbf{x}\,\varphi_{m}^{*}(\mathbf{x})e^{i\mathbf{q}\cdot\mathbf{x}}\varphi_{0}(\mathbf{x}),\\
\hbar\omega_{jm}^{Q}\to\hbar\omega_{m}^{n}&=&\epsilon_{m}+U_{12}\alpha_{0m}(n-1)\end{aligned}$$ where $\alpha_{0m}\equiv\int d^{3}\mathbf{x}\,|\varphi_{0}(\mathbf{x})|^{2}|\varphi_{m}(\mathbf{x})|^{2}$. Since the Hamiltonian for system 1 reduces to $\hat{H}_{1}\to\epsilon_{0}\hat{n}+U_{11}\alpha_{w}\hat{n}(\hat{n}-1)/2$ in this limit, we have replaced $Q$ by $n$ (i.e. the single site many-body number states are just single mode number states $|n\rangle$) and evaluated the correlation function as $$c_{ij}^{Q}(t)\to c_{ij}^{n}(t)=n\exp(i[\epsilon_{0}+U_{11}\alpha_{w}(n-1)]t/\hbar).$$ Using Eq. (\[eq:RQ2\]) with $\rho_{1}=\sum_{nn^{\prime}}|n\rangle\rho_{nn^{\prime}}\langle n^{\prime}|$, we obtain$$\begin{aligned}
R(\mathbf{q},\omega)&=&\sum_{n}\rho_{nn}\, n|A_{m}|^{2}\delta(\omega_{{\rm res}}(n)-\omega),\label{eq:RqwSS} \\
\hbar\omega_{{\rm res}}(n)&=&\epsilon_{m}-\epsilon_{0}+[U_{12}\alpha_{0m}-U_{11}\alpha_{w}](n-1).\end{aligned}$$ We have also assumed that the probe only couples to a particular excited state $m$, with the other states sufficiently far detuned that their contribution is negligible.
Eq. (\[eq:RqwSS\]) shows that the response of the system is proportional to $\rho_{nn}$, and most notably, if $[U_{12}\alpha_{0m}-U_{11}\alpha_{w}]\ne0$, then the response frequency is linearly dependent on the value of $n$ (i.e. the number of atoms at the site). In this regime the Raman spectrum reveals the number distribution at the site. For the case of $^{87}$Rb (the atom of primary interest in bosonic optical lattice experiments) the interactions between the relevant hyperfine states are approximately degenerate (i.e. $U_{11}\approx U_{12}$) , so that the difference $\alpha_{0m}-\alpha_{w}$ will be the primary factor in determining the magnitude of the number dependent shift. As an immediate consequence we note that for the case $U_{11}=U_{12}$ and a Raman pulse coupling to ground vibrational state of system 2, then the spectrum will be independent of $n$ (i.e. for $m=0$ we have $\alpha_{0m}=\alpha_{w}$). Therefore to obtain a number dependent spectral response with $^{87}$Rb will require scattering into excited vibrational states of system 2.
Uniform Mott Insulator
======================
We now consider the more general case of a translationally invariant lattice with $N_{s}$ sites and periodic boundary conditions. We assume that the number of atoms in the system is commensurate with the number of lattice sites and system is deep in the Mott insulating regime, where $U_{11}\alpha_{w}\gg J$. In this regime the many-body ground state is well approximated as $|Q_{n}\rangle=|\ldots,n,n,\ldots\rangle$ (i.e a definite number of atoms at each site). In calculating the response of the system to the Raman probe according to Eq. (\[eq:RQ\]), the summation over $Q$ reduces to this single state. The next order correction to the translationally Mott state is particle hole states [@Rey2005a], which contribute to the ground state with an amplitude $J/U_{11}\alpha_{w}\ll1$, and can be neglected.
To obtain an approximation for the temporal correlation function of system 1 in state $|Q_{n}\rangle$ we ignore the tunneling term in $\hat{H}_{1}$. This approximation amounts to neglecting particle tunneling between sites in system 1 over the time scale of the Raman probe, and should be a good approximation in deep lattices. In this limit the correlation function is $$c_{ij}^{Q_{n}}(t)=n\,\exp(i[\epsilon_{0}+U_{11}\alpha_{w}(n-1)]t/\hbar)\delta_{ij},$$ and making use of the translational invariance in evaluating Eq. (\[eq:RQ2\]) we obtain $$\begin{aligned}
R(\mathbf{q},\omega)&=&\sum_{m}n\, N_{s}|A_{mll}^{Q_{n}}|^{2}\delta(\omega-\omega_{{\rm res}}^{Q_{n}m}),\label{eq:RqLatt}\\
\omega_{{\rm res}}^{Q_{n}m}&=&\omega_{lm}^{Q_{n}}-[\epsilon_{0}+U_{11}\alpha_{w}(n-1)]/\hbar.\end{aligned}$$ Note that due to translational invariance the precise value of $l$ used in Eq. (\[eq:RqLatt\]) is unimportant.
![\[cap:SpecComp\] The percentage of atoms excited after Raman excitation in a 1D lattice of depth (a) $V_{D}=10E_{R}$ (b) $V_{D}=30E_{R}$. In each plot the spectrum of Mott insulator states with filling factors of $n=1$ (dark line), $n=2$ (medium line) and $n=3$ (light line) are shown. The respective single site results for $m=1$ are given as dashed lines. Parameters are $q=\pi/\lambda$, $T_{p}=1.5$ms, $V_{p}=0.05E_{R}$, $\lambda=850$nm, $N_{s}=51$, transverse confinement taken to be harmonic with $f_{\perp}\approx37.6$kHz, and $a_{11}=a_{12}=5.29$nm. ](Fig2.eps){width="12cm"}
We now calculate the response spectrum $\left\langle \hat{N}_{2}\right\rangle $ using Eqs. (\[eq:LinRespforN2\]) and (\[eq:RqLatt\]) for a translationally invariant 1D lattice. We take the lattice potential to be $V_{{\rm ext}}(x)=V_{D}\cos^{2}(x/\lambda),$ arising from counter-propagating lasers fields of wavelength $\lambda$, and we specify the lattice depth $V_{D}$ in units of $E_{R}=h^{2}/2m\lambda^{2}$. The calculation is performed for typical $^{87}$Rb parameters (e.g. see Ref. [@Greiner2002a]) to demonstrate the practicality of our probing scheme. We will not consider the influence of $\mathbf{q}$ on the spectrum in this paper[^3], and for simplicity we have taken $\mathbf{q}$ as being identical to a reciprocal lattice for our calculations.
The results we have obtained are shown in Fig. \[cap:SpecComp\]. For the case of $V_{D}=10E_{R}$ (Fig. \[cap:SpecComp\](a)) the Raman response is shown over a frequency range that includes resonant coupling to the first two excited bands. The superimposed graphs show the spectra for Mott insulating states of various filling factors and clearly exhibit frequency shifts proportional to $n$. Additionally, we notice that in the first excited band ($12$kHz - $18$kHz), the dominant response peak occurs at the low end of the spectral feature (e.g. the peak at $\sim15.5$kHz for $n=1$), adjacent to a broad base. This is feature is also seen in the 2nd excited band ($24$kHz - $36$kHz), however the peaked feature is much less dominant relative to the broad base. The peak originates from an excited band state that is partially localized above the defect site. This localization leads to a strong coupling matrix element $A_{mll}^{Q_{n}}$, and as this state resides at the defect, its energy is lower than the other states in the excited band. We refer to this state as the defect state, which is clearly identified as the most strongly excited feature at the bottom each spectral band. Since the localization is not perfect, the other states in the excited band have appreciable amplitude at the defect site. These states give rise to the broad though more weakly excited, band of states above the resonant peak. In the 2nd excited band, the same features are seen, however as the effective tunneling in this band is much higher, and the defect state is much less localized.
In Fig. \[cap:SpecComp\](b) the response spectrum is shown for a lattice of depth $V_{D}=30E_{R}$. At this depth the response from the first excited band states have shifted up to $30$kHz, and the second excited band has moved out of the frequency range considered. In contrast to the spectrum in Fig. \[cap:SpecComp\](a), we only notice the resonant peak due to the defect state, without a discernible broad base of band states. This arises because at this depth the tunneling rate in the excited band is sufficiently small that the defect state a becomes completely localized. The other states in the excited band are necessarily orthogonal to the defect state and so have vanishing coupling matrix elements $A_{mll}^{Q_{0}}$.
For comparison in Figs. \[cap:SpecComp\](a) and (b) we also show the single site predictions for the spectrum calculated using expression (\[eq:RqwSS\]) with $\omega_{{\rm ho}}$ chosen to match the effective trap frequency at the lattice site minima. The frequency location of the response spectra compares badly with the full lattice solution, arising primarily from the inadequacy of the harmonic approximation for accurately predicting band structure. The location of the spectra can be easily corrected for by calculating the term $\epsilon_{m}-\epsilon_{0}$ in the expression for $\omega_{{\rm res}}(n)$ using the non-interacting band structure result. However, the single site approximation captures many of the salient features of the full lattice solution, such as the magnitude of the $n$-dependent shift in the the response spectrum. In the strongly-localized defect limit (Fig. \[cap:SpecComp\](b)), the single site approximation quantitatively predicts the response amplitude as the role of the band states can be neglected.
![\[cap:SpecDefect\] (a) Spectrum of excited states versus lattice depth. (b) The bandwidth of the first excited band (solid) and the on-site interaction (dashed) versus lattice depth. Other parameters as in Fig. \[cap:SpecComp\].](Fig3.eps){width="12cm"}
To quantify the emergence of the defect state we examine the spectrum of $\omega_{{\rm res}}^{Q_{n}m}$ as a function of $V_{D}$ in Fig. \[cap:SpecDefect\](a). For $V_{D}\gtrsim13.5E_{R}$ a single defect state is observed to drop below the first excited band. The condition for the emergence of a strongly localized defect state is that the energy reduction from localization above the defect in system 1 is large compared to the effective inter-site tunneling in the excited band. To verify this criterion we compare the excited bandwidth (characterizing the excited band tunneling rate) and energy reduction at the defect, approximated by $U_{11}\alpha_{w}$. These two energy scales are seen to cross at $V_{D}\approx13.5E_{R}$. Finally, we note the validity condition for linear response treatment of Raman spectroscopy. This requires the number of atoms excited to remain small compared to the total number of atoms in the system. In terms of the Raman parameters this condition is given as $nV_{p}^{2}T_{p}^{2}/4\hbar^{2}\ll1$, where $n$ is the average number of atoms per site.
We briefly comment on the relationship of the 1D calculations presented here to an equivalent system in a 3D lattice. The primary difference in applying our formalism to a fully three-dimensional system is that Eq. (\[eq:HjQ\]) will need to be solved in 3D. For the case of coupling to excited bands, the tunneling between lattice sites can occur in all directions, but is dominated along the direction in which the vibrational excitation has occurred. This direction will be parallel to the direction of the momentum transfer in the Raman coupling, which we take to be parallel to a lattice vector. The tunneling in the orthogonal directions will be given by the ground band tunneling rate which is typically much smaller. This suggests that the additional shifts in a fully 3D lattice will be of order the ground state tunneling rate and will thus contribute small corrections to the results presented here. A full study in 3D will be the subject of a future investigation.
Conclusions and outlook
=======================
In this work we have proposed and analysed a Raman spectroscopy technique for probing the properties of quantum degenerate bosons in the ground band of an optical lattice. We have observed that for sufficiently deep lattices, localized states in higher vibrational bands play an important role in the system response, and shifts in resonant frequency of excitation are sensitive to the number of particles per site. While our main study has considered the case of a perfect Mott insulator in a translationally invariant lattice, our results suggest that in the limit of strongly localized defect states the response of the system is well-described by the single site result, and thus only depends on the local number distribution at each lattice site. The Raman spectrum may therefore be a useful method for measuring the relative portion of system 1 at sites with filling factor $n$ atoms. We speculate that in the strongly localized limit, the homogeneity of the lattice is rather unimportant, with the existence of the defect states arises from the large difference in the effective potential energy between the site where the particle is excited and the neighbouring site. This suggests that many of the predictions we have made here, and in particular the results of the 1-site model, should qualitatively apply to inhomogeneous lattices, such as the combined harmonic and optical lattice potentials made in experiments. In this case, as well as in the superfluid limit (where significant number fluctuations exist), significant corrections may arise from resonances between neighbouring sites that would allow the defect to be localized over several sites. However, it seems reasonable to expect that these resonances would contribute to the broad background in the Raman spectrum, and the sharp features from Mott-insulating regions (if present) would be clearly visible. Characterizing the role of superfluid fluctuations and the external confining potential will be the subject of future work.
PBB would like to acknowledge valuable discussions with Trey Porto, Crispin Gardiner, and thanks the University of Otago for supporting this research.
References {#references .unnumbered}
==========
[^1]: The observable for Bragg spectroscopy in optical lattices is the energy transferred to the system. This observable is rather difficult to measure accurately, and to obtain a signal experiments necessarily add a large amount of energy, well-beyond the linear response limit.
[^2]: We note that the primary difference of our treatment to that of Konabe *et al.*, is that we include interactions between scattered and unscattered atoms, which are essential for the main results we present here.
[^3]: Generally large $q$ favors coupling to higher bands, though certain choices of $\mathbf{q}$ can suppress coupling efficiency, e.g for $\mathbf{q}=\mathbf{0}$ the response from the first band is zero by symmetry.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
Linear system solving is one of the main workhorses in applied mathematics. Recently, theoretical computer scientists have contributed sophisticated algorithms for solving linear systems with symmetric diagonally dominant matrices (a class to which Laplacian matrices belong) in provably nearly-linear time. While these algorithms are highly interesting from a theoretical perspective, there are no published results how they perform in practice.
With this paper we address this gap. We provide the first implementation of the combinatorial solver by \[Kelner et al., STOC 2013\], which is particularly appealing for implementation due to its conceptual simplicity. The algorithm exploits that a Laplacian matrix corresponds to a graph; solving Laplacian linear systems amounts to finding an electrical flow in this graph with the help of cycles induced by a spanning tree with the low-stretch property.
The results of our comprehensive experimental study are ambivalent. They confirm a nearly-linear running time, but for reasonable inputs the constant factors make the solver much slower than methods with higher asymptotic complexity. One other aspect predicted by theory is confirmed by our findings, though: Spanning trees with lower stretch indeed reduce the solver’s running time. Yet, simple spanning tree algorithms perform in practice better than those with a guaranteed low stretch.
author:
- Daniel Hoske^1^
- Dimitar Lukarski^2^
- Henning Meyerhenke^1^
- Michael Wegner^1^
bibliography:
- 'lit.bib'
title: |
[Is *Nearly-linear* the same in Theory and Practice?\
A Case Study with a Combinatorial Laplacian Solver]{}
---
Introduction {#chap:intro}
============
Solving square linear systems $Ax=b$, where $A\in \real^{n\times n}$ and $x, b \in \real^n$, has been one of the most important problems in applied mathematics with wide applications in science and engineering. In practice system matrices are often *sparse*, [i.e. ]{}they contain $o(n^2)$ nonzeros. Direct solvers with cubic running times do not exploit sparsity. Ideally, the required time for solving sparse systems would grow linearly with the number of nonzeros $2m$. Moreover, approximate solutions usually suffice due to the imprecision of floating point arithmetic. Spielman and Teng [@Spielman2004], following an approach proposed by Vaidya [@Vaidya90], achieved a major breakthrough in this direction by devising a nearly-linear time algorithm for solving linear systems in symmetric diagonally dominant matrices. *Nearly-linear* means $\OO\bigl(m\cdot\polylog(n)\cdot\log(1/\epsilon)\bigr)$ here, where $\polylog(n)$ is the set of real polynomials in $\log(n)$ and $\epsilon$ is the relative error $\Vert x - \xopt\Vert_A / \Vert \xopt \Vert_A$ we want for the solution $x \in \real^n$. Here $\Vert \cdot \Vert_A$ is the norm $\Vert x \Vert_A := \sqrt{x^TAx}$ given by $A$, and $\xopt := A^{+}b$ is an exact solution. A matrix $A = (a_{ij})_{i,j\in \interval} \in \mat$ is *diagonally dominant* if $|a_{ii}| \geq \sum_{j\neq i} |a_{ij}|$ for all $i \in \interval$. Symmetric matrices that are diagonally dominant (SDD matrices) have many applications: In elliptic PDEs [@Boman04], maximum flows [@Christiano:2011], and sparsifying graphs [@Spielman2008]. Thus, the problem of solving linear systems $Ax=b$ for $x$ on SDD matrices $A$ is of significant importance. We focus here on Laplacian matrices (which are SDD) due to their rich applications in graph algorithms, e.g. load balancing [@DBLP:journals/pc/DiekmannFM99], but this is no limitation [@Kelner2013].
#### Related work. {#sec:related .unnumbered}
Spielman and Teng’s seminal paper [@Spielman2004] requires a lot of sophisticated machinery: a multilevel approach [@Vaidya90; @Reif1998] using recursive preconditioning, preconditioners based on low-stretch spanning trees [@Spielman2009] and spectral graph sparsifiers [@Spielman2008; @Koutis2012]. Later papers extended this approach, both by making it simpler and by reducing the exponents of the polylogarithmic time factors.[^1] We focus on a simplified algorithm by Kelner et al. [@Kelner2013] that reinterprets the problem of solving an SDD linear system as finding an electrical flow in a graph. It only needs low-stretch spanning trees and achieves $\OO\bigl(m\log^2\!n \log\log n \log(1/\epsilon)\bigr)$ time.
Another interesting nearly-linear time SDD solver is the recursive sparsification approach by Peng and Spielman [@Peng2013]. Together with a parallel sparsification algorithm, such as the one given by Koutis [@Koutis2014], it yields a nearly-linear work parallel algorithm.
Spielman and Teng’s algorithm crucially uses the low-stretch spanning trees first introduced by Alon et al. [@Alon95]. Elkin et al. [@Elkin2005] provide an algorithm for computing spanning trees with polynomial stretch in nearly-linear time. Specifically, they get a spanning tree with $\OO(m\log^2\!n \log\log n)$ stretch in $O(m\log^2\!n)$ time. Abraham et al. [@Abraham2008; @Abraham2012] later showed how to get rid of some of the logarithmic factors in both stretch and time.
#### Motivation, Outline and Contribution. {#motivation-outline-and-contribution. .unnumbered}
Although several extensions and simplifications to Spielman and Teng’s nearly-linear time solver [@Spielman2004] have been proposed, none of them has been validated in practice so far. We seek to fill this gap by implementing and evaluating an algorithm proposed by Kelner et al. [@Kelner2013] that is easier to describe and implement than Spielman and Teng’s original algorithm. \[sec:outline\] Thus, in this paper we implement the KOSZ solver (the acronym follows from the authors’ last names) by Kelner et al. [@Kelner2013] and investigate its practical performance. To this end, we start in Section \[chap:prelim\] by settling notation and outlining KOSZ. In Section \[chap:implementation\] we elaborate on the design choices one can make when implementing KOSZ. In particular, we explain when these choices result in a provably nearly-linear time algorithm. Section \[chap:evaluation\] contains the heart of this paper, the experimental evaluation of the Laplacian solver KOSZ. We consider the configuration options of the algorithm, its asymptotics, its convergence and its use as a smoother. Our results confirm a nearly-linear running time, but at the price of very high constant factors, in part due to memory accesses. We conclude the paper in Section \[chap:conclusion\] by summarizing the experimental results and discussing future research directions.
Preliminaries {#chap:prelim}
=============
#### Fundamentals. {#fundamentals. .unnumbered}
We consider undirected simple graphs $G = (V, E)$ with $n$ vertices and $m$ edges. A graph is *weighted* if we have an additional function $w\colon E \to \real_{>0}$. Where necessary we consider unweighted graphs to be weighted with $w_e = 1 ~ \forall e \in E$. We usually write an edge $\{u, v\} \in E$ as $uv$ and its weight as $w_{uv}$. Moreover, we define the set operations $\cup$, $\cap$ and $\setminus$ on graphs by applying them to the set of vertices and the set of edges separately. For every node $u \in V$ its *neighbourhood $N_G(u)$* is the set $N_G(u) := \{v\in V: uv \in E\}$ of vertices $v$ with an edge to $u$ and its *degree $d_u$* is $d_u = \sum_{v \in N_G(u)} w_{uv}$. The *Laplacian matrix* of a graph $G = (V, E)$ is defined as $
L_{u,v} := -w_{uv} \text{ if $uv \in E$},
\sum_{x \in N_G(u)} w_{ux} \text{ if $u = v$ and }
0 \text{ otherwise}
$ for $u, v \in V$. A Laplacian matrix is always an SDD matrix. Another useful property of the Laplacian is the factorization $L = B^T R^{-1} B$, where $B \in \real^{E\times V}$ is the *incidence matrix* and $R \in \real^{E\times E}$ is the *resistance matrix* defined by $B_{ab,c} = 1$ if $a = c$, $= -1$ if $b=c$ and $0$ otherwise. $R_{e_1,e_2} = 1/w_{e_1}$ if $e_1 = e_2$ and $0$ otherwise. This holds for all $e_1, e_2 \in E$ and $a, b, c \in V$, where we arbitrarily fix a start and end node for each edge when defining $B$. With $
x^T L x = (Bx)^T R^{-1} (Bx) = \sum_{e \in E} (Bx)_e^2\cdot w_e \geq 0
$ (every summand is non-negative), one can see that $L$ is positive semidefinite. (A matrix $A \in \real^{n\times n}$ is *positive semidefinite* if $x^T Ax \geq 0$ for all $x \in \real^n$.)
#### Cycles, Spanning Trees and Stretch. {#sec:prelim-cycles .unnumbered}
A *cycle* in a graph is usually defined as a simple path that returns to its starting point and a graph is called *Eulerian* if there is a cycle that visits every edge exactly once. In this work we will interpret cycles somewhat differently: We say that a cycle in $G$ is a subgraph $C$ of $G$ such that every vertex in $G$ is incident to an even number of edges in $C$, [i.e. ]{}a cycle is a union of Eulerian graphs. It is useful to define the addition $C_1 \oplus C_2$ of two cycles $C_1, C_2$ to be the set of edges that occur in exactly one of the two cycles, [i.e. ]{}$C_1 \oplus C_2 := (C_1\setminus C_2) \cup (C_2\setminus C_1)$. In algebraic terms we can regard a cycle as a vector $C \subseteq \SF_2^E$ such that $\sum_{v \in N_C(u)} 1 = 0$ in $\SF_2$ for all $u \in V$ and the cycle addition as the usual addition on $\SF_2^E$. We call the resulting linear space of cycles $\mathcal{C}(G)$.
In a *spanning tree* (ST) $T=(V, E_T)$ of $G$ there is a unique path $P_T(u, v)$ from every node $u$ to every node $v$. For any edge $e = uv \in E\setminus E_T$ (an *off-tree-edge with respect to $T$*), the subgraph $e \cup P_T(u, v)$ is a cycle, the *basis cycle induced by $e$*. One can easily show that the basis cycles form a basis of $\mathcal{C}(G)$. Thus, the basis cycles are very useful in algorithms that need to consider all the cycles of a graph. Another notion we need is a measure of how well a spanning tree approximates the original graph. We capture this by the *stretch $\st(e) = \bigl(\sum_{e' \in P_T(u, v)} w_{e'}\bigr)/w_e$ of an edge $e = uv \in E$*. This stretch is the detour you need in order to get from one endpoint of the edge to the other if you stay in $T$, compared to the length of the original edge. In the literature the stretch is sometimes defined slightly differently, but we follow the definition in [@Kelner2013] using $w_e$. The *stretch of the whole tree $T$* is the sum of the individual stretches $\st(T) = \sum_{e\in E} \st(e)$. Finding a spanning tree with low stretch is crucial for proving the fast convergence of the KOSZ solver.
#### KOSZ (Simple) Solver. {#sec:solver .unnumbered}
As illustrated in Figure \[fig:to-resist\] in the appendix, we can regard $G$ as an electrical network where each edge $uv$ corresponds to a resistor with conductance $w_{uv}$ and $x$ as an assignment of potentials to the nodes of $G$. Then $x_v - x_u$ is the voltage across $uv$ and $(x_v - x_u) \cdot w_{uv}$ is the resulting current along $uv$. Thus, $(Lx)_u$ is the current flowing out of $u$ that we want to be equal to the right-hand side $b_u$. These interpretations used by the KOSZ solver are summarized in Table \[tbl:reinterpret\] in the appendix. Furthermore, one can reduce solving SDD systems to the related problem [@Kelner2013]: Given a Laplacian $L=L(G)$ and a vector $b \in \im(L)$, compute a function $f\colon \widetilde{E} \to \real$ with (i) $f$ being a valid graph flow on $G$ with demand $b$ and (ii) the potential drop along every cycle in $G$ being zero, where a valid graph flow means that the sum of the incoming and outgoing flow at each vertex respects the demand in $x$ and that $f(u,v) = -f(v,u) ~\forall uv \in E$. Also, $\widetilde{E}$ is a bidirected copy of $E$ and the potential drop of cycle $C$ is $\sum_{e \in C} f(e) r_e$. The idea of the algorithm is to start with any valid flow and successively adjust the flow such that every cycle has potential zero. We need to transform the flow back to potentials at the end, but his can be done consistently, as all potential drops along cycles are zero.
$T \leftarrow$ a spanning tree of $G$ $f \leftarrow$ unique flow with demand $b$ that is only nonzero on $T$
Regarding the crucial question of what flow to start with and how to choose the cycle to be repaired in each iteration, Kelner et al. [@Kelner2013] suggest using the cycle basis induced by a spanning tree $T$ of $G$ and prove that the convergence of the resulting solver depends on the stretch of $T$. More specifically, they suggest starting with a flow that is nonzero only on $T$ and weighting the basis cycles by their stretch when sampling them. The resulting algorithm is shown as Algorithm \[alg:basic2\]; note that we may stop before all potential drops are zero and we can consistently compute the *potentials induced by $f$* at the end by only looking at $T$.
The solver described in Algorithm \[alg:basic2\] is actually just the in Kelner et al.’s [@Kelner2013] paper. They also show how to improve this solver by adapting preconditioning to the setting of electrical flows. In informal experiments we could not determine a strategy that is consistently better than the , so we do not pursue this scheme any further here. Eventually, Kelner et al. [@Kelner2013] derive the following running time for KOSZ:
[@Kelner2013 Thm. 3.2] \[thm:main\] can be implemented to run in time $O(m \log^2 n \log \log n \log (\epsilon^{-1} n))$ while computing an $\epsilon$-approximation of $x$.
Implementation {#chap:implementation}
==============
While Algorithm \[alg:basic2\] provides the basic idea of the KOSZ solver, it leaves open several implementation decisions that we elaborate on in this section.
#### Spanning trees. {#sec:trees .unnumbered}
As suggested by the convergence result in Theorem \[thm:main\], the KOSZ solver depends on low-stretch spanning trees. Elkin et al. [@Elkin2005] presented an algorithm requiring nearly-linear time and yielding nearly-linear average stretch. The basic idea is to recursively form a spanning tree using a star of balls in each recursion step. We note that we use Dijkstra with binary heaps for growing the balls and that we take care not to need more work than necessary to grow the ball. In particular, ball growing is output-sensitive and growing a ball $B(x, r) := \{v \in V: \text{Distance from $x$ to $v$ is $\leq r$}\}$ should require $\OO(d\log n)$ time where $d$ is the sum of the degrees of the nodes in $B(x, r)$. The exponents of the logarithmic factors of the stretch of this algorithm were improved by subsequent papers (see Table \[tbl:spanning\] in the appendix), but Papp [@Pap2014] showed experimentally that these improvements do not yield better stretch in practice. In fact, his experiments suggest that the stretch of the provable algorithms is usually not better than just taking a minimum-weight spanning tree. Therefore, we additionally use two simpler spanning trees without stretch guarantees: A minimum-distance spanning tree with Dijkstra’s algorithm and binary heaps; as well as a minimum-weight spanning with Kruskal’s algorithm using union-find with union-by-size and path compression.
[0.35]{} ![Special spanning tree with $\OO\bigl(\frac{(n_1+n_2)^2 \log(n_1+n_2)}{n_1 n_2}\bigr)$ average stretch for the $n_1 \times n_2$ grid.[]{data-label="fig:special-rec"}](recursive-st-new.png "fig:")
[0.35]{} ![Special spanning tree with $\OO\bigl(\frac{(n_1+n_2)^2 \log(n_1+n_2)}{n_1 n_2}\bigr)$ average stretch for the $n_1 \times n_2$ grid.[]{data-label="fig:special-rec"}](special-st.png "fig:")
To test how dependent the algorithm is on the stretch of the ST, we also look at a *special ST* for $n_1 \times n_2$ grids. As depicted in Figure \[fig:special-rec\], we construct this spanning tree by subdividing the $n_1 \times n_2$ grid into four subgrids as evenly as possible, recursively building the STs in the subgrids and connecting the subgrids by a U-shape in the middle.
\[prop:low-stretch-st\] The special ST has $\OO\bigl(\frac{(n_1+n_2)^2 \log(n_1+n_2)}{n_1 n_2}\bigr)$ average stretch on an $n_1 \times n_2$ grid.
#### Flows on trees. {#sec:flows .unnumbered}
Since every basis cycle contains exactly one off-tree-edge, the flows on off-tree-edges can simply be stored in a single vector. To be able to efficiently get the potential drop of every basis cycle and to be able to add a constant amount of flow to it, the core problem is to efficiently store and update flows in $T$. More formally, we want to support the following two operations for all $u, v \in V$ and $\alpha \in \real$ on the flow $f$:
- $\query(u, v)$: return the potential drop $\sum_{e \in P_T(u, v)} f(e)r_e$
- $\update(u, v, \alpha)$: set $f(e) := f(e) + \alpha$ for all $e \in P_T(u, v)$
We can simplify the operations by fixing $v$ to be the root $r$ of $T$: $\query(u)$: return the potential drop $\sum_{e \in P_T(u, r)} f(e)r_e$ and $\update(u, \alpha)$: set $f(e) := f(e) + \alpha$ for all $e \in P_T(u, r)$. The itemized two-node operations can then be supported with $\query(u, v) := \query(u) - \query(v)$ and $\update(u, v, \alpha) := \bigl\{\update(u, \alpha) \text{\,and} \update(v, -\alpha)\bigr\}$ since the changes on the subpath $P_T\bigl(r, \lca(u, v)\bigr)$ cancel out. Here $\lca(u, v)$ is the *lowest common ancestor* of the nodes $u$ and $v$ in $T$, the node farthest from $r$ that is an ancestor of both $u$ and $v$. We provide two approaches for implementing the operations, first an implementation of the one-node operations that stores the flow directly on the tree and uses the definitions of the operations without modification. Obviously, these operations require $\OO(n)$ worst-case time and $\OO(n)$ space. With an data structure, one can implement the itemized two-node operations without the subsequent simplification of using one-node operations. This does not improve the worst-case time, but can help in practice. Secondly, we use the improved data structure by Kelner et al. [@Kelner2013] that guarantees $\OO(\log n)$ worst-case time but uses $\OO(n\log n)$ space. In this case the one-node operations boil down to a dot product ($\query$) and an addition ($\update$) of a dense vector and a sparse vector. We unroll the recursion within the data structure for better performance in practice.
#### Cycle selection. {#sec:selection .unnumbered}
The easiest way to select a cycle is to choose an off-tree edge *uniformly at random* in $\OO(1)$ time. However, to get provably good results, we need to weight the off-tree-edges by their stretch. We can use the flow data structure described above to get the stretches. More specifically, the data structure initially represents $f = 0$. For every off-tree edge $uv$ we first execute $\update(u, v, 1)$, then $\query(u, v)$ to get $\sum_{e \in P_T(u, v)} r_e$ and finally $\update(u, v, -1)$ to return to $f = 0$. This results in $\OO(m\log n)$ time to initialize cycle selection. Once we have the weights, we use *roulette wheel selection* in order to select a cycle in $\OO(\log m)$ time after an additional $\OO(m)$ time initialization.
For convenience we summarize the implementation choices for Algorithm \[alg:basic2\] in Table \[tbl:time\] (appendix). The top-level item in each section is the running time of the best sub-item that can be used to get a provably good running time. The convergence theorem requires a low-stretch spanning tree and weighted cycle selection. Note that $m = \Omega(n)$ as $G$ is connected.
Evaluation {#chap:evaluation}
==========
Settings {#sec:eval-bench}
--------
We implemented the KOSZ solver in C++ using NetworKit [@Staudt2014], a toolkit focused on scalable network analysis algorithms. As compiler we use g++ 4.8.3. The benchmark platform is a dual-socket server with two 8-core Intel Xeon E5-2680 at 2.7 GHz each and 256 GB RAM. Only a representative subset of our experiments are shown here. More experiments and their detailed discussion can be found in [@Hoske14experimental]. We compare our KOSZ implementation to existing linear solvers as implemented by the libraries Eigen 3.2.2 [@Eigen] and Paralution 0.7.0 [@Paralution]. CPU performance characteristics such as the number of executed FLOPS (floating point operations), etc. are measured with the PAPI library [@Browne2000].
We mainly use two graph classes for our tests: (i) Rectangular $k \times l$ grids given by $\mathbb{G}_{k,l} :=
\bigl(\interval[k]\times\interval[l],
\bigl\{\{(x_1, y_1), (x_2, y_2)\} \subseteq \binom{V}{2}: |x_1-x_2| = 1 \lor |y_1-y_2|=1\bigr\}\bigr)$. Laplacian systems on grids are, for example, crucial for solving boundary value problems on rectangular domains; (ii) Barabási-Albert [@Barabasi1999] random graphs with parameter $k$. These random graphs are parametrized with a so-called *attachment $k$*. Their construction models that the degree distribution in many natural graphs is not uniform at all. For both classes of graphs, we consider both unweighted and weighted variants (uniform random weights in $[1, 8)$). We also did informal tests on 3D grids and graphs that were not generated synthetically. These graphs did not exhibit significantly different behavior than the two graph classes above.
Results {#sec:eval-comps}
-------
#### Spanning tree. {#sec:comp-spanning .unnumbered}
Papp [@Pap2014] tested various low-stretch spanning tree algorithms and found that in practice the provably good low-stretch algorithms do not yield better stretch than simply using Kruskal. We confirm and extend this observation by comparing our own implementation of Elkin et al.’s [@Elkin2005] low-stretch ST algorithm to Kruskal and Dijkstra in Figure \[fig:stretch-bench\]. Except for the unweighted $100\times100$ grid, Elkin has worse stretch than the other algorithms and Kruskal yields a good ST. For Barabási-Albert graphs, Elkin is extremely bad (almost factor $20$ worse). Interestingly, Kruskal outperforms the other algorithms even on the unweighted Barabási-Albert graphs, where it degenerates to choosing an arbitrary ST. Figure \[fig:stretch-bench\] also shows that our special ST yields significantly lower stretch for the unweighted 2D grid, but it does not help in the weighted case.
#### Convergence. {#sec:eval-conv .unnumbered}
In Figure \[fig:conv-residual\] we plot the convergence of the residual for different graphs and different algorithm settings. We examined a $100\times 100$ grid and a Barabási-Albert graph with $25,\!000$ nodes. While the residuals can increase, they follow a global downward trend. Also note that the spikes of the residuals are smaller if the convergence is better. In all cases the solver converges exponentially, but the convergence speed crucially depends on the solver settings. If we select cycles by their stretch, the order of the convergence speeds is the same as the order of the stretches of the ST (cmp. Figure \[fig:stretch-bench\]), except for the Dijkstra ST and the Kruskal ST on the weighted grid. In particular, for the Elkin ST on Barabási-Albert graphs, there is a significant gap to the other settings where the solver barely converges at all and the special ST wins. Thus, low-stretch STs are crucial for convergence. In informal experiments we also saw this behavior for 3D grids and non-synthetic graphs. We could not detect any correlation between the improvement made by a cycle repair and the stretch of the cycle. Therefore, we cannot fully explain the different speeds with uniform cycle selection and stretch cycle selection. For the grid the stretch cycle selection wins, while Barabási-Albert graphs favor uniform cycle selection. Another interesting observation is that most of the convergence speeds stay constant after an initial fast improvement at the start to about residual $1$. That is, there is no significant change of behavior or periodicity. Even though we can hugely improve convergence by choosing the right settings, even the best convergence is still very slow, [e.g. ]{}we need about $6$ million iterations ($\approx 3000$ sparse matrix-vector multiplications (SpMVs) in time comparison) on a Barabási-Albert graph with $25,\!000$ nodes and $100,\!000$ edges in order to reach residual $10^{-4}$. In contrast, conjugate gradient (CG) without preconditioning only needs $204$ SpMVs for this graph.
#### Asymptotics. {#sec:eval-asymp .unnumbered}
Now that we know which settings of the algorithm yield the best performance for 2D grids and Barabási-Albert graphs, we proceed by looking at how the performance with these settings behaves asymptotically and how it compares to conjugate gradient (CG) without preconditioning, a simple and popular iterative solver. Since KOSZ turns out to be not competitive, we do not need to compare it to more sophisticated algorithms.
In Figure \[fig:bench-grid\] each occurrence of $c$ stands for a new instance of a real constant. We expect the cost of the CG method to scale with $\OO(n^{1.5})$ on 2D grids [@Demmel1997], while our algorithm should scale nearly-linearly. This expectation is confirmed in the plot: Using Levenberg-Marquardt [@Levenberg63] to approximate the curves for CG with a function of the form $ax^b+c$, we get $b \approx 1.5$ for FLOPS and memory accesses, while the (more technical) wall time and cycle count yield a slightly higher exponent $b \approx 1.6$. We also see that the curves for our algorithm are almost linear from about $650\times 650$. Unfortunately, the hidden constant factor is so large that our algorithm cannot compete with CG even for a $1000\times
1000$ grid. Note that the difference between the algorithms in FLOPS is significantly smaller than the difference in memory accesses and that the difference in running time is larger still. This suggests that the practical performance of our algorithm is particularly bounded by memory access patterns and not by floating point operations. This is noteworthy when we look at our special spanning tree for the 2D grid. We see that using the special ST always results in performance that is better by a constant factor. In particular, we save a lot of FLOPS (factor $10$), while the savings in memory accesses (factor $2$) are a lot smaller. Even though the FLOPS when using the special ST are within a factor of $2$ of CG, we still have a wide chasm in running time. The results for the Barabási-Albert graphs are basically the same (and hence not shown in detail): Even though the growth is approximately linear from about $400,\!000$ nodes, there is still a large gap between our algorithm and CG since the constant factor is enormous. Also, the results for the number of FLOPS are again much better than the result for the other performance counters. In conclusion, although we have nearly-linear growth, even for $1,\!000,\!000$ graph nodes, the KOSZ algorithm is still not competitive with CG because of huge constant factors, in particular a large number of iterations and memory accesses.
#### Smoothing. {#sec:eval-smoother .unnumbered}
One way of combining the good qualities of two different solvers is *smoothing*. Smoothing means to dampen the high-frequency components of the error, which is usually done in combination with another solver that dampens the low-frequency error components. It is known that in CG and most other solvers, the low-frequency components of the error converge very fast, while the high-frequency components converge slowly. Thus, we are interested in finding an algorithm that dampens the high-frequency components, a good smoother. This smoother does not necessarily need to reduce the error, it just needs to make its frequency distribution more favorable. Smoothers are particularly often applied at each level of multigrid or multilevel schemes [@Briggs2000] that turn a good smoother into a good solver by applying it at different levels of a matrix hierarchy. To test whether the Laplacian solver is a good smoother, we start with a fixed $x$ with $Lx=b$ and add white uniform noise in $[-1, 1]$ to each of its entries in order to get an initial vector $x_0$. Then we execute a few iterations of our Laplacian solver and check whether the high-frequency components of the error have been reduced. Unfortunately, we cannot directly start at the vector $x_0$ in the solver. Our solution is to use *Richardson iteration*. That is, we transform the residual $r = b - Lx_0$ back to the source space by computing $L^{-1}r$ with the Laplacian solver, get the error $e = x - x_0 = L^{-1}r$ and then the output solution $
x_1 = x_0 + L^{-1}r.
$
\#1\#2\#3
[0.35]{}
[0.35]{}
\#1
Figure \[fig:smoother\] shows the error vectors of the solver for a $32\times 32$ grid together with their transformations into the frequency domain for different numbers of iterations of our solver. We see that the solver may indeed be useful as a smoother since the energies for the large frequencies (on the periphery) decrease rapidly, while small frequencies (in the middle) in the error remain.
In the solver we start with a flow that is nonzero only on the ST. Therefore, the flow values on the ST are generally larger at the start than in later iterations, where the flow will be distributed among the other edges. Since we construct the output vector by taking potentials on the tree, after one iteration $x_1$ will, thus, have large entries compared to the entries of $b$. In subplot (c) of Figure \[fig:smoother\] we see that the start vector of the solver has the same structure as the special ST and that its error is very large. For the $32\times 32$ grid we, therefore, need about $10000$ iterations ($\approx
150$ SpMVs in running time comparison) to get an error of $x_1$ similar to $x_0$ even though the frequency distribution is favorable. Note that the number of SpMVs the $10000$ iterations correspond to depends on the graph size, [e.g. ]{}for an $100\times100$ grid the $10000$ iterations correspond to $20$ SpMVs.
While testing the Laplacian solver in a multigrid scheme could be worthwhile, the bad initial vector creates robustness problems when applying the Richardson iteration multiple times with a fixed number of iterations of our solver. In informal tests multiple Richardson steps lead to ever increasing errors without improved frequency behavior unless our solver already yields an almost perfect vector in a single run.
Conclusions {#chap:conclusion}
===========
At the time of writing, the presented KOSZ [@Kelner2013] implementation and evaluation provide the first comprehensive experimental study of a Laplacian solver with provably nearly-linear running time. Our study supports the theoretical result that the convergence of KOSZ crucially depends on the stretch of the chosen spanning tree, with low stretch generally resulting in faster convergence. This particularly suggests that it is crucial to build algorithms that yield spanning trees with lower stretch. Since we have confirmd and extended Papp’s [@Pap2014] observation that algorithms with provably low stretch do not yield good stretch in practice, improving the low-stretch ST algorithms is an important future research direction. Even though KOSZ proves to grow nearly linearly as predicted by theory, the constant seems to be too large to make it competitive, even compared to the CG method without preconditioner. Hence, our initial question in the paper title can be answered with “yes” and “no” at the same time: The running time is nearly linear, but the constant factors prevent usefulness in practice. While the negative results may predominate, our effort is the first to provide an answer at all. We hope to deliver insights that lead to further improvements, both in theory and practice. A promising future research direction is to repair cycles other than just the basis cycles in each iteration, but this would necessitate significantly different data structures.
Appendix {#appendix .unnumbered}
========
KOSZ Solver Background {#chap:apdx-solver}
======================
Correspondence between Graphs and Laplacian Matrices
----------------------------------------------------
---------------- ------------------------------------------
$e$ edge/resistor $e$
$w_e$ conductance of resistor $e$
$r_e := 1/w_e$ resistance of resistor $e$
$x_u$ potential at node $u$
$(Lx)_u$ current flowing out of node $u$
$b_u$ current required to flow out of node $u$
---------------- ------------------------------------------
: Interpretations given to a Laplacian $L = L(G) \in \real^{n \times n}$ and a vector $x \in \real^n$ where the $w_e$ for each $e \in E$ are the edge weights.[]{data-label="tbl:reinterpret"}
$L$ operates on every vector $x \in \real^n$ via $$\begin{aligned}
(Lx)_u & = - x_u \cdot \sum_{v \in N(u)} w_{uv} + \sum_{v \in N(u)} x_v \cdot w_{uv}\\
& = \sum_{v \in N(u)} (x_v - x_u) \cdot w_{uv} \label{eq:lapl}\end{aligned}$$ for each $u \in V$.
\#1\#2\#3[(\#2) node\[dot node, circle, draw=black, label=\#3:$#1$\] (\#1) ;]{} \#1\#2\#3\#4[(\#1) edge node\[inner sep=1mm, \#4\] [$#3$]{} (\#2);]{}
(4, 1.5) – (5, 1.5);
\#1\#2\#3[(\#2) node\[dot node, circle, draw=black, label=\#3:$#1V$\] (\#1) ;]{} \#1\#2\#3\#4[(\#1) to\[resistor\] node\[inner sep=2.0mm, \#4\] [$1/#3\Omega$]{} (\#2);]{}
(0, -0.5) – node\[inner sep=1mm, below\] [$(5V-1V) / 1\Omega = 4A$]{} (2, -0.5);
Algorithm Components {#sub:components}
--------------------
Spanning Tree Results
=====================
Proof of Proposition \[prop:low-stretch-st\]
--------------------------------------------
We can inductively show that the average stretch $S(n_1, n_2)$ of the special ST on the $n_1 \times
n_2$ grid is in $\OO\bigl((n_1 + n_2)^2 \log(n_1 + n_2) / n_1 n_2 \bigr)$. To do so, we first prove that by the recursive construction the distance of a node on a border of the grid to a corner of the same border is in $\OO(n_1+n_2)$. Thus, the stretches of the $n_1 + n_2 - 3$ off-tree edges between the rows $\lfloor
n_2/2 \rfloor$ and $\lfloor n_2/2 \rfloor + 1$ as well as the columns $\lfloor n_1/2
\rfloor$ and $\lfloor n_1/2 \rfloor + 1$ are in $\OO(n_1+n_2)$ each. Consequently, $$S\bigl(n_1, n_2\bigr) = 4\cdot S\bigl(n_1/2, n_2/2\bigr) + \OO\bigl(n_1+n_2\bigr)^2$$ when disregarding rounding. After solving this recurrence (note that $S(n_1/2, n_2/2)$ is essentially one fourth in size compared to $S(n_1,n_2)$), we get $$S\bigl(n_1, n_2\bigr)
= \OO\bigl((n_1 + n_2)^2 \log(n_1 + n_2)\bigr).$$ Since the number of edges of the grid is $\Theta(mn)$, the claim for the average stretch follows. Note that in case of a square grid ($n_1 = n_2$) with $N = n_1 \times n_2$ vertices, we get $$S(N) = 4S(N/4) + \OO(N) = \OO(N \log N) = \OO(n_1^2 \log (n_1))$$ and thus $\OO(\log n_1)$ average stretch.
Overview of spanning tree algorithms and their stretch
------------------------------------------------------
**Time** **Stretch**
---------------- ------------------------------------ -------------------------------------------------------
[@Alon95] $\OO\bigl(m^2\bigr)$ $m\cdot exp\bigl(\OO(\sqrt{\log n \log\log n})\bigr)$
[@Elkin2005] $\OO\bigl(m\log^2\!n\bigr)$ $m \cdot \OO\bigl(\log^2 \!n\log\log n\bigr)$
[@Abraham2008] $\OO\bigl(m\log^2\!n\bigr)$ $m \cdot \OO\bigl(\log n (\log\log n)^3\bigr)$
[@Koutis2011] $\OO\bigl(m\log n\log\log n\bigr)$ $m \cdot \OO\bigl(\log n (\log\log n)^3\bigr)$
[@Abraham2012] $\OO\bigl(m\log n\log\log n\bigr)$ $m \cdot \OO\bigl(\log n \log\log n\bigr)$
Dijkstra $\OO\bigl((m+n)\log n\bigr)$ No guarantee
Kruskal $\OO\bigl(m\alpha(n)\log n\bigr)$ No guarantee
: Spanning tree algorithms and their guaranteed stretch[]{data-label="tbl:spanning"}
[^1]: Spielman provides a comprehensive overview of later work at <http://www.cs.yale.edu/homes/spielman/precon/precon.html> (accessed on February 10, 2015).
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
Ground- and space-based planet searches employing radial velocity techniques and transit photometry have detected thousands of planet-hosting stars in the Milky Way. With so many planets now discovered, the next step towards identifying potentially habitable planets is atmospheric characterization. While the Sun–Earth system provides a good framework for understanding atmospheric chemistry of Earth-like planets around solar-type stars, the observational and theoretical constraints on the atmospheres of rocky planets in the habitable zones around low-mass stars (K and M dwarfs) are relatively few. The chemistry of these atmospheres is controlled by the shape and absolute flux of the stellar spectral energy distribution, however, flux distributions of relatively inactive low-mass stars are poorly known at present. To address this issue, we have executed a panchromatic (X-ray to mid-IR) study of the spectral energy distributions of 11 nearby planet hosting stars, the [*Measurements of the Ultraviolet Spectral Characteristics of Low-mass Exoplanetary Systems*]{} (MUSCLES) Treasury Survey. The MUSCLES program consists of contemporaneous observations in the X-rays with $Chandra$ and [*XMM-Newton*]{}, ultraviolet observations with $Hubble$, and visible observations from $Hubble$ and ground-based observatories. Infrared and astrophysically inaccessible wavelengths (EUV and Ly$\alpha$) are reconstructed using stellar model spectra to fill-in gaps in the observational data.
In this overview and companion papers describing the MUSCLES survey, we show that energetic radiation (X-ray and ultraviolet) is present from magnetically active stellar atmospheres at all times for stars as late as M5. Emission line luminosities of and are strongly correlated with band-integrated luminosities and we present empirical relations that can be used to estimate broadband FUV and XUV ($\equiv$ X-ray + EUV) fluxes from individual stellar emission line measurements. We find that while the slope of the spectral energy distribution, FUV/NUV, increases by approximately two orders of magnitude form early K to late M dwarfs ($\approx$ 0.01 to 1), the absolute FUV and XUV flux levels at their corresponding habitable zone distances are constant to within factors of a few, spanning the range 10 – 70 erg cm$^{-2}$ s$^{-1}$ in the habitable zone. Despite the lack of strong stellar activity indicators in their optical spectra, several of the M dwarfs in our sample show spectacular flare emission in their UV light curves. We present an example with flare/quiescent ultraviolet flux ratios of order 100:1, where the transition region energy output during the flare is comparable to the total quiescent luminosity of the star $E_{flare}$(UV) $\sim$ 0.3 $L_{*}$$\Delta$$t$ ($\Delta$$t$ = 1 second). Finally, we interpret enhanced $L(line)$/$L_{Bol}$ ratios for and as tentative observational evidence for the interaction of planets with large planetary mass-to-orbital distance ratios ($M_{plan}$/$a_{plan}$) with the transition regions of their host stars.
author:
- 'Kevin France, R. O. Parke Loyd, Allison Youngblood, Alexander Brown, P. Christian Schneider, Suzanne L. Hawley, Cynthia S. Froning, Jeffrey L. Linsky, Aki Roberge, Andrea P. Buccino James R. A. Davenport, Juan M. Fontenla, Lisa Kaltenegger Adam F. Kowalski, Pablo J. D. Mauas Yamila Miguel, Seth Redfield, Sarah Rugheimer Feng Tian, Mariela C. Vieytes Lucianne M. Walkowicz Kolby L. Weisenburger'
bibliography:
- 'ms\_M22\_emapj.bib'
nocite:
- '[@fressin13]'
- '[@cowan15; @berta15]'
- '[@dressing15; @kopparapu13; @bonfils13]'
- '[@rahmati14]'
- '[@hawley91; @osten05; @robinson01]'
- '[@kaltenegger11]'
- '[@shkolnik14]'
- '[@silva11]'
- '[@loyd15; @loyd16; @youngblood15; @youngblood16; @linsky16]'
- '[@shkolnik14b]'
- '[@west08]'
- '[@linsky13; @linsky14]'
- '[@ribas05; @engle11]'
- '[@linsky95; @hawley03b]'
- '[@smith01]'
- '[@chadney15]'
- '[@lecavelier11; @hallinan13; @shkolnik05; @kashyap08]'
- '[@johnson09; @rojas10]'
- '[@poppenhaeger10]'
- '[@wordsworth11; @vonbraun11]'
- '[@poppenhaeger10]'
- '[@vonbraun12; @torres07]'
- '[@perrin88]'
- '[@bailey09; @wittenmyer14]'
- '[@tuomi13; @mayor09]'
title: 'The MUSCLES Treasury Survey I: Motivation and Overview'
---
Introduction
============
The $Kepler$ mission and ground-based planet searches have detected thousands of exoplanets within the Milky Way and have demonstrated that approximately one-in-six main sequence FGK stars hosts an Earth-size planet (with periods up to 85 days; Fressin et al. 2013). One of the highest priorities for astronomy in the coming decades is the characterization of the atmospheres, and possibly the surfaces, of Earth-size planets in the Habitable Zones (HZs, where liquid water may exist on terrestrial planet surfaces) around nearby stars. An intermediate step towards the discovery of life on these worlds is the measurement of atmospheric gases that may indicate the presence of biological activity. These gases are often referred to as biomarkers or biosignatures.
However, the planetary effective surface temperature alone is insufficient to accurately interpret biosignature gases when they are observed in the coming decades. The dominant energy input and chemistry driver for these atmospheres is the stellar spectral energy distribution (SED). The ultraviolet (UV) stellar spectrum which drives and regulates the upper atmospheric heating and chemistry on Earth-like planets, is critical to the definition and interpretation of biosignature gases (e.g., Seager et al. 2013), and may even produce false-positives in our search for biologic activity (Hu et al. 2012; Tian et al. 2014; Domagal-Goldman et al. 2014).
The nearest potentially habitable planet is likely around an M dwarf at $d$ $<$ 3 pc [@dressing15], the nearest known Earth-size planet orbits an M dwarf (GJ 1132b, $R_{p}$ = 1.2 $R_{\oplus}$, $d$ = 12 pc; Berta-Thompson et al. 2015), and the nearest known Super-Earth mass planets in habitable zones orbit M and K dwarfs, making planetary systems around low-mass stars prime targets for spectroscopic biomarker searches (see also Cowan et al. 2015). The low ratio of stellar-to-planetary mass more readily permits detection of lower mass planets using the primary detection techniques (radial velocity and transits). Moreover, the HZ around a star moves inward with decreasing stellar luminosity. These factors make potentially habitable planets easier to detect around M and K dwarfs. The importance of M dwarf exoplanetary systems is underscored by recent $Kepler$ results and radial velocity measurements showing that between $\sim$ 10 – 50% of M dwarfs host Earth-size planets (0.5 – 1.4 $R_{\oplus}$) in their HZs (Dressing & Charbonneau 2015; Kopparapu 2013; Bonfils et al. 2013). Furthermore, approximately 70% of the stars in the Milky Way are M dwarfs, so rocky planets around low-mass stars likely dominate the planet distribution of the Galaxy. Theoretical work has shown that planets around M dwarfs could be habitable despite their phase-locked orbits (Joshi 2003) and dynamic modeling of transiting systems reveals that most systems permit stable orbits of Earth-mass planets in the HZ long enough for the development of life, i.e. $\gtrsim$ 1.7 Gyr (Jones & Sleep 2010).
M and K dwarfs show significantly larger temporal variability and fraction of their bolometric luminosity at UV wavelengths than solar-type stars [@france13], yet their actual spectral and temporal behavior is not well studied except for a few young ($<$ 1 Gyr), active flare stars. The paucity of UV spectra of low-mass stars and our current inability to accurately model the UV spectrum of a particular M or K dwarf without a direct observation limits our ability to reliably predict possible atmospheric biomarkers. Without the stellar UV spectrum, we cannot produce realistic synthetic spectra of Earth-like planets in these systems, a necessary step for interpreting biomarker gases and their potential to diagnose habitability. Therefore, our quest to observe and characterize biosignatures on rocky planets must consider the star-planet system as a whole, including the interaction between the stellar irradiance and the exoplanetary atmosphere.
High-energy Spectra as Photochemical Atmospheric Model Inputs
-------------------------------------------------------------
[*FUV and NUV Irradiance: Photochemistry and Biosignatures*]{} – Spectral observations of O$_{2}$, O$_{3}$, CH$_{4}$, and CO$_{2}$, are expected to be the most important signatures of biological activity on planets with Earth-like atmospheres (Des Marais et al. 2002; Kaltenegger et al. 2007; Seager et al. 2009). The chemistry of these molecules in the atmosphere of an Earth-like planet depends sensitively on the strength and shape of the UV spectrum of the host star [@segura05]. H$_{2}$O, CH$_{4}$, and CO$_{2}$ are sensitive to FUV radiation (912 –1700 Å), in particular the bright HI Ly$\alpha$ line, while the atmospheric oxygen chemistry is driven by a combination of FUV and NUV (1700 – 3200 Å) radiation (Figure 1).
The photolysis (photodissociation) of CO$_{2}$ and H$_{2}$O by Ly$\alpha$ and other bright stellar chromospheric and transition region emission lines (e.g., $\lambda$1335 Å and $\lambda$1550 Å) can produce a buildup of O$_{2}$ on planets illuminated by strong FUV radiation fields. Once a substantial O$_{2}$ atmosphere is present, O$_{3}$ is primarily created through a multi-step reaction whereby O$_{2}$ dissociation (by 1700 – 2400 Å photons) is followed by the reaction O + O$_{2}$ + $\xi$ $\rightarrow$ O$_{3}$ + $\xi$, where $\xi$ is a reaction partner required to balance energy conservation. O$_{3}$ photolysis is then driven by NUV and blue optical photons. Therefore, on planets orbiting stars with strong FUV and weak NUV flux, a substantial O$_{3}$ atmosphere may arise via photochemical processes alone [@segura10; @hu12; @gao15; @sonny15].
This strong photochemical source of O$_{3}$ may be detectable by future space observatories designed to carry out direct atmospheric spectroscopy of rocky planets (e.g., the HDST or LUVOIR mission concepts), and may be misinterpreted as evidence for biologic activity on these worlds. Therefore, characterization of the stellar FUV/NUV ratio is an essential complement to spectroscopy of exoplanet atmospheres to control for potential false-positive “biomarkers”. Furthermore, it has been shown that the abundances of water and ozone, as well as the atmospheric equilibrium temperature, can respond to changes in the stellar flux on timescales ranging from minutes to years [@segura10]. A detailed knowledge of the absolute flux level and temporal behavior stellar spectrum is important for understanding the evolution of potentially habitable atmospheres.
[*X-ray and EUV Irradiance: Atmospheric Heating and Mass-Loss*]{} – The ratios of X-ray to total luminosity of M dwarfs are orders of magnitude higher ($\gtrsim$ 10 – 100 $\times$) than those of the present day Sun (Poppenhaeger et al. 2010), and the smaller semi-major axes of the HZ around M dwarfs means that X-ray effects on HZ planets will likely be more important than on HZ planets orbiting solar-type stars (Cecchi-Pestellini et al. 2009). Soft X-ray heating of planetary atmospheres enhances evaporation and atmospheric escape (Scalo et al. 2007; Owen & Jackson 2012) which may impact the long-term stability of an exoplanetary atmosphere. Recent works suggest that the influences of early evolution of low mass stars and XUV heating could lead to a bi-modal distribution of water fractions on Earth-mass planets in the HZ of M dwarfs [@tian15]. In order to model the atmosphere as a system, we require inputs for both heating (soft X-ray and EUV, see below) and photochemistry (FUV and NUV).
Extreme-UV (EUV; 100 $\lesssim$ $\lambda$ $\lesssim$ 911 Å) photons from the central star are an important source of atmospheric heating and ionization on all types of extrasolar planets. For terrestrial atmospheres, increasing the EUV flux to levels estimated for the young Sun ($\approx$ 1 Gyr; Ayres 1997) can increase the temperature of the thermosphere by a factor of $\gtrsim$ 10 (Tian et al. 2008), potentially causing significant and rapid atmospheric mass-loss. Ionization by EUV photons and the subsequent loss of atmospheric ions to stellar wind pick-up can also drive extensive atmospheric mass-loss on geologic time scales (e.g., Rahmati et al. 2014 and references therein). Estimates of the incident EUV flux are therefore important for evaluating the long-term stability of a HZ atmosphere; however, a direct measurement of the EUV irradiance from an exoplanetary host star is essentially impossible because interstellar hydrogen removes almost all of the stellar EUV flux for virtually all stars except the Sun. The stellar EUV energy budget contains contributions from both the transition region (Lyman continuum as well as helium and metal line emission in the 228 – 911 Å bandpass) and the corona. FUV emission lines (Ly$\alpha$, , and ) are required to estimate the contribution of the transition region to the EUV flux (Fontenla et al. 2011; Linsky et al. 2014), while X-ray data are necessary to constrain the contribution of the corona (e.g., Sanz-Forcada et al. 2011).
Variability on Timescales of Minutes-to-Hours: Atmospheric Abundances
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An important measurement relating to the habitability of extrasolar planets is the time variability of the energetic incident radiation. While most of the quiescent UV emission from M dwarfs comes from emission lines, continuum emission can become the dominant UV luminosity source during flares (Kowalski et al. 2010). The relative UV emission line strengths also vary during flares (e.g., Hawley et al. 2003; Osten et al. 2005; Loyd & France 2014). Thus, molecular species in the atmospheres of HZ planets will be “selectively pumped” during quiescent periods; only species that have spectral coincidences with stellar emission lines will be subject to large energy input from the host star. However, during flares with strong continuum emission, the relative excitation and dissociation rates relative to quiescent periods could change radically. Therefore, temporally and spectrally resolved observations are essential for understanding the impact of time variability on HZ planetary atmospheres. The amplitude and frequency of flare activity on older M-star exoplanetary hosts is completely unexplored, although GALEX NUV imaging observations suggests that flares may significantly alter the steady-state chemistry in the atmospheres of planets in the HZ (Welsh et al. 2006).
Impulsive UV events are also signposts for energetic flares associated with large ejections of energetic particles. Segura et al. (2010) have shown that energetic particle deposition into the atmosphere of an Earth-like planet without a magnetic field during a large M dwarf flare can lead to significant atmospheric O$_{3}$ depletions ($>$ 90% for the most extreme flares). @buccino07 also studied the impact of a series of lower intensity flares from highly active stars. These events could alter the atmospheric chemistry and increase the penetration depth of UV photons that are damaging to surface life. The impact of a single flare may be detrimental to the development and maintenance of life, but the potentially far more significant impact of persistent flare events has not been studied because the temporal behavior of UV flares is unexplored outside of a few extreme M dwarf flare stars (e.g., AD Leo, EV Lac, and AU Mic; e.g., Hawley et al. 1991; Osten et al. 2005; Robinson et al. 2001).
The MUSCLES Treasury Survey: An Energetic Radiation Survey of Exoplanetary Hosts
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With the previously described motivation in mind, the question that arises is “what are the shapes and absolute flux levels of the UV stellar SEDs incident on these planetary systems?” At present, GJ 832 is the only M dwarf for which a semi-empirical atmosphere model has been built and vetted by direct comparison with spectroscopic observations at optical, NUV, FUV, and X-ray wavelengths [@fontenla15]. There are no other stellar atmosphere models for M dwarfs that treat the chromosphere, transition region, and corona in a self-consistent manner, and none that can produce synthetic spectra for the important X-ray and ultraviolet bandpasses (5 - 3000 Å).
Several approaches have been taken in the literature, including assuming that the star has no UV emission (essentially, no magnetically active upper atmosphere; Segura et al. 2005; Kaltenegger et al. 2011), assuming the extreme flare environment of a star like AD Leo [@segura10; @wordsworth10], or using low-S/N observations of the few flaring M dwarfs that could be observed by $IUE$ [@segura05; @buccino07]. Previously available archival data are insufficient for an accurate quantitative analysis of the radiation fields incident on potentially habitable planets orbiting M dwarfs. Low-sensitivity and contamination by geocoronal Ly$\alpha$ emission make $IUE$ observations insufficient for this work (see example in §2). The lack of observational constraints from UV spectra of M dwarfs will have a major impact on how we judge whether the planets in these systems are actually inhabited. While the need for panchromatic data has been realized for Sun-like stars (e.g., Sun in Time; Ribas et al. 2005), M dwarfs have received less attention (see also Guinan & Engle 2009), despite the fact that these systems dominate the planet statistics of the Galaxy.
To address the above question, we have carried out the first panchromatic survey of M and K dwarf exoplanet host stars in the solar neighborhood ($d$ $\lesssim$ 20 pc). We refer to this program as the [*Measurements of the Ultraviolet Spectral Characteristics of Low-mass Exoplanetary Systems*]{} (MUSCLES) Treasury Survey, a coordinated X-ray to NIR observational effort to provide the exoplanet community with empirically-derived panchromatic irradiance spectra for the study of all types of exoplanets orbiting these stars. MUSCLES is largely based on a [*Hubble Space Telescope*]{} Cycle 22 treasury program and makes use of smaller guest observing programs on $HST$, [*XMM-Newton*]{}, $Chandra$, and several ground-based observatories.
Our survey provides a database of the chromospheric, transition region, and coronal properties of low-mass stars hosting exoplanets, providing high-quality input for models of both Jovian and Earth-like planets as vast numbers of these systems are discovered and characterized in the next decade with missions such as $WFIRST$, $Plato$, $TESS$, and $JWST$. While the long-term evolution of the broadband UV luminosity function of M dwarfs can be constrained with large photometric samples from $GALEX$ (e.g., Shkolnik & Barman 2014), a further uncertainty in the temporal behavior of low-mass exoplanet host stars is the variability on time scales of years (the stellar equivalent of the solar cycle). Given the limitations on space observatories, particularly as it is likely that we will be in the post-$HST$ and post-$Chandra$ era within the next 5 – 10 years, it is critically important to identify a set of visible-wavelength tracers (e.g., Gomes da Silva et al. 2011) that can be used to quantify the longer-term (years-to-decades) UV variability of these stars. The MUSCLES dataset will enable us to derive empirical relations between optical, UV, and X-ray fluxes (e.g., FUV luminosity vs. and H-$\alpha$ profiles), as well as their relative behavior during flares, supporting long-term ground-based programs to study the time evolution of the energetic radiation environment.
This paper provides an overview of the motivation for and the design of the MUSCLES Treasury Survey, as well as some initial quantitative results. A detailed example of the need for $HST$ to carry out this work is given is Section 2. Section 3 describes the MUSCLES target list and the description of the observing modes used in the program. Section 4 presents the evolution of the broad-band SED with stellar effective temperature and habitable zone distance, provides scaling relations to estimate the broadband luminosity from individual spectral line measurements, and presents a first look at the intense high-energy flare behavior of these otherwise inactive stars. Section 5 explores the interaction of the planets and host stars in these systems, and compares the UV flux measurements with predictions from coronal models. Section 6 presents a summary of the important results from this paper.
Details on all aspects of the data analysis and scientific results from the program can be found in the companion papers from the MUSCLES team. @youngblood15 present an analysis of the essential reconstruction of the intrinsic Ly$\alpha$ stellar emission lines and the calculations of the EUV irradiance (“Paper II”). @loyd15, “Paper III”, describe the creation of the panchromatic spectral energy distributions (SEDs), quantification of the SEDs for exoplanet atmospheric modeling, and a description of how to download the data in machine-readable format from the Milkulski Archive for Space Telescopes[^1]. @fontenla15 present semi-empirical modeling of M dwarf atmospheres based in part on MUSCLES observations. Detailed descriptions of the flare properties of the MUSCLES sources and a comparison of the UV and X-ray emission from these stars with contemporaneous ground-based photometry and spectroscopy are presented in papers by Loyd (2016 – in prep.) and Youngblood (2016 – in prep.), respectively, and Linsky et al. (2016) describe the kinematics of the stellar atmospheres derived from the UV emission lines in the $HST$ data.
Motivation: The Importance of High-Sensitivity, Spectrally Resolved Data for Atmospheric Modeling
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As mentioned above, the observational and theoretical understanding of the upper atmospheres of “weakly active” M dwarfs [@walkowicz09] is insufficient for a deterministic prediction of the flux in the HZ around an M dwarf host star. $GALEX$ data provide a statistical picture of the evolution of the UV luminosity of M dwarfs [@shkolnik14], but do not include spectral information on the specific intensity (important for accurate photoexcitation and photolysis rates of key atmospheric species), coverage of the brightest FUV and NUV spectral lines (Ly$\alpha$ and , respectively), information about the flare or stellar cycle state of the star, or the connection to the X-ray and optical fluxes of the stars. @shkolnik14b provided scaling relations to calculate the Ly$\alpha$ and fluxes from $GALEX$ broadband fluxes when $HST$ spectroscopy is not available.
To demonstrate the importance of using high-quality UV observations from $HST$, we compare in Figure 2 the observed UV flux of GJ 581 with another $\sim$ M3 (spectral types for GJ 581 range from M2.5 – M5 in the literature) stellar spectrum taken from the $IUE$ archive (the M3V binary star GJ 644, scaled to the distance of GJ 581). One might consider using GJ 644 as a proxy for the GJ 581 stellar radiation field before the MUSCLES data were acquired. No weakly active M dwarfs were bright enough to be observed by $IUE$ outside of flares. Two discrepancies between the spectra shown in Figure 2 are immediately apparent: 1) the “continuum flux” level of GJ 644 is approximately two orders of magnitude larger than GJ 581. The continuum level of the GJ 644 spectrum is approximately 2 $\times$ 10$^{-15}$ erg cm$^{-2}$ s$^{-1}$ Å$^{-1}$, which is consistent with the instrumental background equivalent flux for the $IUE$ FUV low-resolution channel. The $HST$-COS background level is approximately 3 $\times$ 10$^{-17}$ erg cm$^{-2}$ s$^{-1}$ Å$^{-1}$, and no stellar FUV continuum is detected above this level in GJ 581. In Paper III, we show that approximately half of the M dwarfs in the MUSCLES sample have FUV continuum detections at this flux level. Therefore, the $IUE$ spectrum includes a large amount of instrumental noise relative to the true flux upper limits in the FUV continuum. 2) the GJ 644 Ly$\alpha$ emission line is much brighter and broader than the reconstructed Ly$\alpha$ emission from GJ 581. The Ly$\alpha$ emission line in the $IUE$ spectrum is almost entirely geocoronal airglow emission. The large 10$^{''}$ $\times$ 20$^{''}$ oval $IUE$ aperture admits 5000 times more geocoronal emission than STIS E140M observations using the 0.2$^{''}$ $\times$ 0.2$^{''}$ aperture, and the $IUE$ spectral resolution is too low to separate the interstellar absorption component from the stellar and geocoronal emission components. Therefore, $IUE$ spectra cannot be used to compute the Ly$\alpha$ irradiance from faint, low-mass stars.
A comparison of the instrumental background and geocoronal airglow emission with the total 1160 - 1690 Å FUV flux from the $IUE$ spectrum of GJ 644 indicates that $\approx$ 80 % of the recorded counts come from non-stellar sources. This dramatic overestimation of the FUV flux in $IUE$ M dwarf data would lead one to infer FUV/NUV flux ratios $\gtrsim$ 40 when using the GJ 644 spectrum. This is much larger than the FUV/NUV flux ratios of $\sim$ 0.5 – 1 that are found for M dwarf exoplanet host stars using the higher sensitivity, higher resolution $HST$ data (Section 4.1). The MUSCLES database therefore provides researchers modeling atmospheric photochemistry and escape high-fidelity, $HST$-based, host star SEDs for their calculations.
Targets and Observing Program
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MUSCLES Target Stars
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The MUSCLES target list (Table 1) was chosen to cover a broad range of stellar types (K1V – M5V; 4 K dwarfs and 7 M dwarfs), exoplanet masses, and semi-major axes; including most of the known M dwarf exoplanet host stars located within $d$ $\lesssim$ 13pc (7/12), while excluding flare stars (e.g. GJ 674) that require intensive multi-wavelength monitoring to clear $HST$ instrument safety protocols. The M dwarfs span a range of spectral types (from M1 – M5), a range of X-ray luminosity fractions (log$_{10}$($L_{X}$/$L_{Bol}$) $\approx$ $-$5.1 to $-$4.4), an indicator of activity level), and planetary systems ranging from Jupiters (GJ 832) to super-Neptunes (GJ 436) to super-Earths (GJ 1214). About $\sim$65 % of the exoplanet host stars in our sample (7/11) harbor Super-Earths ($M_{plan}$ $<$ 10 $M_{\oplus}$; [**bold**]{} in Table 1). In the brief summaries of the star-planetary systems given in the Appendix, we refer to the [*M sin i*]{} of the planets as their mass as a shorthand.
With the exception of $\epsilon$ Eri, the MUSCLES stars are not traditionally classified as active or flare stars (in contrast with widely studied M dwarf flare stars such as AU Mic or AD Leo). The stars in our sample are considered “optically inactive”, based on their H$\alpha$ absorption spectra [@gizis02]. However, the H and K emission line cores are a more straightforward means of diagnosing chromospheric activity in the optical [@silva11] because the emission line flux increases with activity while H$\alpha$ first shows enhanced absorption before becoming a strong emission line with increasing activity. All of our stars with measured H and K profiles show weak but detectable emission (equivalent widths, W$_{\lambda}$() $>$ 0; note that the references below present emission lines as positive equivalent widths, opposite from the traditional convention), indicating that at least a low level of chromospheric activity is present in these stars [@rauscher06; @walkowicz09]. In addition, @hawley14 have shown that inactive M dwarfs display flares in their $Kepler$ light curves, confirming that chromospheric activity is present on MUSCLES-type stars. Figure 3 shows the K line equivalent width as a function of (B – V) color for six of the M dwarfs in the sample, overplotted on the data from @rauscher06. The K emission is in the range 0.1 $<$ W$_{\lambda}$() $<$ 0.8 Å, approximately an order of magnitude smaller than traditional flares stars EV Lac, AU Mic, AD Leo, and Proxima Cen (also shown in Figure 3). According to the M dwarf classification scheme of @walkowicz09, these stars have intermediate chromospheres and are referred to as “weakly active”. In the Appendix, we present brief descriptions of each of the stars studied here. For more detailed descriptions of the stellar parameters of the MUSCLES target stars, we refer the reader to @loyd15 and @youngblood15.
MUSCLES Observing Strategy
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[*Ultraviolet Observations, 100 – 3200 Å*]{} – In order to obtain a full census of the UV emission incident on the habitable zones of low-mass stars, we require $HST$ spectral coverage from 1150 – 3100 Å: the G130M, G160M, and G230L modes of COS, and the G140M, E140M, E230M, E230H, and G230L modes of STIS provide spectral coverage across this bandpass. This combination of instrumental settings allows us to catalog the stellar emission lines that are relevant to the photoexcitation of the primary atmospheric constituents of the exoplanets in these systems. The Ly$\alpha$ emission line dominates the total UV luminosity of M dwarfs [@france12a]. We use the G140M mode of STIS with the 52 $\times$ 0.1 slit to measure the Ly$\alpha$ profile for our M dwarf targets. We have previously demonstrated that this technique can produce high-quality measurements of the local Ly$\alpha$ flux [@france13]. For the brighter K dwarfs, we employ the STIS E140M mode, using the 0.2 $\times$ 0.06 slit to resolve the intrinsic line profile and minimize contamination by telluric . Resonant scattering of Ly$\alpha$ in the local ISM requires that the line must be reconstructed to provide a reliable measure of the intrinsic Ly$\alpha$ radiation field in these exoplanetary systems. We direct the reader to Paper II for a detailed description of the Ly$\alpha$ reconstruction developed for the MUSCLES Treasury Survey [@youngblood15]. In the FUV (except Ly$\alpha$) we use COS G130M ( $\lambda$1206, $\lambda$1240, $\lambda$1335, $\lambda$1400 lines) and COS G160M ($\lambda$1550, $\lambda$1640, $\lambda$1671 lines). Emission from Ly$\alpha$, , , and is particularly interesting because these lines provide constraints on the Lyman continuum/EUV (200 $\lesssim$ $\lambda$ $\lesssim$ 900 Å) irradiance in these systems (Linsky et al. 2014; Shkolnik & Barman 2014). COS is essential for a moderate spectral resolution FUV line census as the lower effective area and higher detector background of STIS make observations of all but the very brightest emission line ( Ly$\alpha$) prohibitively time consuming. We use the medium resolution (“M”) COS modes to resolve chromospheric emission lines and maximize contrast from narrow spectral features.
At NUV wavelengths, we use STIS G230L ($\lambda$ $>$ 2200 Å) to observe the NUV continuum, $\lambda$2400 and $\lambda$2600, and $\lambda$2800, but take advantage of the superior sensitivity of the COS G230L mode to observe the 1750 – 2200 Å region that is important for the photodissociation of O$_{2}$ and the production of O$_{3}$. For targets that exceed the G230L bright-object limit (the K stars, $\epsilon$ Eri, HD 40307, HD 85512, and HD 97658), we use the higher-resolution E230M mode ($\lambda$1978 + $\lambda$2707 settings), covering the 1800 – 3100 Å bandpass. Target brightness limits dictated that we employ the STIS E230H mode for observations of the emission lines on the brightest K dwarf ($\epsilon$ Eri).
The Ly$\alpha$ fluxes are combined with models of solar active regions [@fontenla11] to estimate the EUV luminosity in the wavelength region 100 – 1170 Å, in 100 Å bins (see Linsky et al. 2014 and Paper II). These EUV calculations were compared to the GJ 832 atmosphere model developed by @fontenla15; we found that the 10 – 100 Å and 100 – 1000Å bands differed by less than a factor of two. Given that no inactive M dwarf EUV observations exist as a reference point, we consider this intermodel agreement to sufficient.
The MUSCLES observing plan was designed to study both the spectral and temporal variability of low-mass exoplanet host stars on the characteristic timescale for UV/optical variability (Kowalski et al. 2009). Using the most sensitive UV photon-counting mode on $HST$ (COS G130M) and the time-tag capability of the COS microchannel plate detector [@green12], we measure the 10$^{4-5}$ K chromospheric and transition region activity indicators (using the , , , and emission lines) in 8-hour intervals (5 contiguous spacecraft orbits for each star). Quasi-simultaneous X-ray observations were coordinated with these visits when possible (see below). While this strategy does not provide continuous coverage due to Earth occultation, it is optimized for constraining the types and frequency of flare behavior on low-mass exoplanet host stars. Since the characteristic timescale for UV/optical flare activity on M dwarfs is thought to be minutes-to-hours (Welsh et al 2007; Kowalski et al. 2009; Loyd & France 2014), our observations are ideal for quantifying the importance of flare activity to the local UV radiation field in these systems.
With the above considerations in mind, we arranged the UV – visible observations for each target into campaigns comprising 3 $HST$ visits: [**1)**]{} COS G130M, [**2)**]{} COS G160M+G230L, [**3)**]{} STIS G/E140M+G230L+G430L, all executed within a day of each other to mitigate uncertainties introduced by month or year timescale variations in the stellar flux. Between 9 and 13 total $HST$ exposures were acquired for each star depending on the target brightness and the observing modes used. For a graphical description of which modes contribute at which wavelengths for each target, we refer the reader to @loyd15.
[*X-ray Observations, 5 – 50 Å (2.5 keV to 0.25 keV)* ]{} – We used quasi-simultaneous $Chandra$ and [*XMM-Newton*]{} observations of the MUSCLES stars to provide temporally consistent SEDs and to explore the wavelength-dependent behavior of stellar flares. The X-ray observations were coordinated with the COS G130M 5 orbit monitoring program (see above) to establish the longest possible simultaneous baseline over which to explore the panchromatic properties of M and K dwarf flares. We observed GJ 667C, GJ 436, GJ 176, and GJ 876 using the $Chandra$ ACIS-S back-illuminated S3 chip. Owing to an $HST$ safing event in 2015 June, the GJ 876 campaign was not simultaneous. We observed $\epsilon$ Eri and GJ832 with [*XMM-Newton*]{} because they are optically bright. $\epsilon$ Eri is by far the brightest X-ray source in our sample and provided a high-quality RGS grating spectrum. $\epsilon$ Eri required use of the pn/MOS “Thick” filters, while GJ832 used the “Medium” filters. We also obtained photometry with the UVM2 filter for GJ 832. HD 85512 and HD 40307 were observed as part of a complementary [*XMM-Newton*]{} program (PI – A. Brown) using the same configuration as for GJ 832. For the stars without contemporaneous X-ray data, we obtained spectra from the $Chandra$ and [*XMM-Newton*]{} archives (Loyd et al. 2016; Brown et al. 2016 – in prep.). The coronal model fit to the X-ray data is extended into the 50 – 100 Å region where no direct observations are available.
[*Optical Observations, 3200 – 6000 Å*]{} – We carried out complementary ground-based observations as close in time as possible to the MUSCLES UV/X-ray observations. Our primary optical spectra came from the ARCES and DIS instruments on the Astrophysical Research Consortium (ARC) 3.5-m telescope at Apache Point Observatory (3700 – 10000 Å, depending on the mode), the 2.15-meter Jorge Sahade telescope at the Complejo Astronomico El Leoncito [@Cincunegui04], and the FLOYDS intrument on the Las Cumbras Observatory Global Telescope Network [@brown13]. Multi-band optical photometry was acquired with the Apache Point Observatory 0.5-m ARCSAT telescope (with the Flarecam instrument) and the LCOGT. The optical data and their correlation with spectral and temporal behavior of the high-energy emission will be presented in a future work by the MUSCLES team [@youngblood16]. In order to calibrate the UV data with respect to visible/IR photospheric models and the ground-based spectra, we also acquired short optical observations with STIS G430L or G430M (depending on target brightness) during each $HST$ campaign.
[*Infrared Extension, $>$ 6000 Å*]{} – Stellar atmosphere models considering only the photosphere (e.g., the PHOENIX models) are able to reproduce the peak and Rayleigh-Jeans tail of the stellar SED given the correct prescription for the effective temperature and abundances of atoms and molecules [@allard95; @husser13]. We employ PHOENIX models, matched to the Tycho B and V band fluxes, to calibrate the optical spectroscopy from $HST$-STIS and fill out the panchromatic SEDs from the red-optical to the mid-IR ($\sim$ 0.6 – 5.5 $\mu$m; Loyd et al. 2016). The red-optical and infrared fluxes are important as they make up the majority of the bolometric luminosity from cool stars, regulate the effective surface temperature of orbiting planets, and provide the reference for the activity indicators (e.g., $L$()/$L$$_{Bol}$) presented in this work.
Results: 5 Å to 5 $\mu$m Irradiance Spectra
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The primary goal of the MUSCLES Treasury Survey is to develop a uniform set of irradiance spectra to support the study of extrasolar planets orbiting low-mass stars. The X-ray through optical observations are described above, a description of the analysis required to complete “missing” parts of the SED (intrinsic Ly$\alpha$ line profiles and EUV fluxes) is presented in Paper II [@youngblood15], and a detailed discussion of the creation of the panchromatic radiation fields is given in Paper III of this series [@loyd15]. In the following subsections, we present both quantitative and qualitative descriptions of the SEDs, including the behavior of the broadband fluxes as a function of stellar effective temperature and habitable zone location, the distribution of stellar emission lines observed in the $HST$ spectra, the use of individual spectral tracers to estimate the broadband energetic radiation fluxes, and an initial description of the temporal variability of the MUSCLES Treasury dataset.
For the results presented below, we used both broadband and emission line fluxes. The broadband fluxes are simply integrals of the panchromatic SED over the wavelength region of interest. Ly$\alpha$ fluxes are the wavelength-integrated reconstructed line-profiles obtained by @youngblood15. We measured the metal emission lines by employing a Gaussian line-fitting code that takes into account the line-spread function of the instrument used to create that portion of the spectrum. For instance, and lines are fit with a single-component Gaussian emission line convolved with the appropriate wavelength-dependent $HST$-COS line-spread function[^2] [@france12b]. Some emission lines are clearly better described by a two-component model [@wood97], but this only applies to a small number of emission lines in the sample and we assume a single Gaussian for the results presented in this paper. emission line fluxes assume an unaltered Gaussian line-shape for both COS and STIS observations. As most of the observations were made with the low-resolution G230L modes of COS and STIS, an interstellar absorption correction is not possible for all targets, so the line fluxes are presented as-measured. Figures 4 and 5 show sample FUV (1328 – 1410 Å) and NUV (2789 – 2811 Å) spectral regions at full resolution with the prominent emission lines labeled.
Broadband Fluxes
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Figure 6 shows a montage of the full 5 Å – 5 $\mu$m SEDs for the 11 MUSCLES exoplanet host stars, binned to 5 Å pixel$^{-1}$ for display and divided by the bolometric luminosity. The bolometric luminosity used in this work is simply the integral of the complete SED for each star (including a calculated Rayleigh-Jeans tail at $\lambda$ $>$ 5.5 $\mu$m; Loyd et al. 2016). Many of the component spectra have spectral resolutions much higher than this (resolving powers of $R$ $>$ 15,000 for all of the FUV data and up to $R$ = 114,000 for STIS E230H observations of $\epsilon$ Eri), and both full resolution and binned data sets are available on the MAST HLSP archive[^3]. As discussed in the introduction, the broadband behavior of the stellar SED can give general insights into the relevant heating and photochemical rates for all types of exoplanet atmospheres.
We define the FUV flux, $F$(FUV), as the total stellar flux integrated over the 912 – 1700 Å bandpass, including the reconstructed Ly$\alpha$ emission line, FUV = ($F$(Ly$\alpha$) + $F$(1170 – 1210 + 1220 – 1700) + $F$(912 – 1170)). $F$(Ly$\alpha$) is the reconstructed Ly$\alpha$ line flux, $F$(1170 – 1210 + 1220 – 1700) is the non-Ly$\alpha$ flux directly measured by $HST$ COS and STIS, and $F$(912 – 1170) is the “Lyman Ultraviolet” emission constructed from the solar active region relations for low-mass stars presented by @linsky14. Ly$\alpha$ contributes on average 83% ($\pm$ 5%) to the total FUV flux. $F$(NUV) are the combined $HST$ spectra integrated over 1700 – 3200 Å. The fractional luminosities in each band as well as key emission lines are given in Table 3.
Figure 7 shows the FUV/NUV flux ratio as a function of stellar effective temperature. The FUV/NUV flux ratio is in the range 0.5 – 0.7 for the latest M stars (GJ 1214 and GJ 876), is in the range 0.2 – 0.4 for M1 to M3 stars, declines to 0.04 by mid-K, and is $\lesssim$ 0.01 for spectral types K2 and earlier. @france13 have previously shown that the FUV/NUV ratio drops to $\sim$ 10$^{-3}$ for solar-type stars. The trend in FUV/NUV with effective temperature is largely driven by the large increase in photospheric emission moving into the NUV spectral bandpass with increasing effective temperature. Large FUV fluxes dissociate O$_{2}$ to generate O$_{3}$. The photospheric NUV photons destroy photochemically produced ozone and keep the atmospheric O$_{3}$ mixing ratios low on Earth-like planets around solar-type stars, in the absence of a disequilibrium process such as life.
The XUV flux (5 – 911 Å), the combination of the observed and modeled soft X-ray flux (usually 2.5 keV – 0.125 keV, or $\approx$ 5 – 100 Å[^4]) and the calculated EUV flux (100 – 911 Å), is an important heating agent on all types of planets. XUV irradiance is particularly important for short-period planets [@lammer09]. Figure 8 shows the relative XUV and FUV luminosities as a function of effective temperature for the MUSCLES stars. The total XUV and FUV luminosities are shown to be well-correlated with stellar effective temperature (and therefore stellar mass) and the fractional XUV and FUV luminosities are in the range 10$^{-5}$ $\lesssim$ $L$(band)/$L_{bol}$ $\lesssim$ 10$^{-4}$ with no dependence on stellar effective temperature. For comparison, the disk-integrated quiet Sun [@woods09] has $L$(XUV)/$L_{Bol}$ = 2 $\times$ 10$^{-6}$ and $L$(FUV)/$L_{Bol}$ = 1 $\times$ 10$^{-5}$, respectively. Note however the solar $L$(FUV) contains a contribution from the photosphere, whereas the photospheric contribution in negligible for M and K dwarfs. From the perspective of integrated planetary atmosphere mass-loss over time, our measurements represent only a conservative lower-limit as the stellar XUV radiation was likely factors of 10 – 100 times higher during the star’s younger, more active periods. The active periods of M stars are more prolonged than the equivalent “youthful magnetic exuberance” [@ayres10b] period of solar-type stars (e.g., West et al. 2008). Additionally, it is likely that the stellar mass-loss rates are higher earlier in their evolution [@wood05b], increasing the potential planetary mass-loss.
We can combine these results into a comprehensive picture of the energetic radiation environment in the HZs around the low-mass stars. Figure 9 shows the FUV/NUV ratios, the total FUV fluxes, and the total XUV fluxes at the habitable zone orbital distances for each of our targets. The plots show the average habitable zone distance for each star, $\langle$$r_{HZ}$$\rangle$, computed as the mean of “runaway greenhouse” and “maximum greenhouse” limits to the HZ presented by @kopparapu14. The error bars on $\langle$$r_{HZ}$$\rangle$ represent these extrema. One observes the two orders-of-magnitude decline of the FUV/NUV ratio from 0.1 – 0.7 AU, mainly driven by the stellar effective temperature dependence described above. The FUV HZ fluxes show a weak trend of increasing flux with increasing $r_{HZ}$, however both the M dwarf and K dwarf samples have a factor of roughly 5 dispersion at a given HZ distance. For example, while it is possible for the absolute FUV flux to be a factor of 10 greater at 0.7 AU than 0.15 AU, it is also possible that the FUV flux at 0.15 AU is greater than at 0.7 AU. A similar dispersion is seen in the XUV fluxes, and there is no statistically significant change in the XUV flux across the habitable zone (Table 2). Therefore, the average FUV and XUV fluxes in the HZs of M and K dwarfs are 10 – 70 erg cm$^{-2}$ s$^{-1}$. We emphasize that the activity level of the individual star must be considered, and direct observations are preferable when available. In Section 4.3, we will discuss the correlation of the broadband fluxes with specific emission line measurements to simplify the characterization of the broadband fluxes from single emission line flux measurements.
1150 – 3200 Å Stellar Emission Lines: The Ubiquity of UV Emission
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Chromospheric and transition region emission lines are observed in $all$ of the MUSCLES spectra, arguing that all exoplanet host stars with spectral type M6 and earlier have UV-active atmospheres (Figures 4 and 5). This seems to rule out photosphere-only models of cool stars and indicates that the chromospheric, transition region, and coronal emission must be included for accurate modeling of the atmospheres of planets orbiting these stars. Bright emission lines in the MUSCLES spectra with chromospheric formation temperatures in the range $\sim$ (4 – 30) $\times$ 10$^{3}$ K include Ly$\alpha$, $\lambda$1264, 1526, 1808, $\lambda$$\lambda$1334,1335, $\lambda$1671, multiplets near 2400 and 2600 Å, and $\lambda$$\lambda$2796,2803 (see e.g., the M dwarf contribution functions presented by Fontenla et al. 2015). We observe many transition region lines with formation temperatures from $\sim$ (40 – 200) $\times$ 10$^{3}$ K, including the 1175 multiplet, $\lambda$1206, $\lambda$1218, 1371, $\lambda$$\lambda$1239,1243, \] $\lambda$1401, $\lambda$$\lambda$1394,1403, $\lambda$$\lambda$1548,1550, and $\lambda$1640. The coronal iron lines, $\lambda$1242 and $\lambda$1354 [@ayres03], are observed in a fraction of $HST$-COS observations (7/11 stars show emission lines while only GJ 832 and GJ 876 show emission lines). Taken together with the X-ray observations, these highly ionized iron lines demonstrate the presence of coronal gas in the atmospheres of all of our stars and enable an alternative calculation of the EUV irradiance using emission measure techniques [@forcada11; @chadney15]. We will present a detailed discussion of the atmospheric kinematics derived from emission line parameters in an upcoming work (Linsky et al. – in preparation).
A major complication in previous attempts to assemble panchromatic radiation fields, particularly of M dwarfs, is the flux variability between observations separated by years (e.g., comparing $IUE$ spectra from the 1980s with $ROSAT$ X-ray data from the 1990s and $HST$ observations of Ly$\alpha$ from the 2000s, Linsky et al. 2013, 2014). One of the goals of the MUSCLES Treasury Survey is to obtain multi-wavelength observations close in time to minimize this large systematic uncertainty. In the MUSCLES observing strategy, emission lines with similar formation temperatures were acquired on different $HST$ visits (owing to their inclusion on different $HST$ grating modes) separated by 18 – 48 hours. The overlap in formation temperature and spectral coverage between adjacent modes facilitates scaling over calibration variations [@loyd15] and smaller day-to-day variations.
This approach has been successful – Figure 10 shows the relationship of the fractional luminosity in the transition region ions and . These lines have formation temperatures within a factor of two of each other (between (1 – 2) $\times$ 10$^{5}$ K) and have been shown to be tightly correlated in numerous astrophysical plasmas, including the atmospheres and accretion columns around young stars [@oranje86; @ardila13; @france14]. The versus correlation is very well maintained over the MUSCLES sample despite the non-simultaneous observations, with a Pearson correlation coefficient of 0.90 ($\rho$ in the legend of Figure 10) and a probability of no correlation of 6.0 $\times$ 10$^{-5}$ ($n$ in the legend of Figure 10). Because we do not exclude discrete impulsive flares, the MUSCLES spectra can be considered to be an accurate snapshot of the average stellar spectrum (averaged on the timescale of hours-to-days). Variability on the timescales of the solar cycle (or stellar cycles, years) will be addressed by developing XUV – optical tracer correlations and carrying out long-term monitoring from ground-based facilities (Section 1.3).
The emission lines in the MUSCLES stars also show an evolution of decreasing fractional luminosity of the transition region with increasing effective temperature. Figure 11 ($top$) shows the fractional luminosity as a function of effective temperature. Given that the total XUV and FUV factional luminosities are approximately constant with effective temperature (Figure 8, $lower$), this suggests that the upper transition region activity (traced by ) declines faster than the rest of the upper atmosphere (e.g., the chromosphere and corona), possibly relating to the pressure-density structure of the stellar atmosphere changing with mass [@fontenla15]. While the Pearson correlation coefficient is only $-$0.60 ($n$ = 3.5 $\times$ 10$^{-2}$), this is skewed by the inclusion of $\epsilon$ Eri, the only active star in the sample. Excluding $\epsilon$ Eri, the coefficient becomes $-$0.65 ($n$ = 9.1 $\times$ 10$^{-3}$). The middle plot in Figure 11 shows the total luminosity with the stellar rotation rate, suggesting a period-activity relation analogous to the well-studied relationship in solar-type stars (Ribas et al. 2005; see also Engle & Guinan 2011 for M dwarf X-ray evolution). The notable exception is GJ 876, with a luminosity approximately an order of magnitude larger than expected based on its rotational period. This is partially the result of the strong flare activity on this star (see Section 4.4), and also suggests that the period is not well-determined for this star. Excluding GJ 876, the Pearson and Spearman coefficients are $-$0.62 and $-$0.92, respectively. If GJ 876 fell on this trend, we would expect a rotation period closer to $\sim$ 40 days. Interestingly, there is no trend in the fractional luminosity with stellar rotation rate (Figure 11, $bottom$).
The Correlation Between and Fluxes and Broadband Luminosities
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Another goal of the MUSCLES Treasury Survey is to identify individual spectral tracers that can serve as proxies for the broadband fluxes from low-mass stars so that large, resource-intensive projects may not be required to obtain accurate estimates for the energetic radiation environments around exoplanets in the future. This anticipates a time beyond the current suite of UV and X-ray observatories (e.g., $XMM$, $Chandra$, and $HST$) capable of making these measurements, and a time when the sheer number of potentially habitable planets around low-mass stars precludes a detailed panchromatic characterization of every target (possibly after the $TESS$ mission). In this case, simple relationships for acquiring reasonably accurate broadband stellar flux measurements will be essential for accurately modeling the atmospheric spectra from these worlds when they are acquired by future flagship missions in the 2020s, 2030s, and 2040s. Below, we describe the relationship between the XUV and FUV luminosities of the MUSCLES stars with the most prominent FUV and NUV emission lines, and [^5]. In a future work, we will explore ground-based tracers of the UV radiation environment (e.g., H & K fluxes and equivalent widths from our contemporaneous ground-based observations; Youngblood et al. 2016a – in preparation).
Figure 12 shows the relationship between the broadband luminosities $L$(XUV) and $L$(FUV) and the emission line luminosities $L$() and $L$(). The figures show a strong correlation between all of these quantities, with Pearson correlation coefficients of \[0.83, 0.86, 0.87, 0.90\] for the relationships between \[FUV – , FUV – , XUV – , XUV – )\], respectively. The probability of a non-correlation is $<$ 3 $\times$ 10$^{-3}$ for all four curves. We present quantitative log-log relations[^6] in Table 2. The RMS scatter around the \[FUV – , FUV – , XUV – , XUV – )\] fit, ($L$(band) - $L$(fit))/$L$(band), is \[153%, 39%, 109%, 57%\], respectively. We have selected and because they are the two most readily-observable emission lines in the FUV and NUV bandpasses. shows a tighter correlation, but as we begin to probe exoplanetary systems at greater distances from the Sun, this relationship will become compromised by the additional contribution from interstellar absorption components with radial velocities coincident with the stellar radial velocities [@redfield02] as well as the possibility of gas-rich circumstellar environments fueled by mass-loss from short period gaseous planets [@haswell12; @fossati15]. By contrast, is essentially free from interstellar extinction out to the edge of the Local Bubble where dust opacity begins to contribute appreciably. There are rare exceptions where hot gas in the Local Bubble can contribute small amounts of attenuation [@welsh10], but the effect is considerably less than for .
Temporally Resolved Spectra and Energetic Flares
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As discussed in the introduction, temporal variability of the energetic radiation is considerably higher for M dwarfs than for solar type stars [@mitra05], with active M stars showing disk-integrated flux increases of an order-of-magnitude or more during large flares [@hawley91; @hawley03; @osten05]. The MUSCLES pilot program showed that even inactive M dwarfs could show impulsive flare behavior in their UV light curves [@france12a]. The UV monitoring component of the MUSCLES Treasury Survey was designed to present a uniform database of flares from exoplanet host stars so the flare frequency-amplitude relations could be derived for these stars and the impact of impulsive events on the atmospheres or orbiting planets could be assessed. For cases where the X-ray observations were scheduled simultaneously, the overlapping observatory coverage was planned for the HST 5-orbit monitoring campaigns.
The intention of this section is not to give a thorough quantitative description of the flare catalog produced in the survey; that work will be presented in a follow-on paper by @loyd16. This section is intended to introduce the variability data and present one example of an optically quiet M dwarf host star that is among the most UV/X-ray active sources ever observed. Figure 13 shows the light curves of GJ 876, recorded in four bright emission lines, $\lambda$$\lambda$1334, 1335 ($T_{form}$ $\sim$ 3 $\times$ 10$^{4}$ K), $\lambda$1206 ($T_{form}$ $\sim$ 4 $\times$ 10$^{4}$ K), $\lambda$$\lambda$1394, 1403 ($T_{form}$ $\sim$ 6 $\times$ 10$^{4}$ K), and $\lambda$$\lambda$1239, 1243 ($T_{form}$ $\sim$ 2 $\times$ 10$^{5}$ K). The data are binned to a 30 second cadence and displayed relative to the start time of the first observation. The individual exposures are labeled for reference. Several flares are immediately apparent, the strongest being during the last two orbits of the monitoring campaign. The COS detector background, measured at the same dispersion direction location as the lines but offset below the spectral trace in the cross-dispersion direction, is also shown (orange squares) to demonstrate the stability of the instrument during these measurements.
Figure 14 ($top$) shows a zoom on the brightest UV flare, binned to a 10 second cadence, occurring near $T_{exp}$ = 17,400s. Each light curve is normalized to unity during a pre-flare window (17,100 – 17,250s) to enable a comparison of the relative flare responses of each line. The flare/quiescent flux increase in the brightest line, is $\sim$ 110. and show flux increases of order $\sim$ 50, and and the FUV continuum increase by factors of $\sim$ 5. This level of luminosity increase indicates the largest relative UV flare ever directly detected in a disk-integrated observation of a star, despite the relative inactivity of GJ 876 suggested by the activity index. One observes that the lightcurves are line-dependent, with intermediate temperature ions showing a larger relative flux increase while the higher temperature ion () does not respond as strongly but shows a decay time several times longer than , , and (Figure 13). Figure 14, ($bottom$) shows time-resolved line ratios in the GJ 876 lightcurves. Several emission lines are ratioed to and normalized to their pre-flare line ratios, so their relative change is meaningful. In this way, we can place constraints on the atmospheric temperature regime where most of the $observable$ energy is deposited (our observations do not contain information about energy deposited in cooler or optically thick atmospheric layers). The / ratio increases during the flare, while the / and / ratios, which sample gas both hotter and colder than are depressed. This argues that the flare energy distribution is peaked near the formation temperature, roughly (4 – 5) $\times$ 10$^{4}$ K (with both low and high energy tails), and evolves during the flare.
In order to understand the impact of these flares on the planets in these systems, we need to convert the observed emission line flares to an estimate of the broadband UV flare energy. Converting the raw spectral light curves into flux units by comparing the orbit-averaged count rates and stellar emission line fluxes, we can create flux-calibrated lightcurves [@loyd14]. The total flare energy in a given emission line is then $$E(\lambda)~=~4 \pi d^{2} \int_{t_{start}}^{t_{end}} F(\lambda) dt$$ where $F$($\lambda$) is the line flux in (erg cm$^{-2}$ s$^{-1}$), $d$ is the stellar distance, and $t_{start}$ and $t_{end}$ are the initial flare rise times and the time when the flare returns to the quiescent level, respectively. For the bright flare considered here, the exposure ends before the ionic lines return to the quiescent levels, so we set $t_{end}$ as end of the orbit in this case. The and emission lines are representative of the upper chromosphere and transition region emission formed between $\sim$ 30 – 80 kK; we use these lines to estimate the total flare emission from these regions. The total energies in these lines are log$_{10}$ $E$() = 29.17 erg and log$_{10}$ $E$() = 29.08 erg. We use the M dwarf stellar atmosphere model of @fontenla15 to estimate the total emission in the 300 – 1700 Å range that contains the majority of transition region emission. Emission at $\lambda$ $<$ 300 Å is mainly coronal in origin while emission at $\lambda$ $>$ 1700 Å is mainly cooler chromospheric gas.
The ionized silicon flux energies can be converted to a broadband XUV + FUV (300 – 1700Å) energy by computing the fractional flux emitted in these lines and the fractional radiative cooling rate that is contributed by 30 – 80 kK gas. The total broadband energy is then $$\begin{split}
E(300 - 1700 \AA)~=~E(Si III + Si IV) \\
\times~( f_{300 - 900} + f_{900 - 1200} + f_{1200 - 1700} ) \\
\times C_{30-80kK}
\end{split}$$ where $f_{300 - 900}$ is the ratio of stellar flux in the 300 – 900Å band to the combined + flux, $F$(300 – 900Å)/$F$( + ). Similarly, $f_{900 - 1200}$ = $F$(900 – 1200Å)/$F$( + ) and $f_{1200 - 1700}$ = $F$(1200 – 1700Å)/$F$( + ). The $f_{300 - 900}$, $f_{900 - 1200}$, and $f_{1200 - 1700}$ ratios from the model atmosphere of @fontenla15 are 32, 14, and 290, respectively. $C_{30-80kK}$ is the fraction of the total radiative cooling rate from the upper stellar atmosphere (6000 K – 10$^{6}$ K) contributed by 30 – 80 kK gas, $C_{30-80kK}$ = $\Gamma$(30 – 80 kK)/$\Gamma$(6000 K – 10$^{6}$ K), where $\Gamma$ is the radiative cooling rate in units of erg cm$^{-3}$ s$^{-1}$. Using the cooling rates for this model atmosphere [@fontenla15], we compute $C_{30-80kK}$ = 16.8%. Note that these are the equilibrium cooling rates for the quiescent model atmosphere, and may be different during the post-reconnection heating associated with the flare. Combining these elements, we estimate that the total UV flare energy associated with this event is log$_{10}$ $E$(300 – 1700Å) = 31.18 erg, comparable to the total quiescent luminosity of the star $E_{flare}$(UV) $\sim$ 0.3 $L_{*}$$\Delta$$t$ ($\Delta$$t$ = 1 second). In a future work, @loyd16 will show that $HST$ monitoring observations are able to measure and quantify the M dwarf flare amplitude–frequency distribution. Critically, $HST$ provides the sensitivity and spectral resolution to analyze light curves covering a range of formation temperatures, and capture the short duration events that dominate the flare distribution. A dedicated $HST$ spectroscopic flare monitoring program is currently the best avenue for understanding the energy and temporal distribution of flares and their potential influence on low-mass planets
### X-ray Lightcurves
When possible, X-ray observations were taken in concert with the UV light curves described here. However, this was only possible for about half of the MUSCLES observations. Figure 15 shows the non-simultaneous X-ray light curve of GJ 876[^7], finding another large flare on this object. The X-ray spectra of both the quiescent and flare periods have been fitted, and the overall X-ray luminosity increased by a factor of $\sim$ 10 during the flare. The actual luminosity increase was likely larger but diluted by the cadence adopted to provide sufficient S/N in each time bin. Given the similarity of the UV and X-ray light curves of dMe stars observed by $XMM$ [@mitra05], it seems likely that these large flares are similar events and occur with regularity on GJ 876. We note that @poppenhaeger10 also noted a high level of X-ray activity on GJ 876 and @france12a observed a large UV flare in the very first MUSCLES observations. This apparently inactive star displays a flare outburst almost every time it is observed at wavelengths below the atmospheric cut-off. While we have elected to focus on the most UV active star in the MUSCLES data set, roughly half of our targets displayed UV flare activity. This will be explored in greater detail in @loyd16.
Discussion
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Chromospheric and Transition Region Activity of GJ 1214: On, Off, or Variable?
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We use GJ 1214, the only star in our sample with effective temperature $<$ 3000 K, to investigate the evolution of upper atmosphere activity towards the stellar/sub-stellar boundary. In the MUSCLES pilot study, we presented an upper limit to the Ly$\alpha$ flux of GJ 1214 based on earlier STIS G140M observations. GJ 1214 has a spectral type of M4.5V, with $T_{eff}$ = 2949 $\pm$ 30 K (Kundurthy et al. 2011; consistent with the 2935 K derived from the photospheric model fitting described in Paper III). It was the most distant source in the MUSCLES pilot study, and the only star for which Ly$\alpha$ was not detected. This non-detection was approximately a factor of 10 below the expected flux level based on an extrapolation of the F(Ly$\alpha$) versus F() relation [@france13] and was made more surprising by the solid detection of $\lambda$ 1548 emission. This led us to speculate about the possibility of a high molecular fraction atmosphere that suppresses atomic hydrogen emission or possibly an ‘on/off’ behavior where the basal flux level is very low and the chromospheric and coronal emission can only be observed during flares (as has been noted for the M8 star VB 10 by Linsky et al. 1995, however more recent observations demonstrated an ‘always on’ behavior; Hawley & Johns-Krull 2003). Subsequently, X-rays from GJ 1214 were detected by [*XMM-Newton*]{} [@lalitha14]. All of the source counts during the $XMM$ observations occurred in a single time bin, suggesting that the ‘on/off’ scenario may be correct.
The new MUSCLES observations instead argue for a weak but persistent high-energy flux from GJ 1214. We have detected the full complement of FUV and NUV emission lines from GJ 1214 with the deeper MUSCLES treasury data (including direct observation of Ly$\alpha$, see Paper II; Table 3). While the Ly$\alpha$ reconstruction is uncertain owing to the low-S/N and the small fraction of the line profile that is detected, we can compare the observed flux from the star over multiple epochs. The Ly$\alpha$ (STIS G140M) and + (COS G160M + STIS G230L) observations in @france13 were separated by roughly 15 months, while the 2015 MUSCLES Treasury data were obtained over a period of $<$ 2 days. We find that the observed flux level from GJ 1214 in 2015, $F_{2015}$(Ly$\alpha$) = 1.8 ($\pm$ 0.3) $\times$ 10$^{-15}$ erg cm$^{-2}$ s$^{-1}$, is consistent with the upper limit presented previously, $F_{2013}$(Ly$\alpha$) $\leq$ 2.4 $\times$ 10$^{-15}$ erg cm$^{-2}$ s$^{-1}$. The reconstructed flux is approximately twice this value $F_{2015,recon}$(Ly$\alpha$) = 5.5 $\times$ 10$^{-15}$ erg cm$^{-2}$ s$^{-1}$, which is still roughly a factor of three lower than the expected reconstructed Ly$\alpha$ flux based on the $F$(Ly$\alpha$) versus $F$() relation [@youngblood15]. We find that the total brightness has increased by a factor of 2 compared to our earlier study ($F_{2015}$() = 5.2 ($\pm$ 1.2) $\times$ 10$^{-16}$ erg cm$^{-2}$ s$^{-1}$ versus $F_{2013}$() = 2.6 ($\pm$ 0.5) $\times$ 10$^{-16}$ erg cm$^{-2}$ s$^{-1}$), while the flux is approximately constant ($F_{2015}$() = 1.7 ($\pm$ 0.3) $\times$ 10$^{-15}$ erg cm$^{-2}$ s$^{-1}$ versus $F_{2013}$() = 2.2 ($\pm$ 0.2) $\times$ 10$^{-16}$ erg cm$^{-2}$ s$^{-1}$).
GJ 1214 displays fractional luminosities that are typical for the mid/late-M dwarfs in the MUSCLES survey (Table 3; Figures 10 and 11). It has the lowest absolute levels of high-energy radiation (Figure 8) owing to its low mass. The low intrinsic luminosity level and the star’s relatively large distance ($d$ = 13 pc) can explain the non-detection of Ly$\alpha$ in the previous work. It appears that GJ 1214 is simply a scaled-down version of the typical planet hosting mid-M dwarf, and not in a fundamentally different state of chromospheric activity. While GJ 1214 clearly has flares, we have now shown that the basal flux level of this star is similar to other low-mass exoplanet host stars. Combining this result with the high activity levels on M5 stars such as GJ 876, we conclude that special time-dependent photochemistry (other than the incorporation of impulsive flares and longer term variability; Section 1) is not necessary down to spectral type M5.
Comparison of UV Observations to Coronal Models
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The MUSCLES panchromatic SED creation relies on direct observation at all wavelengths except at red/IR wavelengths ($\lambda$ $>$ 6000 Å) and the XUV/FUV region between 50 – 1170 Å. As discussed in the introduction, the EUV regulates heating and mass-loss in planetary atmosphere, particularly for short-period planets around stars with large EUV and X-ray fluxes. However, obtaining a complete EUV spectrum of any cool star other than the Sun is currently impossible owing to attenuation by the ISM. Our approach to filling in this observationally inaccessible gap is to employ a coronal model from 50 – 100 Å (a single or two-temperature APEC model, Smith et al. 2001) and models of solar active regions from 100 – 1170 Å [@fontenla11; @linsky14]. This approach assumes a scaling between the chromospheric Ly$\alpha$ emission and the chromospheric + transition region + coronal flux that contributes to the EUV bands, particularly between 100 – 400 Å.
An alternative approach has been taken by @forcada11, who used X-ray observations of the coronae of exoplanet host stars and an emission measure distribution technique to predict the EUV and part of the FUV spectra of low-mass host stars. The synthetic spectral output of coronal models for three of the MUSCLES stars ($\epsilon$ Eri, GJ 436, and GJ 876) are available on the [X-exoplanets]{} website[^8], and provide overlap with the high S/N MUSCLES data in the 1150 – 1200 Å region. We downloaded these synthetic spectra to compare the coronal model fluxes with the data for the bright $\lambda$1175 multiplet. These lines are formed in the transition region at T$_{form}$ $\sim$ 6 $\times$ 10$^{4}$ K, and therefore provide a good diagnostic for how well coronal models are able to reproduce the UV emission throughout the FUV. The [X-exoplanets]{} synthetic spectra are provided in units of photons s$^{-1}$ cm$^{-2}$ bin$^{-1}$, with a variable bin size. The spectra are multiplied by a factor of (bin size)$^{-1}$ and then integrated over wavelength to compute integrated line photons s$^{-1}$ cm$^{-2}$. The MUSCLES data were converted to photons and integrated over the same wavelength interval (1174 – 1177 Å).
We find that the coronal models systematically underpredict the UV emission line strengths by factors of a few to tens (Table 4). For $\epsilon$ Eri, we find $F_{data}$() = 1.8 $\times$ 10$^{-2}$ photons s$^{-1}$ cm$^{-2}$ versus $F_{Xexoplanets}$() = 6.6 $\times$ 10$^{-4}$ photons s$^{-1}$ cm$^{-2}$. For GJ 436, we find $F_{data}$() = 4.2 $\times$ 10$^{-5}$ photons s$^{-1}$ cm$^{-2}$ versus $F_{Xexoplanets}$() = 8.0 $\times$ 10$^{-6}$ photons s$^{-1}$ cm$^{-2}$. For GJ 876, we find $F_{data}$() = 6.5 $\times$ 10$^{-4}$ photons s$^{-1}$ cm$^{-2}$ versus $F_{Xexoplanets}$() = 1.9 $\times$ 10$^{-5}$ photons s$^{-1}$ cm$^{-2}$. The coronal models underpredict these transition region fluxes by factors of approximately 27, 5, and 33 for $\epsilon$ Eri, GJ 436, and GJ 876, respectively. This difference underscores the importance of having empirical inputs for transition region and chromospheric emission in the calculations of the EUV flux from cool stars (see also the discussion presented in Linsky et al. 2014). The systematic underprediction of the transition region emission brings into question the accuracy of the coronal models in the 400 – 900 Å EUV region that is dominated by chromospheric and transition region line and continuum spectra. Another important factor is the time variability in the coronal and chromospheric emission from these stars, which may not be the same. The MUSCLES database provides an excellent resource for emission measure distribution-based stellar atmosphere models that simultaneously take into account both the intermediate and high-temperature regions of the stellar atmosphere (e.g., Chadney et al. 2015) and may provide more accurate EUV flux estimates for the exoplanet community.
Star-Planet Interactions Observed in Transition Region Emission Lines
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It has been suggested that stellar and exoplanetary magnetic fields can interact in exoplanetary systems [@shkolnik03; @lanza08], possibly manifesting as enhanced stellar activity relative to nominal age-rotation-activity relationships for isolated main sequence stars [@barnes07; @mamajek08]. The magnitude of this interaction, as measured by the energy dissipated in the stellar atmosphere, may be proportional to the strength of the stellar magnetic field, the planetary magnetic field, and the relative speed of the planet’s orbital velocity compared to the stellar magnetic rotation rate. While the stellar magnetic field and orbital velocity can be readily measured, exoplanet magnetic fields have proven notoriously hard to detect (see, e.g., Lecavelier des Etangs 2011 and Hallinan et al. 2013), with very few possible detections from low-frequency radio observations [@lecavelier13] and early-ingress measurements of NUV and optical transit light curves [@fossati10; @lai10; @vidotto10; @llama11; @cauley15]. If the above dependencies on the magnetic field and velocities are valid, then a general trend of enhanced energy dissipation, which could be observed as enhanced emission from the stellar corona (X-rays; Kashyap et al. 2008) or chromosphere (; Shkolnik et al. 2005), should correlate with $M_{plan}$/$a_{plan}$, where $M_{plan}$ is the planetary mass and $a_{plan}$ is the semi-major axis. Because magnetic field strength increases with planetary mass in the solar system, one would expect that the most massive, closest-in planets in exoplanetary systems should produce the largest signal on their host stars. However, almost every claimed detection of a modulation in activity on exoplanet host stars or connection with orbital phase has not been confirmed by follow up observation or re-analysis of the same data set (e.g., Shkolnik et al. 2008; Poppenhaeger et al. 2010).
The search for star-planet interaction (SPI) has predominantly focused on hot Jupiters orbiting F, G, and K stars. The MUSCLES database allows us to examine potential star-planet interactions for a range of planetary masses and semi-major axes on a new class of optically inactive low-mass star. Furthermore, the MUSCLES database allows us to explore SPI as a function of emission line formation temperature, which can constrain the possible location of magnetic field line connection in the stellar atmosphere and subsequent location of the plasma heating. Figure 16 shows a correlation between the fractional luminosity and $M_{plan}$/$a_{plan}$. The – $M_{plan}$/$a_{plan}$ relation has a Pearson correlation coefficient of 0.83 and a statistical non-correlation likelihood of 0.035. This suggests that the systems with close-in, massive planets may indeed be generating enhanced transition region activity, as probed by this $\sim$ 2 $\times$ 10$^{5}$ K gas[^9]. On the other hand, we do not observe a correlation with the lower-temperature chromospheric gas traced by ($T_{form}$ $\sim$ (1 – 2) $\times$ 10$^{4}$ K). The – $M_{plan}$/$a_{plan}$ relation, for instance, displays a correlation coefficient and probability of a null-correlation, $-$0.3 and 0.06, respectively. Figure 17 presents the Pearson correlation coefficients for the relation between the fractional luminosity and $M_{plan}$/$a_{plan}$ for several lines in the MUSCLES spectra. While the null-hypothesis cannot be ruled out with high confidence for individual hot gas lines (i.e., the $n$ values are not very low), the trend with line formation temperature suggests that the connection between hot gas emission enhancement and $M_{plan}$/$a_{plan}$ may be real. This suggests that the SPI correlation would not be observed in , which has a comparable formation temperature distribution as in M dwarfs [@fontenla15].
The correlation of the hot gas lines in the MUSCLES sample is somewhat surprising given the null result found by @shkolnik13 for $GALEX$ fluxes. As the $GALEX$ observations are broader band measurements, there is an uncertain and variable contribution from the stellar photosphere that must be subtracted, lines from a range of formation temperatures are simultaneously included, and the highest temperature lines (e.g., ) are excluded. Therefore, potential SPI signals are significantly diluted in UV imaging surveys relative to the MUSCLES survey, which focuses on lower mass stars with less continuum emission and spectrally resolved line-profiles that can be robustly measured. This result argues that very low spectral resolution ($R$ $<$ 500) observations at FUV or NUV wavelengths are not a promising avenue for discovering SPI-driven activity enhancements in low-mass stars.
We emphasize that given the small sample size, the detection of SPI in the MUSCLES sample should viewed with caution. One explanation for these results is a simple scaling with stellar effective temperature. Our inactive K dwarfs have low fractional luminosities and low-mass planetary systems, driving one end of the correlations in some cases. A larger sample that is better controlled for systematics is needed to confirm the results presented here. A future spectroscopic survey of G, K, and M exoplanet host stars with the COS G130M mode (to cover , , and ) could be a promising data set with which to characterize enhanced magnetospheric stellar emission generated by SPI.
Summary
=======
We have presented the first panchromatic survey of M and K dwarf exoplanet host stars from X-ray to UV to optical to IR wavelengths. The MUSCLES Treasury Survey is built upon contemporaneous $Chandra$ (or $XMM$), $HST$, and ground-based data. The 5 Å to 5 $\mu$m SEDs have been assembled and hosted as high-level science products on the MAST website. The main purpose of this paper was to present the motivation and overview of the MUSCLES Treasury Survey; more detailed analyses of the spectrally and temporally resolved SEDs, their impact of atmospheric photochemistry, and the suitability of M dwarfs as habitable planet hosts will be addressed by our team in future publications.
The main results of this work are:
1. All stars show energetic radiation (X-ray through UV) at all times during the observations. Chromospheric, transition region, and coronal emission is directly observed from all targets. Despite all but one of our targets having H$\alpha$ in absorption (that is, “inactive” in the traditional optical sense), all of the MUSCLES stars are X-ray and UV active.
2. We have provided empirically-derived relations to compute the FUV and XUV stellar luminosity from and emission line fluxes.
3. The FUV/NUV flux ratio, an indicator for the potential abiotic formation of O$_{2}$ and O$_{3}$, declines with increasing stellar effective temperature by more than two orders of magnitude from $T_{eff}$ = 3000 – 5000 K. Consequently, the FUV/NUV flux ratio declines by more than two orders of magnitude as habitable zone orbital distances increase from 0.1 – 0.7 AU. The total FUV radiation field strength increases by factors of 2 – 3 over this distance, while the XUV radiation field strength is approximately constant. The average FUV and XUV fluxes in the habitable zones of all K and M dwarfs studied are $\approx$ 10 – 70 erg cm$^{-2}$ s$^{-1}$. [*The spectral energy distribution of the radiation field changes dramatically for different habitable zone distances around low-mass stars, but the intensity of the high-energy radiation field does not.*]{}
4. Despite their weak optical activity indicators (e.g., emission core equivalent widths), several of our stars display extremely strong UV and X-ray flares. Flare/quiescent flare increases by a factor of $\sim$ 10 are common on at least half of our stars with the strongest flares showing $E_{flare}$(UV) $\sim$ 0.3 $L_{*}$$\Delta$$t$.
5. Emission measure distribution models based on X-ray (coronal) data alone underestimate the FUV transition region flux by factors of $\sim$ 5 – 30, meaning that these models should not be used for calculating the FUV radiation field of exoplanet host stars. The MUSCLES database provides an excellent resource for emission measure modeling of the stellar atmosphere where lines formed in the chromosphere, transition region, and corona can all be taken into account.
6. We present tentative evidence for star-planet interaction by measuring the fractional emission line luminosity as a function of the star-planet interaction strength, $M_{plan}$/$a_{plan}$. Only the high-temperature transition region lines ( and ) show a positive correlation. No correlation exists for lines with formation temperatures below 10$^{5}$ K, suggesting the interaction takes place primarily in the transition region or corona. Moderate resolution FUV spectroscopy appears to be a promising avenue to further characterize star-planet interaction, while narrow-band FUV and NUV imaging are of less utility for characterizing the interactions of stars and planets owing to bandpass and spectral resolution limitations.
The data presented here were obtained as part of the $HST$ Guest Observing programs \#12464 and \#13650 as well as the COS Science Team Guaranteed Time programs \#12034 and \#12035. This work was supported by STScI grants HST-GO-12464.01 and HST-GO-13650.01 to the University of Colorado at Boulder. Data for the MUSCLES Treasury Survey were also acquired as part of $Chandra$ and $XMM$ guest observing programs, supported by $Chandra$ grants GO4-15014X and GO5-16155X from Smithsonian Astrophysical Observatory and NASA $XMM$ grant NNX16AC09G to the University of Colorado at Boulder. This work is based in part upon observations obtained with the Apache Point Observatory 3.5-meter and 0.5-meter telescopes, which are owned and operated by the Astrophysical Research Consortium. KF thanks Evgenya Shkolnik for enjoyable discussions about low-mass stars and Jorge Sanz-Forcada for assistance with the absolute flux levels of the [X-exoplanets]{} model spectra. The MUSCLES team also thanks STScI program coordinator Amber Armstrong for her long hours spent scheduling these complicated coordinated observations. PCS gratefully acknowledges an ESA Research Fellowship. SLH acknowledges support from NSF grant AST 13-11678. FT is supported by the National Natural Science Foundation of China (41175039), the Startup Fund of the Ministry of Education of China (20131029170), and the Tsinghua University Initiative Scientific Research Program.
Author Affiliations:\
$^{1}$Laboratory for Atmospheric and Space Physics, University of Colorado, 600 UCB, Boulder, CO 80309; [email protected]\
$^{2}$Center for Astrophysics and Space Astronomy, University of Colorado, 389 UCB, Boulder, CO 80309\
$^{3}$European Space Research and Technology Centre (ESA/ESTEC), Keplerlaan 1, 2201 AZ Noordwijk, The Netherlands\
$^{4}$Department of Astronomy, University of Washington, Box 351580, Seattle, WA 98195, USA\
$^{5}$Department of Astronomy, C1400, University of Texas at Austin, Austin, TX 78712\
$^{6}$JILA, University of Colorado and NIST, 440 UCB, Boulder, CO 80309\
$^{7}$Exoplanets and Stellar Astrophysics Laboratory, NASA Goddard Space Flight Center, Greenbelt, MD 20771\
$^{8}$Instituto de Astronomía y Física del Espacio (UBA-CONICET) and Departamento de Física (UBA) ,CC.67, suc. 28, 1428, Buenos Aires, Argentina\
$^{9}$Department of Physics & Astronomy, Western Washington University, Bellingham, WA 98225\
$^{10}$NSF Astronomy and Astrophysics Postdoctoral Fellow\
$^{11}$NorthWest Research Associates, 3380 Mitchell Lane, Boulder, CO 80301-2245\
$^{12}$Carl Sagan Institute, Cornell University, Ithaca, 14850, NY, USA\
$^{13}$Department of Astronomy, University of Maryland, College Park, MD 20742, USA\
$^{14}$Laboratoire Lagrange, Universite de Nice-Sophia Antipolis, Observatoire de la Cote d’Azur, CNRS, Blvd de l’Observatoire, CS 34229, 06304 Nice cedex 4, France\
$^{15}$Astronomy Department and Van Vleck Observatory, Wesleyan University, Middletown, CT 06459-0123, USA\
$^{16}$Department of Earth and Environmental Sciences, Irvine Building, University of St Andrews, St Andrews KY16 9AL, UK\
$^{17}$Ministry of Education Key Laboratory for Earth System Modeling, Center for Earth System Science, Tsinghua University, Beijing 100084, China\
$^{18}$Instituto de Astronomía y Física del Espacio (UBA-CONICET) and UNTREF,CC.67, suc. 28, 1428, Buenos Aires, Argentina\
$^{19}$The Adler Planetarium, 1300 S Lakeshore Dr, Chicago IL 60605
MUSCLES Targets
===============
[*GJ 1214*]{} – GJ 1214 is a late M dwarf (M4.5V) at a distance of 14.6 pc, making it the coolest and most distant M dwarf in the MUSCLES Treasury Survey. It has an age of 6 $\pm$ 3 Gyr [@charbonneau09], a roughly solar metallicity (\[Fe/H\] = +0.05; Neves et al. 2014), and shows signs of optical flare activity [@kundurthy11]. Coronal emission from GJ 1214 was recently observed by [*XMM-Newton*]{} [@lalitha14] with log$_{10}$$L_{X}$ $=$ 25.87 erg s$^{-1}$. GJ 1214b is a transiting super-Earth ($M_{plan}$ $\approx$ 6.5 $M_{\oplus}$, $a_{plan}$ = 0.014 AU; Charbonneau et al. 2009), possibly harboring a dense, water-rich [@bean10; @desert11] and likely cloudy [@kreidberg14] atmosphere.
[*GJ 876*]{} – GJ 876 is an M5 dwarf at a distance of 4.7 pc. GJ 876 has super-solar metallicity (\[Fe/H\] = 0.14; Neves et al. 2014); differing estimates on the stellar rotation period (40 $\leq$ $P_{*}$ $\lesssim$ 97 days) result in large uncertainties in the age estimate for this system, 0.1 – 5 Gyr [@rivera05; @rivera10; @correia10]. While the star would be characterized as weakly active based on its H$\alpha$ absorption spectrum, UV and X-ray observations show the presence of an active upper atmosphere (Walkowicz et al. 2008; France et al. 2012a; log$_{10}$$L_{X}$ $=$ 26.48 erg s$^{-1}$; Poppenhaeger et al. 2010). GJ 876 has a rich planetary system with four planets ranging from a super-Earth (GJ 876d, $M_{plan}$ $\approx$ 6.6 $M_{\oplus}$) in a short-period orbit ($a_{plan}$ = 0.02 AU; Rivera et al. 2010) to two Jovian-mass planets in the HZ (GJ 876b, $M_{plan}$ $\approx$ 2.27 $M_{Jup}$, $a_{plan}$ = 0.21 AU; GJ 876c, $M_{plan}$ $\approx$ 0.72 $M_{Jup}$, $a_{plan}$ = 0.13 AU; Rivera et al. 2010).
[*GJ 581*]{} – GJ 581 is an M5 dwarf at a distance of 6.3 pc. It is estimated to have an age of 8 $\pm$ 1 Gyr [@selsis07] and a somewhat subsolar metallicity, \[Fe/H\] = $-$0.20 [@neves14]. GJ 581 was not detected in early X-ray surveys (log$_{10}$$L_{X}$ $<$ 26.89 erg s$^{-1}$; Poppenhaeger et al. 2010), however X-rays were subsequently detected, log$_{10}$$L_{X}$ $=$ 26.11 erg s$^{-1}$, by $Swift$ [@vitale13]. Its optical spectrum displays H$\alpha$ in absorption, therefore chromospheric and coronal activity are thought to be low for this target. GJ 581 has one of the richest known planetary systems, with possibly up to six planets (four confirmed) including several with Earth/super-Earth masses [@mayor09; @tuomi11]. GJ 581d is a super-Earth ($M_{plan}$ $\approx$ 6 $M_{\oplus}$) that resides on the outer edge of the habitable zone (HZ; $a_{plan}$ = 0.22 AU; Wordsworth et al. 2011; von Braun et al. 2011).
[*GJ 176*]{} – GJ 176 is an M2.5 dwarf at a distance of 9.4 pc. It is estimated to have an age between 0.56 - 3.62 Gyr [@kiraga07; @forcada11] and a solar metallicity, \[Fe/H\] = $-$0.01 [@neves14]. GJ 176 has an archival X-ray luminosity of (log$_{10}$$L_{X}$ $=$ 27.48 erg s$^{-1}$; Poppenhaeger et al. 2010) and a 39 day rotational period [@robertson15]. GJ 176 b was initially detected as a 24 $M_{\oplus}$ planet in a 10 day orbit [@endl08], but subsequent analysis of the stellar light-curve has refined this estimate to a super-Earth mass planet (8.3 $M_{\oplus}$) in an 6.6 day orbit [@butler09; @forveille09].
[*GJ 436*]{} – GJ 436 is an M3.5 dwarf star located at a distance of 10.3 pc. It has a 45 day rotation period, a relatively old age ($\sim$ 6$^{+4}_{-5}$ Gyr; Torres 2007), and may have a super-solar metallicity (\[Fe/H\] = 0.00 – 0.25; Neves et al. 2014; Johnson & Apps 2009; Rojas-Ayala et al. 2010). GJ 436 does show signs of an active corona with log$_{10}$$L_{X}$ $=$ 27.16 erg s$^{-1}$ (Poppenhaeger et al. 2010), and its chromospheric Ly$\alpha$ emission was previously observed by @ehrenreich11 and @kulow14. GJ 436 is notable for its well-studied transiting Neptune mass planet [@butler04; @pont09], orbiting at a semi-major axis of $\approx$ 0.03 AU, interior to its HZ (0.16 – 0.31 AU; von Braun et al. 2012). The impact of Ly$\alpha$ on the atmosphere of GJ 436b has been studied by @miguel15. Additional low-mass planets may also be present in this system [@stevenson12].
[*GJ 667C*]{} – GJ 667C (M1.5V) is a member of a triple star system (GJ 667AB is a K3V + K5V binary) at a distance of 6.9 pc. This 2 – 10 Gyr M dwarf [@anglada_escude12] is metal-poor (\[Fe/H\] = $-$0.50 Neves et al. 2014, and a similar value of $-$0.59 $\pm$ 0.10 based on an analysis of GJ 667AB; Perrin et al. 1988). GJ 667C may host as many as three planets, including a super-Earth mass planet (GJ 667Cc, $M_{plan}$ $\approx$ 4.5 $M_{\oplus}$, $a_{plan}$ = 0.12 AU) orbiting in the HZ (0.11 – 0.23 AU; Anglada-Escud[é]{} et al. 2012).
[*GJ 832*]{} – GJ 832 is an M1.5 dwarf at $d$ = 4.9 pc. GJ 832 is not as well characterized as other targets in our sample; an age determination for this star is not available. Coronal X-rays have been detected from GJ 832 with log$_{10}$$L_{X}$ $=$ 26.77 erg s$^{-1}$ (Poppenhaeger et al. 2010). This subsolar metallicity star (\[Fe/H\] = $-$0.17; Neves et al. 2014) hosts two known exoplanets: “b”, a 0.7 $M_{Jup}$ Jovian planet at $a_{b}$ = 3.6 AU (Bailey et al. 2009) and “c”, a 5.4 $M_{\oplus}$ super-Earth planet in the Habitable Zone ($a_{c}$ = 0.16 AU; Wittenmyer et al. 2014).
[*HD 85512*]{} – HD 85512 is a K6 dwarf at a distance of 11.2 pc. It is estimated to have an age of 5.6 $\pm$ 0.1 Gyr and a subsolar metallicity, \[Fe/H\] = $-$0.26 [@tsantaki13], based on its 47 day rotation period. There is no previously published X-ray detection of HD 85512, the [*XMM-Newton*]{} luminosity is log$_{10}$$L_{X}$ $\sim$ 26.5 erg s$^{-1}$, (Loyd et al. 2016, Brown et al. 2016). HD 85512 b is a super-Earth mass planet (3.6 $M_{\oplus}$) orbiting with a semi-major axis of 0.26 AU [@pepe11].
[*HD 40307*]{} – HD 40307 is a K2.5 dwarf at a distance of 12.9 pc. It is estimated to have an age of 4.5 Gyr [@barnes07] and a subsolar metallicity, \[Fe/H\] = $-$0.36 [@tsantaki13]. HD 40307 has a 48 day rotation period [@mayor09] and an archival X-ray luminosity of log$_{10}$$L_{X}$ $=$ 26.99 erg s$^{-1}$ (Poppenhaeger et al. 2010). HD 40307 hosts somewhere between 3 and 6 planets, including several of super-Earth mass (Table 1; Mayor et al. 2009, Tuomi et al. 2013).
[*$\epsilon$ Eri*]{} – $\epsilon$ Eri is a K2 dwarf at a distance of 3.2 pc. It is one of the best-studied active K stars [@dring97; @ness02; @jeffers14]. It is a relatively young star with an age $\approx$ 0.44 Gyr [@barnes07], an 11.68 day rotation period [@rueedi97], and a slightly subsolar metallicity, \[Fe/H\] = $-$0.15 [@tsantaki13]. $\epsilon$ Eri displays magnetic cycles that suggest it may have been in a relatively “inactive” state during the MUSCLES observations [@metcalfe13]. Archival spectra from [*XMM-Newton*]{} show an X-ray luminosity in the 0.2 – 2.0 keV energy band of log$_{10}$$L_{X}$ $=$ 28.22 erg s$^{-1}$ (Poppenhaeger et al. 2010) that grows to log$_{10}$$L_{X}$ $=$ 29.32 erg s$^{-1}$ when the full range covered by the $Chandra$ LETGS is included (0.07 – 2.5 keV; Ness et al. 2002). $\epsilon$ Eri hosts a $\sim$ 1.1 $M_{Jup}$ planet in a 3.4 AU semi-major axis orbit [@hatzes00; @butler06].
[*HD 97658*]{} – HD 97658 is a K1 dwarf at a distance of 21.1 pc. It is estimated to have an age of 3.8 $\pm$ 2.6 Gyr [@bonfanti15] and a subsolar metallicity \[Fe/H\] = $-$0.26 [@valenti05]. There are no published X-ray data of HD 97658 in the literature, but X-ray observations are scheduled as part of $Chandra$’s Cycle 16. HD97658 hosts a super-Earth mass planet (7.9 $M_{\oplus}$) in a short-period orbit ($a$ = 0.08 AU; Dragomir et al. 2013).
---------------- ------ ------ ----------- ----------- ---------------------------------- --------------------------- ---------------------- ------------
Star Type T$_{eff}$ P$_{rot}$ $\langle$$r_{HZ}$$\rangle$$^{a}$ Semi-major Axis Ref.$^{b}$
(K) (days) (AU) (AU)
GJ 1214 14.6 M4.5 2935 53 0.096 [**6.4**]{} 0.0143 1,2,3
GJ 876 4.7 M5 3062 96.7 0.178 615, 194, 0.208, 0.130, 3,4,5
[**5.7**]{}, 12.4 0.021, 0.334
GJ 581 6.3 M5 3295 94.2 0.146 15.9, [**5.4**]{}, 0.041, 0.073, 3,4,6
[**6.0**]{}, [**1.9** ]{} [**0.218**]{}, 0.029
GJ 436 10.3 M3.5 3281 48 0.211 23 0.0287 3,4,7
GJ 176 9.4 M2.5 3416 38.9 0.262 [**8.3**]{} 0.066 3,4,8
GJ 667C 6.9 M1.5 3327 105 0.172 [**5.7**]{}, [**4.4**]{} 0.049,[**0.123**]{} 3,1,9
GJ 832 4.9 M1.5 3816 40$^{c}$ 0.235 203, [**5.4**]{} 3.6, [**0.16**]{} 3,4
HD 85512 11.2 K6 4305 47.1 0.524 [**3.5**]{} 0.26 3,10,11
HD 40307 12.9 K2.5 4783 48 0.674 [**4.1**]{}, [**6.7**]{}, 0.047, 0.080, 3,10,12
9.5, [**3.5**]{}, 0.132, 0.189,
[**5.1**]{}, [**7.0** ]{} 0.247, 0.600
$\epsilon$ Eri 3.2 K2 5162 11.7 0.726 $\sim$400 3.4 3,10,13
HD 97658 21.1 K1 5156 38.5 0.703 [**6.4**]{} 0.080 3,14,15
---------------- ------ ------ ----------- ----------- ---------------------------------- --------------------------- ---------------------- ------------
\
$^{a}$ – Habitable zone distance is defined as the average of the runaway greenhouse and maximum greenhouse limits (Kopparapu et al. 2014).\
$^{b}$ – References: 1. @neves14, 2. @berta11, 3. Simbad parallax distance, 4. @gaidos14, 5. @rivera10, 6. @robertson14, 7. @demory07, 8. @kiraga07,9. @anglada13,10. @tsantaki13,11. @pepe11,12. @mayor09, 13. @donahue96,14. @valenti05,15. @henry11\
$^{c}$ – GJ 832 has no published rotation period, we assume a relatively short period due to the persistence of UV flare activity in this star.\
$y$ $x$ $m$ $b$
------------------- ---------------------------- -------------------- --------------------
$L$(FUV)$^{a}$ $L$() 1.12 $\pm$ 0.27 $-$1.01 $\pm$ 6.86
$L$(FUV) $L$() 0.75 $\pm$ 0.06 7.56 $\pm$ 1.52
$L$(XUV) $L$() 0.97 $\pm$ 0.19 2.72 $\pm$ 4.91
$L$(XUV) $L$() 0.58 $\pm$ 0.08 11.98 $\pm$ 2.12
FUV/NUV $\langle$$r_{HZ}$$\rangle$ $-$2.42 $\pm$ 0.30 $-$2.31 $\pm$ 0.18
HZ FUV Flux$^{a}$ $\langle$$r_{HZ}$$\rangle$ 0.51 $\pm$ 0.16 1.71 $\pm$ 0.10
HZ XUV Flux$^{a}$ $\langle$$r_{HZ}$$\rangle$ $-$0.01 $\pm$ 0.22 1.41 $\pm$ 0.14
$L$()/$L_{Bol}$ $M_{plan}$/$a_{plan}$ 0.64 $\pm$ 0.24 $-$8.53 $\pm$ 0.56
: Emperical Log-log Relations, log$_{10}$$y$ = ($m$ $\times$ log$_{10}$$x$) + $b$ \[lya\_lines\]
\
$^{a}$ – All luminosities in units of (erg s$^{-1}$).\
$^{b}$ – Habitable Zone fluxes in units of (erg cm$^{-2}$ s$^{-1}$), $\langle$$r_{HZ}$$\rangle$ and $a_{plan}$ in units of (AU), $M_{plan}$ in units of (M$_{\oplus}$).
Target $L_{Bol}$ $f$(XUV) $f$(FUV) $f$(NUV) $f$(Ly$\alpha$) $f$() $f$() $f$()
---------------- ----------- ---------- ---------- ---------- ----------------- ------- ------- -------
GJ1214 31.15 -4.50 -4.45 -4.20 -4.52 -6.48 -6.03 -5.53
GJ176 32.12 -4.46 -4.40 -3.81 -4.51 -6.61 -6.01 -4.91
GJ436 32.02 -4.41 -4.52 -4.02 -4.59 -6.93 -6.54 -5.33
GJ581 31.64 -4.50 -4.84 -4.18 -4.90 -7.23 -6.70 -5.69
GJ667C 31.70 -4.21 -4.17 -3.86 -4.23 -7.11 -6.47 -5.33
GJ832 31.79 -4.30 -4.29 -3.60 -4.36 -6.80 -6.46 -5.07
GJ876 31.69 -4.04 -4.53 -4.33 -4.68 -6.24 -5.91 -5.77
HD40307 32.98 -4.57 -4.31 -2.41 -4.37 -8.20 -7.21 -5.15
HD85512 32.78 -4.82 -4.45 -3.01 -4.51 -7.79 -6.89 -4.93
HD97658 33.12 -4.61 -4.36 -2.05 -4.43 -7.86 -6.97 -5.02
$\epsilon$ Eri 33.08 -4.15 -4.10 -1.97 -4.20 -6.97 -6.13 -4.30
: MUSCLES Broadband and Emission Line Luminosity$^{a,b,c}$. \[lya\_lines\]
\
$^{a}$ – Flux measurements are averaged over all exposure times for individual observations, and broadband SEDs are constructed as described in @loyd15.\
$^{b}$ – All quantities presented as log$_{10}$($L_{Bol}$) or $f$(band) = log$_{10}$($L$(band)/$L_{Bol}$).\
$^{c}$ – Broadband bandpasses are defined as: Bol $\Delta$$\lambda$ = 5Å – $\infty$, XUV $\Delta$$\lambda$ = 5 – 911Å, FUV $\Delta$$\lambda$ = 912 – 1700Å (including Ly$\alpha$), NUV $\Delta$$\lambda$ = 1700 – 3200Å.
Star F(), MUSCLES F(), [X-exoplanets]{}
---------------- ------------------------ ------------------------
GJ 876 6.5 $\times$ 10$^{-4}$ 1.9 $\times$ 10$^{-5}$
GJ 436 4.2 $\times$ 10$^{-5}$ 8.0 $\times$ 10$^{-6}$
$\epsilon$ Eri 1.8 $\times$ 10$^{-2}$ 6.6 $\times$ 10$^{-4}$
: Comparison of MUSCLES $\lambda$1175 emission and [X-exoplanets]{} spectral synthesis prediction.
\
$^{a}$ – All fluxes in units of (photons s$^{-1}$ cm$^{-2}$).
[^1]: https://archive.stsci.edu/prepds/muscles/
[^2]: The COS line-spread function experiences a wavelength dependent non-Gaussianity due to mid-frequency wave-front errors produced by the polishing errors on the $HST$ primary and secondary mirrors; [http://www.stsci.edu/hst/cos/documents/isrs/]{}
[^3]: https://archive.stsci.edu/prepds/muscles/
[^4]: We do not include hard X-rays in our panchromatic SEDs as the observations are not easily obtainable and the photoionization cross-sections for most of the relevant atmospheric constituents decline at energies higher than EUV + soft-X-rays
[^5]: The stellar Ly$\alpha$ emission line is the brightest FUV line in M dwarf spectra [@france13], however the total emission line flux cannot be retrieved without significant modeling analysis [@wood05; @youngblood15]
[^6]: log$_{10}$$L$(band) = $m$ $\times$ log$_{10}$$L$(line) + $b$
[^7]: This observations was scheduled simultaneously, but $HST$ entered safe mode during the scheduled period during early June 2015 and the $Chandra$ observations executed without $HST$. The $HST$ observations were rescheduled about a month later, without supporting X-ray observations.
[^8]: http://sdc.cab.inta-csic.es/xexoplanets/jsp/homepage.jsp
[^9]: For systems with additional, lower-mass planets, we also computed the correlation between $L$(line)/$L_{Bol}$ and the summed SPI integration strength for all $j$ planets in the system, $\Sigma_{j}$ ($M_{j}$/$a_{j}$), and find that the correlation is somewhat weakened, Pearson coefficient = 0.82 and $n$ = 0.2.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Lyman break analogues (LBAs) are local galaxies selected to match a more distant (usually $z\sim3$) galaxy population in luminosity, UV-spectral slope and physical characteristics, and so provide an accessible laboratory for exploring their properties. However, as the Lyman break technique is extended to higher redshifts, it has become clear that the Lyman break galaxies (LBGs) at $z\sim3$ are more massive, luminous, redder, more extended and at higher metallicities than their $z\sim5$ counterparts. Thus extrapolations from the existing LBA samples (which match $z=3$ properties) have limited value for characterising $z>5$ galaxies, or inferring properties unobservable at high redshift. We present a new pilot sample of twenty-one compact star forming galaxies in the local ($0.05<z<0.25)$ Universe, which are tuned to match the luminosities and star formation volume densities observed in $z\ga5$ LBGs. Analysis of optical emission line indices suggests that these sources have typical metallicities of a few tenths Solar (again, consistent with the distant population). We also present radio continuum observations of a subset of this sample (13 sources) and determine that their radio fluxes are consistent with those inferred from the ultraviolet, precluding the presence of a heavily obscured AGN or significant dusty star formation.'
author:
- |
Elizabeth R. Stanway$^{1}$[^1] and Luke J. M. Davies$^{2,3}$\
$^{1}$Department of Physics, University of Warwick, Gibbet Hill Road, Coventry, CV4 7AL, UK\
$^{2}$H. H. Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol, BS8 1TL, UK\
$^{3}$ICRAR, The University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia
title: Establishing an analogue population for the most distant galaxies
---
\[firstpage\]
galaxies: evolution – galaxies: high redshift – galaxies: star formation - radio continuum: galaxies – ultraviolet: galaxies
Introduction
============
Lyman break analogues (LBAs) are a class of compact, UV-luminous galaxies in the local universe that are selected to match the more distant Lyman break galaxy (LBG) population in luminosity, UV-spectral slope and physical characteristics. Lyman break galaxies are named for their dominant spectral feature - a strong rest-frame ultraviolet continuum, which demonstrates a significant flux decrement at the wavelengths corresponding to 912Å and 1216Å in the galaxy rest-frame. Below 912Å photons are sufficiently energetic to ionise neutral Hydrogen in the intergalactic medium and so are efficiently scattered from the line of sight before reaching the observer. Between 912Å and 1216Å the same intergalactic medium can also scatter light in narrow absorption lines representing electron transitions in the Lyman series of Hydrogen (particularly Lyman-$\alpha$). Given that the density of gas clouds in the intergalactic medium rises sharply with increasing redshift, and each cloud absorbs at wavelengths corresponding to its own redshift, the resulting ‘forest’ of redshifted lines can lead to a significant amount of flux lost from the line of sight.
At $z\sim3$, the cumulative effect of scattering in Lyman-$\alpha$ forest leads to a sharp break in the spectrum, such that about half the flux shortwards of 1216Å is lost and below the second break at 912Å reduces to negligible levels. Thus a source will appear absent in a photometric image taken in a filter lying below this wavelength and is said to have “dropped-out”. Therefore $z\sim3$ Lyman break galaxies, identified by their broadband imaging, are termed “U-dropouts” or sometimes “U-drops”. At higher redshifts still, the Lyman-$\alpha$ forest is denser, with the decrement across 1216Å in the rest frame exceeding 90%, and the break has moved to higher observed wavelengths. Thus $R-$drops can be selected at $z\sim5$, $I-$drops at $z\sim6$ and so on.
While rest-frame ultraviolet-selected LBGs are our primary source for understanding galaxy formation and the ‘normal’ galaxy population in the distant Universe, our knowledge of them is necessarily limited by their faint apparent magnitudes, small projected sizes, and extreme redshifts. The interpretation of these galaxies can be greatly enhanced by the consideration of local analogues for which more extensive and more detailed data can be obtained. The study of local galaxies as potential LBAs has developed rapidly over recent years. Heckman and collaborators have built on their early work [@2005ApJ...619L..35H; @2007ApJS..173..441H], identifying a sample of ultra-compact, ultraviolet-luminous galaxies that are a close mirror to $z\sim3$ LBGs. Such sources are selected in data from the Galaxy Evolution Explorer (GALEX) survey and subsequent studies have explored their properties in infrared photometry , X-rays [@2011ApJ...731...55J], ultraviolet spectroscopy [@2011ApJ...730....5H] and morphology [e.g. @2011ApJ...726L...7O].
However, work by @2012ApJ...745...96H and others has shown the importance of consistency in selection techniques across redshift. Recent colour selected samples of $z\sim2$ candidates such as the BX/BM samples have been described as ‘Lyman break galaxies’ but are in fact selected based on rest-frame optical, rather than ultraviolet colours [@2004ApJ...604..534S]. As @2012ApJ...745...96H found, this difference in selection technique, while identifying galaxies at the predicted redshift, biases galaxy samples towards a quantitatively different population. Similarly $z=0.2$ ‘Green Peas’ [star forming galaxies selected for their strong rest-frame optical emission lines and compact morphology, @2009MNRAS.399.1191C] are distinct from analogues selected from rest-UV photometry at the same redshift [e.g. @2011ApJ...730....5H]. While all the galaxies identified by these techniques are actively star-forming, their derived properties such as star formation rate, dust extinction, stellar population age, physical size and hence physical conditions can be very different. Given that each of these populations is deemed, in certain ways, characteristic of its redshift, it becomes increasingly difficult to interpret comparisons between these discrepant populations as indicative of evolution either in the building blocks of today’s massive galaxies or of wider volume-averaged cosmological properties. It is vital that the criteria used to identify galaxy populations to be compared at different redshifts are as close as possible to being identical.
As the Lyman break technique is extended to ever higher redshifts [e.g. @2010MNRAS.409.1155D; @2009MNRAS.400..561D; @2012ApJ...754...83B; @2007ApJ...670..928B], it has become increasingly clear that the LBA population identified by Heckman and co-workers, and the ‘Green Pea’ selection of Cardamone et al, provides a poor match for the properties of high redshift ($z>5$) UV-selected galaxies. Selected to match the typical properties of $z\sim3$ LBGs, they are too massive and often too red to provide a good comparison to existing $5<z<8$ samples, which are less luminous [L$*_{z=6}\sim0.3$L$*_{z=3}$, e.g. @2007ApJ...670..928B], younger and less massive [$\sim30$Myr, M$_\mathrm{stellar}\sim$few$\times10^9$M$_\odot$, e.g. @2007MNRAS.377.1024V although c.f. Oesch et al 2013] and at lower metallicities [Z$\sim0.2-0.5$Z$_\odot$, e.g. @2010MNRAS.409.1155D] than their $z\sim3$ counterparts. Curiously, galaxies that may act as plausible analogues for exceptionally low-metallicity star-forming galaxies (touted as being appropriate for the very earliest galaxies at $z>9$, but yet to be observationally confirmed) have received more attention than those at a few tenths of solar, as appropriate to $z\sim5-7$. LBGs at $z>5$ are also significantly more compact than their lower redshift counterparts [@2010ApJ...709L..21O], driving the star formation density (both in terms of volume and per unit stellar mass) to levels not seen in the equivalent $z=3$ population. This higher star formation density is a key observational driver for searching for better analogues. Higher UV-photon density changes the physical conditions within the galaxies. Dust temperatures are likely to be higher, the intergalactic medium warmer, and hence the collapse of molecular clouds into stars, the ionization of the intergalactic medium and potentially the mode of star formation itself rather different. Thus analogues with lower star formation rate density (as typical for the $z=3$ population) simply [*cannot*]{} provide good physical models with which to understand galaxies in the very distant universe. If LBAs are to inform our understanding at $z\geq5$, or extrapolate from the UV-luminous component, then a specialized $z\sim5$-equivalent LBA population must be established.
Through a combination of archival work and proprietary observations, we have developed a sample of spectroscopically-confirmed, star-forming, $z=0.05-0.20$ galaxies identified in the UV and optical (using GALEX DR6 and the SDSS DR7), that we are exploring in detail as analogues for the $z>5$ population. In this paper we discuss the properties of a plausible candidate Lyman Break analogue galaxy sample, tuned to match the physical properties of the observed $z\geq5$ galaxy population. In section \[sec:cand\] we identify the sample under discussion, exploring their properties in the SDSS optical survey in section \[sec:sdss\]. In section \[sec:radio\] we present new radio imaging of a subsample of objects and discuss the results, before extending the discussion to the implications of the sample more generally in section \[sec:conc\].
Throughout, optical magnitudes are presented in the AB system. Where necessary, we use a standard $\Lambda$CDM cosmology with $H_0=$70kms$^{-1}$Mpc$^{-1}$, $\Omega_M=0.3$ and $\Omega_\Lambda=0.7$.
Candidate Selection {#sec:cand}
===================
Motivation
----------
While it has become clear that the intrinsic properties of the Lyman break galaxy population evolves with redshift, it has not been clear whether the Lyman break analogue candidate samples already suggested or investigated is sufficiently broad to encompass a subset of good $z>5$ analogue systems.
We explored this question using the largest, best developed LBA sample: that identified by @2005ApJ...619L..35H and expanded in @2007ApJS..173..441H[^2] and related work. These sources were selected based on a combination of UV luminosity and UV-optical colour to mirror the properties of the $z=3$ Lyman break galaxy population, with the primary selection based on UV flux (and hence inferred star formation rate) exceeding $\sim$0.3$L_\odot$ for $z=3$ LBGs [@2007ApJS..173..441H]. The galaxies occupy a redshift range of $0.08<z<0.30$, with a mean of $z=0.20$. No constraint was placed on the angular size of the systems. However, projected size (after deconvolution with the seeing) was available from SDSS data.
One of the primary drivers for identifying analogues to distant sources is to explore the effects of very high volume-averaged star formation densities, as discussed above. In figure \[fig:lum-size\], we calculate the inferred star formation density (assuming that the observed star formation is concentrated within the observed half-light radius of the galaxy) for the catalogue of @2007ApJS..173..441H and compare it to the derived typical values for Lyman break galaxy samples at high redshift, using the Schecter parameter M$^\ast$ [from @2007ApJ...670..928B] as a typical luminosity and the mean physical half light radius at each redshift from @2010ApJ...709L..21O. Star formation rates are calculated from the rest-frame ultraviolet flux density, using the conversion factor of @1998ApJ...498..106M, $L_{UV}=8.0\times10^{27}$(SFR/M$_\odot$/yr)ergss$^{-1}$Hz$^{-1}$, assuming a Salpeter IMF. As is often the case for work on Lyman break galaxies at $z\ga5$, no correction is made for dust extinction of the ultraviolet flux (which is essentially unknown at $z>5$) or any change in the IMF with redshift. Given that dust extinction at $z\sim3$ can be as much as factor of a few, the inferred star formation rates are considered to be lower limits, and the star formation densities may be still higher than shown - however observations in the thermal far-infared and submillimeter would be required to reliably determine a total star formation rate - challenging even for ALMA at $z\sim5$. The rationale here is to compare like for like - observed ultraviolet luminosity for observed ultraviolet luminosity - and investigate dust properties when a sample for investigation has been defined. As the figure clearly demonstrates, very few of the Hoopes et al analogue sample ($\sim$3%) have a star formation density comparable to that observed in distant Lyman break galaxies - a discrepancy that only becomes exacerbated as the Lyman break samples are pushed to ever higher redshifts.
Figure \[fig:lum-size\] also illustrates a second concern with the use of previously identified local, luminous starbursts as Lyman break analogues: the increasingly low luminosities being probed by deep and distant surveys. The typical luminosity of Lyman break galaxies is firmly established to evolve with increasing redshift, being some 1.5 magnitudes fainter at $z\sim7$ than at $z\sim3$ [e.g. @2011MNRAS.417..717W]. Even setting aside evolution in the physical size distribution of these sources, it is clear that finding analogues for $z\geq5$ galaxies identified in surveys such as CANDELS [@2011ApJS..197...35G which is probing 120arcmin$^{2}$ to a uniform depth of $\sim$27.5 in ten broad wavebands] will require the study of less luminous and more compact galaxies than previously considered.
![The star formation density as function of luminosity for LBG samples at high redshift (diamonds), the LBA sample of Hoopes et al (2007, crosses) and the sample discussed in this work (asterisks). The current sample are shown as lower limits since all are essentially unresolved in SDSS data. We calculate the inferred star formation density from the observed 1500Å flux and SDSS morphology, assuming for simplicity that the observed star formation is concentrated within a cube with a scale length given by the observed half-light radius of the galaxy (note that a spherical assumption would change each value by a factor of 4$\pi$). For LBG samples, we use the typical absolute magnitudes determined by Bouwens et al (2007), and typical sizes from Oesch et al (2010) from deep, space-based dropout samples. The horizontal ranges on these points indicate a factor of 2 in luminosity, either side of the typical luminosity M$^\ast$. We also show the limits in absolute magnitude corresponding to the ‘Deep’ CANDELS survey target depths of 27.5AB. The standard conversion factor of Madau et al 1998 ($L_{UV}=8.0\times10^{27}$(SFR/M$_\odot$/yr)ergss$^{-1}$Hz$^{-1}$, using a Salpeter IMF) is assumed.\[fig:lum-size\]](figure1){width="\columnwidth"}
Selection criteria
------------------
We have explored the public data releases from the Sloan Digital Sky Survey [DR7, @2009ApJS..182..543A] in the optical ($ugriz$ bands), and GALEX (GR6[^3]) in its two ultraviolet bands (FUV and NUV, centred at 1538Å and 2315Å respectively) for systems that might satisfy this requirement. In order to tune our sample to better match the high redshift population, we apply several selection criteria:
*Colour:* Where @2005ApJ...619L..35H use the fairly liberal UV-to-optical criterion of $FUV-r<2$, suitable for identifying continuous star formation over timescales of several hundred years, we do not rely on optical colours but rather apply a tight constraint in the UV - analogous to a high redshift LBG selection. Beyond $z=5$, the rest-frame optical is shifted well into the infrared and difficult to observe. Hence candidate selection is performed through a combination of a strong spectral break at 1216Å [due to absorption in the intervening intergalactic medium, e.g. @2007ApJ...670..928B; @2004MNRAS.347L...7B; @2003MNRAS.342..439S] and a flat or relatively blue spectral slope observed as a small photometric colour between bands in the rest-frame ultraviolet [@2013MNRAS.430.2885W; @2005MNRAS.359.1184S].
To mirror this, we require that the rest-UV slope is very close to flat (i.e. $FUV-NUV<0.5$), targeting galaxies with young starbursts of $<$200Myr (see figure \[fig:col-col\]). We must, of course, allow for some range of dust extinction in the population. @2007MNRAS.377.1024V found that the spectral energy distribution of $z\sim5$ galaxies are best fit with $A_V\sim0.3$mag, a little lower than that measured at $z\sim3$ ($A_V\sim0.5$mag, Shapley et al, 2003). Evidence from the rest-frame ultraviolet slopes of candidate samples suggest Lyman break galaxies at higher redshift may have still lower extinctions, but with considerable uncertainties [see @2013MNRAS.430.2885W for a detailed discussion]. To accommodate this variation, we do not explicitly constrain the $NUV-r$ band colour, allowing for variations in stellar population age within the sample, and moderate dust extinction. However we note that no plausible candidates had colours redder than $NUV-r=2.2$.
*Luminosity:* We tune our selection window to FUV absolute magnitudes which are equivalent to those in existing $z>5$ LBG samples. This requires that we probe galaxies with L$_{UV}$=$0.1-5$L$*_{z=6}$, where M$^*_{UV}$=-20.24 at $z=6$ [@2007ApJ...670..928B].
*Size:* We also require that this star formation occurs in a compact region [distant galaxies have a projected half-light radius $<$ 2kpc, @2010ApJ...709L..21O], so as to ensure a similar star formation volume density in the local analogue systems to that observed in the high redshift sources. Given the typical seeing of SDSS images, deconvolution of very compact sources to determine their physical size is problematic. At $z>0.2$, a physical size of 2kpc is identified as only a small perturbation on the light profile due to average seeing. Thus the requirement that the SDSS pipeline is able to constrain the size effectively provides an upper limit on our redshift distribution. Given the uncertainties on deconvolution of these objects, we apply the reasonably liberal criteria that the best fit deconvolution corresponds to a physical scale $r<3.5$ kpc, and that the SDSS $g$-band petrosian radius is $<1.2\arcsec$. While this may admit a fraction of sources more extended than those at high redshift, as figure \[fig:lum-size\] shows these lower limits select a population with star formation densities comparable to those observed in the distant Universe. Note that we must constrain the physical size in the optical, since GALEX does not have sufficient resolution in the UV.
*Additional Constraints:* In order to identify a useful pilot sample for further study we require that our candidates have SDSS spectroscopy, and that this spectroscopy identifies their UV luminosity as arising from star formation rather than AGN [the AGN fraction in distant galaxy samples is known to be very small, @2007MNRAS.376.1393D; @2005MNRAS.360L..39N]. In the interests of further, ground-based follow-up, we also place a declination constraint (dec$<-8.5\deg$) to optimize availability to southern telescopes (e.g. ATCA, VLT, ALMA etc.). We eliminate catalogue sources with obvious problems with their photometry (e.g. small clumps which form part of a much larger, more extended system, or objects whose photometry is affected by near neighbours - a particular problem for these faint targets in the GALEX data).
These criteria identify a sample of just 21 sources to investigate in detail, as a subset of the larger, unrestricted declination, population. As figure 1 (and also section \[sec:sdss\]) demonstrates, these occupy a region of parameter space distinct from that considered by previous studies - both in luminosity and in star formation density. Unsurprisingly these are both fainter in apparent magnitude, and at a slightly lower typical redshift than those of the @2007ApJS..173..441H sample (see figure \[fig:zfuv\]).
![Ultraviolet-optical colours expected of young starburst stellar populations at $0<z<0.3$ (increments of $\Delta z=0.1$ are marked with crosses on each evolutionary track). We use @2005MNRAS.362..799M stellar population synthesis models for a declining initial starburst to determine the expected colour at four ages, and two metallicities and also show the effects of dust reddening with an A$_V$=0.2, 0.4, assuming a @2000ApJ...533..682C extinction law. Increasing stellar population age primarily effects the UV-optical colour, while the effects of dust are larger in the ultraviolet. A selection criterion of $FUV-NUV<0.5$ encompasses the bulk of young stellar populations and reasonable dust extinctions in the local Universe. \[fig:col-col\]](figure2){width="\columnwidth"}
![The redshift-apparent magnitude distribution of our targets, compared to the ultraviolet-luminous galaxy sample of Hoopes et al (2007). While there is some overlap, our targets are typically fainter, and at a slightly lower mean redshift. \[fig:zfuv\]](figure3){width="\columnwidth"}
Properties in the GALEX/SDSS data {#sec:sdss}
=================================
UV slope and optical morphology
-------------------------------
The selection criteria discussed in section \[sec:cand\] essentially select sources which are unresolved in the optical, as figure \[fig:morph\] shows. However a few objects are selected for which the properties of a dominant central core satisfy our selection criteria, but which consist of multiple compact components or possess a slightly more extended, low surface brightness tail. Given that multiple components are often seen in $z\sim5$ samples [e.g. @2009MNRAS.400..561D], and that low surface brightness emission may well be undetectable at high redshifts, we retain these sources in our sample, in order to explore the full properties of this colour-selected population. Further observations (using either adaptive optics or space-based data) will be required to fully constrain the morphology of these sources.
![Examples of the essentially unresolved optical morphology (SDSS $g$ band) of these candidates. Boxes are 10$\arcsec$ on a side. A few objects (for example, the second here) show more extended structure around a central core, and these may prove not to be ideal high redshift analogues, but they are included in the sample so as to explore the range of galaxies identified in catalogue data. Deconvolution suggests a typical half-light radius of $<$0.5arcsec.\[fig:morph\]](figure4){width="\columnwidth"}
Each source in our sample has a full suite of UV-optical photometry. We use GALEX photometry (at 1300 and 2400Å) and redshift information from SDSS spectroscopy, to explore the observed rest-frame ultraviolet spectral slope of our sample. The observed spectral slope is primarily dependent on a combination of stellar population age and dust extinction, with the bluest colours ($f_\lambda\propto\lambda^\beta$, $\beta<-2$) only possible with zero dust and very low metallicities (see figure \[fig:col-col\], and also Wilkins et al 2013 for a fuller discussion). In common with the high redshift samples with which we ultimately wish to compare, we are unable to straightforwardly disentangle the effects of dust and stellar population from a single colour, but instead consider the observed range of values, with a full analysis of dust deferred until more data is available (see discussion later). Even without separating dust and stellar population age, their combined effect is an important indicator of the properties of a galaxy, and has drawn recent attention.
Some authors [e.g. @2013arXiv1306.2950B; @2012ApJ...754...83B; @2010ApJ...724.1524O] have suggested that galaxies at the highest redshifts ($z>7$) are systematically bluer than those observed at later times and can have extreme spectral slopes, with $\beta<-3$. Such a blue slope is difficult to produce using normal stellar population models and may imply the presence of very low metallicity or Population III stars. However this observation is subject to significant uncertainties and the colours are likely less extreme than originally suggested [see @2013MNRAS.430.2885W for detailed discussion]. @2012ApJ...756..164F, considering the same data as @2012ApJ...754...83B, suggested that the spectral slope, while evolving to bluer colours between $z\sim4$ and $z\sim7$, does not require such exotic stellar populations. By contrast, @2013MNRAS.432.3520D have suggested that there’s no strong evolution in the slope beyond $z\sim4$ and that while the colours are typically blue ($\beta\sim-2$), they can be straightforwardly explained with low dust, or slightly sub-solar metallicity, populations. Interestingly, some studies at [intermediate]{} redshift ($z<4$) have found [relatively]{} little evidence for a systematic and linear colour evolution with redshift. [Studying stacked mid-infrared (Herschel) data for ultraviolet-selected galaxies in photometric redshift bins at $z\sim1.5, 3$ and 4, @2014MNRAS.437.1268H identified a strong evolution in their dust extinction with stellar mass, but negligible evolution with redshift. If UV spectral slope is interpreted as varying due purely to dust (as opposed to stellar population age, or metallicity, then this might suggest that somewhere around $z\sim4-5$ (i.e. when the first galaxies are about 1Gyr old), the colour evolution stabilises as older stellar populations (and the dust they generate) begin to contribute significantly to the observed colours.]{}
![A comparison of the $z\sim3$ LBA population (crosses), with the sample discussed here (shown as asterisks). The sample presented here forms a distinct and separate galaxy population, probing the lower luminosities, higher star formation densities and sometimes bluer rest-UV colours typical of the high-z population. Shaded regions indicate the range of parameter space that has been estimated by varied authors for LBG samples at $z=5$ and $z=7$ [see @2013MNRAS.430.2885W].\[fig:lum\]](figure5){width="\columnwidth"}
[Given that studies such as @2014MNRAS.437.1268H and @2011ApJ...734L..12B [ who studied individually Herschel-detected $z<2$ examples] suggest a strong luminosity dependence on inferred dust extinction at moderate redshift, while at the highest redshifts @2013arXiv1306.2950B [@2012ApJ...754...83B] and others have suggested a similar luminosity dependence, it is informative to examine our sample for any similar trend.]{} In figure \[fig:lum\] we plot the rest-frame ultraviolet spectral slope for our sample, as a function of absolute magnitude. As can be seen, the sample probes slopes in the range $-2.5<\beta<-1.0$, comparable to the typical slopes seen at $z\sim5-7$ [@2011MNRAS.417..717W; @2013MNRAS.430.2885W; @2010MNRAS.409.1155D; @2005MNRAS.359.1184S], and shows a similar trend towards bluer slopes at lower luminosities to that suggested in the distant universe. This trend is, admittedly, rather weak in our small pilot sample. Nonetheless, we note that six of our sources (29%) have spectral slopes with $\beta<-1.5$, substantially steeper than that seen in the $z=3$ population and its analogues [$\beta\sim-0.9$, @2003ApJ...588...65S].
Emission Line indices
---------------------
All the targets in this sample were identified as star-forming galaxies on the basis of their SDSS spectra, and show the typical strong and narrow optical emission lines (see figure \[fig:spec\]). In order to evaluate the effectiveness of this sample as LBAs, an important property that must be considered is the metallicity of the dominant stellar population. This is still somewhat poorly constrained at higher redshifts where optical spectroscopy is beyond the grasp of current instruments. As mentioned in the previous section, extremely low metallicities ($<0.001$Z$_\odot$) have been proposed for the most distant sources, although it is not clear these are strictly necessary.
![An example of the emission line spectrum typical of these galaxies. Several prominent emission lines related to star formation and metallicity fall into the observed frame optical spectrum. This example lies at $z=0.097$.\[fig:spec\]](figure6){width="\columnwidth"}
At $z\sim5-6$ the metallicity of typical Lyman break galaxies is slightly better known. Fitting of the full spectral energy distribution of photometrically selected galaxies suggest that reasonable fits can be obtained using synthetic stellar populations with metallicities of a few tenths solar [e.g. @2007MNRAS.377.1024V], while solar metallicity models tend to suggest implausibly old stellar populations at these early times. Similarly, the blue spectral slopes at high redshift, if interpreted as a metallicity indicator yield approximate values of $\sim$0.25Z$_\odot$ [@2010MNRAS.409.1155D]. This is consistent with the observed metallicity of absorption line systems in the interstellar medium which is seen to decrease by $\sim1$ dex between $z=0$ and $z=5$ [and perhaps more rapidly at still higher redshifts, e.g. @2012ApJ...755...89R]. Similarly, Gamma Ray Burst host galaxies at $z>4$ [which are believed to be very low mass, @2012ApJ...754...46T, but must be star-forming in order to generate the short-lived GRB progenitors], have been found to have typical metallicities $<0.15$Z$_\odot$ [see compilation in @2013MNRAS.428.3590T].
Thus we would expect good analogues for the distant galaxy population to have metallicities of a few tenths of solar, perhaps decreasing as we search for analogues of galaxies at either higher redshifts or lower luminosities. This is, of course, a crude estimate, and will depend somewhat on the method and species used to make the measurement. We might expect good analogues (which should be young starbursts) to show an enhancement in $\alpha$-process elements, for example [e.g. @2008ApJ...681.1183K].
![The variation in optical emission line ratios, established as metallicity indicators, in our sample. The solid line shows the mean metallicity as a function of emission line ratio strength in the SDSS galaxy population, calibrated by Maiolino et al 2008 - note that the SDSS galaxy population shows considerable scatter around this mean relation. We adopt a Solar metallicity of 12+log(O/H)=8.69. Galaxies with radio data (see section \[sec:radio\]) are shown in red, the remainder of the sample in blue. Formal errors on the measured equivalent widths (propagated to the line ratios) are too small to plot (with a few exceptions), however we note that uncertainty in the continuum for these relatively faint sources may introduce errors at the $<10$% (0.05 dex) level. The sources in this sample probe moderate metallicities, with a few outliers.\[fig:metallicity1\]](figure7){width="\columnwidth"}
In figure \[fig:metallicity1\], we plot the classic metallicity indicators $R_{23}$=(\[OII\]+\[OIII\])/H$_\beta$ and log(\[OIII\]/\[OII\]) for our sample. Line fluxes were extracted from the SDSS spectroscopic catalog for our candidate sources, and tested for consistency against the OSSY determinations by @2011ApJS..195...13O[^4]. Overplotted on figure \[fig:metallicity1\], we show the calibrated empirical relation between these quantities and metallicity from . As can be seen, the use of the oxygen ratio can break the well-known metallicity degeneracy in the $R_{23}$ index, placing all of our candidate sample on the upper arm of the metallicity relation. The bulk of our sample (17/21 objects) shows metallicities in the range $8.00<$12+log(O/H)$<8.69$ (i.e. $0.2<$Z$_\odot<1.0$, a few tenths Solar). A few outliers in the sample either have higher metallicities or poorly constrained metallicity indices (due to redshifting of one or more emission lines into regions of low signal to noise in the spectroscopy). We note that the blue colours targeted in our sample are clearly possible even at super-Solar metallicities, and will consider the effects of this higher metallicity in individual objects during later analysis. However, as figure \[fig:zbeta\] demonstrates, at least half (11/21 objects) of our sample have measured optical spectral indices consistent with the metallicity range $0.2<$Z$_\odot<0.5$ that may be appropriate for high redshift analogues.
![The variation in rest-UV spectral slope $\beta$ with metallicity (indicated here by the ratio of oxygen emission line strengths) based on the calibration of Maiolino et al 2008 in figure \[fig:metallicity1\]. The bulk of this sample probes $0.2<$Z$_\odot<0.6$, exploring the same metallicity range inferred for the $z\sim5$ population. There is tentative evidence for bluer spectral slopes (or at least more dispersion in the spectral slope) with decreasing metallicity.\[fig:zbeta\]](figure8){width="\columnwidth"}
Evolution in the rest-frame ultraviolet spectra of distant sources is expected to be driven, at least in part, by evolution in the cosmic mean metallicity at early times. This has been observed between $z=5$ and $z=3$ (e.g. Douglas et al 2010), and inferred photometrically at higher redshifts (although with some uncertainty, see Wilkins et al 2013). Figure \[fig:zbeta\] suggests an intriguing trend is present in our sample. While the number statistics are small, there appears to be a trend towards bluer rest-frame ultraviolet spectral slopes with decreasing metallicity within our sample. Certainly there is a larger dispersion in the spectral slopes for sources at Z$<0.5$Z$_\odot$ than for sources around Solar metallicity. However we caution that this sample is too small to disentangle the effects of trends of luminosity (see figure \[fig:lum\]) and stellar population age from a pure metallicity evolution. Thus we do not quantify the trend here, due to the small sample statistics, but will investigate this further in future work.
Optical Colour
--------------
In addition to the UV-selected LBA samples discussed above, a second category of local object has been suggested as Lyman break analogues: the optically-selected Green Pea population [@2009MNRAS.399.1191C]. These sources are identified from their broad-band photometry as having an excess of flux in the $r$-band, interpreted as indicative of strong \[OIII\] emission at $0.112<z<0.360$. Hence this selection is designed to identify strongly star-forming galaxies in the local universe.
In figure \[fig:greenpeas\], we compare the optical colours of our candidate sample with Green Peas presented by @2009MNRAS.399.1191C. Our sources do not satisfy the two-fold colour criteria required for identification as Green Peas. Although their spectra are generally blue (as required in the $u-r-z$ colour plane), they do not have the extreme excesses in the $r$-band identified by the $g-r-i$ colour selection. This is not entirely surprising. The requirement for a very high equivalent width from emission lines in the $r$-band, but not in the $g$-band skews the distribution of emission line ratios in the Green Pea sample, and yields galaxies with a mean metallicity of 12-log(O/H)$\sim8.7$, rather higher than is typical for our sample, and comparable to Solar metallicity[^5]. It also biases the sample to very high star formation rates - with the required line widths giving a typical star formation rate of $\sim$30M$_\odot$yr$^{-1}$. Thus these sources are typically more metal rich and also forming stars at $\sim3-10$ times the rate of our targets. Given that star formation rate directly influences rest-UV flux, we would also expect the majority of Green Peas to fall outside our ultraviolet luminosity selection, were GALEX data for the whole sample to be available.
Green Peas are selected to be compact in SDSS data ($g$-band petrosian radius $<2.0\arcsec$, c.f. $<1.2\arcsec$ for our sample), although this requirement translates to a less strict constraint at the high redshift end of their sample. Nonetheless, there would likely be overlap between our sample and the Green Pea selection criteria were we to admit more UV-luminous sources. Those Green Peas which lie towards the low metallicity and lower redshift end of the Cardamone et al and similar selections may well be good analogues for luminous ($L_{UV}>L*_{z=6}$) Lyman break galaxies at high redshift in terms of star formation density. However, the $z\sim5$ LBA sample presented here remains distinct from the established Green Pea population and is more closely tuned to the properties of relatively faint galaxies currently being identified in deep surveys (see figure \[fig:lum-size\]).
![The optical colours of our candidate sample (asterisks), compared to those of the optically-selected ‘Green Pea’ population identified by Cardamone et al (2009, crosses). Photometric errors are typically smaller than the data points. Our candidates do not satisfy the green pea criteria. Although their spectra are generally blue (as required in the $u-r-z$ colour plane), they do not have the extreme excesses in the $r$-band identified by the $g-r-i$ colour selection.\[fig:greenpeas\]](figure9a "fig:"){width="0.495\columnwidth"} ![The optical colours of our candidate sample (asterisks), compared to those of the optically-selected ‘Green Pea’ population identified by Cardamone et al (2009, crosses). Photometric errors are typically smaller than the data points. Our candidates do not satisfy the green pea criteria. Although their spectra are generally blue (as required in the $u-r-z$ colour plane), they do not have the extreme excesses in the $r$-band identified by the $g-r-i$ colour selection.\[fig:greenpeas\]](figure9b "fig:"){width="0.495\columnwidth"}
Radio Continuum Follow-Up {#sec:radio}
=========================
Observations at radio wavelengths have the potential to further elucidate the properties of galaxies such as these in two important ways. They can reveal the presence of dust-extincted emission, i.e. a more extended or intense emission source than that seen in the ultraviolet bands where this sample was selected. Detection of a significant excess of emission in radio wavelengths over that predicted from the ultraviolet star formation rate could indicate that the ultraviolet sources are small regions of intense star formation and low extinction embedded in a much larger and more evolved galaxy [@2010ApJ...710..979O see also Stanway et al 2010, Davies et al 2012, 2013] and so suggest that these sources would be more analoguous to ULIRG-like super-starbursts than Lyman break galaxies. Alternately, they may indicate the presence of an obscured active galactic nucleus which could substantially effect the evolution and broadband photometry of the system. These two effects are potentially distinguishable based on the radio-frequency spectral slope of any detected emission.
Observations
------------
We have undertaken an examination of the radio continuum flux at 3 and 6cm wavelengths in a subset of thirteen galaxies, randomly selected from our larger sample of twenty-one targets.
We obtained continuum measurements of the selected galaxies over the period 2012 Aug 27-29, using the Australia Telescope Compact Array (ATCA) in its 6A configuration. The six antennae at the ATCA were aligned along an East-West axis and the longest baselines were 6km in length. We tuned the Compact Array Broadband Backend (CABB) correlator such that one 2GHz IF was centered at 5500MHz (6cm) and the second at 9000MHz (3cm), with full polarization information collected simultaneously at both frequencies. Measurements of PKS1934-638, the standard calibrator at the ATCA, were used for absolute flux calibration. A bright, compact point source close to each science target was used for atmospheric phase calibration. Our observations were associated with observing programme C2695 (PI: Stanway).
Data were reduced using the dedicated software package [miriad]{} [@1995ASPC...77..433S] and radio frequency interference carefully flagged as a function of time on a channel-by-channel basis. Each frequency band had 2048 oneMHz-wide spectral channels allowing interference to be restricted to a few distinct channels. This is particularly important at 9000MHz where powerful interference spikes occurred frequently in certain narrow frequency ranges. Multifrequency synthesis images were constructed from the 2GHz bandwidth at each frequency. In each image a number of additional sources (often NVSS and occasionally 2MASS objects) were detected. The images were cleaned using a Clark algorithm, constrained to cut off at a level twice the noise standard deviation in the uncleaned images. Uniform weighting was used to optimize suppression of side-lobes in the imaging. All targets were expected to be point sources at the angular resolution of the ATCA.
Short tracks were taken across the full range of possible hour angles, in order to optimize coverage of the $uv$-plane, with a total on-source integration time of 2hours on each target. These sources are relatively northern for the ATCA. As a result, sources are below the horizon for part of each potential twelve hour earth rotation synthesis track, and the resulting synthesised beam is heavily asymmetric. Typical beam sizes were $\sim2\arcsec$ in right ascension and $\sim15-20\arcsec$ in declination at 5500MHz, with a beam position angle close to zero. Flux uncertainty naturally scales with beam-size for a point source and the noise is heavily correlated between adjacent pixels due to uncertainties in the reconstruction of an incompletely-sampled $uv$-plane.
The [miriad]{} task ‘imfit’ was used to measure the flux and noise levels at the location of the galaxy, which was placed close to the centre of the primary beam (8.5 FWHM at 5500MHz). Measured fluxes, beam sizes and uncertainties for each target are given in table \[tab:radio\].
Results
-------
Eight sources are detected at better than $3\sigma$ significance in our 5500MHz data. Of these, the brightest three sources were independently detected in two or more observing sessions, each separated by at least one day. Visual inspection suggests that the weakest detection (in object 05083) is marginal and may or may not be reliable, while a second (object 71294) is similarly faint and has a flux measurement that may be affected by a brighter neighbour. All others appear to be robust detections. None of the eight detected sources are measurably resolved in the radio imaging.
The 9000MHz band is more heavily affected by noise than that at 5500MHz. Observations were taken simultaneously in the two bands, and those at 9000MHz are naturally slightly shallower, although the synthesised beam is smaller by a factor of two. The four brightest sources at 5500MHz were also detected in the higher frequency data. No other target was significantly detected at 9000MHz. The results of our radio observations for all targets are given in table \[tab:radio\].
Object ID z 5500MHz Flux S/N Beam size 9000 MHz S/N SFR $\alpha$
----------- ------- -------------- ----- ------------------------ -------------- ----- --------------- ------------------
23734 0.108 220 $\pm$ 34 6.4 16$\times$2.0$\arcsec$ 164 $\pm$ 31 5.2 5.5 $\pm$ 0.9 -0.60 $\pm$ 0.15
10880 0.169 86 $\pm$ 24 3.6 18$\times$2.0$\arcsec$ 43 $\pm$ 34 1.2 5.5 $\pm$ 1.5 $<$-0.04
19220 0.188 60 $\pm$ 82 0.7 17$\times$2.0$\arcsec$ 44 $\pm$ 32 1.4 $<$13.3
54061 0.074 215 $\pm$ 38 5.6 19$\times$2.0$\arcsec$ 136 $\pm$ 34 4.0 2.5 $\pm$ 0.4 -0.93 $\pm$ 0.29
60392 0.120 54 $\pm$ 26 2.1 19$\times$1.9$\arcsec$ 73 $\pm$ 41 1.8 $<$1.6
27473 0.083 57 $\pm$ 23 2.5 16$\times$2.0$\arcsec$ 47 $\pm$ 31 1.5 $<$0.7
71294 0.146 72 $\pm$ 21 3.4 15$\times$2.0$\arcsec$ 79 $\pm$ 53 1.5 3.4 $\pm$ 1.0 $<$1.25
16911 0.097 91 $\pm$ 20 4.6 18$\times$2.0$\arcsec$ 109 $\pm$ 27 4.1 1.8 $\pm$ 0.4 -0.61 $\pm$ 0.12
10045 0.137 69 $\pm$ 26 2.6 15$\times$2.0$\arcsec$ 112 $\pm$ 46 2.4 $<$2.2
24784 0.113 151 $\pm$ 27 5.6 21$\times$1.9$\arcsec$ 93 $\pm$ 43 2.2 4.2 $\pm$ 0.7 $<$-0.70
62100 0.136 61 $\pm$ 25 2.4 20$\times$1.9$\arcsec$ 67 $\pm$ 40 1.7 $<$2.0
05083 0.146 76 $\pm$ 22 3.4 20$\times$1.9$\arcsec$ 80 $\pm$ 32 2.5 3.6 $\pm$ 1.0 $<$0.13
08755 0.164 159 $\pm$ 22 7.1 20$\times$1.9$\arcsec$ 163 $\pm$ 33 5.0 9.6 $\pm$ 1.3 0.05 $\pm$ 0.01
Radio spectral slope
--------------------
Table \[tab:radio\] also presents the radio spectral index for those sources with robust detections in one or more wavebands. Where a source is not detected at 9000MHz, we estimate a limit on the slope based on a limit of 2.5 times the noise level for a point source (at which point detection begins to be plausible in the images).
Radio continuum emission arises as a result either of AGN emission or star formation. [While synchrotron emission, arising from electrons accelerated in a strong magnetic field, is ubiquitous in radio sources of all types, the presence of other thermal or non-thermal processes can contribute different emission components, modifying the continuum spectral slope. For example, in ionised hydrogen clouds there is a substantial contribution from free-free or bremstrahlung radiation which arises when unbound electrons are scattered by the magnetic field of nearby ions [see @2002ira..book.....B for a full explanation].]{} Hence, where AGN emission dominates, a relatively steep spectral slope is expected, theoretically becoming as steep as $\alpha\sim-2$ for bright quasars with no star formation contribution [@2002ira..book.....B], and observed values steeper than $\alpha\sim-1$ [e.g. @2012arXiv1201.3922G; @2013ApJ...768...37C and references therein]. By contrast thermal free-free absorption and emission in star formation regions are expected to flatten the spectrum (particularly at low and high frequencies respectively) by contributing flux with $\alpha\sim-0.1$ and the majority of star forming systems have a fairly shallow slope ($\alpha\sim-0.5-0.6$) . Therefore we expect galaxies with ongoing star formation to have moderately negative spectral slopes, and AGN or old stellar populations to be steeper.
Measurements of the spectral index are possible for the four sources detected in both bands of our observations (albeit with substantial uncertainty). Of these, three galaxies have spectral slopes consistent with $-\alpha\sim0.6-0.8$, as expected for faint star-forming sources [see @2003ApJ...588...99B].
The above assumes that we are observing in a frequency regime described by a single, unbroken power law - usually a safe assumption. However aging of the synchrotron-emitting (i.e. relativistic electron) population tends to steepen the radio spectrum above a time-dependent critical value, since high energy electrons decay more quickly than those at lower energies . This leads to a break in the spectrum which increases in observed frequency with age. In all but the youngest star forming galaxies this break will lie above the frequencies considered here. One source in our sample (object 08755) has a substantially flatter spectral slope than the others, with $\alpha=0.05$. Spectral slopes this flat are relatively unusual, and may indicate the presence of a spectral break at around $\sim$6-8GHz, which would in turn suggest that the source of non-thermal synchrotron emission in this source is very young [@2012arXiv1201.3922G].
None of the galaxies in this sample have spectral slopes steep enough to preclude star formation dominated emission, but deeper observations, and observations at more frequencies (both long-wards and short-wards of 5500MHz) will be necessary to study the radio spectral energy distribution of these sources in more detail.
Star formation rates
--------------------
Where robustly detected, and with the possible exception of one object, the radio spectral indices in this sample are consistent with the flux arising from electrons accelerated by supernovae and their remnants. Hence they should track the supernova (and so also star formation) rate.
As a result, radio continuum flux can be straightforwardly converted to an inferred star formation rate given an empirically determined conversion factor. This factor is almost constant for stellar populations forming stars at a constant rate and aged more than about $10^8$years, but can vary for younger stellar populations . We calculate the star formation rates inferred from our continuum measurements at 5500MHz using the same prescription as @2003ApJ...588...99B and @2002ApJ...568...88Y. Following @2003ApJ...588...99B, we set $\alpha=-0.6$, appropriate for faint radio sources and consistent with those measured in our sample. We use $T_d=58$K and $\beta_\mathrm{FIR}=1.35$, respectively the dust temperature and emissivity index [again fixed to the values of @2002ApJ...568...88Y for comparison with previous studies]. At these wavelengths, the thermal dust emission makes a negligible contribution, and setting $T_d=35$K and $\beta_\mathrm{FIR}=2$ [as found at $z\sim3$, @2013MNRAS.433.2588D] has no measurable effect.
![Star formation rates inferred from the GALEX UV (as in figure \[fig:lum-size\]) and ATCA 5500MHz continuum flux (described in section \[sec:radio\]) for the subsample of our targets with ATCA imaging. 2$\sigma$ limits are shown where a source is undetected. Error bars show uncertainties due to photometric errors, but not systematic uncertainties in the star formation rate conversion factors. The rates are consistent to within a factor of a few both for detected sources and limits. Object 19220, for which relatively weak radio limits were obtained, is omitted from this figure. The star formation rates in these sources are similar to those typical in star forming galaxies at $z>5$ observed in deep fields. The non-detections at 5500MHz may indicate young stellar populations, yet to establish a SNe-driven radio continuum in some cases, or that these sources have an unexpectedly steep radio spectral slope.\[fig:radio\_sfr\]](figure10){width="\columnwidth"}
.
Figure \[fig:radio\_sfr\] illustrates the resultant radio-inferred star formation rates (or limits thereupon) for our sample, in comparison to the star formation rates inferred from their ultraviolet flux[^6]. Of the sources with radio detections, the inferred star formation rates derived using the two methods are consistent within a factor of a few, with one source, object 08755, showing a 4.3$\sigma$ excess in radio continuum, suggesting moderate dust extinction may be reducing the observed ultraviolet flux. We note that this source is an outlier in the sample in several ways: not only does it show a radio excess, it also has a significantly flatter radio spectral slope than those measured in other sources, and is also amongst the reddest sources in the ultraviolet (with $\beta\sim-1$). The sources without radio detections (5/13 targets) are also broadly consistent with their ultraviolet-inferred star formation rates at the 3$\sigma$ level, although in one case (object 19220), the constraint is relatively weak and this source not considered further here.
However, both the detected fluxes and the limits on undetected objects hint that the radio flux in the typical member of this population may actually be somewhat deficient with respect to their ultraviolet flux. A point source with a flux at least 2.5$\sigma$ times the background level should have been detected in three of the four cases, and would also likely be accessible to visual inspection, even if formally undetected. Similarly, of the sources that were detected, half (4 out of 8) have measured radio fluxes below those predicted from their UV luminosity (see figure \[fig:ratio\]). This is a tentative, but potentially interesting result if supported by future observations.
![The ratio of star formation rates derived from 5.5GHz flux to those derived from the ultraviolet (i.e. a source with dusty star formation, or other sources of excess radio emission, would have a ratio greater than one). Where a source is undetected in the radio, 2$\sigma$ limits are shown. Object 19220, for which relatively weak radio limits were obtained, is omitted from this figure. Four out of eight detections, and three radio limits, are consistent with a deficit in radio flux relative to the ultraviolet. \[fig:ratio\]](figure11){width="\columnwidth"}
Given the uncertainties in the calibrations applied to derive star formation rates here, it is useful to consider whether alternative indicators might be useful. There is currently no deep near-infrared data available for these sources (which will be surveyed by VISTA over the next few years). As a result, SED fitting becomes ill-constrained and will be explored in a later paper. Data from the WISE survey in the mid-infrared does exist and, particularly in the 22$\mu$m W4 band, can also used as a measure of star formation rate over a wide range in luminosity. However, without a deep $K$-band image to constrain the location of red sources, deconvolution of blended sources in the relatively shallow and low resolution mid-infrared images becomes a substantial problem. As a result, non-detections and extended sources in the WISE catalogues are unlikely to provide useful limits on our targets. Nonetheless, we identify four of our radio-observed sample as relatively isolated and cleanly detected point sources in the “ALLWISE” data release of the W4-band all-sky catalog[^7].
Perhaps unsurprisingly, all four detected sources are also well detected in the radio data. We provide a comparison between ultraviolet-, 22$\mu$m- and radio-derived star formation rates for these four sources in figure \[fig:wise\_sfr\], using the star formation rate conversion for the W4 band derived by @2012JApA...33..213S. As the figure shows, the infrared-derived star formation rates suggest that these sources contain relatively little warm dust, exceeding the ultraviolet-derived values by only $\sim$25%. Interestingly, object 08755 (which has the highest star formation rate in the sample and a flatter than expected radio spectral slope, as discussed in the previous section), does not present as an outlier in this respect, but is consistent with the sample as a whole.
Given that the sources with strong radio detections might usually be expected to be the most massive, and often the dustiest, of our sample, the bulk of the non-detections and blended sources are likely to prove similarly spare of warm dust and until data is available to form a full SED fitting analysis, the ultraviolet-derived star formation rates do not require a substantial dust correction.
![Star formation rates inferred from the WISE 22$\mu$m band compared to their ultraviolet and 5500MHz continuum derived values for the four sources with good WISE detections. Error bars show uncertainties due to photometric errors, but not systematic uncertainties in the star formation rate conversion factors.\[fig:wise\_sfr\]](figure12){width="\columnwidth"}
As mentioned above, we convert radio flux to star formation using an assumed radio continuum spectral slope $\alpha=-0.6$, consistent with those we measure. Using a steeper spectral slope, $\alpha=-0.75$, results in star formation rate estimates 25% higher. However, even given an adjustment of this magnitude, six of the sample still show inferred star formation rates lower in the radio than the ultraviolet. Since the few sources bright and isolated enough for WISE detections suggest that the ultraviolet star formation rates might also increase by 25%, the relative deficiency in radio flux remains.
A radio flux in excess of that predicted from the ultraviolet photometry can be straightforwardly interpreted as indicating the presence of dust-obscured star formation. A deficit in the radio emission is more challenging to explain. Any star forming region from which ultraviolet light can escape should present no impediment to the escape of radio photons. Similarly, if the emission was influenced by the presence of an AGN, the radio flux would be expected to exceed that escaping in the ultraviolet. An intriguing possibility is that a deficit in the radio may indicate a very young stellar population. The ultraviolet flux in star forming systems arises from the photospheres of the hottest, most massive stars, with a main sequence lifetime of a few tens of Myr. By contrast, the radio flux is established by supernovae and their remnants at the end of stellar lifetimes, and thus takes longer to stabilize to the standard conversion factors applied.
A young stellar population ($<$100Myr, approximately) would be deficient in radio flux . While this could conceivably arise at the onset of continuous star formation, it could also indicate a recent burst, or an exponentially rising star formation rate with cosmic time in these sources [as has been suggested at high redshift by recent modeling and simulation, e.g. @2010MNRAS.407..830M]. Any scenario featuring a young, hot stellar population is likely to lead to a bluer intrinsic ultraviolet spectral slope, and thus runs counter to the tentative trend observed in figure \[fig:ratio\], which is intriguing. The sources with the bluest spectral slopes in fact show least evidence, one way or the other, for a deficit in radio flux. Since emission from any regions of moderately-dust extincted radio emission should, redden the observed spectral slope, the relatively red ultraviolet colours of the sources with the highest radio excesses are unsurprising. The red colours of the sources with the largest radio *deficits* are rather more so and warrants more investigation. Disentangling the effect of stellar population age and reddening on the spectral slopes in these galaxies will require a full analysis of the ultraviolet through infrared spectral energy distribution of these sources and their spectra and will be investigated in future work.
It is, of course, possible that the radio-deficient galaxies lie just marginally below the detection level in our ATCA observations. Further observations will be required to determine whether these sources are indeed substantially lower in radio flux than expected. If they are indeed young systems, this will strengthen their interpretation as analogues for the generally young systems observed at the highest redshift, and make them potentially useful models for predicting the properties of distant galaxies at submillimetre/radio wavelengths.
Discussion and Conclusions {#sec:conc}
==========================
In this paper we have presented a new pilot sample of 21 local star forming galaxies that present potential analogues for star forming galaxies in the distant universe - that is, are potential Lyman break analogues for the $z>5$ galaxy population.
We have demonstrated that these sources provide a good match to the established or expected properties of the distant galaxy population, in terms of star formation density, physical size, ultraviolet luminosity and metallicity, and that they populate a region of these parameter spaces either sparsely populated or unpopulated by existing LBA samples. They show a weak trend towards blue ultraviolet colours at low luminosities and low metallicities that mirrors those seen in the high redshift galaxy population.
Radio continuum investigations preclude the presence of a strong obscured AGN in any of a subsample of 13 sources. Eight sources are detected at a flux level consistent with that expected from their ultraviolet luminosity, assuming that both radio and ultraviolet flux arises from the same star forming population. The remaining sources are undetected in the radio. The relatively low fraction of sources with robust radio detections is intriguing, and, if supported by future observations and forthcoming analysis of their spectral energy distributions, may indicate that these sources are entirely dominated by a very young stellar population.
LBAs such as this may prove extremely useful in interpreting the limited data accessible through observations of faint and heavily redshifted galaxies in the distant Universe. Deep surveys such as CANDELS, and still more so the Hubble Ultra Deep Field campaign and Frontier Fields, are probing a luminosity range, colour selection and star formation density regime that isn’t well explored by existing LBA samples. While the number density of these very distant sources, and their photometric colours in the rest-frame ultraviolet, are straightforward to determine, very little detailed information can be extracted from their photometry, and spectroscopy in most cases is limited to Lyman-alpha line emission and weak detections of a low resolution spectral continuum. By contrast, LBAs can be investigated at a full range of wavelengths, from the ultraviolet through to centimetre wavelengths, and detailed abundances and dust properties extracted from optical through near-infrared photometry.
A logical next step in our investigations is to explore the integrated stellar populations in these galaxies - both through their optical spectroscopy (which is inaccessible in high redshift galaxies) and through analysis of their photometric spectral energy distribution (which is directly comparable to the most commonly-applied technique for analysis of high redshift samples). Forthcoming data from approved LABOCA observations and the VISTA public surveys in the near-infrared, forming a powerful combination with SDSS optical, GALEX ultraviolet photometry and WISE (3-22$\mu$m) imaging should allow tight constraints on dust content, stellar populations and mass to be obtained through fitting of the spectral energy distributions. We also plan to explore their radio, millimetre and far-infrared properties in more detail, now that radio detections have been secured on an initial subsample, and are pursuing an approved programme of integral field spectroscopy in the infrared using SINFONI on the ESO VLT on a subset of objects. This information, will allow us to determine the gas content, dust properties and physical conditions within these young, compact systems, putting them in the wider context of star forming galaxies at low redshifts, and comparing them to the models commonly applied for high redshift star formation.
Acknowledgments {#acknowledgments .unnumbered}
===============
LJMD acknowledges post-doctoral funding from the UK Science and Technology Facilities Council.
This paper is based in part on data obtained at the Australia Telescope Compact Array associated with programme C2695. The Australia Telescope Compact Array is part of the Australia Telescope National Facility which is funded by the Commonwealth of Australia for operation as a National Facility managed by CSIRO.
It is also based in part on public data from the Sloan Digital Sky Survey DR7. 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.
Also based in part on public data from GALEX GR6. The Galaxy Evolution Explorer (GALEX) satellite is a NASA mission led by the California Institute of Technology.
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\[lastpage\]
[^1]: E-mail: [email protected]
[^2]: While the Hoopes et al targets were primarily identified as ultraviolet-luminous galaxies (UVLGs), they were nonetheless motivated as an LBA sample.
[^3]: Available at http://galex.stsci.edu/GR6/
[^4]: http://gem.yonsei.ac.kr/$\sim$ksoh/wordpress/?page\_id=18
[^5]: 12-log(O/H)$_\odot=8.69$ [@2001ApJ...556L..63A]
[^6]: Using the same conversion factor applied in figure \[fig:lum-size\]
[^7]: http://wise2.ipac.caltech.edu/docs/release/allwise/
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{
"pile_set_name": "ArXiv"
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---
author:
- Michael Xuelin Huang
- Andreas Bulling
bibliography:
- 'references.bib'
title:
-
-
---
<ccs2012> <concept> <concept> <concept\_id>10003120.10003121</concept\_id> <concept\_desc>Human-centered computing Human computer interaction (HCI)</concept\_desc> <concept\_significance>500</concept\_significance> </concept> </ccs2012>
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"pile_set_name": "ArXiv"
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author:
- Mateusz Fedoryszak
- Brent Frederick
- Vijay Rajaram
- Changtao Zhong
bibliography:
- 'bib/kdd\_evd.bib'
title: 'Real-time Event Detection on Social Data Streams'
---
<ccs2012> <concept> <concept\_id>10002951.10003227.10003351.10003444</concept\_id> <concept\_desc>Information systems Clustering</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10002951.10003227.10003351.10003446</concept\_id> <concept\_desc>Information systems Data stream mining</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10002951.10003260.10003282.10003292</concept\_id> <concept\_desc>Information systems Social networks</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10002951.10003260.10003282.10003296</concept\_id> <concept\_desc>Information systems Crowdsourcing</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10010147.10010257.10010258.10010260.10003697</concept\_id> <concept\_desc>Computing methodologies Cluster analysis</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10010147.10010257.10010258.10010260.10010229</concept\_id> <concept\_desc>Computing methodologies Anomaly detection</concept\_desc> <concept\_significance>300</concept\_significance> </concept> </ccs2012>
The authors thank Guy Hugot-Derville for his guidance and his initial drive to research this idea. We also thank Alexandre Lung-Yut-Fong, Jeff de Jong, Samir Chainani, Lei Wang, David Blackman, Gilad Buchman, Prakash Rajagopal, Volodymyr Zhabiuk, Lu Gao, Dennis Zhao, Ashish Bansal, and Ajeet Grewal for their support.
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{
"pile_set_name": "ArXiv"
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abstract: 'Despite intensive research, the physical origin of the speed-up offered by quantum algorithms remains mysterious. No general physical quantity, like, for instance, entanglement, can be singled out as the essential useful resource. Here we report a close connection between the trace speed and the quantum speed-up in Grover’s search algorithm implemented with pure and pseudo-pure states. For a noiseless algorithm, we find a one-to-one correspondence between the quantum speed-up and the maximal trace speed that occurs during the algorithm operations. For time-dependent partial depolarization and for interrupted Grover searches, the speed-up can be bounded by the maximal trace speed. Our results quantify the quantum speed-up with a physical resource that is experimentally measurable and related to multipartite entanglement and quantum coherence.'
author:
- Valentin Gebhart
- Luca Pezzé
- Augusto Smerzi
bibliography:
- './trace\_paper.bib'
title: 'Quantifying Computational Advantage of Grover’s Algorithm with the Trace Speed'
---
Understanding and quantifying the key resource for the speed-up of quantum computations [@nielsen2002quantum; @kaye2007introduction] has been a highly disputed topic over the past few decades [@vedral2010elusive]. There has been particular interest in the role played by entanglement [@jozsa1997entanglement; @ekert1998; @Lloyd1999; @linden2001good; @jozsa2003role; @Vidal2003; @Biham2004; @kenigsberg2006quantum; @Horodecki2007; @VandenNest2013]. It is known that exponential speed-ups of quantum algorithms implemented with pure states require multipartite entanglement [@jozsa2003role; @Vidal2003]. However, it was shown that polynomial advantage in query complexity can be achieved without entanglement [@bernstein1997quantum]. Also, it is an open question whether exponential quantum advantage can be reached in mixed-state algorithms in absence of entanglement. Here, other quantum correlations such as quantum discord have been indicated as possible candidates for computational resources [@Datta2008; @vedral2010elusive]. Furthermore, it was shown that several entanglement measures do not recognize useful entanglement in quantum algorithms [@VandenNest2013]. Other possible resources have been considered such as coherence [@Hillery2016; @Ma2016; @Matera2016], distinguishability [@vedral2010elusive], contextuality [@Howard2014], tree size [@cai2015] and interference [@Stahlke2014].
Quantum statistical speeds [@Wootters1981; @Petz1996; @Spehner2014; @Braunstein1994; @Gessner2018] offer a possible approach to quantify useful resources in quantum technology tasks. As a major example, the quantum Fisher information [@helstrom1976quantum; @Braunstein1994], which is the quantum statistical speed associated with the Bures distance [@Braunstein1994], was shown to fully characterize metrologically useful entanglement [@pezze2009; @hyllus2012fisher; @toth2012], that is, the entanglement necessary for sub-shot-noise phase estimation sensitivities [@Pezze2016; @Toth2014]. One might conjecture that different statistical speeds may be useful to characterize the performances of different quantum tasks. Here, we use the trace speed ($\operatorname{TS}$), namely, the statistical speed associated to the trace distance [@nielsen2002quantum; @Petz1996], to quantify the speed-up in Grover’s algorithm [@grover1997quantum] in both absence and presence of dephasing. We show that in the cases without dephasing, the maximal $\operatorname{TS}$ occurring during the search algorithm completely determines the speed-up. For general pseudo-pure dephasing models [@braunstein2000speed], we prove that the $\operatorname{TS}$ bounds the speed-up, rendering it a necessary resource for quantum advantage. The $\operatorname{TS}$ is an experimentally relevant measure of quantum coherence (asymmetry) [@marvian2014extending; @Streltsov2017] and witnesses multipartite entanglement [@Gessner2018]. To our knowledge, this is the first result for a physical resource in Grover’s algorithm that generalizes to mixed state versions. This can pave the way to a new approach to investigate useful resources in quantum computations.
*Grover’s algorithm.* Grover’s search algorithm [@grover1997quantum] is one of the most important protocols of quantum computation [@nielsen2002quantum; @kaye2007introduction]. It searches an unstructured database of $N$ elements for a target $\omega$. The target is marked in the sense that we are given a test function $f$ that vanishes for all elements but $\omega$. The task is to identify $\omega$ with as few function calls as possible. In the quantum version of the algorithm, a function call can be used as a measurement or as an application of a corresponding unitary, the so-called oracle unitary. As we will discuss shortly, Grover’s algorithm admits a quadratic advantage to classical search algorithms. To utilize the exponential size of the dimensionality of composite quantum systems [@Lloyd1999], we encode all different elements $x$ of the register into computational basis states of $n=\log_2 N$ qubits, $x\in \left\{0,1\right\}^{ n}$. Grover’s algorithm is performed by preparing the system in the register state $\left| \psi_\mathrm{in} \right\rangle=1/\sqrt{2^n}\sum_x \left| x \right\rangle$, where $\left| x \right\rangle$ are the computational basis vectors, followed by $k$ applications of the Grover unitary $G=U_d U_\omega$. Here, the oracle unitary $U_\omega=1-2\left| \omega \right\rangle\left\langle \omega \right|$ represents a function call and the Grover diffusion operator is defined as $U_d=2\left| \psi_\mathrm{in} \right\rangle\left\langle \psi_\mathrm{in} \right|-1$. After $k$ iterations of the Grover unitary, the state of the system is given by [@kaye2007introduction] $$\left| \psi_k \right\rangle = \sin[(2k+1)\theta]\left| \omega \right\rangle+\cos[(2k+1)\theta]\left| \omega^\perp \right\rangle,\label{eq:pureevolution}$$ where $\theta=\arcsin(1/\sqrt{2^n})$ and $\left| \omega^\perp \right\rangle=1/\sqrt{2^n-1}\sum_{x\neq\omega}\left| x \right\rangle$ is the projection of the initial state on the subspace orthogonal to $\left| \omega \right\rangle$. This yields a probability $p_k$ of finding the target state as $p_k=\sin^2 [(2k+1)\theta]$. After $k_\mathrm{Gr}\approx (\pi/4) \sqrt{2^n}$ iterations one finds the target state $\left| \omega \right\rangle$ with probability $p_{k_\mathrm{Gr}}=1-\mathcal{O}(1/2^n)$ [@kaye2007introduction].
One defines the cost $C$ for a general search algorithm as the average number of applications of the test function $f$ (or its corresponding oracle unitary) required to find the target state [@braunstein2000speed]. Simply counting the oracle applications is also known as query complexity [@kaye2007introduction], while other complexities such as time complexity are usually not considered in Grover’s algorithm (see [@Lloyd1999] for a discussion).
In the classical search algorithm, the query application can be thought of as opening one of $2^n$ boxes, where each box represents one state of the register. For an unstructured search algorithm, i.e., in each iteration one randomly opens one of the $2^n$ boxes. The average number of steps needed to find the target state is given by $C_\mathrm{cl}=2^n$. If one remembers the outcome of all previous searches, the cost can be reduced to $C_\mathrm{cl}=2^{n}/2+\mathcal{O}(1)$ [@braunstein2000speed]. Note that $C_\mathrm{cl}$ for both structured and unstructured searches scales with $2^n$.
In a quantum search algorithm, one uses $k$ oracle unitaries and a final oracle measurement yielding the target with probability $p_k$, such that the cost is given by [@braunstein2000speed] $$C_\mathrm{qu}(k)=\dfrac{k+1}{p_k}. \label{eq:cost}$$ Hence, the optimal cost is obtained by minimizing $C_\mathrm{qu}(k)$ over the number of oracle applications, $C_\mathrm{qu}=\min_k(k+1)/(p_k)$. Let us emphasize that this definition of the cost does not distinguish between applying the oracle as a unitary or as a measurement observable.
In Grover’s algorithm, the cost function Eq. (\[eq:cost\]) is not necessarily minimal for the highest success probability $p_k$ of one single search [@zalka1999]. However, the optimal number of steps $\tilde{k}_\mathrm{Gr}$ and the optimal cost $C_\mathrm{qu}$ for large $n$ still scales as $\tilde{k}_\mathrm{Gr}= r \sqrt{2^n}$ and $C_\mathrm{qu}=K \sqrt{2^n}$, where $r$ is the solution of $\tan(2r)=4r$ and $K=r/\sin^2(2r)$, yielding the quadratic speed-up over $C_\mathrm{cl}$. It was shown that this speed-up is optimal [@bennett1997strengths; @zalka1999].
Grover’s algorithm can be executed on a single multimode system and, therefore, simply makes use of superposition and constructive interference [@Lloyd1999; @Caves2004; @bhattacharya2002implementation]. However, in order to reduce exponential overhead in space, time or energy, one usually considers a system composed of many qubits [@Lloyd1999; @Caves2004]. In this case, different measures of bipartite and multipartite entanglement have been used to detect entanglement during Grover’s algorithm [@bruss2011multipartite; @Meyer2002; @Fang2005; @Rungta2009; @Rossi2013]. Genuine multipartite entanglement was shown to be present already after the first step of the noiseless algorithm [@bruss2011multipartite]. However, the quantitative relationship between these measures and speed-up was not resolved. In particular, the methods could not be easily applied to any mixed state generalization of Grover’s algorithm. Quantum coherence [@Shi2017; @anand2016coherence; @Pan2019] and quantum discord [@Shi2017] have been considered as resources in the noiseless algorithm as well.
*Trace speed.* The $\operatorname{TS}$ is the susceptibility of a quantum state $\rho$ to unitary displacements generated by a generic Hamiltonian $H$ [@marvian2014extending]. That is, the $\operatorname{TS}$ quantifies the distinguishability between $\rho$ and $\rho(t)=e^{-iHt}\rho e^{iHt}$ for small $t$. It is defined as [@nielsen2002quantum; @marvian2014extending; @Gessner2018] $$\operatorname{TS}(\rho,H) = \left\lVert \partial_t \rho(t) \big|_{t=0} \right\rVert_1 = \left\lVert \left[\rho, H \right] \right\rVert_1 ,$$ where $[\cdot,\cdot]$ is the commutator and $\left\lVert \cdot \right\rVert_1$ is the $l_1$-norm, defined as $\left\lVert A \right\rVert_1 = \operatorname{tr}\left[ \sqrt{A^\dag A} \right]$ for a generic operator $A$.
In general, $\operatorname{TS}$ is a measure of coherence, in this case usually referred to as asymmetry [@marvian2014extending]: a state with no coherence with respect to $H$, namely a classical mixture of its eigenstates, will not change under phase displacements, while off-diagonal matrix elements (coherences) of $\rho$ are responsible for a finite susceptibility to phase displacements. The $\operatorname{TS}$ is upper bounded by the quantum Fisher information [@Petz1996]. If the system is a composite system of $n$ qubits and $H$ is the sum of local Hamiltonians $H_i$, $H=\sum_{i=1}^n H_i$ with $\operatorname{spec}(H_i)=\left\{-1/2,1/2\right\}$ and $\operatorname{TS}(\rho,H)>\sqrt{nr}$, it follows that $\rho$ has to be at least $(r+1)$-partite entangled [@hyllus2012fisher; @toth2012; @Gessner2018]. Since the value of $\operatorname{TS}$ depends on the generating Hamiltonian $H$, we consider the optimization over all Hamiltonians of the above form. When the whole evolution is restricted to the completely symmetric subspace, it suffices to perform this optimization over collective spin Hamiltonians, $H_i=\mathbf{n}\cdot\boldsymbol{\sigma}^{(i)}/2$, where $\mathbf{n}$ is a point on the unit sphere and $\boldsymbol{\sigma}^{(i)}$ are the Pauli operators for the $i$-th qubit. For pure states $\left| \psi \right\rangle$, the optimized $\operatorname{TS}$ coincides with the square root of the largest eigenvalue of the matrix $\Gamma_{ij}=\operatorname{Re}[\left\langle J_iJ_j\right\rangle]-\left\langle J_i\right\rangle\left\langle J_j\right\rangle$ [@hyllus2012fisher]. Here, $J_m=\sum_{i=1}^n\mathbf{e}_m\cdot\sigma^{(i)}/2$ is the coherent spin operator in $\mathbf{e}_m$-direction, $m=x,y,z$, and $\langle \cdot \rangle$ is the expectation value with respect to the state $\left| \psi \right\rangle$.
*Pure state algorithm.* Let us first discuss the $\operatorname{TS}$ for the standard version of Grover’s algorithm implemented with pure states and unitary evolution, as introduced above. Without loss of generality, we can take $\left| \omega \right\rangle=\left| 0 \right\rangle^{\otimes n}$ [^1]. Since $\left| \psi_\mathrm{in} \right\rangle$ and $\left| 0 \right\rangle^{\otimes n}$ are elements of the completely symmetric subspace and $G$ commutes with all permutations of the qubits, the complete evolution is restricted to the symmetric subspace. This reduces the dimensionality of the Hilbert space from $2^n$ to $n+1$, facilitating the computation of $\operatorname{TS}$. By neglecting terms in $\mathcal{O}(1/2^n)$, one can exactly compute the largest eigenvalue of $\Gamma_{ij}$ at any step $k$ [^2], yielding the optimized $\operatorname{TS}$. In Fig. (\[fig:QFIpure\]), we show the optimized $\operatorname{TS}(k)$ for $n=30$ qubits. The initially separable state $\left| \psi_\mathrm{in} \right\rangle$ evolves into a multipartite entangled state already after the first oracle operation. Multipartite entanglement further increases until reaching a maximal value of $$\operatorname{TS}^\mathrm{pure}_\mathrm{max}=\sqrt{\dfrac{n(n+1)}{2} }$$ which occurs at $k=k_\mathrm{Gr}/2$. This detects $(n/2+1)$-partite entanglement during the pure state Grover algorithm. For $k>k_\mathrm{Gr}/2$, multipartite entanglement detected by the $\operatorname{TS}$ decreases until the algorithnm reaches the separable target state $\left| \omega \right\rangle$.
![The dependence of the optimized trace speed $\operatorname{TS}$ on the iteration step $k$ in the pure state Grover’s algorithm (solid line). The dashed lines indicate thresholds above which $\operatorname{TS}$ detects bipartite ($\sqrt{n}$), three-partite ($\sqrt{2n}$) and ($n/2+1$)-partite ($\sqrt{n^2/2}$) entanglement. Here, $n=30$, $k_\mathrm{Gr}\approx(\pi/4)\sqrt{2^n}$.[]{data-label="fig:QFIpure"}](./Fig1.eps){width="\linewidth"}
*Mixed register state.* We now consider Grover’s algorithm with the register initialized in a pseudo-pure state, while the algorithm is still implemented with unitary operations. For a pure $n$-qubit state $\left| \psi \right\rangle$, the corresponding pseudo-pure state $\rho_{\psi,\epsilon}$ with purity parameter $\epsilon$ is defined as $$\rho_{\psi,\epsilon}=\epsilon\left| \psi \right\rangle \left\langle \psi \right| +\dfrac{1-\epsilon}{2^n}\mathbb{I}.\label{eq:pseudopure}$$ Pseudo-pure states represent, for small purities, an approximation to the thermal state of the system and therefore arise naturally in liquid-state NMR (see for instance Ref. [@havel1997ensemble]). We replace the pure initial state $\left| \psi_\mathrm{in} \right\rangle$ with the pseudo-pure state $\rho_{\psi_\mathrm{in},\epsilon}$ such that, after $k$ Grover iterations, the state of the system is given by $\rho_k=\epsilon\left| \psi_k \right\rangle \left\langle \psi_k \right|+(1-\epsilon)\mathbb{I}/2^n$, with $\left| \psi_k \right\rangle$ defined in Eq. (\[eq:pureevolution\]). The probability $p_k$ of finding the target state after $k$ steps is $p_k=\epsilon\sin^2 ((2k+1)\theta)+(1-\epsilon)/2^n$. Here we observe that for $\epsilon=\mathcal{O}(1/2^n)$, it becomes more efficient to just measure the state without any iteration because the probability contribution due to the Grover iteration is no longer dominant [@linden2001good]. However, if $\epsilon$ does not decrease exponentially with $n$, one can neglect the second term in $p_k$. Hence, the minimum of Eq. (\[eq:cost\]) occurs after the same number of steps as in the pure state algorithm while its minimal value $C_\mathrm{qu}$ is simply $C_\mathrm{qu}=C_\mathrm{qu,pure}/\epsilon$, where $C_\mathrm{qu,pure}$ is the cost of the pure state algorithm.
The $\operatorname{TS}$ for a pseudo-pure state $\rho_{\psi,\epsilon}$ is given by $\operatorname{TS}(\rho_{\psi,\epsilon},H)=\epsilon \operatorname{TS} (\left| \psi \right\rangle,H)$ because $\left[H,\operatorname{id}\right]=0$. Therefore, the maximal $\operatorname{TS}$ during the algorithm is $\operatorname{TS}_\mathrm{max}=\epsilon \sqrt{n(n+1)/2}$. Hence, for the Grover algorithm with a pseudo-pure initial state with purity parameter $\epsilon$, $\operatorname{TS}$ detects $\epsilon (n+1)/2$-partite entanglement. We combine the results for the cost function and the $\operatorname{TS}$ to obtain the direct dependence of the cost function on the maximal $\operatorname{TS}$ as $$C_\mathrm{qu}(n,\operatorname{TS}_\mathrm{max})=K\sqrt{2^n}\frac{\operatorname{TS}^\mathrm{pure}_\mathrm{max}}{\operatorname{TS}_\mathrm{max}},$$ where $K=r/\sin^2(2r)\approx 0.69$ with $r$ being the solution of $\tan(2r)=4r$. The quantum speed-up $S=C_\mathrm{cl}/C_\mathrm{qu}$ is thus given in terms of $\operatorname{TS}$ as $$S= \frac{\sqrt{2^n}}{2 K}\frac{\operatorname{TS}_\mathrm{max}}{\operatorname{TS}^\mathrm{pure}_\mathrm{max}}\label{eq:speeduppseudo}$$
Note that for $\epsilon<2/(n+1)$, $\operatorname{TS}$ does not detect entanglement anymore. It was already observed that for purities $\epsilon>1/2^{n/2}$ the algorithm still offers a speed-up [@Biham2002; @vedral2010elusive; @Kay2015], indicating that entanglement detected by $\operatorname{TS}$ is not necessary for quantum speed-up. We should emphasize that the form of Eq. (\[eq:speeduppseudo\]) suggests that similar results can possibly be found connecting the speed-up to other measures of coherence or other quantum properties. Importantly, the choice of the $\operatorname{TS}$ was suggested by the fact that its value for pseudo-pure states has a simple dependence on its pure state value and that it also detects multipartite entanglement for mixed states. We have also considered different quantum statistical speeds, other than the $\operatorname{TS}$, such as the generalized quantum Fisher information [@Gessner2018]. However, after including time-dependent depolarization, the $\operatorname{TS}$ proved to have the closest connection to the speed-up for pseudo-pure state algorithms.
*Partial depolarization.* The results of pseudo-pure initial states can be generalized to search dynamics subject to time-dependent partial depolarization (see Refs. [@cohn2016grover; @vrana2014fault] for earlier investigations). In this case, the state after $k$ steps of the algorithm is given by $$\rho_k=\epsilon(k)\left| \psi_k \right\rangle \left\langle \psi_k \right|+\dfrac{1-\epsilon(k)}{2^n}\mathbb{I},\label{eq:pseudopureevolution}$$ where the now time-dependent decreasing purity $\epsilon(k)$ represents both initial impurity and partial depolarization during the algorithm. As can be seen in Fig. (\[fig:QFIdeph\]), different purity functions $\epsilon(k)$ with the same final purity can lead to different maximal $\operatorname{TS}$ during the iteration. While the one-to-one correspondence between the $\operatorname{TS}$ and the speed-up is generally lost, as shown below, we can still bound the speed-up using $\operatorname{TS}$.
![Purities $\epsilon(k)$ (solid lines) and trace speeds $\operatorname{TS}(k)$ (dashed lines) for an initial pseudo-pure state without dephasing (blue), an initial pure state with linearly decaying purity (yellow) and an initial pure state with exponentially decaying purity (green). Here, $n=30$, $\epsilon_f=0.3$, $k_\mathrm{Gr}\approx(\pi/4)\sqrt{2^n}$.[]{data-label="fig:QFIdeph"}](./Fig2.eps){width="\linewidth"}
For a partial depolarization during the algorithm it turns out that, in general, it is optimal to stop the iterations and perform the final measurement already at earlier steps $k_\mathrm{int}<\tilde{k}_\mathrm{Gr}$ [@cohn2016grover]. Let us first consider interrupting the iteration at a time $k_\mathrm{int}\geq k_\mathrm{Gr}/2$, i.e., after the pure state algorithm would have already reached its maximal $\operatorname{TS}$, see Fig. (\[fig:QFIpure\]). In this case, the cost can be bounded by $C_\mathrm{qu}\geq K \sqrt{2^n}/\epsilon(k_\mathrm{int})$. This is because if one would completely stop the dephasing, one could reduce the cost until reaching the optimal value of $K \sqrt{2^n}/\epsilon(k_\mathrm{int})$. With $\epsilon(k_\mathrm{int})\leq \epsilon(k_\mathrm{Gr}/2)$ and $\operatorname{TS}(k_\mathrm{Gr}/2)\leq \operatorname{TS}_\mathrm{max}$, one can then bound $C_\mathrm{qu}\geq K \sqrt{2^n}/\epsilon(k_\mathrm{Gr}/2) \geq (K \sqrt{2^n} \operatorname{TS}_\mathrm{max}^\mathrm{pure})/(\operatorname{TS}_\mathrm{max})$, yielding the following bound $$S \leq \frac{\sqrt{2^n}}{2 K}\frac{\operatorname{TS}_\mathrm{max}}{\operatorname{TS}^\mathrm{pure}_\mathrm{max}}.\label{eq:finalbound}$$
The case $\epsilon(k_\mathrm{int})\leq \epsilon(k_\mathrm{Gr}/2)$ corresponding to strong dephasing becomes more technical since, in the early regime, the maximal $\operatorname{TS}$ is not simply bounded by $\epsilon(k)\operatorname{TS}^\mathrm{pure}_\mathrm{max}$. However, as we show in the appendix, the bound Eq. (\[eq:finalbound\]) still holds. These results for the case of an interruption of the iteration due to minimization of the cost can also be applied to the case of a general interruption of the iteration. Stopping the algorithm at any time will yield an average speed-up which is always bounded by the maximal $\operatorname{TS}$ occurring before the interruption.
*Discussion.* By studying the $\operatorname{TS}$, we have been able to relate the computational speed of Grover’s algorithm to both multipartite entanglement and quantum coherence. It should be noticed that the relation with multipartite entanglement depends on the $n$-qubit implementation that we have considered, while the algorithm can also be implemented with a single $2^n$-level system [@Lloyd1999]. Indeed, as mentioned above, the operating principle of the algorithm and the number of queries used (which determines the cost) do not depend on which implementetion we use. Therefore, multipartite entantanglement cannot be considered as the key resource for the quantum speed-up. We thus argue that the correct interpretation of our result is the evidence that the resource for speed up in query complexity is quantum coherence as captured by the $\operatorname{TS}$. However, multipartite entanglement is crucial to reduce other costs such as space or energy [@Lloyd1999]. We point out that the interpretation of the $\operatorname{TS}$ as quantum coherence holds for any implementation of the algorithm.
The role of quantum coherence during the noiseless Grover’s algorithm has already been investigated in Refs. [@Shi2017; @anand2016coherence]. These works found a one-to-one correspondence between the $l_1$-norm of coherence which is decreasing during the algorithm and the increasing success probability. Both approaches have not been generalized to mixed state versions of the algorithm. In our case, a different measure of coherence, namely the $\operatorname{TS}$, is connected to the average cost of the algorithm. It reaches its maximal value during the algorithm and offers a physical resource also for pseudo-pure generalizations. In Refs. [@Shi2017; @anand2016coherence], the $l_1$-norm of coherence and the relative entropy of coherence are used which detect different states as highly coherent as $\operatorname{TS}$ would. For instance, while the $l_1$-norm detects the initial state $\left| \psi_\mathrm{in} \right\rangle=1/\sqrt{2^n}\sum_x \left| x \right\rangle$ as maximally coherent, $\operatorname{TS}$ would detect $(\left| 0 \right\rangle^{\otimes n}+\left| 1 \right\rangle^{\otimes n})/\sqrt{2}$ as maximally coherent. For a discussion of these so-called speakable and unspeakable coherence, see for instance Ref. [@Marvian2016].
Finally, we emphasize that the $\operatorname{TS}$ can be measured or efficiently bounded experimentally. Following Refs. [@Strobel2014; @Pezze2016a], one measures the Kolmogorov distance between the probability distribution of $\rho(0)$ and $\rho(t)$, for a given measurement observable. A quadratic series expansion of the Kolmogorov distance for sufficiently small $t$ yields the Kolmogorov speed which is a lower bound to the TS and depends on the considered measurement observable. The TS is obtained by maximizing the Kolomogorov speed over all possible observables [@nielsen2002quantum].
*Conclusions.* We showed that both in the pure state version of Grover’s search algorithm and a general pseudo-pure generalization, the trace speed $\operatorname{TS}$ can be used to quantify and bound the possible quantum speed-up. These results offer an unprecedented connection between the speed-up in Grover’s algorithm and a physical resource beyond the case of ideal, noiseless quantum algorithms. The $\operatorname{TS}$ offers a new and experimentally feasible approach to the analysis of quantum advantages. This might inspire further investigations of the still unanswered search for the origins and quantification of quantum advantage. In particular, one could check the importance of the $\operatorname{TS}$ and other quantum statistical speeds for other oracle-based quantum algorithms such as, e.g., the Deutsch-Jozsa algorithm or Simon’s algorithm, or general quantum technology tasks. Also, whether or not the $\operatorname{TS}$ is a necessary resource in different noisy variations of Grover’s algorithm, merits further investigation. More general dephasing models or unitary noise could be considered that render the analysis more cumbersome. Overall, our results suggest that quantum statistical speeds can be used to recognize useful properties of quantum states for different quantum technology tasks.
Appendix {#appendix .unnumbered}
========
Proof of the bound for strong dephasing {#proof-of-the-bound-for-strong-dephasing .unnumbered}
---------------------------------------
Let us consider the case of interrupting the search algorithm at an early step $k\leq k_\mathrm{Gr}/2$, i.e., at a step where the pure state algorithm would not have reached its maximal trace speed $\operatorname{TS}$ yet. The cost for stopping the algorithm at step $k$ is given by $C_\mathrm{qu}(k)=(k+1)/(\epsilon(k)\sin^2((2k+1)\theta))$, see Eq. (\[eq:pureevolution\]). The $\operatorname{TS}$ at step $k$ still fulfills $\operatorname{TS}(k)=\epsilon(k)\operatorname{TS}_\mathrm{pure}(k)$, where $\operatorname{TS}_\mathrm{pure}(k)$ is the $\operatorname{TS}$ in the pure state algorithm, see Fig. (\[fig:QFIpure\]). Therefore, we have $$\begin{aligned}
C_\mathrm{qu}(k)&=\frac{k+1}{\sin^2 [(2k+1)\theta]}\frac{\operatorname{TS}_\mathrm{pure}(k)}{\operatorname{TS}(k)}\notag \\
&\geq \frac{k+1}{\sin^2 [(2k+1)\theta]}\frac{\operatorname{TS}_\mathrm{pure}(k)}{\operatorname{TS}_\mathrm{max}} \notag \\
&=\frac{a(k)}{\operatorname{TS}_\mathrm{max}}\end{aligned}$$ where $\operatorname{TS}_\mathrm{max}$ is the maximal $\operatorname{TS}$ until the interruption step $k$ and we regrouped all other factors into $a(k)$. To further examine this expression, we use the exact form of the pure state $\operatorname{TS}$, $\operatorname{TS}_\mathrm{pure}(k)$, during the algorithm which is given by $$\begin{gathered}
\operatorname{TS}_\mathrm{pure}(k) \qquad\qquad\qquad\qquad\qquad\qquad \qquad\qquad\qquad\qquad\qquad \notag \\=\sqrt{\frac{1}{8}n\left(4+n-f(k)n+\sqrt{8[1+f(k)]+n^2[1-f(k)]^2}\right)}\end{gathered}$$ with $f(k)=\cos[4(2k+1)/\theta]$. We can then compare the factor $a(k)$ with the factor $b=K\sqrt{2^n}\sqrt{n(n+1)/2}$ from the current bound, Eq. (\[eq:finalbound\]). By writing $x=k/\sqrt{2^n}$, one finds for large $n$ $$a(k)- b = \frac{n\sqrt{2^n}}{\sqrt{8}\sin^2(2x)}\left[K(\cos(4x)-1)+2x\sin4x\right].$$ For $0\leq x\leq \pi/8$ ($0\leq k\leq \pi/8 \sqrt{2^n} = k_\mathrm{Gr}/2$) and using $K\approx 0.69$, one finds that $a(k)-b>0$. Therefore, the bound of Eq. (\[eq:finalbound\]) still holds for the regime $k\leq k_\mathrm{Gr}/2$.
[^1]: We can transform any target state $\left| \omega \right\protect\rangle$ into $\left| 0 \right\protect\rangle^{\otimes n}$ by applying local $\sigma_x$ unitaries. These local spin flips commute with $U_d$ and do not alter $\left| \psi_\mathrm{in} \right\protect\rangle$. Also, notice that local unitaries do not change the optimized $\operatorname{TS}$ of a state.
[^2]: After a lengthy but straight-forward calculation, one obtains $\operatorname{TS}$ at step $k$ as $\operatorname{TS}_\mathrm{pure}(k)=\sqrt{n\left(4+n-f(k)n+\sqrt{8[1+f(k)]+n^2[1-f(k)]^2}\right)/8}$ with $f(k)=\cos[4(2k+1)/\theta]$
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|
---
abstract: 'Recently, it was shown that differential rotation is an unavoidable feature of nonlinear *r*-modes. We investigate the influence of this differential rotation on the detectability of gravitational waves emitted by a newly born, hot, rapidly-rotating neutron star, as it spins down due to the *r*-mode instability. We conclude that gravitational radiation may be detected by the advanced laser interferometer detector LIGO if the amount of differential rotation at the time the *r*-mode instability becomes active is not very high.'
author:
- 'Paulo M. Sá'
- Brigitte Tomé
date: 'May 31, 2006'
title: |
The influence of differential rotation on the detectability of gravitational waves\
from the *r*-mode instability
---
\[sect-introduction\]Introduction
=================================
The new kilometer-scale laser interferometer gravitational-wave detectors LIGO and Virgo, or their advanced versions, operating at a broad frequency band (between about 10 and $10^4$ Hz), may detect gravitational waves from a variety of sources (for a recent review see Ref. [@KS2006]). One such source could be a spinning neutron star emitting gravitational radiation due to the *r*-mode instability.
First studied more than twenty years ago [@PP1978], *r*-modes are non-radial oscillation modes of rotating stars that have the Coriolis force as their restoring force and a characteristic frequency comparable to the angular velocity of the star. These modes have been attracting increasing attention since it was discovered that they are driven unstable by gravitational radiation emission [@A1998; @FM1998] and that, for a large range of relevant temperatures and angular velocities of neutron stars, the driving effect of gravitational radiation is stronger than the damping effect of viscosity [@LOM1998; @AKS1999].
Using a phenomenological model for the evolution of the *r*-mode instability, Owen *et al.* [@OLCSVA1998] studied the detectability of gravitational waves emitted by newly born, hot, rapidly-rotating neutron stars, arriving at the conclusion that such waves could be detected by the enhanced version of LIGO if sources were located at distances up to 20 Mpc from Earth. However, a deeper understanding of this issue requires taking into account nonlinear effects in the evolution of the *r*-mode instability. Arras *et al.* [@AFMSTW2003] considered the nonlinear interaction between modes, arriving at the conclusion that enhanced laser interferometer detectors could see gravitational radiation from *r*-modes if the sources were much closer to Earth, namely, at distances smaller than about 200 kpc. In this paper we will investigate the issue of detectability of gravitational radiation emitted by newly born neutron stars as they spin down due to the *r*-mode instability, but taking into account another nonlinear effect: differential rotation.
Rezzolla *et al.* [@RLS2000] were the first to suggest that *r*-modes induce a drift of fluid elements along azimuthal directions and derived an approximate analytical expression for these drifts. Soon afterwards, numerical studies, both in general relativistic [@SF2001] and Newtonian hydrodynamics [@LTV2001], confirmed the existence of such drifts. Recently, an exact *r*-mode solution, representing differential rotation of kinematic nature that produce large scale drifts of fluid elements along stellar latitudes, was found within the nonlinear theory up to second order in the mode’s amplitude in the case of a Newtonian, barotropic, nonmagnetized, perfect-fluid star [@Sa2004]. This differential rotation, which is an unavoidable feature of nonlinear *r*-modes, contributes to the physical angular momentum of the mode [@Sa2004] and, therefore, plays an important role in the nonlinear evolution of the *r*-mode instability [@ST2005]. In particular, the amplitude of the *r*-mode saturates in a natural way at a value that depends on the amount of differential rotation at the time the instability becomes active; this saturation value can be much smaller than unity [@ST2005].
In this paper, using the model of evolution of Ref. [@ST2005], we investigate how differential rotation induced by *r*-modes influences the detectability, by the laser interferometer detectors LIGO and Virgo, of gravitational waves emitted by a newly born, hot, rapidly-rotating neutron star. In Sect. \[sect-role-dif-rot\] we briefly review the evolution model of the *r*-mode instability. The main results of the paper are presented in Sect. \[sect-detectability\], where we derive expressions for the maximum value of the gravitational wave amplitude and for the frequency-domain gravitational waveform. We also compare, for different values of differential rotation, the characteristic amplitude of the signal with the rms strain noise in the initial LIGO, Virgo and advanced LIGO gravitational-wave detectors and compute the signal-to-noise ratio. Finally, Sect. \[sect-conclusions\] is devoted to a discussion of the results and conclusions.
\[sect-role-dif-rot\] Nonlinear evolution of the *r*-mode instability
=====================================================================
In a recent paper [@ST2005], we have investigated the role of differential rotation in the evolution of the $l=2$ *r*-mode instability in a newly born, hot, rapidly-rotating neutron star, using a simple phenomenological model adapted from the one proposed in Ref. [@OLCSVA1998]. The main difference between our modified model and the one proposed in Ref. [@OLCSVA1998] is that the former takes into account that the full physical angular momentum of the *r*-mode perturbation also includes a contribution from differential rotation. Namely, at second order in the mode’s amplitude $\alpha$, the physical angular momentum was taken to be [@Sa2004] $$\begin{aligned}
\delta^{(2)}\! J = \frac12 \alpha^2 \Omega (4K+5) \tilde{J} M R^2,
\label{phys-ang-mom}\end{aligned}$$ where $\Omega$, $R$ and $M$ denote the angular velocity, the radius and the mass of the star, respectively, and $\tilde{J}\equiv\int_0^R \rho r^6 dr/(MR^4)=1.635\times 10^{-2}$. In the above expression, it was assumed that the star’s mass density $\rho$ and pressure $p$ are related by a polytropic equation of state $p=k\rho^2$ with $k$ such that $M=1.4 M_{\odot}$ and $R=12.53 \mbox{ km}$. In Eq. (\[phys-ang-mom\]), $K$ is a constant fixed by initial data, giving the initial amount of differential rotation associated with the *r*-mode.
Within our model (see Ref. [@ST2005] for details), the total angular momentum is given by the sum of the angular momentum of the unperturbed star and the angular momentum of the *r*-mode perturbation, $$\begin{aligned}
J=I\Omega+\delta^{(2)} \! J(\Omega,\alpha),
\label{total-ang-mom}\end{aligned}$$ where $I=(8\pi/3)\int_0^R\rho r^4 dr=\tilde{I}MR^2$ ($\tilde{I}=0.261$) is the momentum of inertia of the unperturbed star. Assuming that the total angular momentum of the star decreases due to the emission of gravitational radiation and that the angular momentum of the perturbation increases due to the emission of gravitational radiation and decreases due to the dissipative effect of viscosity, we arrive at a system of differential equations determining the evolution of the star’s angular velocity $\Omega(t)$ and the *r*-mode’s amplitude $\alpha(t)$, namely, $$\begin{aligned}
\frac{d\Omega}{dt} &=& \frac83 (K+2)Q \alpha^2
\frac{\Omega}{\tau_{GR}} + \frac83 \left( K+\frac54 \right)Q
\alpha^2 \frac{\Omega}{\tau_{V}}, \label{eq-dif-omega(t)-c-visc}
\\
\frac{d\alpha}{dt} &=& -\left[ 1 + \frac43 (K+2)Q \alpha^2 \right]
\frac{\alpha}{\tau_{GR}} \nonumber \\
& & - \left[ 1 + \frac43 \left( K+\frac54 \right)Q \alpha^2
\right] \frac{\alpha}{\tau_{V}}, \label{eq-dif-alpha(t)-c-visc}\end{aligned}$$ where $Q\equiv3\tilde{J}/(2\tilde{I})=0.094$ and the value of $K$ is chosen to lie in the interval $-5/4\leqslant K\ll 10^{13}$. The upper limit for $K$ results from the fact that one wishes to impose the condition that the initial value of the angular momentum of the *r*-mode is much smaller than the angular momentum of the unperturbed star, i.e., $\delta^{(2)}\!J_0 \ll
I\Omega_0$ implies that $K\ll 10^{13}$ if we choose $\alpha_0=10^{-6}$. The lower limit for $K$ results from the fact that we do not want to saturate the amplitude of the mode by hand, a procedure needed for the case $K<-5/4$ in order to avoid that the total angular momentum of the star becomes negative (see Ref. [@ST2005] for details).
In Eqs. (\[eq-dif-omega(t)-c-visc\]) and (\[eq-dif-alpha(t)-c-visc\]) the gravitational-radiation and viscous timescales[^1] are given, respectively, by [@LOM1998] $$\begin{aligned}
\frac{1}{\tau_{GR}} &=& \frac{1}{\tilde{\tau}_{GR}} \left(
\frac{\Omega}{ \sqrt{\pi G \bar{\rho}} } \right)^6,
\label{gr-timescale}
\\
\frac{1}{\tau_V} &=& \frac{1}{\tilde{\tau}_S} \left(
\frac{10^9\mbox{K}}{T} \right)^2 + \frac{1}{\tilde{\tau}_B} \left(
\frac{T}{10^9\mbox{K}} \right)^6 \left( \frac{\Omega}{\sqrt{\pi G
\bar{\rho}}} \right)^2 \! \!, \label{visc-timescale}\end{aligned}$$ with the fiducial timescales $\tilde{\tau}_{GR}=-3.26 \mbox{ s}$, $\tilde{\tau}_S = 2.52\times10^8 \mbox{ s}$ [@LOM1998] and $\tilde{\tau}_B=2.01\times10^{11} \mbox{ s}$ [@LMO1999]. The temperature of the star is assumed to decrease due to the emission of neutrinos via a modified URCA process, $$\begin{aligned}
\frac{T(t)}{10^9\mbox{ K}}=\left[ \frac{t}{\tau_c} +\left(
\frac{10^9\mbox{ K}}{T_0} \right)^6 \right]^{-1/6}, \label{T(t)}\end{aligned}$$ where $T_0$ is the initial temperature of the star and $\tau_c=1\mbox{ yr}$ characterizes the cooling rate [@OLCSVA1998].
The viscous timescale given above was derived for a simple model of a neutron star with shear and bulk viscosity. The consideration of other types of viscosity (as, for instance, viscosity at the core-crust boundary in neutron stars with a crust [@BU2000] or hyperon bulk viscosity [@LO2002]) leads to expressions for the viscous timescale that differ substantially from Eq. (\[visc-timescale\]). However, because of the complexity of the involved processes it is not clear yet which viscous timescale is more appropriate to describe a real neutron star. In this paper we choose to restrict ourselves to the viscous timescale given by Eq. (\[visc-timescale\]).
Let us assume that the temperature, the mode’s amplitude and the star’s angular velocity take the initial values $T_0=10^{11}\mbox{
K}$, $\alpha_0=10^{-6}$ and $\Omega_0=\Omega_K$, where $\Omega_K=(2/3)\sqrt{\pi G\bar{\rho}}=5612 \mbox{ s}^{-1}$ is the Keplerian angular velocity at which the star starts shedding mass at the equator (these initial values will be used throughout the paper). Then, in the first moments of the evolution, bulk viscosity dominates the dynamics of Eqs. (\[eq-dif-omega(t)-c-visc\]) and (\[eq-dif-alpha(t)-c-visc\]). But this lasts just a fraction of a second. Indeed, the temperature of the star falls so rapidly that the gravitational-radiation driving effect almost immediately dominates over the bulk viscosity damping effect. This preponderance of gravitational radiation continues during most of the evolution, ending between $t_b=3.6\times 10^6 \mbox{ s}$ (for $K=-5/4$) and $t_b=7.1\times 10^6 \mbox{ s}$ (for $K\gg 1$) . Afterwards, shear viscosity dominates the dynamics of the evolution. At $t=1~\mbox{ yr}$, which corresponds to a temperature $T=10^9 \mbox{ K}$, we stop evaluating Eqs. (\[eq-dif-omega(t)-c-visc\]) and (\[eq-dif-alpha(t)-c-visc\]), since at this temperature superfluid effects are expected to become important, rendering invalid our assumptions about the viscous timescale [@LM1995].
In Fig. \[fig:omega-temp\] the numerical solution of the system of equations (\[eq-dif-omega(t)-c-visc\]) and (\[eq-dif-alpha(t)-c-visc\]), for different values of the constant $K$, is represented in a $(\Omega,T)$ diagram.
![\[fig:omega-temp\] Angular velocity of the star as a function of the temperature. As the star cools and emits gravitational radiation, its angular velocity decreases to values which are quite insensitive to the value of $K$. The dotted line represents the stability curve, $\tau_{GR}^{-1}(\Omega)+
\tau_V^{-1}(\Omega,T) = 0$, i.e., the set of points for which the damping effect of viscosity balances exactly the driving effect of gravitational radiation.](omegat.eps){width="8.6"}
As we can see there, for a newly born, hot, rapidly-rotating neutron star, there is an interval of relevant temperatures and angular velocities of the star for which the *r*-mode instability is active. For such values of temperature and angular velocity (corresponding to $0 \lesssim t < t_b$), the influence of viscosity on the evolutionary equations (\[eq-dif-omega(t)-c-visc\]) and (\[eq-dif-alpha(t)-c-visc\]) is so small, as compared with the influence of gravitational radiation, that $\Omega(t)$ and $\alpha(t)$ can be determined, in a very good approximation, by the system of equations: $$\begin{aligned}
\frac{d\Omega}{dt} &=& \frac83 (K+2)Q \alpha^2
\frac{\Omega}{\tau_{GR}}, \label{eq-dif-omega(t)-s-visc}
\\
\frac{d\alpha}{dt} &=& -\left[ 1 + \frac43 (K+2)Q \alpha^2 \right]
\frac{\alpha}{\tau_{GR}}. \label{eq-dif-alpha(t)-s-visc}\end{aligned}$$ Therefore, in what follows, Eqs. (\[eq-dif-omega(t)-s-visc\]) and (\[eq-dif-alpha(t)-s-visc\]), which can be solved analytically, are used to determine $\Omega(t)$ and $\alpha(t)$, while Eqs. (\[eq-dif-omega(t)-c-visc\]) and (\[eq-dif-alpha(t)-c-visc\]), which we solve numerically, are used just to determine $t_b$, i.e., the time at which the damping effect of viscosity becomes equal to the driving effect of gravitational radiation. As mentioned above, the value of $t_b$, which depends on $K$, lies in the range $3.6\times 10^6 \mbox{
s}\leqslant t_b \leqslant 7.1\times 10^6 \mbox{ s}$.
As shown in Ref. [@ST2005], Eqs. (\[eq-dif-omega(t)-s-visc\]) and (\[eq-dif-alpha(t)-s-visc\]) yield the following solution: $$\begin{aligned}
& & \hspace{-5mm} -\frac{2}{\tilde{\tau}_{GR}}\left(
\frac{\Omega_0}{\sqrt{\pi G\bar{\rho}}}
\right)^6 \left[ 1+\frac43(K+2)Q\alpha_0^2 \right]^6 t \nonumber \\
& & = \ln \frac{\Omega_0}{\Omega} + \ln
\frac{1+\frac43(K+2)Q\alpha_0^2 -
\frac{\Omega}{\Omega_0}}{\frac43(K+2)Q\alpha_0^2} \nonumber \\
& & \quad +\; \sum_{n=1}^5
\frac{\left[1+\frac43(K+2)Q\alpha_0^2\right]^n}{n} \left[ \left(
\frac{\Omega_0}{\Omega} \right)^n-1 \right],
\label{omega(t)-s-visc}
\\
& & \hspace{-5mm} -\frac{1}{\tilde{\tau}_{GR}}\left(
\frac{\Omega_0}{\sqrt{\pi G\bar{\rho}}} \right)^6 \left[
1+\frac43(K+2)Q\alpha_0^2 \right]^6 t \nonumber
\\
& & =\ln \frac{\alpha}{\alpha_0} + \sum_{n=1}^5
\frac{5!}{2n(5-n)!n!}\left[\frac43(K+2)Q\right]^n \nonumber
\\
& & \quad \times \left( \alpha^{2n}- \alpha_0^{2n} \right).
\label{alfa(t)-s-visc}\end{aligned}$$
An analysis of the above solution makes it clear that the evolution of the *r*-mode instability proceeds in two stages.
In the first stage of the evolution, the above solution is well approximated by $$\begin{aligned}
\frac{\Omega(t)}{\Omega_0} &\approx& 1- \frac43(K+2)Q\alpha_0^2
\exp \left\{ \left( \frac{\Omega_0}{\sqrt{\pi G\bar{\rho}}}
\right)^6
\frac{2t}{|\tilde{\tau}_{GR}|} \right\}, \label{omega-1-fase} \nonumber \\
\\
\alpha(t) &\approx& \alpha_0 \exp \left\{ \left(
\frac{\Omega_0}{\sqrt{\pi G\bar{\rho}}} \right)^6
\frac{t}{|\tilde{\tau}_{GR}|} \right\}, \label{alpha-1-fase}\end{aligned}$$ i.e., both the angular velocity of the star and the amplitude of the mode evolve on the gravitational-radiation timescale.
In the second stage of evolution, the solution (\[omega(t)-s-visc\]) and (\[alfa(t)-s-visc\]) is well approximated by $$\begin{aligned}
\hspace{-4mm} \frac{\Omega(t)}{\Omega_0} &\approx& 0.63 \left(
\frac{\Omega_0}{\sqrt{\pi G\bar{\rho}}} \right)^{-6/5} \left(
\frac{t}{|\tilde{\tau}_{GR}|} \right)^{-1/5}, \label{omega-2-fase}
\\
\hspace{-4mm} \alpha(t) &\approx& \frac{3.56}{\sqrt{K+2}} \left(
\frac{\Omega_0}{\sqrt{\pi G\bar{\rho}}} \right)^{3/5} \left(
\frac{t}{|\tilde{\tau}_{GR}|} \right)^{1/10}, \label{alpha-2-fase}\end{aligned}$$ i.e., due to nonlinear effects (the presence of differential rotation), the amplitude of the mode saturates at a value that depends crucially on the parameter $K$, namely, $\alpha_{sat}\propto (K+2)^{-1/2}$, and the angular velocity decreases to a final value which is quite insensitive to the value of $K$ (it depends on $K$ through $t_b$).
The smooth transition between the two stages of evolution occurs a few hundred seconds after the instability becomes active. The moment of occurrence of this transition can be defined more precisely by introducing the transition time $t_a$, corresponding to the moment, determined by the condition $d^2\alpha/dt^2(t_a)=0$, when the mode’s amplitude changes from an exponential to a much slower power-law growth. Using Eqs. (\[gr-timescale\]), (\[eq-dif-omega(t)-s-visc\]) and (\[eq-dif-alpha(t)-s-visc\]), the above condition yields $\alpha(t_a)=[12(K+2)Q]^{-1/2}$ or, using the relation $\Omega
\approx \Omega_0 [1+\frac43(K+2)Q\alpha^2]^{-1}$, $\Omega(t_a)=0.9\Omega_0$. Inserting $\alpha(t_a)$ into Eq. (\[alfa(t)-s-visc\]), or $\Omega(t_a)$ into Eq. (\[omega(t)-s-visc\]), and taking into account that $4(K+2)Q\alpha_0^2/3 \ll 1$ and $\alpha_0\ll\alpha(t_a)$, one obtains $$\begin{aligned}
t_a\approx [521.0-18.5\ln(K+2)]\mbox{ s}.\end{aligned}$$
\[sect-detectability\] Detectability of gravitational waves from the *r*-mode instability
=========================================================================================
The gravitational radiation emitted by a newly born neutron star, as it spins down due to the *r*-mode instability, could in principle be detected by the laser interferometer detectors LIGO and Virgo using appropriate detecting strategies.
The gravitational wave amplitude, averaged over source and detector orientation, is given by [@OLCSVA1998] $$\begin{aligned}
|h(t)|=1.3\times 10^{-24} \alpha(t) \left(
\frac{\Omega(t)}{\Omega_K} \right)^3 \left( \frac{20\mbox{
Mpc}}{D} \right), \label{h(t)}\end{aligned}$$ where $D$ is the distance to the source.
In Fig. \[fig:h-time\] the gravitational wave amplitude is shown for different values of $K$ and a distance to the source of $D=20\mbox{ Mpc}$. During the first stage of evolution, $|h(t)|$ grows exponentially \[see Eqs. (\[omega-1-fase\]) and (\[alpha-1-fase\])\], while during the second stage of evolution it decreases slowly, as $t^{-1/2}$ \[see Eqs. (\[omega-2-fase\]) and (\[alpha-2-fase\])\]. The maximum value $h_{max}\equiv|h(t_*)|$ is achieved when $d|h|/dt(t_*)=0$, which, using Eqs. (\[eq-dif-omega(t)-s-visc\]) and (\[eq-dif-alpha(t)-s-visc\]), gives the mode’s amplitude $\alpha(t_*)=\sqrt{3/[20(K+2)Q]}$, or, using the relation $\Omega
\approx \Omega_0 [1+\frac43(K+2)Q\alpha^2]^{-1}$, the star’s angular velocity $\Omega(t_*)=(5/6)\Omega_K$. Substituting these values of $\alpha$ and $\Omega$ into Eq. (\[h(t)\]), we obtain $$\begin{aligned}
h_{max}=\frac{9.5\times 10^{-25}}{\sqrt{K+2}} \left(
\frac{20\mbox{ Mpc}}{D} \right). \label{h-max}\end{aligned}$$ If $\alpha(t_*)$ is inserted into Eq. (\[alfa(t)-s-visc\]), or $\Omega(t_*)$ into Eq. (\[omega(t)-s-visc\]),we obtain that the gravitational wave amplitude reaches its maximum value at $t_*\approx[543.1-18.5\ln(K+2)]\mbox{ s}$, i.e., shortly after the transition between the first and second stages of evolution of the *r*-mode instability.
![\[fig:h-time\] Gravitational wave amplitude $|h(t)|$ for different values of $K$ and distance to the source $D=20\mbox{
Mpc}$. At $t_*\approx[543.1-18.5\ln(K+2)]\mbox{ s}$ the gravitational wave amplitude reaches its maximum value, $h_{max}=9.5\times 10^{-25}(K+2)^{-1/2}$.](hgrav.eps){width="8.6"}
Let us now derive an analytical expression for the frequency-domain gravitational wave amplitude $|\tilde{h}(f)|$. Using the stationary phase approximation, namely, $|\tilde{h}(f)|=|h(t)|/\sqrt{|df/dt|}$, where $df/dt$ is straightforwardly obtained from Eq. (\[eq-dif-omega(t)-s-visc\]), $$\begin{aligned}
\frac{df}{dt}&=& -8.1(K+2)\alpha^2(t) \left( \frac{f}{f_{max}}
\right)^7 \mbox{Hz}/\mbox{s}, \label{dfdt}\end{aligned}$$ we can easily obtain the frequency-domain gravitational wave amplitude $|\tilde{h}(f)|$, $$\begin{aligned}
|\tilde{h}(f)| = \frac{4.6\times 10^{-25}}{\sqrt{K+2}}
\sqrt{\frac{f_{max}}{f}} \left( \frac{20 \mbox{ Mpc}}{D} \right)
\mbox{ Hz}^{-1},
\label{htilde}\end{aligned}$$ where $f=2\Omega/(3\pi)$ is the frequency of the emitted gravitational wave.
Equation (\[htilde\]) applies to both stages of evolution, for frequencies ranging from $f_{min}=(77-80) \mbox{ Hz}$ to $f_{max}=1191\mbox{ Hz}$, where the maximum frequency corresponds to the initial value of the angular velocity of the star, $\Omega_K=5612\mbox{ s}^{-1}$, and the minimum frequency corresponds to the final angular velocity of the star, $\Omega(t_b)$, which lies between $0.065 \Omega_K$ (for $K=-5/4$) and $0.067 \Omega_K$ (for $K\gg 1$).
Let us mention that in Ref. [@OLCSVA1998] an expression for the frequency-domain gravitational wave amplitude $|\tilde{h}(f)|$ was derived based on the assumption that $dJ/df\propto I$, where $J$ is the total angular momentum of the star, $f$ is the frequency of the emitted gravitational wave and $I$ is the moment of inertia of the unperturbed star. Since within the model of Ref. [@OLCSVA1998], the condition $dJ/df\propto I$ only applies during the second stage of evolution, the expression obtained there for $|\tilde{h}(f)|$ is valid just for this stage of evolution. Within our model, however, it can be easily shown that the condition $dJ/df\propto I$ holds, not only during the second stage of evolution, but also during the first stage. Indeed, from Eqs. (\[phys-ang-mom\]) and (\[total-ang-mom\]), we obtain that $$\begin{aligned}
\frac{dJ}{d\Omega}=I \left[ 1+ \frac13 \left( 4K+5 \right) Q
\alpha^2 \left( 1+2\frac{\Omega}{\alpha} \frac{d\alpha}{d\Omega}
\right) \right]. \label{dJdf}\end{aligned}$$ Now we can use Eqs. (\[eq-dif-omega(t)-s-visc\]) and (\[eq-dif-alpha(t)-s-visc\]) to relate $d\Omega$ and $d\alpha$, namely, $$\begin{aligned}
\frac{d\Omega}{\Omega} =
-\frac{\frac83(K+2)Q\alpha}{1+\frac43(K+2)Q\alpha^2}d\alpha.\end{aligned}$$ Inserting this expression into Eq. (\[dJdf\]) and using $f=2\Omega/(3\pi)$ we obtain $$\begin{aligned}
\frac{dJ}{df}=\frac{9\pi I}{8(K+2)}. \label{conditionOwenetal}\end{aligned}$$ Since the above condition holds during both stages of the evolution, we could use it to derive, in the manner proposed in Ref. [@OLCSVA1998], the frequency-domain gravitational wave amplitude given by Eq. (\[htilde\]).
The gravitational wave amplitude $|\tilde{h}(f)|$ is represented in Fig. \[fig:htilde-freq\] for different values of $K$ and for gravitational-wave frequencies lying between $f_{min}$ and $f_{max}$.
![\[fig:htilde-freq\] Gravitational wave amplitude in the frequency domain, $|\tilde{h}(f)|$, for different values of $K$ and distance to the source $D=20\mbox{ Mpc}$.](htilde.eps){width="8.6"}
It is worth mentioning that within the model of evolution proposed in Ref. [@OLCSVA1998], the frequency-domain gravitational wave amplitude has a spike at high frequencies, due to the fact that during the first stage of evolution the angular velocity of the star evolves very slowly on the viscous timescale, leading to a quasi-monochromatic gravitational wave emission during the first 500 s of evolution. However, as we have seen, if one takes into account the influence of differential rotation, namely, the fact that it contributes to the physical angular momentum of the *r*-mode perturbation, then the angular velocity of the star evolves in the gravitational-radiation timescale already in the first stage of evolution \[see Eq. (\[eq-dif-omega(t)-c-visc\])\]. As a consequence, the gravitational wave amplitude $|\tilde{h}(f)|$ in the first stage of evolution is also given by Eq. (\[htilde\]) and we observe no spike.
Let us now analyze the possibility of detecting the emitted gravitational waves with laser interferometer detectors LIGO and Virgo. Because of the complexity of real neutron stars, our knowledge of the evolution of the *r*-mode instability is insufficient to predict the gravitational waveform with such an accuracy that a matched filtering signal-processing technique would be feasible. However, we can use matched filtering in order to estimate the detectability of the gravitational-wave signal. For that purpose, the characteristic amplitude of the signal, $$\begin{aligned}
h_c(f)&\equiv&f |\tilde{h}(f)| \nonumber \\
&=& \frac{5.5\times 10^{-22}}{\sqrt{K+2}}
\sqrt{\frac{f}{f_{max}}} \left( \frac{20 \mbox{ Mpc}}{D} \right),\end{aligned}$$ is compared with the rms strain noise in the detector, $$\begin{aligned}
h_{rms}(f)\equiv\sqrt{f S_h(f)},\end{aligned}$$ where $S_h(f)$ is the noise power spectral density of the detector.
For frequencies in the interval $50\mbox{ Hz}\leqslant f\leqslant
1200\mbox{ Hz}$, the curves for the noise power spectral densities of the initial LIGO [@LIGO], Virgo [@Virgo] and advanced LIGO [@AdvLIGO] detectors are well approximated by the following analytical expressions, respectively: $$\begin{aligned}
S_h(f)=S_1\left[ \left( \frac{f_1}{f} \right)^4 +\left(
\frac{f}{f_1} \right)^2 \right], \label{shfligo}\end{aligned}$$ where $S_1=3.4\times10^{-46}\mbox{ Hz}^{-1}$ and $f_1=142.0\mbox{
Hz}$, $$\begin{aligned}
S_h(f)=S_2\left[ 1+ \frac16 \left( \frac{f_2}{f} \right)^2 +
\frac16 \left( \frac{f}{f_2} \right)^2 \right], \label{shfvirgo}\end{aligned}$$ where $S_2=1.5 \times10^{-45}\mbox{ Hz}^{-1}$ and $f_2=249.6\mbox{
Hz}$, and $$\begin{aligned}
S_h(f) &=& S_3\left\{ 1+ \left( \frac{f_3}{f} \right)^7 -
\frac{10}{3} \left( \frac{f}{f_4} \right) \left[ 1- \left(
\frac{f}{f_4} \right) \right. \right. \nonumber \\
& & \qquad + \left. \left. \frac{3}{50} \left( \frac{f}{f_4}
\right)^2 \right] \right\}, \label{shfadvligo}\end{aligned}$$ where $S_3=2.2 \times10^{-47}\mbox{ Hz}^{-1}$, $f_3=52.8\mbox{
Hz}$ and $f_4=421.3\mbox{ Hz}$.
In Fig. \[fig:hcharac-hrms\] the curves corresponding to the rms strain noises in the initial LIGO, Virgo and advanced LIGO detectors are compared with the curves corresponding to the characteristic amplitude of the gravitational wave signal for different values of $K$ and for a distance to the source of $D=20
\mbox{ Mpc}$. For such a distance, which includes the Virgo cluster of galaxies, a few supernovae per year are expected.
![\[fig:hcharac-hrms\] The dotted lines correspond to the rms strain noises in the initial LIGO, VIRGO and advanced LIGO detectors. The solid lines correspond to the characteristic amplitude for different values of $K$. The distance to the source is taken here to be $D=20\mbox{ Mpc}$.](hcvinte.eps){width="8.6"}
The most striking feature of Fig. \[fig:hcharac-hrms\] is that the detectability of gravitational waves from the *r*-mode instability of newly born neutron stars is drastically reduced as the initial value of differential rotation associated with *r*-modes increases. Indeed, for $K\gtrsim100$, even the advanced LIGO detector would not have enough sensitivity to see such sources (for $D=20\mbox{ Mpc}$).
The visual comparison, in Fig. \[fig:hcharac-hrms\], between the characteristic amplitude of the signal and the rms strain noise in the detector gives us a qualitative measure of the signal-to-noise ratio for matched filtering. A quantitative determination of the signal-to-noise ratio is obtained from $$\begin{aligned}
\left( \frac{S}{N} \right)^2=2\int\limits_{f_{min}}^{f_{max}}
\frac{df}{f} \left( \frac{h_c}{h_{rms}} \right)^2, \label{snr}\end{aligned}$$ which, for $f_{min}=(77-80)\mbox{ Hz}$ and $f_{max}= 1191\mbox{
Hz}$, yields $$\begin{aligned}
\frac{S}{N} \approx\frac{1}{\sqrt{K+2}} \frac{20 \mbox{
Mpc}}{D}\times
\left\{ \begin{array}{rl}
0.9 & (\mbox{initial LIGO}) \\
0.7 & (\mbox{Virgo}) \\
12.9 & (\mbox{advanced LIGO})
\end{array} \right. \! \! \! \! .
\label{snr2}\end{aligned}$$
For the initial LIGO and Virgo detectors, even for small initial differential rotation ($K\approx0$), the signal-to-noise ratio is not significant for $D=20 \mbox{ Mpc}$. Of course, this ratio can be increased if we consider sources located at smaller distances, but at the cost of decreasing the number of expected supernova events per year and, hence, the probability of a detection.
For the advanced LIGO detector the situation improves: for small values of $K$, the signal-to-noise ratio is significant even for $D=20 \mbox{ Mpc}$. Therefore, and since within such a distance several supernovae per year are expected, one could hope that advanced LIGO detectors would see gravitational radiation from the *r*-mode instability of young neutron stars. However, such hope is based on the assumption that neutron stars are born with small initial differential rotation associated with *r*-modes. But this may not be the case. And if a neutron star is born with substantial differential rotation associated with *r*-modes (say $K\gtrsim100$), then the emitted gravitational waves will not be seen even by advanced LIGO. Of course, as already mentioned above, one could consider smaller distances to the source, but then the number of events per year would decrease, decreasing the probability of a detection. In Fig. \[fig:hcharac-hrms-30kpc\] the characteristic amplitude for high values of $K$ is compared with the rms strain noise in the advanced LIGO detector for a distance to the source of $30 \mbox{
kpc} $, i.e., within our Galaxy, in which about 2 supernovae are expected every hundred years [@D2006].
![\[fig:hcharac-hrms-30kpc\] The dotted line corresponds to the rms strain noise in the advanced LIGO detector. The solid lines correspond to the characteristic amplitude for different values of $K$. The distance to the source is taken here to be $D=30\mbox{ kpc}$.](hctrinta.eps){width="8.6"}
The fact that an increase of the initial amount of differential rotation associated with *r*-modes makes it more difficult to detect gravitational waves from these modes (for a given distance to the source ) can be easily understood from angular momentum considerations. Indeed, it was shown in Ref. [@OL2002], based on an unpublished general argument of Blandford, that the signal-to-noise ratio given by Eq. (\[snr\]) is well approximated by $$\begin{aligned}
\left( \frac{S}{N} \right)^2\approx \frac{2G}{5\pi c^3 D^2}
\frac{\Delta J}{h_{rms}^2},
\label{snr-delta-J}\end{aligned}$$ where $\Delta J\equiv J_0-J(t_b)$ is the total amount of angular momentum carried away by gravitational waves, $J(t)$ is the total angular momentum of the star and $J_0\approx I\Omega_0$ is the initial angular momentum of the star. Using Eqs. (\[phys-ang-mom\]) and (\[total-ang-mom\]), the total amount of angular momentum carried away by gravitational waves can be written as $$\begin{aligned}
\frac{\Delta J}{J_0}\approx
1-\frac{\Omega(t_b)}{\Omega_0}-\frac13 (4K+5)Q
\frac{\Omega(t_b)}{\Omega_0} \alpha^2(t_b),\end{aligned}$$ or, since $\Omega\approx\Omega_0[1+\frac43(K+2)Q\alpha^2]^{-1}$, as $$\begin{aligned}
\frac{\Delta J}{J_0} \approx \frac{3}{4(K+2)}\left(
1-\frac{\Omega(t_b)}{\Omega_0} \right),\end{aligned}$$ where $\Omega(t_b)=(0.065-0.067)\Omega_0$. Thus, for $K\gg1$, only a small part of the initial angular momentum of the star is carried away by gravitational waves[^2] (see Fig. \[fig:ang-mom-away\]) and, consequently, detection of these waves becomes a more difficult task.
![\[fig:ang-mom-away\] Angular momentum carried away by gravitational waves, $\Delta J$, as a function of time for different values of $K$. For $K\gg1$, only a small part of the initial angular momentum of the star is carried away by gravitational waves.](deltaj.eps){width="8.6"}
\[sect-conclusions\]Conclusions
===============================
In this paper we have investigated the influence of differential rotation on the detectability of gravitational radiation emitted by a newly born, hot, rapidly-rotating neutron star, as it spins down due to the *r*-mode instability.
A model of evolution of the *r*-mode instability that takes into account differential rotation [@ST2005] has been used to derive the gravitational wave amplitude $|h(t)|$ and its Fourier transform $|\tilde{h}(f)|$. We have shown that the maximum value of the gravitational wave amplitude, $h_{max}$, depends on the amount of differential rotation at the time the *r*-mode instability becomes active, namely, $h_{max}\propto(K+2)^{-1/2}$. We have also shown that the frequency-domain gravitational wave amplitude, $|\tilde{h}(f)|$, has no spike at high frequencies, as opposed to the results of Ref. [@OLCSVA1998]. There, the spike of $|\tilde{h}(f)|$ is due to the fact that during the first stage of evolution of the *r*-mode instability the angular velocity of the star evolves very slowly on the viscous timescale, leading to a quasi-monochromatic gravitational wave emission during the first few hundred seconds of evolution. However, as we have seen, if one takes into account the influence of differential rotation, then the angular velocity of the star evolves in the gravitational-radiation timescale already in the first stage of evolution and, as a consequence, during this stage of evolution the gravitational wave is not monochromatic and $|\tilde{h}(f)|$ has no spike.
We have assumed matched filtering in order to investigate the detectability of the gravitational wave signal. However, our knowledge of the evolution of the *r*-mode instability is insufficient to predict the gravitational waveform with such an accuracy that an optimal matched filtering signal-processing technique would be feasible. But non-optimal signal-processing strategies could be developed such that they would yield results not much different from the ones obtained with matched filtering (see, for instance, the hierarchical search strategies proposed in [@BC2000]). Therefore, our results concerning the detectability of gravitational waves from *r*-modes can be considered to be a good approximation.
Assuming matched filtering, the characteristic amplitude of the signal $h_c(f)$ is compared with the rms strain noise in the initial LIGO and Virgo detectors, as well as with the expected rms strain noise in the advanced LIGO detector \[see Figs. \[fig:hcharac-hrms\] and \[fig:hcharac-hrms-30kpc\] and Eq. (\[snr2\])\]. We conclude that the detectability of gravitational waves from the *r*-mode instability of newly born neutron stars is drastically reduced as the initial value of differential rotation associated with *r*-modes increases. For the initial LIGO and Virgo detectors, the signal-to-noise ratio obtained with matched filtering for sources located at 20 Mpc is smaller than unity even for $K\approx 0$. For advanced LIGO, if neutron stars are born with significant differential rotation associated with *r*-modes, then detection of gravitational waves would be possible only if the distance to these neutron stars is considerably smaller than 20 Mpc. For instance, if $K=10^5$, a signal-to-noise ratio greater than 10 requires a source located no more than about 80 kpc away \[see Eq. (\[snr2\])\], i.e., within a sphere containing the Milky Way, the Magellanic Clouds and a few more small galaxies, in which just a few supernovae per century are expected. However, if the initial value of differential rotation associated with *r*-modes is small ($K\approx0$), then $S/N\gtrsim 10$ can be obtained for $D=20 \mbox{ Mpc}$. Since within a volume of radius 20 Mpc several supernovae per year are expected, it is quite possible that advanced LIGO will detect gravitational radiation from the *r*-mode instability of young neutron stars.
This work was supported in part by the Fundação para a Ciência e a Tecnologia (FCT), Portugal.
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[^1]: The gravitational-radiation timescale $\tau_{GR}$ for *r*-modes was first obtained using an expression for the time evolution of the physical energy of the *r*-mode perturbation, assuming that the imaginary part of the frequency of the mode, $Im(\omega)\equiv 1/\tau_{GR}$, is related to the time derivative of the energy by $dE/dt=-2E/\tau_{GR}$ [@LOM1998]. But $\tau_{GR}$ can also be obtained by solving *explicitly* the hydrodynamic equations in the presence of the gravitational radiation reaction force [@DS2005].
[^2]: As shown in Ref. [@ST2005], for $K\gg1$, most of the angular momentum of the unperturbed star is transferred to the *r*-mode perturbation, due to the fact that the fluid develops a strong differential rotation. Indeed, after a few hundred seconds of evolution of the *r*-mode instability, the average differential rotation increases rapidly, saturating at high values relatively to the initial angular velocity of the star.
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{
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